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Summer 2011

Fisher's zeros in lattice gauge theoryDaping DuUniversity of Iowa

Copyright 2011 Daping Du

This dissertation is available at Iowa Research Online: https://ir.uiowa.edu/etd/1217

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Recommended CitationDu, Daping. "Fisher's zeros in lattice gauge theory." PhD (Doctor of Philosophy) thesis, University of Iowa, 2011.https://doi.org/10.17077/etd.bfnqfycu

FISHER’S ZEROS IN LATTICE GAUGE THEORY

by

Daping Du

An Abstract

Of a thesis submitted in partial fulfillment of therequirements for the Doctor of Philosophy

degree in Physics in theGraduate College of The

University of Iowa

July 2011

Thesis Supervisor: Professor Yannick Meurice

1

ABSTRACT

In this thesis, we study the Fisher’s zeros in lattice gauge theory. The

analysis of singularities in the complex coupling plane is an important tool to un-

derstand the critical phenomena of statistical models. The Fisher’s zero structure

characterizes the scaling properties of the underlying models [40, 33] and has

a strong influence on the complex renormalization group transformation flows

in the region away from both the strong and weak coupling regimes [24]. By

reconstructing the density of states, we try to develop a systematical method to

investigate these singularities and we apply the method to SU(2) and U(1) lattice

gauge models with a Wilson action in the fundamental representation.

We first take the perturbative approach. By using the saddle point approx-

imation, we construct the series expansions of the density of states in both of the

strong and weak regimes from the strong and weak coupling expansions of the

free energy density[5, 4]. We analyze the SU(2) and U(1) models. The expansions

in the strong and weak regimes for the two models indicate both possess finite

radii of convergence, suggesting the existence of complex singularities. We then

perform the numerical calculations. We use Monte Carlo simulations to construct

the numerical density of states of the SU(2) [22] and U(1) models [9]. We also

discuss the convergence of the Ferrenberg-Swendsen’s method [32] which we use

for the SU(2) model and propose a practical method to find the initial values that

improve the convergence of the iterations. The strong and weak series expan-

sions are in good agreement with the numerical results in their respective limits.

The numerical calculations also enable the discussion of the finite volume effects

which are important to the weak expansion.

We calculate the Fisher’s zeros of the SU(2) and U(1) models at various

2

volumes using the numerical entropy density functions. We compare different

methods of locating the zeros. By the assumption of validity of the saddle point

approximation, we find that the roots of the second derivative of the entropy

density function have an interesting relation with the actual zeros and may pos-

sibly reveal the scaling property of the zeros. Using the analytic approximation

of the numerical density of states, we are able to locate the Fisher’s zeros of the

SU(2) and U(1) models. The zeros of the SU(2) stabilize at a distance from the

real axis, which is compatible with the scenario that a crossover instead of a

phase transition is expected in the infinite volume limit [56, 16]. In contrast, with

the precise determination of the locations of Fisher’s zeros for the U(1) model

at smaller lattice sizes L = 4, 6 and 8, we show that the imaginary parts of the

zeros decrease with a power law of L−3.07 and pinch the real axis at β = 1.01134,

which agrees with results using other methods [50]. Preliminary results at larger

volumes indicate a first-order transition in the infinite volume limit.

Abstract Approved:Thesis Supervisor

Title and Department

Date

FISHER’S ZEROS IN LATTICE GAUGE THEORY

by

Daping Du

A thesis submitted in partial fulfillment of therequirements for the Doctor of Philosophy

degree in Physics in theGraduate College of The

University of Iowa

July 2011

Thesis Supervisor: Professor Yannick Meurice

Copyright byDAPING DU

2011All Rights Reserved

Graduate CollegeThe University of Iowa

Iowa City, Iowa

CERTIFICATE OF APPROVAL

PH.D. THESIS

This is to certify that the Ph.D. thesis of

Daping Du

has been approved by the Examining Committeefor the thesis requirement for the Doctor ofPhilosophy degree in Physics at the July 2011graduation.

Thesis Committee:Yannick Meurice, Thesis Supervisor

Andreas S. Kronfeld

Vincent G. J. Rodgers

Wayne N. Polyzou

Craig Pryor

To my wife, Hui-Yun Wu and my family.

ii

ACKNOWLEDGMENTS

First, I would like to thank Professor Yannick Meurice, my Ph.D. thesis

advisor, for his tremendous help in my entire graduate career at the University

of Iowa. He has been encouraging and helping me in my study and research. He

taught me the knowledge as well as the ways of utilizing the knowledge in the

research. He is a good friend in addition to a good advisor. The thesis would

have not been possible without his generous help, continuous encouragement

and meticulous guidance.

I am indebted to my colleagues and friends Alan Denbleyker, Yuzhi Liu,

Haiyuan Zou, Alexei Bazavov and Alex Velytsky. Alan has been helping me with

the data generation and partial proof reading of this paper. Yuzhi and Haiyuan

has provided me with a lot of helps and suggestions. Alex has written the SU(2)

codes which made the numerical calculations possible. Alexei made all the U(1)

data available and I benefited from several discussions with him. I have also got

a lot of help from my friends Ran Lin, Juan Chen and Nan Chen.

I would also like to thank Andreas Kronfeld for his help during my visit at

Fermilab which brought me with new understanding on the applications of lattice

gauge theory and partially expedited the writing of the paper. I also benefited

considerably from the discussions and lectures of Professor Vincent Rodgers,

Professor Wayne Polyzou, Professor Craig Pryor, Professor Mary Hall Reno and

other respected faculties in the physics department.

Finally and importantly, I owe too much to my wife and my family. My

wife has been supportive of me ever since we were married. Without her effort

in it, I could never make it finished. I am grateful to my parents and my siblings

for their encouragement and support.

iii

ABSTRACT

In this thesis, we study the Fisher’s zeros in lattice gauge theory. The

analysis of singularities in the complex coupling plane is an important tool to un-

derstand the critical phenomena of statistical models. The Fisher’s zero structure

characterizes the scaling properties of the underlying models [40, 33] and has

a strong influence on the complex renormalization group transformation flows

in the region away from both the strong and weak coupling regimes [24]. By

reconstructing the density of states, we try to develop a systematical method to

investigate these singularities and we apply the method to SU(2) and U(1) lattice

gauge models with a Wilson action in the fundamental representation.

We first take the perturbative approach. By using the saddle point approx-

imation, we construct the series expansions of the density of states in both of the

strong and weak regimes from the strong and weak coupling expansions of the

free energy density[5, 4]. We analyze the SU(2) and U(1) models. The expansions

in the strong and weak regimes for the two models indicate both possess finite

radii of convergence, suggesting the existence of complex singularities. We then

perform the numerical calculations. We use Monte Carlo simulations to construct

the numerical density of states of the SU(2) [22] and U(1) models [9]. We also

discuss the convergence of the Ferrenberg-Swendsen’s method [32] which we use

for the SU(2) model and propose a practical method to find the initial values that

improve the convergence of the iterations. The strong and weak series expan-

sions are in good agreement with the numerical results in their respective limits.

The numerical calculations also enable the discussion of the finite volume effects

which are important to the weak expansion.

We calculate the Fisher’s zeros of the SU(2) and U(1) models at various

iv

volumes using the numerical entropy density functions. We compare different

methods of locating the zeros. By the assumption of validity of the saddle point

approximation, we find that the roots of the second derivative of the entropy

density function have an interesting relation with the actual zeros and may pos-

sibly reveal the scaling property of the zeros. Using the analytic approximation

of the numerical density of states, we are able to locate the Fisher’s zeros of the

SU(2) and U(1) models. The zeros of the SU(2) stabilize at a distance from the

real axis, which is compatible with the scenario that a crossover instead of a

phase transition is expected in the infinite volume limit [56, 16]. In contrast, with

the precise determination of the locations of Fisher’s zeros for the U(1) model

at smaller lattice sizes L = 4, 6 and 8, we show that the imaginary parts of the

zeros decrease with a power law of L−3.07 and pinch the real axis at β = 1.01134,

which agrees with results using other methods [50]. Preliminary results at larger

volumes indicate a first-order transition in the infinite volume limit.

v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Motivation and Overview . . . . . . . . . . . . . . . . . . . . . 11.2 Introduction to Lattice Gauge Models . . . . . . . . . . . . . . . 2

1.2.1 Feynman Path Integral . . . . . . . . . . . . . . . . . . . 31.2.2 U(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Fisher’s Zeros and Critical Phenomena . . . . . . . . . . . . . . 8

2 THE DENSITY OF STATES . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 The Density of States . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Saddle Point Approximation . . . . . . . . . . . . . . . . . . . . 162.3 Series Expansions of n(S) . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Strong Coupling Expansions . . . . . . . . . . . . . . . . 192.3.2 Weak Coupling Expansions . . . . . . . . . . . . . . . . 27

2.4 Numerical Calculation of n(S) . . . . . . . . . . . . . . . . . . . 352.4.1 The Computational Setup . . . . . . . . . . . . . . . . . 372.4.2 Histogram Reweighting . . . . . . . . . . . . . . . . . . 402.4.3 Volume Dependence of the density states . . . . . . . . 54

3 FISHER’S ZEROS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.1 Single Point Reweighting Method . . . . . . . . . . . . . . . . . 563.2 Quasi-Gaussian Distribution: A Toy Model . . . . . . . . . . . 58

3.2.1 Example 1. λ′3 = 0.1, λ′4 = 0.01 . . . . . . . . . . . . . . . 613.2.2 Example 2: λ′3 = 0.01, λ′4 = 0.002 . . . . . . . . . . . . . 62

3.3 Approximation with Analytic Functions . . . . . . . . . . . . . 633.3.1 LargeNp limit with a fixed f (x) . . . . . . . . . . . . . . 643.3.2 Chebyshev Approximation . . . . . . . . . . . . . . . . 673.3.3 The Ellipse of Convergence . . . . . . . . . . . . . . . . 693.3.4 The Moment Tests . . . . . . . . . . . . . . . . . . . . . . 723.3.5 Evaluation of the Partition Function with Numerical

Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 743.3.6 Localization of Zeros Using Cauchy’s Theorem . . . . . 76

3.4 SU(2) Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.4.1 Zeros From Single-Point Reweighting . . . . . . . . . . 793.4.2 Histograms Analysis . . . . . . . . . . . . . . . . . . . 80

vi

3.4.3 Zeros using Analytic Approximation . . . . . . . . . . 823.5 U(1) Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.5.1 Volume Dependence of the Double Peak . . . . . . . . . 903.5.2 Zeros using the Discrete Reweighting Method . . . . . 923.5.3 Zeros from analytic approximation . . . . . . . . . . . . 953.5.4 The Scaling of Zeros . . . . . . . . . . . . . . . . . . . . . 973.5.5 The Derivatives of f (x) . . . . . . . . . . . . . . . . . . . 98

4 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

APPENDIX

A Two-Data-Point Reweighting . . . . . . . . . . . . . . . . . . . . . . 104

B Ferrenberg-Swendsen’s Formula for the Weight . . . . . . . . . . . . 105

C Trapezoidal Integration for Quasi-Gaussian Functions . . . . . . . . 106

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

vii

LIST OF TABLES

2.1 SU(2) Strong coupling expansion coefficients. . . . . . . . . . . . . 22

2.2 U(1) Strong coupling expansion coefficients. . . . . . . . . . . . . . 22

2.3 The weak coupling expansion for U(1) gauge fields. . . . . . . . . . 31

2.4 Weak coupling coefficients defined in Sec. 2.3.2 for the SU(2)model. The choice of b1 corresponds to V = 64 . . . . . . . . . . . . 34

2.5 SU(2) data setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.6 U(1) Density of States. . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1 The results of the zeros with the increasing ”volumes”. . . . . . . . 67

3.2 Both of the actual zeros and the f ′′(x) zeros are shown with dif-ferent ranges of fitting for the volume 44. The semi-major of thecorresponding ellipses are also given ( the focus length is half therange). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.3 Both of the actual zeros and the f ′′(x) zeros are shown with differentranges of fitting for the volume 64. . . . . . . . . . . . . . . . . . . . 86

3.4 The actual zeros of various orders of approximation for the twovolumes 44 and 64 discussed in the text. . . . . . . . . . . . . . . . . 87

3.5 The zeros with the increasing ”volumes” using the density of statesf4 and f6 of the 44 and 64 lattices. . . . . . . . . . . . . . . . . . . . . 88

3.6 βS, s1 and s2 defined in the text for L = 4, 6 and 8. . . . . . . . . . . 91

3.7 Real part of the first three zeros for L = 4, 6 and 8. . . . . . . . . . . 93

3.8 Imaginary part of the first three zeros for L = 4, 6 and 8. . . . . . . 94

3.9 The lowest three zeros in the three volumes 44,64 and 84. Column 1-4 are, the real parts of the zeros, the estimate error σs from differentseeds of Monte Carlo runs and the error σc due to the orders ofChebyshev interpolation( we used three different orders 40,44 and50 for all three volumes). Same for the imaginary part. . . . . . . . 95

viii

LIST OF FIGURES

2.1 Weak and strong coupling expansions of the average plaquetteP for SU(2) at various orders in the weak and strong couplingexpansion compared to the numerical values. . . . . . . . . . . . . 19

2.2 Numerical value of f (x) compared to the strong coupling expansionat successive orders for the SU(2) lattice gauge fields. . . . . . . . . 23

2.3 Numerical value of f (x) compared to the strong coupling expansionat successive orders for the U(1) lattice gauge fields. . . . . . . . . 23

2.4 The expansion coefficients h2m and g2m in the log scale are plottedversus the order 2m. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Logarithm of the absolute difference between the numerical dataand the strong coupling expansion of P at successive orders forthe SU(2) model (error diagram). The numerical data are takenfrom the volume 64. The error on numerical P are obtained from50 bootstraps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 The error diagram of P at successive orders for the U(1) Model.The numerical data are from the volume 44. The error is estimatedthrough 20 seeds of data. . . . . . . . . . . . . . . . . . . . . . . . . 26

2.7 The error diagram of f (x) at successive orders for the SU(2) Model.The numerical data are from the volume 64. The error on numericalf are obtained from 50 bootstraps. . . . . . . . . . . . . . . . . . . . 26

2.8 The error diagram of f (x) at successive orders for the U(1) Model.The numerical data are from the volume 84. The error is estimatedthrough 20 seeds of data. . . . . . . . . . . . . . . . . . . . . . . . . 27

2.9 The difference of f (x) for two different volumes 44 and 64 is plottedas a function of x = S/Np. We see that in the weak coupling region(small x), the volume effect is apparent and cannot be ignored inthe expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.10 The successive orders of f (x) expansion from U(1) weak couplingexpansion are plotted in contrast with the numerical f (x) frommulti-canonical simulations. . . . . . . . . . . . . . . . . . . . . . . 32

2.11 The error diagram of f (x) at successive orders of weak couplingexpansion for the U(1) model. . . . . . . . . . . . . . . . . . . . . . 32

ix

2.12 The error diagram of P at successive orders of weak coupling ex-pansion for the U(1) model. . . . . . . . . . . . . . . . . . . . . . . . 33

2.13 Numerical value of f (x) compared to the weak coupling expan-sion at successive orders. The coefficients are calculated from thevolume 64. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.14 The error diagram of f at successive orders for the SU(2) model. . 35

2.15 The error diagram of P at successive orders for the SU(2) model(above).The error for the numerical P is calculated using 50 bootstraps ofthe data. The graph below is the case without the contribution ofthe zero mode in b1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.16 The log-log plot of the autocorrelation vs t at β = 2.20 for a 44 SU(2)lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.17 The integrated autocorrelation time is plotted verse β for four dif-ferent volume V = 44, 64, 84, 104. The peak of the distribution isfound at around β = 2.20(for 44), 2.34 (for 64) and 2.40 (for 84). Thepeak is not obvious for the volume 104. . . . . . . . . . . . . . . . . 40

2.18 The overlapping of neighboring data sets. . . . . . . . . . . . . . . 41

2.19 A comparison between the trapezoidal and the generalized Simp-son’s rules that are used to obtained the initial values of F (βα).The data is from the 44 lattice and skipping every 5 datasets. ∆χ2

is the χ-square difference between successive iterations ( definedin the text ), the slope of which indicates the convergent rate of theiteration. It is obvious that when the datasets are sparse, the gener-alized Simpson’s rule gives better initial values and improves theconvergence significantly. . . . . . . . . . . . . . . . . . . . . . . . . 47

2.20 This plot is a comparison of reweighting with various x bin number:200(circles), 500(squares) and 1000(diamonds). They all result inthe same convergence down to the noise level of the data. . . . . . 48

2.21 Two slightly different sets of initials are fed to the reweighting ofa SU(2) data on a 44 lattice and reach the same convergence. Theplot is ∆ f (x) at iteration 10000 (middle two curves) and 200(outertwo curves). Here ∆ f means the difference of f (x) to the stabilizedvalue ( at iteration 15000). . . . . . . . . . . . . . . . . . . . . . . . . 49

x

2.22 An example of the iteration process. All iterations are comparedwith the convergent values f (x)con. Two different initials are com-pared here. For the first one we started with initial values f (x) = 0,for all x. The convergence is slow (upper five curves). For thesecond we used the integration method to get the initials. Themethod is efficient for the iterations (lower two curves). . . . . . . 50

2.23 Convergent iterations may not always lead to satisfactory results.Reweightings with different numbers of β’s are compared. Theresulting f (x)’s are then subtracted from the result with 284 β’s(normalized at x = 1). The solid, dashed, double-dashed curvesare using 142,57 and 29 β’s respectively. Although all lead to con-vergence, the reweighting with fewer datasets show much biggerundesired fluctuations. . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.24 The convergence is sensitive to the number of datasets. The reweight-ing using the same data set while including different numbers ofdatasets at volume 64. The plot is showing the chi2 varies withiterations. The upper one which is using 246 β’s shows obviousdivergence at iteration around 6000, while the lower one with 449datasets indicates an strong sign of convergence. Similar difficultyappears with higher volumes such as 84 and 104. . . . . . . . . . . 52

2.25 The overlapping plots of the previous example. The boxes in eachgraph show two-σ spreading (centered at the average) of each datapoint. The vertical increments have no meaning but to illustratethe neighboring overlapping in a clearer manner. The left graphcorresponds to the diverging reweighting and showing in sufficientoverlapping among the data. . . . . . . . . . . . . . . . . . . . . . . 52

2.26 The errors of the density of states due to the different seeds. . . . . 53

2.27 The collapse of the difference of f (x) of three different volumes,∆ fV4,V8 and ∆ fV6,V8 with the overall constant subtracted (describedin the text). The differences are then divided by their correspondinginverse volume difference. . . . . . . . . . . . . . . . . . . . . . . . 55

2.28 The difference of the differences ∆ fV4,V8 and ∆ fV6,V8 . The flat partcorresponds to the overall constant 0.035. . . . . . . . . . . . . . . . 55

3.1 The plot shows the contours of both real and imaginary part of thepartition function. The intersections of these curves are the zerosof the partition function. . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 The probability distribution of total action in the Gaussian toymodel described in the text. . . . . . . . . . . . . . . . . . . . . . . . 59

xi

3.3 Logarithm of the autocorrelation versus the time series distancefor the original set of 1,600,000 values. . . . . . . . . . . . . . . . . . 60

3.4 Zeros of the real (circles) and imaginary (crosses) part for 40,000Gaussian configurations. The solid lines are the circle of confidenceand the hyperbolas of the normal distribution. . . . . . . . . . . . . 61

3.5 Zeros of the real (circles) and imaginary (crosses) part for 40,000configurations corresponding to the first example. The small dotsare the accurate values for the real (gray) and imaginary (black)parts. The exclusion region boundary for d = 0.12 is representedby boxes (red). The solid line is the circle of confidence of theGaussian approximation. . . . . . . . . . . . . . . . . . . . . . . . . 62

3.6 Zeros of the real (circles) and imaginary (crosses) part for 40,000configurations corresponding to the second example. The smalldots are the accurate values for the real (gray) and imaginary(black) parts. The exclusion region boundary for d = 0.15 is rep-resented by boxes. The solid line is the circle of confidence of theGaussian approximation. . . . . . . . . . . . . . . . . . . . . . . . . 63

3.7 The left are the imaginary parts of the zeros in the toy modeldescribed in the text in different ”volumes” plotted against thef ′′(x) = 0 zeros which corresponds to the zeros in infinite ”volume”limit. The right is the log-log plot of the difference of the zeros tothe f ′′ = 0 zero vs the volume. We found that Im(βL − β∞) ∝ L−2.6457. 68

3.8 The logarithm of the absolute coefficients of the Chebyshev ap-proximation discussed in the text versus the order number n. . . . 71

3.9 The ellipse of convergence of a Chebyshev series ( described in thetext) and the roots of f ′(x) = βwith the values given in the text. Thetwo saddle points which are obviously inside the ellipse merge tothe root of f ′′(x) = 0 (red cross) as the value of β approaches to thevalue corresponding to the root of f ′′(x) = 0. . . . . . . . . . . . . . 71

3.10 The ellipse of convergence of a Chebyshev series and the rootsof f ′′(x) = 0(empty circles). The empty squares are the roots off ′(x) = βwhere β is given by f ′(rootsof f ′′ = 0) ( the cross ). . . . . . 72

3.11 The second moment of the SU(2) model at volume 44. The numer-ical result is plotted against various orders of Chebyshev approxi-mation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.12 The third moment. Similar to Fig. (3.11). . . . . . . . . . . . . . . . 73

3.13 Path of Steepest Descent. . . . . . . . . . . . . . . . . . . . . . . . . 74

xii

3.14 The plot is showing the correspondence of a loop in the β-plane tothe loop in the x-plane. . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.15 An actual loop in which two zeros are presented. The path isdetoured to get a better integration. . . . . . . . . . . . . . . . . . . 79

3.16 The contour plot shows the zeros of a SU(2) gauge model on a 44

lattice. Blue and green lines are the zero curves of the imaginaryand real part of the partition function from the Monte Carlo sim-ulation at β = 2.18. The crosses and circles are the counterpartsfrom a quasi-Gaussian approximation which overlaps the data inthe bottom part. Both show isolated zeros of the partition func-tion. However they are all lying outside the radius of confidence,or more strictly, they are all above the level of confidence ( thesquares). Therefore these zeros are artificial. The Monte Carlodata of a neighboring β doesn’t indicate the consistent locations ofthese zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.17 Distribution of eros of the real part of the partition function in thecomplex β plane and regions of confidence described in the text. . 81

3.18 The residue distribution after subtracting the Gaussian part withβ = 2.18 on a 44 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.19 The residue distribution after subtracting the Gaussian with β =2.18 on a 64 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.20 The residue distribution after subtracting the Gaussian with β =2.348 on a 64 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.21 The plot shows a typical pattern of the SU(2) zeros. The zerosof two different volumes, 44(filled squares) and 64 (filled circles),are plotted against their respective f ′′(x) = 0 zeros (empty squaresfor the 44 and empty circles for the 64). The density of states areapproximated using Chebyshev Polynomial of order 44. The fittingrange is over x ∈ [0, 2]. . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.22 The plot is the zeros (empty cirdles) of two volumes: 44 and 64.Their f ′′(x) = 0 zeros (filled circles) are also plotted for comparison.Both of the actual zeros and the f ′′(x) = 0 zeros tend to approachto the same limit which is distant away from the real axis. . . . . . 86

3.23 The lowest zeros obtained using various orders of approximationat two different volumes: 44 and 64. . . . . . . . . . . . . . . . . . . 87

3.24 The locations of the zeros calculated using the entropy densityfunction f at volume 44 scale with Np = 6 × L4. The red pointcorresponds to the f ′′ = 0 zero. . . . . . . . . . . . . . . . . . . . . . 89

xiii

3.25 The zeros βL obtained using variousNp = 6 × L4 using the entropydensity function from two different volumes: 44,64. The plot showsthe distance from βL to the f ′′ = 0 zero β∞ versus L in the logarithmicscale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.26 f (s) − βs for β = 1.00175, 100177 and 1.00179 on a 64 lattice. Thehorizontal lines is drawn to indicate the asymmetry of the heights.The error bars are provided with the same scale as f (s) − βs. . . . . 91

3.27 The double peak distribution of three volumes: 44, 64 and 84. . . . 92

3.28 Zeros of the real (point +) and imaginary (point x) part of Z forU(1) using the density of states for 44 and 64 lattices. . . . . . . . . 93

3.29 The shift in the imaginary part of the zeros due to the cuts of theintegration described in the text. . . . . . . . . . . . . . . . . . . . . 94

3.30 The lowest zeros from three volumes 44,64 and 84(from left to right).The error bars have taken account of both of the Monte Carlo sta-tistical error(seeds) and the Chebyshev interpolation error(orders).The three guidelines are the fits for the first, second and third low-est zeros, using only the zeros of 64 and 84. They intersect the realaxis approximately at the same point β = 1.01134. The diamondson the real axes(Imβ=0) are the double-peak β’s from Table.(3.6). . 96

3.31 The log-log plot of the imaginary part of the lowest zero vs thelattice size L = 4, 6, 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.32 TheNp scaled zeros discussed in the text. . . . . . . . . . . . . . . . 99

3.33 The orders of power law from the fits are plotted versus the lowestsize L that is included in the fit. . . . . . . . . . . . . . . . . . . . . . 99

3.34 The first derivative of f (x) from different volumes. . . . . . . . . . 100

3.35 The locations of the two real roots of f ′′(x) = 0 are plotted versusthe lattice size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

C.1 The log-log plot of log(∆I/I)− 12A as a function of the number of grids

used in the Trapezoidal integration. We use A = 0.004069 and cutthe integral range beyond the precision goal at ym = 582.692 Thefit gives a linear relation of −4.248355148205 + 2.0000000000000x,which agrees with ln((2π2)/(Ay2

m)) = −4.2483551482054. . . . . . . 108

xiv

1

CHAPTER 1INTRODUCTION

1.1 Motivation and Overview

One very important property of the strong interactions is that the coupling

is running with the energy scale at which the interaction takes place [70, 35]. It

was discovered that for models like QCD (quantum chromodynamics) with the

SU(3) gauge group, the beta function, which describes how fast the coupling is

running with the scale of reference, is negative with 16 or less flavors of quarks

(asymptotically free). There has been a renewed interest [71, 36] in the scenario

of the beta function vanishing at a nontrivial infrared fixed point, in addition

to the ultraviolet fixed point at the zero coupling. This results in the loss of

confinement and the recovery of conformality. It was speculated [49, 6] that

the generalization of the fixed points to the complex coupling plane can help to

describe such a scenario. Along the same consideration, the generalization of

the renormalization group (RG) transformation flows to the complex plane was

proposed [24] and it was found out [61] that the flows, coming out of the weakly

coupled fixed point, are controlled by the ”gate” formed by the complex zeros of

the partition function (Fisher’s zeros). It was observed that if the zeros remain

at a finite distance from the real axis when the volume increases, the flows end

at the strongly coupled fixed point and it leads to a confinement theory for the

non-Abelian gauge fields on a lattice. Complex zeros that are approaching the

real axis will block the flow and therefore, a deconfining phase will appear. So to

understand the confining or deconfining nature of the gauge models, the analysis

of the Fisher’s zeros in lattice gauge theory plays an important role. As the two

simplest but contrastingly different examples, the SU(2) and U(1) lattice gauge

theories with a fundamental Wilson action are the ideal candidates for such a

2

purpose. The SU(2) model is believed [56, 16] to be free of a deconfining phase

transition in the continuum limit while the U(1) model was shown [47, 18] to

have a phase transition but the order of which remains controversial. A clear

picture on the structure of the Fisher’s zeros of the SU(2) pure gauge theory at

zero temperature will put us in a better position to understand the physics when

finite temperature or multiple favors of quarks are introduced.

The locations of the Fisher’s zeros will help the understanding of the RG

flows, while the distances of these zeros will help to understand the nature

of critical behaviors, for instance, the order or strength of phase transitions.

Although the U(1) gauge theory on a lattice has been investigated extensively in

the last decade and it is commonly accepted to possess a phase transition, there

are still arguments regarding the order of the transition [50, 47, 46, 18]. Fisher’s

zero provides an efficient tool for such investigations. It was shown [45, 43] that

the critical exponents which will characterize the order of the phase transition can

be computed through the finite size scaling of the Fisher’s zeros and the strength

of the transition can be revealed by the angle that the zeros pinches the real axis.

So it is very interesting to know, with precise calculations of the Fisher’s zeros for

the U(1) gauge model, whether we can end the controversy of the order of the

transition.

1.2 Introduction to Lattice Gauge Models

In this section, the basic formalism of Lattice Gauge Theory will be briefly

given for the sake of convenience in the following discussions. We will mainly

follow the conventional notations. More general and intensive discussions can

be found in [53, 52, 48, 76, 17, 69, 41, 42, 80].

In Lattice Gauge Theory, a lattice is usually referred to a discrete approxi-

mation of the space-time continuum. In particular, for the discussion that does

3

not involve finite temperature where the temporal direction is treated differently,

a lattice will typically mean a symmetric grid L×L×L×L in the four dimensional

hyper cubic lattice where L denotes the number of sites with the same equal

spacing a in each direction.

The Lattice is not a physical reality, but rather a treatment that builds the

theoretical models on a solid mathematical basis. It has many advantages. First

it has a built-in cutoff which removes the ultraviolet divergence in the quantum

field theory. Second, it makes the degrees of freedom countable and allows us

to use the methods of statistical mechanics, which provides a deep connection

between these two disciplines. Thirdly and more importantly, it naturally results

in a confining behavior in the strong coupling limit. However, the lattice spacing

a is artificial. Actual physics should be expected when the continuum limit a→ 0

and the infinite volume limit are taken.

1.2.1 Feynman Path Integral

In quantum mechanics, the transition amplitude of a particle from (t1, q1) to

(t1, q2) can be described by a statistical-like formula

⟨q2|e−iH(t2−t1)|q1⟩ =∫ q2

q1

D[q] eiS, (1.1)

where the action S(q) =∫Ldt is playing the role of Boltzmann factor. The inte-

gration∫D[q] on the right hand side of the equation means summing over all the

possible paths. However the definition is not complete unless an implementation

of ”summing all the possible paths” is given. Time and space extends can be split

into equal spacing grids so that the enumeration of paths is possible.

It can be generalized to the quantum fields where we can define the vacuum

4

to vacuum expectation to be

Z ≡ ⟨0|e−iHt|0⟩ =∫D[ϕ] eiS, (1.2)

where the integral now is summing over all the possible fields configurations

which is well defined if these fields are now confined on the sites of the space

time lattice. Therefore the action can be defined as S = a4 ∑xL. Z is called

partition function. The physical observables become the expectation which is

averaging over all possible field configurations, for example,

⟨0|T[ϕ(x1)ϕ(x2)...ϕ(xn)]|0⟩ = 1Z

∫D[ϕ]ϕ(x1)ϕ(x2)...ϕ(xn) exp(iS). (1.3)

To complete this analogy with statistical mechanics, we will need to appeal

to the Euclidean time, i.e., τ = −it which differs the physical time t by a Wick

rotation, and will replace iS by −S′.

1.2.2 U(1)

Inspired by Wegner’s Ising lattice gauge theories[74, 53], K. Wilson[76]

generalized it to the continuous gauge groups. Instead of a spin that admits only

value ”up” and ”down” in the Ising model, an internal planar angle parameter

θ(n) is attached to each site of the lattice, n. To compare this angle at different sites,

any two neighboring sites are then bridged by tagging the U(1) group elements

Uµ(n) = exp(iθµ(n)) to the link (n,n + µ) where µ is one of the four positive

directions departing from site n. In the negative direction on the same link, we

define θ−µ(n + µ) = −θµ(n) and it follows that U†µ(n) = U−µ(n + µ).

A plaquette is the smallest square consisting of four links in a plane spanned

by any two directions µ, ν. In the following, we will only consider on a finite,

4-dimensional symmetric lattice L4 with periodic boundary condition and the

total number of plaquettes isNp = 6L4.

5

With the difference operator

∆µθ(n) := θ(n + µ) − θ(n),

it can be easily shown that the following expression which is the sum of all four

links of the plaquette (n, µν),

θµν(n) := ∆µθν(n) − ∆νθµ(n) (1.4)

= θµ(n) + θν(n + µ) + θ−µ(n + µ + ν) + θ−ν(n + ν)

is gauge invariant under the local gauge transformation θ(n)→ θ(n)+χ(n) where

χ is an arbitrary function. To see that, with the transformation, the link variable

will transform like eiθµ(n) → e−iχ(n)eiθµ(n)eiχ(n+µ) or θµ(n) → θµ(n) + ∆µχ(n). This is

in analogy to electrodynamics: Aµ ∼ θµ and Fµν ∼ θµν.

Now we can define the action

S =∑n,µν

(1 − Re(UUUU)) (1.5)

=∑n,µν

(1 − Re exp iθµν(n)

)=

∑n,µν

(1 − cosθµν(n)).

With the continuum limit being taken∑

n,µν →∫

d4x/a4 and a weak coupling

approximation 1 − cosθµν ≈ θ2µν/2 1, the Euclidean action of electrodynamics is

1No Einstein automatic summation rule here

6

recovered1g2 S =

14

∫d4xFµνFµν, (1.6)

where we have defined θµ = agAµ and θµν = a2gFµν and g is the coupling strength.

Note the auto-summation in FµνFµν gives a factor 2 because (n, µν) and (n, νµ) are

actually the same plaquette.

The partition function for the U(1) lattice gauge fields can then written as

Z(β) =∏

l

( ∫ 2π

0

dθl

2π

)exp(−βS), (1.7)

where β = 1/g2 and the product of the integration covers all the links l on the

lattice.

1.2.3 SU(2)

The construction can be extended to the non-Abelian case, specifically, the

SU(2) gauge fields. Instead of having a planar angle parameter sitting at every

site as in the U(1) case, we assign a ”vector” θ(n) in the internal space spanned

by the three Pauli matrices

τ1

2=

0 1

1 0

, τ22 =

0 −i

i 0

, τ3

2=

1 0

0 −1

, (1.8)

say, θ(n) = 12agτiAi(n) where Ai are the three parameters that uniquely determine

the vector. τi/2 are the three elements of the su(2) Lie algebra and satisfy [τi, τ j] =

2iϵi jkτk.

In analogy to the case of U(1), to specify the relative orientation of these

vectors at two neighboring sites, we need to define a SU(2) group element

Uµ(n) = exp[12 iagτiAi

µ(n)] at the corresponding link in the positive direction,

while Uµ(n)† = U−µ(n + µ) for the negative direction.

7

We can define the plaquette element Up ≡ Uµν(n)

Up = Uµ(n)Uν(n + µ)U−µ(n + µ + ν)U−ν(n + ν) (1.9)

= Uµ(n)Uν(n + µ)U†µ(n + ν)U†ν(n)

= exp[iθµ(n)] exp[iθν(n + µ))] exp[−iθµ(n + ν)] exp[−iθν(n)]..

Under the local rotation exp(−iϑ(n)) (where ϑ = τiχi/2) in the internal

space, the link variable undergoes Uµ(n)→ exp(−iϑ(n)Uµ(n) exp(iϑ(n+µ), which

implies that the trace of the plaquette element TrUp is invariant. So the action can

be defined as

S =∑

p

[1 − 12

Re TrUp], (1.10)

where the summation is over all plaquettes.

To compare with the physics in the continuum, we can expand Eq. (1.9)

with respect to a, taking only the non-vanishing lowest order of a,

Up = exp[ia2gFµν +O(a2)], (1.11)

where we have substituted

Fµν = ∂µAν − ∂νAµ + ig[Aµ,Aν] (1.12)

and Aµ ≡ 12τiAi

µ(n). We can modify Eq.(1.11) further and replace the summation

by an integral and Eq. (1.10) becomes

4g2 S =

∫d4x

12

FµνFµν, (1.13)

which is the standard Yang-Mills action in the Euclidean space.

So we can write down the partition function for the SU(2) lattice gauge fields

using the action in (1.7),

Z(β) =∏

l

( ∫dUl

)exp(−βS), (1.14)

where β = 4/g2 for SU(2). The integration is over all the link variables Ul and dUl

8

is the Haar measure which depends on the parametrization of the SU(2) group.

SU(2) With an Adjoint Term

There are various evidences showing that there should be no phase tran-

sition between the strong and weak coupling regions [56, 16]. Since the action

should be a real function of the plaquette variable, it can be expanded using the

character expansion with all the irreducible representations of SU(2). If we only

consider the fundamental representation ( j = 1/2) and the adjoint representation

( j = 1), the modified total action is [12]

βS + βASA ≡ β∑

p

(1 − 12

Re TrUp) + βA

∑p

(1 − 1

3Re TrU(A)

p

), (1.15)

where U(A) is the adjoint representation of the fundamental representation which

is satisfying U†τkU = [UA]klτl. For SU(2), it is a three dimensional representation.

It can be shown that the fundamental and the adjoint representation are related

by

TrU(A)p = |TrUp|2 − 1. (1.16)

The partition function is just [12, 34]

Z(β) =∏

l

( ∫dUl

)exp(−βS − βASA). (1.17)

1.3 Fisher’s Zeros and Critical Phenomena

In general, the physical phenomena that the statistical mechanics formalism

is dealing with can be categorized in two different kinds. In the first kind,

the equilibrium descriptions of the thermodynamic functions are smooth and

continuous in responding to the changes of parameters. In the second kind, it

is characterized by irregularities or discontinuities. The system often exhibits a

cooperative nature and may undergo drastic changes from one state to another of

totally different type, for instance, the phase transition from a liquid to a gas. One

9

would suspect that such phenomena can be naturally described by the formalism

( or its extension) of the first kind where the statistical functions are smooth and

free of singularities. For example, the grand partition function is a polynomial of

the fugacity and will not admit a real zero for a finite volume. Singularities only

happen in the V,N → ∞ limit. Lee and Yang showed remarkably [57, 79] that

it is possible to connect the phase transition of the system to the distribution of

zeros of the grand partition function in the complex fugacity plane. For the Ising

model with ferromagnetic interactions, the zeros lie on the unit circle |z| = 1 where

z = e−4βH which is in analogy to the fugacity ( H is the external field). Singular

behavior of the free energy density in the V,N→∞ limit might be interpreted by

the accumulation of complex zeros near the real axis.

As a natural generalization, it was proposed by Fisher [33] that similar

phenomenon may also be true with the partition function zeros in the complex

temperature plane. He showed it with the evidence from the zero-field Ising

model on the two dimensional lattice with a nearest-neighbor coupling K. The

partition function zeros are lying on double circles | tanh(K ± 1)| =√

2 when the

thermodynamic limit is taken. The logarithmic divergence of the specific heat

is connected to the linearly vanishing density of zeros near the real axis [33, 63],

which was confirmed in various other models [62, 72]. The partition function

zeros in the complex temperature ( or coupling) plane are usually called Fisher’s

zeros, to be distinguished from the Lee-Yang zeros.

The finite size scaling of both of the Lee-Yang and Fisher’s zero were in-

vestigated by Itzykson, Pearson and Zuber [40] through the scaling properties

of the distances among the zeros. They showed that the critical exponents were

governing the scaling law of the distances. Following this, Janke et al [43, 44]

10

analyzed the roles of the zero density function in the scaling of the singular phe-

nomena in various discrete models and proposed a way to determine the order

of the phase transitions as well as the transition strength through scaling of the

the zero density. With all these theoretical setups, it is interesting to know how

the Fisher’s zeros can help us to understand the critical behaviors in the discrete

models as well as the continuous field models.

The analysis using Monte Carlo technique on the U(1) lattice gauge field was

pioneered by Creutz [18]. He approached the continuous U(1) gauge field through

the ZN symmetric groups which take U(1) as the limit of N →∞. He found that a

phase transition of order higher than 1 persisted for N ≥ 6 and survived the U(1)

limit. However a hysteresis structure was observed in [47] on a L = 16 lattice

which suggested a first-order transition. It was revisited in [46] through the

simulations on the spherical lattice using finite size scaling of the Fisher’s zeros.

They found the critical exponent ν = 0.365(8) which is corresponding to a L−2.74

scaling of the imaginary part of the zero, apparently excluding the possibility of a

first-order transition. However a high-statistic analysis on the cumulants showed

[50] that the critical exponents ν is size-dependent and is actually ”rolling” toward

a first-order value (0.25). Therefore a reliable classification of such a transition

using improved methodology is yet to be seen to put the controversy at rest.

It has long been known that the pure SU(2) lattice gauge field with a Wilson

action in the fundamental representation is free from a deconfining phase transi-

tion, which suggests a possible theory of uniform confinement including both of

the weak and strong coupling regimes. In a series of papers [28, 27, 29], Falcioni

et al. studied the location of the complex zeros using single-point reweighting

and the high-order strong coupling expansion method. They spotted the zero

around β = 2.225 + i0.155 on a 44 lattice. What remains interesting is how the

11

zeros depend on the volume and what is the structure of the zero that is charac-

terizing such a system baring no transition in the continuum limit. It was shown

[24] with numerical evidences from the finite-volume large-N O(N) model and

the hierarchical Ising model that the Fisher’s zeros are located at the boundary

of the complex basin of attraction of the infra-red fixed points. With the general-

ization of the renormalization flow to the complex β-plane, it was shown that the

Fisher’s zeros serve as a ”gate” to control the flows which are starting from the

weak coupling fixed point. A confinement theory may be recovered if the zeros

are not blocking the flows from reaching the strong coupled fixed point.

In this thesis, we intend to address part of the questions raised above by

studying the Fisher’s zeros using the density of states method. The thesis is

organized as follows.

In Chapter 2, we give the definition of the density of states and analyze its

general properties. Based on the saddle point approximation, we reconstruct the

density of states from series expansion ( strong and weak coupling expansions).

We analyze the limitation of the perturbative approaches, with the presence

of complex singularities. We then appeal to the numerical calculations using

the Monte Carlo method to approximate the gauge integrations. We collect

sampling data indexed by β. For the SU(2) model, we reconstruct the numerical

representation of the density of states from these spectra of data through the

Ferrenberg-Swendsen’s reweighting method. For the U(1) gauge model, the

density of states are obtained using the Multi-canonical simulations (calculated

by Alexei Bazavov). We also study the volume dependence of these density of

states.

Chapter 3 is devoted to the Fisher’s zeros. We start with the single point

reweighting which is the conventional method to locate the zeros. We point out

12

its limitation for the SU(2) model by showing the fact the zeros are mostly lying

outside the so-called circle of confidence [2]. We examine this carefully with

quasi-Gaussian toy models. By the assumption of saddle point approximation

which is valid for the SU(2) model, we find that there exists a simple connection

between the Fisher’s zeros and the roots of the second order derivative of the

entropy density function. The latter corresponds to the zeros when an infinite

number of plaquettes is taken while using the finite-volume entropy density

function. We develop a series of tools such as the numerical evaluation of the

partition function and the Cauchy’s loop integration method to find Fisher’s zeros

where the entropy density function is approximated by analytic functions. We

discuss the stability of the polynomial approximation and present the results of

the zeros for the SU(2) model at two different volumes 44 and 64. We apply both

of the polynomial approximation method and the discrete reweighting method

to find the lowest three zeros of the U(1) gauge model at three different volumes

44, 64 and 84. We then discuss the scaling properties of these zeros.

The conclusions are provided in Chapter 4.

13

CHAPTER 2THE DENSITY OF STATES

2.1 The Density of States

In the partition functions of the U(1) and SU(2) model, the integration is

over all the link variables. We are mainly interested in the global properties of

the gauge fields on a lattice, in particular, the zeros of the partition function. It is

convenient to work with the density of states which is defined as the number of

configurations per unit interval in the total action S space. The density of states

wraps all the link variables into one, the total action S. The degrees of freedom

of the system are reduced from 4L4 to 1 which makes the problem into a one

dimensional problem. So now the partition function has a form [22]

Z(β) =∫ 2Np

0dS n(S) e−βS, (2.1)

where Np is the total number of plaquettes. We use the function of density of

states n(S) to abstract the complex integration over the links by using the delta

function,

n(S) =∏

l

∫dθl

2πδ(S −

∑p

(1 − cosθp))

(2.2)

and β = 1/g2 for the case of U(1);

n(S) =∏

l

∫dUl δ

(S −

∑p

(1 − 12

Re TrUp))

(2.3)

and β = 4/g2 for the case of SU(2).

The same concept has been used in the discussions for spin models[3] and

gauge models [2] where it is sometimes called the spectral density. It is often

more convenient to use the average action x ≡ S/Np instead of the total action S

as the variable. Correspondingly we will often use f (x), the logarithm of n(S),

n(S) = eNp f (x). (2.4)

14

In this paper, we will call f (x) the entropy density function. Note that in Eq. (2.1),

the integration bound is from 0 to 2Np. This is true for both U(1) and SU(2) lattice

with an even number of sites in each direction. The first bound is quite obvious

since it is the case when all links are taken to be the identity 1 of the group. The

second bound is not so trivial. Since both U(1) and SU(2) contain the Z2 group

as a subgroup and therefore have the element −1. When L is even, it is possible

to construct a path A (not necessarily connected) which touches each plaquette

once and only once [60]. Changing U→ −U on each link variable along this path

Awhile leaving rest of the links unchanged will result in TrUp → −TrUp for each

plaquette, which makes S to be 2Np. This property also implies a symmetry of

Z(β). For instance, in the partition function (1.14), by switching the sign of β, we

have

Z(−β) = e2Npβ∏

l

( ∫dUl

)exp

[− β

∑p

(1 + (1/n)Re TrUp)]. (2.5)

By changing variable Ul → Ul ∗ (−1) at each link along the path A, the integral

is not affected. But this will result in TrUp → −TrUp which makes the expression

after the product sign in Eq. (2.5) identical to Z(β), i.e.,

Z(−β) = e2Npβ Z(β). (2.6)

Similar discussion applies to the case of U(1) gauge fields and in principle any

gauge group which has the Z2 group as a subgroup. This symmetry has an

interesting relation with the Dyson instability [8] (see also [67]). As a consequence,

there is an obvious symmetry in n(S), say,

n(2Np − S) = n(S), (2.7)

or, by our definition of f (x),

f (x) = f (2 − x) . (2.8)

To analyze the property of f (x), we should take a look at the simplest example,

15

the SU(2) fields on a single plaquette.

SU(2) on One Plaquette

A simple but inspiring example is from the SU(2) gauge fields on a single

plaquette, the partition of which, by suitable choice of gauge, can be reduced to a

integration over just one link [60]. We can easily write out their density of states

by

n1pl.(S) =2π

√S(2 − S) . (2.9)

The behavior of the partition function at large β is related to the property of n(x)

near x = 0. In this one-plaquette example, the function n(x) ∝√

x where x is

small. The entropy density function f (x) has a logarithmic singularity at x = 0.

The same thing happens on the other end x = 2. By expanding√

2 − x in n(x) and

do the integration term by term, one can expand the partition function Z(β) in

terms of 1/β explicitly [58]. The large order behavior of the expansion is, of course,

determined by the large order behavior of the expansion in n(x) and is dictated by

the branch cut at x = 2. However, if the integration bounds are pushed to infinity

which is usually taken for mathematical simplicity, the expansion over 1/β will

suffer a zero radius of convergence [25, 60]. If we stick with the finite bounds

of integration [0, 2], we end up with a theory with finite radius of convergence

[58], but the trade-off is that the coefficients need to be expressed in terms of the

incomplete gamma functions. From the perturbative point of view, it is more

economical to expand n(x) instead of the underlying partition function.

The region in 1 < x < 2 is hardly explored in the physical world simulation

because any average plaquette ⟨x⟩ in this region corresponds to a negative β. The

simulations near β = 0 may generate certain number of configurations near x ≥ 1

region. But the bound x = 2 is beyond physical reach in reality. However, due

to the symmetry, this region can be investigated by taking β → −∞ [60], which

16

falls in the common agreement that the large order behavior of the weak coupling

series can be approached in terms of the behavior at small and negative coupling.

2.2 Saddle Point Approximation

With the notation of the average action x, we can rewrite the partition

function into

Z(β) =∫ 2

0dx exp

[Np( f (x) − βx)

]. (2.10)

The total number of plaquettes isNp = 6×L4, which is typically large and justifies

the saddle point approximation. By definition, the saddle point x0 is given by

f ′(x0) = β. (2.11)

x0 depends on the value of β and can be complex. If the saddle point is lying

on the real axis, then the approximation is straightforward. One can expand the

function f (x)−βx at x0 ( assuming there is only one saddle point in the integrating

path) by

f (x) − βx = f (x0) − βx0 + f ′′(x0)(x − x0)2/2 + ....

We ignore the higher orders and make it into a Gaussian distribution which works

well forNp →∞. We then have

Z(β) ≈ eNp[ f (x0)−βx0]

√2π

−Np f ′′(x0), (2.12)

where f ′′(x0) < 0. The same formula holds for the case when x0 is a complex

number if Re f ′′(x0) < 0. Let us discuss the case that only a saddle point and no

pole is presented in the region close to the real axis, to avoid complicating the

discussion. By Cauchy’s theorem, we can deform the integration path through

the saddle point along the so called path of steepest descent. It is a path along

which Im[ f ′′(x0)(x− x0)2/2] keeps to be a constant and along which Re ( f (x)− βx)

reaches maximum at x0. Let x − x0 = teiθ and f ′′(x0) = | f ′′(x0)|eiθ0 , then the path is

17

determined by

2θ + θ0 = nπ, n = 0, 1, 2, ... (2.13)

In practice, instead of finding the curve of the steepest descent, we take a linear

approximation, by integrating only along the straight line that is tangent to the

path of the steepest descent at x0. The real part is then approximated by a Gaussian

function, and then similarly we have

Z(β) ≈ eNp[ f (x0)−βx0]e−i(θ0−π)/2

√2π

−Np| f ′′(x0)| , (2.14)

but that is just Eq. (2.12).

In many occasions we want to work with the free energy density

F (β) ≡ − 1V

ln Z(β) (2.15)

where V is the total number of lattice sites V = L4. The average plaquette is

defined by

P(β) ≡ ⟨x⟩ = − 1Np

∂ ln Z(β)∂β

=∂(F /6)∂β

. (2.16)

With the saddle point approximation, the free energy density can be written

as

F /6 ≈ βx0 − f (x0) +1

2Npln

[− f ′′(x0)

]+ const.. (2.17)

Note that f ′′(x0) = ∂β/∂x0. In Eq(2.16), we replace the derivative over β

using the relation∂∂β=

1f ”(x0)

∂∂x0, (2.18)

then the average plaquette can be approximated by

⟨x⟩ = P(β) ≈ x0 +1

2Np

f ′′′(x0)f ”(x0)2 . (2.19)

The second central moment or fluctuation can also be calculated in a similar

18

manner,

⟨∆x2⟩ = − 1Np

∂2(F /6)∂β2 (2.20)

≈ − 1Np f ”(x0)

[1 − 1

2Np

f (4) f ” − 2 f ′′′2

f ”(x0)3

],

where ∆x = x − ⟨x⟩. In general, for n > 2

⟨∆xn⟩c ∝[

f ”(x0)2]3−2nN1−n

p , (2.21)

where ⟨∆xn⟩c are the central moments.

2.3 Series Expansions of n(S)

When it comes to a theory with interactions, the perturbative expansions

with respect to the interacting strength often play an important role. The most

significant discovery about the strong interaction is that quarks which are in-

teracting inside a hadron or meson are asymptotically free and the coupling is

running. The perturbative expansions in weak and strong coupling limits are

both of great interest[76, 77, 4].

In the pure SU(2) and SU(3) gauge models where no quarks are present,

there is no numerical evidence of a phase transition between the weak and strong

coupling regime, which makes us to think that there is a way to match the strong

and the weak coupling expansions. However there is a difficulty. Fig. (2.1)

[22] shows a plot of the average plaquette P as a function of β = 4/g2 which is

computed through three different methods: the weak and strong coupling series

expansions as well as the numerical simulation results. Neither the strong nor

the weak expansions works in the region around β ∼ 2, constrained by their finite

radius of convergence. We know that a finite region of convergence is defined by

a singularity on the boundary. So it should be the singularities in the complex

β-plane that prevent both of the expansions to go further and give a overlapping

19

coverage. [51]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5 1 1.5 2 2.5 3 3.5

P

β

Plaquette

weak order 3weak order 4weak order 5

strong order 3strong order 5strong order 7

numerical

Figure 2.1: Weak and strong coupling expansions of the average plaquette P forSU(2) at various orders in the weak and strong coupling expansion compared tothe numerical values.

In this section, we will investigate these expansions in the context of the

density of states which can be compared with the simulation results closely.

Strong and weak expansions for both of the SU(2) and U(1) gauge fields are

discussed.

2.3.1 Strong Coupling Expansions

The strong coupling (small β) expansions of various lattice gauge theories

including U(1) and SU(2) have been intensively studied in [4] and a follow-up

correction [5]. They gave the results of the free energy density F up to the order

of β16. Note that by our convention, their βB = β/2 and their action is shifted by

20

a constant Np. As a result, we have an additional term β in the expression of the

free energy density, F /6. We shall write it out explicitly,

F /6 ≈ const. + β +∑m=1

a2mβ2m , (2.22)

where the coefficients a2m have been converted accordingly and summarized

in Table. (2.1). From the relation to the average plaquette P = ∂F /∂β, we

automatically have the expansion version of P

P(β) ≈ 1 +∑m=1

2ma2mβ2m−1 . (2.23)

When β is small, the most probable region is around x = 1. So strong

coupling approximation is equivalent to the expansion of f (x) about x = 1. Thus

it is convenient to use the variable y which is defined by y ≡ x − 1 and the

corresponding entropy density function g(y) ≡ f (1 + y). The symmetry in Eq.

(2.8) ensures that g(y) is an even function in −1 < y < 1 and therefore only even

orders of the expansions present, explicitly,

g(y) ≈∑m=0

g2my2m , (2.24)

We will base our discussion on the approximation for P(β) in Eq. (2.19)

which is justified in the large volume limit. We should start with the following

two relations

g′(y0) = β, (2.25)

P(β) ∼ x0 = 1 + y0(β), , (2.26)

where the first equation is the saddle point condition and the saddle point y0 is a

function of β implicitly. Let’s look at the lowest order. Eq. (2.24) and Eq. (2.25)

imply that 2g2y0 ≈ β. Then combined with Eq. (2.23) and relation Eq. (2.26), it

21

gives

y0 ≈ β/2g2 ≈ 2a2β , (2.27)

which means g2 = 1/(4a2). In general, Eq. (2.25) can be inversed perturbatively so

that the saddle point y0 can be taken as a series function of β, say y0(β). We then

can compare this expansion with the strong coupling expansion Eq. (2.23) order

by order through the relation Eq. (2.26). We match the coefficients of the two

sides and will end up with a set of relations between the expansion coefficients

g2m and a2m. We should write the leading several orders below,

g4 = −a4/(16a42),

g6 = (4a24 − a2a6)/(64a7

2),

g8 = (−24a34 + 12a2a4a6 − a2

2a8)/(256a102 ),

...

We can solve these equations starting from the lowest orders and hence will

be able to determine all the g2m by the values of a2m from the strong coupling

expansions which have been worked out in [5]. The results can be found in

Table.(2.1) [22] and Table.(2.2) for the SU(2) and U(1) model respectively.

Fig. (2.2) and Fig. (2.3) [22] show various orders of these strong coupling

expansions compared to the numerical simulation results which will be discussed

in Section. 2.4. Worse agreement between the expansion and numerical data in

the region away from x = 1 appears in the U(1) model, which implies a smaller

region of convergence. This is arising from the closer singularities to the real axis

in the U(1) model than the case of SU(2).

Similar to the one-plaquette model, in general, n(S) vanishes at S = 0 and

S = 2Np which correspond to the logarithmic divergence of f (x) at x = 0 and 2 (

22

m a2m g2m h2m

1 − 18 −2 −5

4

2 1384 − 2

3 − 724

3 − 79216

209

8936

4 31184320 −16

45 − 121720

5 − 445188473600 − 16816

2025 −660498100

6 26488317836277760

3197368505

256639368040

7 − 40365183235962880 −3724816

297675 −147716891190700

8 18260178731095860905574400 − 163150033

255150 − 26100178034082400

Table 2.1: SU(2) Strong coupling expansion coefficients.

m a2m g2m h2m

1 −14 −1 − 3

4

2 164 − 1

4 − 18

3 − 13576

4336

2318

4 77949152 − 19

192 − 7192

5 − 11819614400 −7343

1800 − 72531800

6 201737310168320

46533125920

46641125920

7 − 202242911734082560 − 983357143

1693440 − 9832966631693440

8 5775175013202937204736 −201757201579

46448640 − 20175575005946448640

Table 2.2: U(1) Strong coupling expansion coefficients.

or g(y) at y = ±1). A more general expression of f (x) which takes this logarithmic

divergence into account would be

h(y) ≡ g(y) − A(ln(1 − y2)) (2.28)

23

-2

-1.5

-1

-0.5

0

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

f(x)

x

SU(2) strong coupling 64

order 2order 4order 6order 8order 10order 12order 14order 16

numerical

Figure 2.2: Numerical value of f (x) compared to the strong coupling expansionat successive orders for the SU(2) lattice gauge fields.

-1

-0.8

-0.6

-0.4

-0.2

0

0.4 0.6 0.8 1

f(x)

x

U(1) strong coupling 64

order 2order 4order 6order 8order 10order 12order 14order 16

numerical

Figure 2.3: Numerical value of f (x) compared to the strong coupling expansionat successive orders for the U(1) lattice gauge fields.

where A is associated with the coefficient of the logarithmic term in the weak

coupling expansion. We will derive the value of A in Section. 2.3.2 and just cite

24

the result here. In the V → ∞ limit, AU(1) = 1/4 for U(1) and ASU(2) = 3/4 for

SU(2). The expansion of h(y) is

h(y) ≃∑m=0

h2my2m . (2.29)

We can easily write down the coefficients h2m which are merely the g2m with

the expansion coefficients of the logarithm term subtracted. The values of h2m

are also listed in Table.(2.1) and Table.(2.2). In the case of SU(2), the coefficients

g2m and h2m are compared on a logarithmic scale in Fig. (2.4) [22]. The two sets

of coefficients merge rapidly to the same order of magnitude, indicating that the

singularities are hardly affected by these logarithmic poles.

-2

-1

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18

Ln|coeffs.|

2m

strong coupling expansion

ln|g2m|: ln|h2m|:

Figure 2.4: The expansion coefficients h2m and g2m in the log scale are plottedversus the order 2m.

The goodness of the strong coupling expansion can be seen from the error

diagrams of different orders compared with the numerical results. First let us

25

look at the average plaquette P(β). The error diagrams of P for the SU(2) and

U(1) models are shown in Fig. (2.5) and Fig. (2.6) which display the logarithm of

the difference between perturbative P of successive orders and those calculated

directly from data for variousβ’s. The similar error diagrams can also be examined

from how the expansion of f (x) departs itself from the numerical data. The data

here means the numerical construction of the density function f (x) from Monte

Carlo simulations using either simple patching method or the multi-histogram

reweighting method ( which will be covered at length in Section 2.4.2). To remove

the dependence of the overall constant, we normalize the numerical f (x) at x = 1

by setting f (1) = 0. The successive orders of expansion of f (x) subtracting from

the numerical f (x) are plotted in the logarithmic scale in Fig. (2.7) and Fig. (2.8)

for SU(2) and U(1) respectively.

-15

-10

-5

0

5

10

15

0 0.5 1 1.5 2 2.5 3 3.5 4

Log|

∆P|

β

SU(2) 64 Strong Coupling

order 2order 4etc...

order 14order 16

num. error

Figure 2.5: Logarithm of the absolute difference between the numerical dataand the strong coupling expansion of P at successive orders for the SU(2) model(error diagram). The numerical data are taken from the volume 64. The error onnumerical P are obtained from 50 bootstraps.

The properties on the convergence of the expansion of P and f can be

26

-20

-15

-10

-5

0

5

10

0 0.5 1 1.5 2

Log|

∆P|

β

U(1) 44 Strong Coupling

order 2order 4etc...

order 14order 16

num. error

Figure 2.6: The error diagram of P at successive orders for the U(1) Model. Thenumerical data are from the volume 44. The error is estimated through 20 seedsof data.

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

0.4 0.6 0.8 1

Log|

∆f|

x

SU(2) 64 strong coupling

order 2order 4etc...

order 14order 16

num. error

Figure 2.7: The error diagram of f (x) at successive orders for the SU(2) Model.The numerical data are from the volume 64. The error on numerical f are obtainedfrom 50 bootstraps.

27

-20

-15

-10

-5

0

5

0.4 0.6 0.8 1

Log|

∆f|

x

U(1) 84 strong coupling

order 2order 4order 6order 8

order 10order 12

num. error

Figure 2.8: The error diagram of f (x) at successive orders for the U(1) Model. Thenumerical data are from the volume 84. The error is estimated through 20 seedsof data.

analyzed from these error diagrams. From Fig. (2.5) [22], we see that the larger

orders get worse error beyond β ∼ 2 and improve in the region β < 2.0. The

curves make a cross between β = 1.5 and 2. This is a sign of finite radius of

convergence [59]. Similarly, the larger order errors for f cross at some x between

0.4 and 0.6 which are close to the values of P at which the error curves cross. So

both of the graphs show the evidence of finite radius of convergence and suggest

there are singularities near the crossing region of the error diagrams. A similar

discussion applies to the U(1)’s error diagrams (Fig. (2.6) and Fig. (2.8) ) around

β ∼ 1 and P ∼ 0.5.

2.3.2 Weak Coupling Expansions

Now let’s continue the discussion to the weak coupling (large β) expansion

of f (x). For the sake of convenience, we switch the notation back to x, f (x). Again,

the discussion is based on the saddle point approximation which should be valid

28

in the infinite volume limit. We shall start with the expansion of the average

plaquette P using the form

P(β) ≃∑m=1

bmβ−m . (2.30)

As we have seen, f (x) has two logarithmic singularities at x = 0 and 2 where the

weak coupling is expanded about. Thus for f (x) around x = 0 we have the ansatz

f (x) ≃ A ln(x) +∑m=0

fmxm . (2.31)

Eq. (2.25) and Eq. (2.26) are still valid here in the infinite volume limit. For the

lowest order (zeroth order) where we only keep the logarithmic term, the saddle

point condition Eq. (2.26) implies that x0 = A/β. On the other hand, Eq. (2.25)

gives P(β) ≈ x0. As a direct consequence, we have A = b1. Disregarding the

volume correction, the perturbation can be performed order by order without

much difficulty. Thus all the coefficients fm can establish relations to the known

coefficients bm. The lowest several orders of equations are like

f1 = b2/b1, (2.32)

f2 = (b3b1 − b22)/(2b3

1),

f3 = (2b32 − 3b1b2b3 + b2

1b4)/(3b51).

In the infinite volume limit, these coefficients can be worked out easily if we have

the weak coupling expansion coefficients bm.

However we need to take into account that at finite volume, the density

function f (x) receives sizable corrections from the volume in the weak coupling

region (small x). Fig. (2.9) [22] shows such a fact. The saddle point approximation

of P should also be corrected with the 1/V effects. The relation Eq. (2.26) should

be replaced by the one with the first-order volume correction, explicitly

P ≈ x0 +1

12Vf ′′′(x0)f ′′(x0)2 , (2.33)

where V = Np/6. To compare the expansion results with the data which are

29

-0.001

0

0.001

0.002

0.003

0.004

0.005

0.2 0.4 0.6 0.8 1

∆ln n(S)/Np

S/Np

SU(2) Diff 44 and 64

Figure 2.9: The difference of f (x) for two different volumes 44 and 64 is plottedas a function of x = S/Np. We see that in the weak coupling region (small x), thevolume effect is apparent and cannot be ignored in the expansion.

calculated at finite volumes, we have to take the volume correction into account.

Assuming the weak coupling expansion coefficients with volume correc-

tions have been worked out as

bm = b̄m + b′m/V + b′′m/V2 + ...

(2.34)

We should examine how this will make a difference in the coefficients A and

fm. At the lowest order, we have f (x) = A ln x and by Eq. (2.33), we have

P ≈ x0(1+ 1/(6VA)). Additionally, with Eq. (2.25), we have x0 = A/β. Combining

with the first order of the expansion P ≈ bm/β, we can have the expression for A

A = b̄1 + (b′1 −16

)/V. (2.35)

For higher orders, we can follow the same virtue to work out all the relations

between fm and bm order by order. The lowest several order will now be corrected

30

as

b1 = A +1

6V, (2.36)

b2 = (A +1

6V) f1 ,

b3 = (A +1

6V)( f 2

1 + 6A f2) − 4A2 f2 ,

...

U(1)

In [38], the average plaquette P(1/β) 1, was expanded up to the fourth order

of 1/β where the coefficients of the first and the second orders are exact due to

the simple loop diagrams. The third and fourth order involve two diagrams

for which they applied numerical calculations. The first order coefficient reads

b1 = (1/4) − 1/(4V). By Eq. (2.35), we can easily see that

A ≈ 1/4 − (5/12)(1/V) . (2.37)

The second order with correction is b2 = (1/32) − (1/16)(1/V), so with Eq. (2.37),

we have

f1 ≈ 1/8 − (1/8)(1/V) . (2.38)

For order three and four, numerical results exist for the volume 44 and 64. The

results are summarized in Table (2.3). The coefficients of the infinite volume limit

as well as that of the two lowest volumes 44 and 64 are given. The upper half is

the list of the expansion coefficients of the average plaquette P with respect to 1/β

[38]. The lower half is the corresponding expansion coefficients of f (x). Volume

effects from 44 and 64 are also given.

1They used a notation which differs from ours by a constant 1 due the to different conventionin the action.

31

V = ∞ V = 64 V = 44

b114

12955184

2551024

b2132

2171747375208971104256

650252097152

b3 0.01311 0.01309 0.01296

b4 0.00752 0.00749 0.00739

A 14

388315552

7633072

f118

129510368

2552048

f2 0.07363 0.07359 0.07314

f3 0.07638 0.07605 0.07515

Table 2.3: The weak coupling expansion for U(1) gauge fields.

To compare with the numerical results, there is a overall constant f0 in the

expansion which cannot be determined using the analysis discussed above, and

hence need to be fitted from the data. We take the numerical f (x) at some small x

to fix f0. The curves of successive orders are plotted against the numerical data in

Fig. (2.10). The error diagrams of f and P are given in Fig. (2.11) and Fig. (2.12).

SU(2)

An expression of the lowest order of the weak coupling expansion b1 on the

SU(2) lattice can be found in [37, 15]. For the case Nc = 2 and D = 4, we have

explicitly

b1 = (1/4)(1 − 1/(3V)) . (2.39)

Similarly, using Eq. (2.35) we have

A ≈ 3/4 − (5/12)(1/V) . (2.40)

32

-1

-0.8

-0.6

-0.4

-0.2

0

0 0.2 0.4 0.6

f(x)

x

U(1) weak coupling 84

order 0order 1order 2order 3

numerical

Figure 2.10: The successive orders of f (x) expansion from U(1) weak couplingexpansion are plotted in contrast with the numerical f (x) from multi-canonicalsimulations.

-14

-12

-10

-8

-6

-4

-2

0

0 0.2 0.4 0.6 0.8 1

ln|Error|

x

U(1) f(x): weak, 4x4x4x4

order 0order 1order 2order 3

num. error

Figure 2.11: The error diagram of f (x) at successive orders of weak couplingexpansion for the U(1) model.

The first order volume correction (−1/(3V)) in the expression of b1 comes from

the absence of zero mode (−1/V) in a sum given in [37] plus the contribution of

33

-14

-12

-10

-8

-6

-4

-2

0

0 1 2 3 4 5 6 7 8 9

ln|Error|

β

U(1) Plaquette 4x4x4x4

order 1order 2order 3order 4

num. error

Figure 2.12: The error diagram of P at successive orders of weak coupling expan-sion for the U(1) model.

the zero mode with periodic boundary conditions (+2/(3V)) calculated in [15].

No analytical expression is available for orders greater than 1. Numerical values

for the higher orders are available for some specific volume in the literature. An

estimate of b2 can be found in [37] while b3 was given in [1]. Unfortunately, in

these references, the volume 64 is not yet calculated. So rough extrapolations

from the existing data is needed. It shows that for V = 64 uncertainties can be

controlled to be as small as 0.0002 for b2 and 0.0008 for b3. For β ≥ 3, these effects

are close to the numerical errors of P for large β. In the following, we will use the

approximate values b2 = 0.1511 and b3 = 0.1427 for the particular volume V = 64.

We are not aware of any calculation of bm for m ≥ 4 for SU(2). The results are

summarized in Table (2.4). We did a numerical extrapolation for the high order

coefficients and compare the successive orders in Fig. (2.13).

34

m bm fm

1 0.7498 0.2015

2 0.1511 0.0999

3 0.1427 0.0796

4 0.1747 0.0791

Table 2.4: Weak coupling coefficients de-fined in Sec. 2.3.2 for the SU(2) model.The choice of b1 corresponds to V = 64 .

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

f(x)

x

SU(2) 64 weak coupling

order 1order 2

etc.order 15

numerical

Figure 2.13: Numerical value of f (x) compared to the weak coupling expansionat successive orders. The coefficients are calculated from the volume 64.

We can also compare the error diagrams for successive orders in the weak

coupling expansion of P with the numerical values in the case V = 64. The results

are shown in Fig. (2.15) [22]. In the region where the curves are smooth, the error

decreases with the order and appears to accumulate. This is very similar to the

35

-12

-10

-8

-6

-4

-2

0

0 0.2 0.4 0.6 0.8

ln|

∆f|

x

SU(2) 64 weak coupling

order 1order 2

etc.order 15

num. error

Figure 2.14: The error diagram of f at successive orders for the SU(2) model.

case of SU(3) [2]. However, the results show dependence on the estimates for

m ≥ 4 in the SU(2) model. We should note that for large β, the noise in the error

is similar to the numerical error on P. However, this would not be the case if we

had not included the contribution of the zero mode to b1 as shown in the second

part of Fig. 2.15 which is clearly indicating a persisting discrepancy.

We now look at the error diagram of the f (x) compared to the numerical

values at the volume V = 64. The results are shown in Fig. 2.14. In these graphs

we have taken f0 = −0.14663 which maximizes the length of the accumulation

line on the left of Fig. (2.14).

2.4 Numerical Calculation of n(S)

In this section, we will describe the numerical methods that we used to

construct the density of states, which is essential to our discussion of the Fisher’s

zeros.

We used Monte Carlo simulation to approximate the gauge integration

36

-14

-12

-10

-8

-6

-4

-2

0

2 3 4 5 6 7 8

ln|Error|

β

SU(2) Plaquette 6x6x6x6

order 1order 2

etc.order 15

num. error

-14

-12

-10

-8

-6

-4

-2

0

2 3 4 5 6 7 8

ln|

∆P|

β

SU(2) 64 no zero mode

order 1order 2

etc.order 15

num. error

Figure 2.15: The error diagram of P at successive orders for the SU(2)model(above). The error for the numerical P is calculated using 50 bootstraps ofthe data. The graph below is the case without the contribution of the zero modein b1.

37

over the links. We generated configurations at various values of β for multiple

volumes. The program for the SU(2) gauge fields was written by our collaborator

Alex Velytsky. The data were generated, with the collaborating effort by Alan

Denbleyker, for four different lattices: L = 4, 6, 8, 10 in four dimensions. We

then used the Ferrenberg-Swendsen’s multi-histogram reweighting method to

reconstruct the density of states through a spectrum of data. For the U(1) gauge

field, the numerical density of states were provided by our collaborator Alexei

Bazavov using multi-canonical method [11] in three different lattice sizes: L =

4, 6, 8. We used relatively small volumes because considering the precision we

desired, we need larger data samples which is demanding for our relatively small

cluster.

In this section, the numerical setup will be briefly discussed. It is followed

by a detailed description on the implementation of the Ferrenberg-Swendsen’s

method for numerical reconstruction of the density of states for the SU(2) gauge

field. The results are compared with the series expansion that we discussed in

the previous section. The volume dependence of f (x) for the U(1) model will also

be studied.

2.4.1 The Computational Setup

For the SU(2) gauge field, we used Monte Carlo simulation method to

generate the configurations which are the important samples of the full set of

gauge link configurations. We have generated the data for the volumes V =

44, 64, 84 and 104 with various seeds of the random numbers. The spectrum of

β range from 0.001 to 100 and we have used multiple spacings. Small steps

∆β = 0.0015, 0.02, 0.025 is taken for small β, while for large β such as β > 10, big

steps ∆β > 1 are taken to save the computation hours. Each data set consists of

100,000 configurations with the first 200 equilibrium steps removed. Table (2.5)

38

shows the numerical data lineup for the SU(2) gauge field.

L Num. of β’s Unit Configs seeds

4 284 100,000 20

6 447 100,000 15

8 937 100,000 2

10 980 100,000 1

Table 2.5: SU(2) data setup.

We should look at the autocorrelation which is important for the discussion

on the histogram reweighting method. The correlation is defined as

Γ(t) = ⟨(xi − ⟨x⟩)(xi+t − ⟨x⟩)⟩. (2.41)

Fig. (2.16) shows the typical correlation near the critical region of β. The

absolute value of the slope will give the autocorrelation length which is equal to

the integrated autocorrelation if Γ(t) satisfies perfectly an exponential law. The

integrated autocorrelation can be computed by

1 + 2τ =∞∑

t=−∞

Γ(t)Γ(0). (2.42)

Fig. (2.17) shows how the integrated autocorrelation is distributed with

respect to β for the four volumes V = 44, 64, 84, 104. We can see that the 1 + 2τ

reaches a peak at volume 64. As the volume increases, the distribution becomes

flat and hence there is no sign of divergent tendency of the autocorrelation length

which is an important signature for the critical behavior. We should expect no

phase transition for the SU(2) model which has been widely accepted. However

39

çççççççççççççççççççççççççççççççççççççç

ççççççççççççç

ççççççç

ç

çç

ç

çççç

ç

ççç

çç

ç

Slope = 0.091

0 10 20 30 40 50 60 70

-16

-14

-12

-10

-8

tlnHÈG

tÈL

Figure 2.16: The log-log plot of the autocorrelation vs t at β = 2.20 for a 44 SU(2)lattice.

it is still interesting to know how the distribution of the zeros ( as we have

known there must be a singularity near the transition region of the strong and

weak coupling domains ) will characterize such a system. From the graph, we

can see that the distribution peaks at around β = 2.20, 2.34, 2.40 for the volume

V = 44, 64, 84 (there is no obvious peak for the volume V = 104). This gives us a

hint on where the singularity might be located along the real axis.

For the U(1) gauge field, Alexei Bazavov has obtained the density of states

using Multi-Canonical method [11] for three different volumes: V = 44, 64, 84.

Each volume is simulated with 20 seeds of random numbers to provide a partial

set of statistical variety. Each seed will be weighted through the number of

tunnelings between the two attractors (the double-peak nature). The sampling

importance is evaluated by the tunneling numbers. The full set of tunneling

numbers as well as the fluctuations estimated from the seeds can be found in [9].

40

0

4

8

12

16

20

1.8 2 2.2 2.4 2.6 2.8

1+2

τ c

β

Correlation Time : SU(2)

44

64

84

104

Figure 2.17: The integrated autocorrelation time is plotted verse β for four dif-ferent volume V = 44, 64, 84, 104. The peak of the distribution is found at aroundβ = 2.20(for 44), 2.34 (for 64) and 2.40 (for 84). The peak is not obvious for thevolume 104.

V Num. of bins seeds

44 1000 20

64 1000 20

84 1000 20

Table 2.6: U(1) Density of States.

2.4.2 Histogram Reweighting

Monte Carlo data are usually collected by varying parameters like tempera-

ture or coupling strength in discrete steps. The Metropolis algorithm often results

in configurations distributed narrowly around the average value. Hence a single

41

run of such simulations will only cover a small piece of the configuration space

(or phase space) and gives zero statistics for the rest of the space. Since the parti-

tion function always plays a role as the normalization constant, the conventional

method is not capable to address the connection between the runs and therefore

is not able to measure statistical quantities such as free energy or entropy.

However, when these Monte Carlo runs have overlaps in the configuration

space, it is possible to assign weights to these runs according to their statistical

significances at each position in the configuration space and make connections

between the neighboring datasets [10, 32, 31, 55]. The approach is often called the

Ferrenberg-Swendsen’s method. There is also an alternative algorithm developed

by Wang and Landau [73]. However we will focus on Ferrenberg-Swendsen’s

method and provide a practical approach to obtain the initial values which are

important for the convergence of the iterations.

Figure 2.18: The overlapping of neighboring data sets.

Consider the simplest case where there are only two distributions involved.

Let S denote a point in the phase space, for example, the total action for a particular

configuration, and the partition function is Z(β) =∑

S exp[−βE(S)]. Generically we

can write the effective partition combining the two runs (distributions associated

42

with β1 and β2) into ∑S

W(S) exp(−β1E(S) − β2E(S)),

where W(S) is the weight factor at S, connecting both of the distributions.

If there is no overlapping be between the distributions, like the situation

shown in the left graph in Fig. (2.18), W(S) can be taken to be exp[β2E(S)] for all S

on the left to the middle point and exp[β1E(S)] on the right, then the description

remains consistent. However it is not so trivial when the two distributions have

an overlapping region. The weight W(S) has to satisfy certain form to make the

two distributions logically compatible to each other and not violating the fact that

they are both satisfying the Boltzmann assumption Z(β) =∑

S exp[−βE(S)].

The statistical quantity that is connecting the two distributions is the free

energy A ≡ − ln Z(β), and the difference of the free energy can be written as

∆A = A2 − A1 = lnZ(β1)Z(β2)

= lnZ(β1)Z(β2)

∑S We−β1E(S)−β2E(S)∑S We−β1E(S)−β2E(S)

,

= ln⟨We−β1E(S)⟩⟨We−β2E(S)⟩ .

Suppose that there are n1 (n2) configurations in the distribution associated with

β1 (β2). We should consider the continuous version by taking∑

S exp[−βE(S)] →∫dqn(q) exp[−U(q)] where U(q) = βq and we absorb n(q) in the weight W(q).

Requiring the fluctuation of this free energy difference to be stationary with the

weight function, i.e., δ⟨δ2∆A⟩/δW = 0, will result in the fact that W(q) has to

satisfy [10] ( see also Appendix)

W(q) =const

Z0n0

e−U0 + Z1n1

e−U1. (2.43)

Notice that to compute this weight function, one will need the implicit

information about the partition function Z(β) itself. So the equation is rather a

consistent constraint than a calculable implementation.

43

This formula can be generalized to the case of a number of distributions

[31, 55]. Let’s brief the setup of our actual calculations. We can carry out a series

of Monte Carlo simulations at β = β1, β2, ...βR. There are Nα configurations in

the αth set, with an integrated autocorrelation τα. Their configurations are then

binned into multiple histograms, Hβα(S) ≡ number of configurations in bin S. In

each histogram, if we divide it into a large number of sub bins, then each bin will

either contain zero or one configuration while the total expectation is a constant.

Thus the number of configurations in each histogram should fall in a Poisson

distribution and the errors can be estimated by,

⟨δ2Hα(S)⟩ = gα⟨Hα(S)⟩, (2.44)

where the gα = 1 + 2τα.

The partition function, simplified as the integration over S, is

Z(β) =∫

dS n(S)e−βS, (2.45)

where the density of states can be written as

n(S) = e f (S) (2.46)

Writing it into the form with probability density, one will have

1 =∫

P(S)dS =∫

n(S)e−βα,S

Z(βα)dS (2.47)

So the probability in a particular bin can be estimated by the histogram of a

dataset βα at S,

P(S)∆S =⟨Hα(S)⟩

Nα, (2.48)

where ∆S is the bin width of bin S. Comparing the above two equations, we

easily have

n(S) =⟨Hα(S)⟩Nα∆S

eβαS−Fα , (2.49)

where we have defined the average free energy Fα = − ln Z(β) .

44

The right hand side of last equation should be independent of choice of any

particular dataset βα. However this property is only valid in principle. For S far

away from the average value ⟨S⟩α of a particular dataset, the exponential term is

almost infinitely large. To keep n(S) finite, we need an infinitely large number

Nα of configurations. With one dataset β, it is impossible to have histograms that

cover all the S due to the limited data size. Actually one simulation at βwill give

empty histograms for most of the S, leaving only those near ⟨S⟩β non-vanishing.

To find the density of state near S we need the Monte Carlo run at a β such

that ⟨S⟩β is close to S. Multiple Monte Carlo runs at nearby β’s will give us piece-

wise information about n(S). As the case of two datasets, in the region where the

datasets overlap, we can combine the contributions from each of these datasets

by assigning a proper weight (function of S) in the following manner,

n(S) =R∑α=1

Wα(S) nα(S),

where nα(S) is calculated in Eq. (2.49) using dataset βα and by normalization,∑Wα(S) = 1.

By minimizing the error in n(S), it can be found that the weight should

satisfy [32] ( see also Appendix)

Wα(S) =Nα/gα exp(Fα − βαS)∑α′ Nα′/gα′ exp(Fα′ − βα′S)

(2.50)

and we can write out the density of state explicitly (we assume each dataset

consists of the same number of configurations)

n(S) =∑α(Hα(S)/gα)∑

α exp(Fα − βαS). (2.51)

Notice that it depends on the input of the free energy Fα which is not explicitly

available, but it can be calculated by the relation

exp(−Fα) =∑

S

n(S) ∆S e−βαS. (2.52)

45

So the determination of n(S) (or Fα) has to go through iterations. It is clear from

these two equations that the fluctuations in either of Fα or n(S) will increase after

it goes through the iteration cycle. So the initial values of the iterations need to

be chosen carefully otherwise it can easily result in divergence. Because of the

positive feedback feature, if the iteration converges, it converges slowly.

It is very important to start with a precise set of initial values. We found

that it is effective to use the average plaquettes to calculate the initial values of

the free energy Fα. Let’s fix the overall constant at β0 by Z(β0) = 1, or F0 = 0, then

we have the relation

F(β) =∫ β

β0

dF(β) =∫ β

β0

(−Z′(β)Z(β)

)dβ =∫ β

β0

⟨S⟩βdβ . (2.53)

where the average plaquettes P = ⟨S⟩β can be calculated from the data. By

integrating through the datasets from the lowest β, we have a spectrum of initial

values for the free energy. We found that in most occasions, this approach results

in precise values and leads to convergence. We shall discuss this method in detail

below.

Detailed Implementation For the SU(2) Model

First, each dataset is binned into certain number of histograms. It was

argued that there is an optimized choice on the number of histograms ( or bin

width), however we didn’t find it is really decisive. Typically, the width of the

distribution varies significantly with β. We make sure that the bin width is not

too large such that distributions of large β’s are not reduced to only several bins.

Second, we need to prepare the initial values for the iterations. The differ-

ence between successive β’s are usually optimized to maximize the overlapping

between the neighboring datasets, and hence may not be selected to be equally

46

spaced. To get the initial free energy density F (βα) through numerical integra-

tion, an algorithm of integration with non-equal steps should be used to improve

over the simple trapezoidal integration. we will use the generalized Simpson’s

Rule [66]. We set F (β0) = 0, then the higher F (βα) can the obtained by∫ βα

β0

Pdβ ≈ 12

[ f1 + fα +α−2∑i=0

f (βi, βi+1, βi+2)], (2.54)

where

f1 =12

(P1 + P0)(β1 − β0),

fα =12

(Pα + Pα−1)(βα − βα−1),

f (βi, βi+1, βi+2) = (βi+2 − βi)[Pi +

βi+2 − βi

βi+1 − βi

Pi+1 − Pi

2

]+

16

(2β2i+2 − βiβi+2 − β2

i + 3βiβi+1 − 3βi+1βi+2)[Pi+2 − Pi+1

βi+2 − βi+1− Pi+1 − Pi

βi+1 − βi

].

(2.55)

We found that the integration is effective and the iteration converges much faster.

Thirdly, to do the iteration, using the entropy density function f (x) =

(1/Np) ln n(x) is convenient. To prevent numerical overflow, the summation in

both of the equations involved is performed using the logarithmic formalism,

i.e., factoring out the major term and treating the rest as small contributors to the

log function. In order to determine the convergence of the iteration, we monitor

the average Chi-Square of the average plaquettes with 90% of the datasets,

χ2(i) =1

ncut

ncut∑α=0

(⟨x⟩βα(i) − ⟨x⟩βα)2

σ2α

, (2.56)

where ⟨x⟩βα and σα are the average plaquettes and its fluctuations from the data,

while ⟨x⟩βα(i) is calculated using the density of states at iteration i. We choose a

ncut to ignore the largest 10% of β’s because typically we use large β steps there

and the overlapping of the datasets is low, which makes the f (x) fluctuating but

causing no notable effect to the small-β region.

47

1e-007

1e-006

1e-005

0.0001

0.001

0.01

0.1

100 1000 10000

∆χ2

Iterations

Obtaining Initials: Itegration Methods Compare

TrapezoidalGen. Simpson

Figure 2.19: A comparison between the trapezoidal and the generalized Simp-son’s rules that are used to obtained the initial values of F (βα). The data is fromthe 44 lattice and skipping every 5 datasets. ∆χ2 is the χ-square difference be-tween successive iterations ( defined in the text ), the slope of which indicatesthe convergent rate of the iteration. It is obvious that when the datasets aresparse, the generalized Simpson’s rule gives better initial values and improvesthe convergence significantly.

Finally and most importantly, the iteration should be determined to be con-

vergent. The density of states over the histograms forms a non-linear dynamical

system of huge number of degrees of freedom. The convergence is hard to de-

termine and the criterion for the convergence is not yet available. However what

can be seen is that this system is not uniformly stable, i.e., the convergence is

conditional, depending on the histogram bin width, initial values, β steps, and

the overlapping between the datasets. The convergence, if it exists, is typically

reached slowly (Fig. (2.22)), which is a characteristic of such a system. Although

not proved, the convergence can be judged empirically by the asymptotic behav-

ior of the χ2 ∼iterations curve. In practice, we monitor the difference between

the successive χ2’s and look for a plateau over the iterations. In general for a

48

large enough number of iterations, if the absolute difference of the successive χ2

is decreasing over the iterations, i.e., the convergent rate is decreasing, it usually

results in a convergence.

Comments:

• If convergent, the iterations usually do not depend on the histogram bin

width ( or bin number of x ). However the bin width should be monitored

and adjusted so that it optimizes the smoothness/noise of the histograms

while it is reasonable for the computational convenience.

-4e-005

-2e-005

0

2e-005

4e-005

6e-005

8e-005

0.0001

102 103 104

∆ χ2

Iteration Number

Successive Average χ2 Difference: SU(2) 44

2005001000

Figure 2.20: This plot is a comparison of reweighting with various x bin num-ber: 200(circles), 500(squares) and 1000(diamonds). They all result in the sameconvergence down to the noise level of the data.

• Although the initial values may affect the convergence, if two sets of initial

values for such a system both result in convergence respectively, then they

will approach to the same convergent values if they do not differ too much

initially (Fig. (2.21)).

49

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.2 0.4 0.6 0.8 1

∆f(x)

x

Reweight with different initials: 44

init 1:iters=10kinit 2:iters=10kinit 1:iters=200init 2:iters=200

Figure 2.21: Two slightly different sets of initials are fed to the reweighting of aSU(2) data on a 44 lattice and reach the same convergence. The plot is ∆ f (x) atiteration 10000 (middle two curves) and 200(outer two curves). Here ∆ f meansthe difference of f (x) to the stabilized value ( at iteration 15000).

• The most decisive factor that will affect the reweighting is the overlapping

between the datasets.

First, even a convergence is reached, the insufficient overlapping between

different β’s may result in incorrect values, since the reweighting is to reach

the compatibility between overlapping data sets based on their statistical

levels, a sparse data lineup doesn’t have enough constraints to decide the

density to the desired precision. See Fig. (2.23) for an example.

Second, insufficient overlapping often leads to uncontrollable divergences.

This usually becomes worse as the volume increases because the distribu-

tions of the data become more and more narrow and therefore result in

less overlapping. A detail example is shown in Fig. (2.24) and Fig. (2.25)

where fewer datasets make the iteration diverge, and the iteration is made

50

-22

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

0.4 0.6

ln |f(x)-f(x)con|

x

Iterations with diff. initials

5200...2400

optim 20optim 200

Figure 2.22: An example of the iteration process. All iterations are comparedwith the convergent values f (x)con. Two different initials are compared here. Forthe first one we started with initial values f (x) = 0, for all x. The convergence isslow (upper five curves). For the second we used the integration method to getthe initials. The method is efficient for the iterations (lower two curves).

convergent with more datasets.

To increase the overlapping, we can either add intermediate datasets or

increase the statistics of each dataset. However, either way will be expensive

and hence an optimized selection of the values of β’s is often designed.

51

-0.0003

-0.0002

-0.0001

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.2 0.4 0.6 0.8 1

∆f(x)

x

Reweight with Different Numbers of β: 44

n=142n=57n=29

Figure 2.23: Convergent iterations may not always lead to satisfactory results.Reweightings with different numbers of β’s are compared. The resulting f (x)’sare then subtracted from the result with 284 β’s (normalized at x = 1). The solid,dashed, double-dashed curves are using 142,57 and 29 β’s respectively. Althoughall lead to convergence, the reweighting with fewer datasets show much biggerundesired fluctuations.

52

0.6

0.65

0.7

0.75

0.8

101 102 103 104 105

Average

χ2

Iteration Number

Convergent Overlapping : SU(2) 64

246 βs449 βs

Figure 2.24: The convergence is sensitive to the number of datasets. The reweight-ing using the same data set while including different numbers of datasets atvolume 64. The plot is showing the chi2 varies with iterations. The upper onewhich is using 246 β’s shows obvious divergence at iteration around 6000, whilethe lower one with 449 datasets indicates an strong sign of convergence. Similardifficulty appears with higher volumes such as 84 and 104.

Figure 2.25: The overlapping plots of the previous example. The boxes in eachgraph show two-σ spreading (centered at the average) of each data point. Thevertical increments have no meaning but to illustrate the neighboring overlappingin a clearer manner. The left graph corresponds to the diverging reweighting andshowing in sufficient overlapping among the data.

53

The error analysis of the Ferrenberg-Swendsen’s method was studied by

the same authors in [30]. However in this paper, we will use a simple method

to estimate the uncertainty due to the reweight procedure. For the two small

volumes 44 and 64, we generate 20 seeds of data which are independent replica of

the same observables. Then we use the reweighting method to find the density

of states for each of these replica, and estimate the uncertainty (Fig. (2.26)).

-0.0002

-0.0001

0

0.0001

0.0002

0.0003

0 0.2 0.4 0.6 0.8 1

∆f(x)

x

Error Due to Different seeds: 44, 19 seeds

Figure 2.26: The errors of the density of states due to the different seeds.

54

2.4.3 Volume Dependence of the density states

With the density of states precisely determined at various volumes, we can

investigate its volume dependence. We discussed the weak coupling expansion

in Section 2.3.2, however, here we are only interested in the volume effect in those

expansion coefficients. We should follow the same notation there, by expanding

f (x) in the following way

fV(x) = A ln(x) + ( f0 +f0′

V) + ( f1 +

f1′

V)x + ( f2 +

f2′

V)x2, (2.57)

where we already know A = 3/4−(5/12)(1/V) for SU(2) and A = 1/4−(5/12)(1/V)

for U(1). If we have f (x) calculated at two different volumes, say V1 and V2, then

by taking their difference, we can get rid of the volume-independent part of the

expansion coefficients and be able to to extract first order volume effect by

∆ f∆(1/V)

=fV1(x) − fV2(x)1/V1 − 1/V2

(2.58)

= − 512

ln(x) + f0′ + f1′ x + f2′ x2 +O(max(1/V21, 1/V

22)).

We make use of three different volumes: VL = L4, where L = 4, 6, 8 for

the U(1) model. We calculated the following two differences, ∆ fV4,V8 , ∆ fV6,V8

and do the data collapse using Eq. (2.58). The curves of ∆ fV4,V8 and ∆ fV6,V8 ,

with the small overall constant removed, overlap nicely (Fig. (2.27)). Fitting

the curves gives close results: f ′0 = 0.555, f ′1 = −0.105, f ′2 = 1.122 for ∆ fV4,V8 and

f ′0 = 0.520, f ′1 = −0.096, f ′2 = 1.137 for ∆ fV6,V8 . The difference between the f ′0 ’s is

about 0.035 which corresponds to the height of the flat part in Fig. (2.28).

55

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.1 0.2 0.3

∆f/

∆(1/V)

x

Volume dependence of f(x): U(1)

L=4,8L=6,8

Figure 2.27: The collapse of the difference of f (x) of three different volumes,∆ fV4,V8 and∆ fV6,V8 with the overall constant subtracted (described in the text). Thedifferences are then divided by their corresponding inverse volume difference.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.1 0.2 0.3 0.4 0.5

x

The difference of ∆fV4,V8 and ∆fV6,V8

Figure 2.28: The difference of the differences ∆ fV4,V8 and ∆ fV6,V8 . The flat partcorresponds to the overall constant 0.035.

56

CHAPTER 3FISHER’S ZEROS

3.1 Single Point Reweighting Method

A Monte Carlo simulation at a fixed value of the parameter β gives a partial

description of the density of state ( or partition function) in the vicinity of β.

This ”vicinity” includes the real neighborhood in the parameter space as well

as the analytic continuation into the complex plane . This idea leads to local

approximations of Z(β). Each dataset gives a normalized statistical description.

Therefore, it is possible to analyze the Fisher’s Zeros if they are within the reach

of such approximations [2, 29]. Suppose that Monte Carlo data are collected at a

value β0 and for a close-by value β = β0 +∆β the partition function can be written

as [2, 23, 19]

Z(β0 + ∆β) = Z(β0)⟨exp(−∆βS)⟩β0 , (3.1)

where S is the total action. We should be aware that when admitting a complex

β, Z(β0 +∆β) is an oscillating function with its frequency scaled by the number of

plaquettes. So it is convenient to use the reduced form which is defined as [19]

z(∆β) =Z(β0 + ∆β)

Z(β0)exp(∆β⟨S⟩β0) = ⟨exp(−∆β(S − ⟨S⟩β0)]⟩β0 (3.2)

z(β) should share the same zeros as Z(β0 + ∆β).

We write explicitly ∆β = βR + iβI, then the exponential reduces to the real

part ZR = ⟨exp(−βR∆S) cos(βI∆S)⟩ ( here ∆S = S − ⟨S⟩β0) and the imaginary part

ZI = ⟨− exp(−βR∆S) sin(βI∆S)⟩ which are both oscillating functions. Note that

all these expectations are taken with the data collected at β0. We then scan a

region (for example, a square) near β0 in the β-plane vertically and monitor the

successive values of the real and imaginary functions ZR, ZI. If the value of the

function, say the real part, switches sign then it means it reaches the value of 0

57

in between. By following this methodology, we can produce the zero contours of

both ZR and ZI. The intersections of these are the zeros of the total expectation,

and hence are the complex zeros of the partition function Z(β). Fig. (3.1) shows

an example of these contours.

0.04

0.06

0.08

0.1

0.12

2.25 2.3 2.35 2.4

Im

β

Re β

Finding Zeros: Method of Contour Plot

ReZ(β)=0 ImZ(β)=0

Figure 3.1: The plot shows the contours of both real and imaginary part of thepartition function. The intersections of these curves are the zeros of the partitionfunction.

This method can be generalized to multiple-point reweighting by assigning

weights to the neighboring datasets and combining them together [2]. This

method is proved to be effective when the complex zeros are close to the real axis,

especially for the statistical models that have a phase transition in the infinite

volume limit. Later we will demonstrate that this method works well in the U(1)

case.

However, the method is limited if the zeros have large imaginary compo-

nents, to be specific, if the zeros lie outside the circle of confidence [2]. This circle

of confidence is determined by the statistical level of the data. For a model with

58

a Gaussian-like distribution, the radius of the circle can be estimated by using

the Gaussian approximation. For example, in the SU(2) model, the probability

density of a particular data is narrowly distributed along S and resembles almost

the shape of a Gaussian distribution. For a model with an exact Gaussian dis-

tribution, the fluctuation on the reduced partition function z(β) = Z(β0 + β)/Z(β0)

can be calculated by [2]

σ2(|z(β)|) = exp(β2

R

A) − |z(β)|2, (3.3)

where A corresponds to the quadratic coefficient in the Gaussian distribution

exp(−AS2). A model with an exact Gaussian distribution is free of zero. So we

should only trust the region such that the Gaussian z(β) is at least one sigma

distinctive away from zero, which means that we have 84% confidence to exclude

a fake zero due to the fluctuation. Specifically, σ(|z̄(β)|) = σ(z(β))/√

(N) ≤ |z(β)|

where N is the data size. This will define a circle [2], β2R + β

2I ≤ A ln(N + 1).

Generally for a non-Gaussian model, this circle of confidence is replaced by a

level of confidence which can be calculated by the criteria [19, 20]

σz(β)/|z′(β)| < d, (3.4)

where d is a fraction of the typical distance between the zero contours of the

real and imaginary parts. One should be cautious with the zeros obtained with

the contour plot method. Due to the fluctuation of the data, it will often produce

intersections of the real and imaginary curves, but only those well within the

confidence region are candidates of the actual zeros. So it is always be good to

verify them with other independent methods.

3.2 Quasi-Gaussian Distribution: A Toy Model

We know that for the SU(2) model its probability distributions of the action

are very close to Gaussian. For a partition function with its probability density

59

resembling exactly a Gaussian, there should be no zero. So the zeros must arise

from the deviation from the Gaussian distribution. We should look at a simple

toy model for which we can calculate the result precisely.

We use the Metropolis algorithm to generate random numbers obeying a

normal distribution as follows [20]

P(S) = (2π)−1/2 exp(−S2/2) . (3.5)

Figure 3.2: The probability distribution of total action in the Gaussian toy modeldescribed in the text.

We generated 1,600,000 values of S. The average of this set gives 0.00087.

The values were then binned into 100 histograms as is displayed in Fig. (3.2)

which shows a very small deviation from the Gaussian distribution. The data

is highly correlated. To remove the autocorrelation, we simply skip a couple of

times of the correlation length which is around 8 from the inverse of the slope of

Fig. (3.3).

With the skipped data which consists of 40, 000 independent configurations,

we can now explore its zero using the afore-mentioned single-point reweighting

method. The zero-level contours of the real and imaginary parts of< exp(−β∆S) >

60

Figure 3.3: Logarithm of the autocorrelation versus the time series distance forthe original set of 1,600,000 values.

are shown in Fig. (3.4) [20]. The circle of convergence (Reβ)2+(Imβ)2 = ln(40000) ≈

3.262 is also plotted. The zero closest to the real axis is around β ≈ 1.5 + i3.0,

but it is outside the radius of confidence defined above, which fits well in our

expectation that the Gaussian Model should be free of zeros. The solid lines

in Fig. (3.4) correspond to the hyperbolas: βRβI = 2nπA (imaginary part) and

βRβI = 2(n+ 1)πA (real part). The hyperbolas will never intersect with each other

and hence a partition function due to a Gaussian distribution should not possess

a zero in the complex plane. We can see that these contours coincide well with

the hyperbolas.

We now look at the quasi-Gaussian distribution by adding small perturba-

tions with a cubic and a quartic term,

P′(S′) ∝ exp(−(1/2)S′2 − λ′3S′3 − λ′4S′4). (3.6)

Due to the perturbations, the variance of the distribution may differ from unity

61

Figure 3.4: Zeros of the real (circles) and imaginary (crosses) part for 40,000Gaussian configurations. The solid lines are the circle of confidence and thehyperbolas of the normal distribution.

as designed. So we will perform the following transformation to normalize the

distribution,

S = (S′ − ⟨S′⟩)/σS′ . (3.7)

Using the new variables, the probability density can be re-expressed as

P(S) ∝ exp(−λ1S − λ2S2 − λ3S3 − λ4S4) (3.8)

where the λ’s are related to the old λ′ by plugging Eq. (3.7) into Eq. (3.6).

We could then study how the zeros structure responds to the changes of the

controlling parameters λ′3, λ′4.

3.2.1 Example 1. λ′3 = 0.1, λ′4 = 0.01

We first consider the case λ′3 = 0.1, λ′4 = 0.01. It has been chosen in such a

way that we have both zeros inside and outside the Gaussian region of confidence.

With all these setup, the locations of all the imaginary and real zeros can be rather

precisely determined using numerical integration, which we will call the ”actual”

zeros. The actual zero level curves are compared with the MC ones in Fig. (3.5).

62

Figure 3.5: Zeros of the real (circles) and imaginary (crosses) part for 40,000configurations corresponding to the first example. The small dots are the accuratevalues for the real (gray) and imaginary (black) parts. The exclusion regionboundary for d = 0.12 is represented by boxes (red). The solid line is the circle ofconfidence of the Gaussian approximation.

We found zeros at β = 1.3735 + i1.7104 and β = 1.3735 + i2.9478. The variance

of the original variable S′ is 1.0988, so the zeros in the original frame need to be

rescaled with this value. For the lowest zero, the MC zero is very close to the

actual zeros which is located near 1.3+i1.6 and the second lowest zero which is

at 0.9+i2.8, it shows a significant difference to the MC result, and is nevertheless

excluded by the circle of confidence and level of confidence in Fig. (3.5).

3.2.2 Example 2: λ′3 = 0.01, λ′4 = 0.002

In the second example we use λ′3 = 0.01, λ′4 = 0.002, which are much smaller

perturbations and the lowest actual zero should appear at β = 1.207 + i5.241, far

outside the Gaussian circle of confidence(Fig. (3.5)) [20]. But the Monte Carlo

method produces fake zeros with much smaller imaginary parts. However they

are all excluded by the circle of confidence. The result is further confirmed by the

level of confidence.

63

Figure 3.6: Zeros of the real (circles) and imaginary (crosses) part for 40,000configurations corresponding to the second example. The small dots are theaccurate values for the real (gray) and imaginary (black) parts. The exclusionregion boundary for d = 0.15 is represented by boxes. The solid line is the circleof confidence of the Gaussian approximation.

3.3 Approximation with Analytic Functions

Theoretically, once the density of states of a system is known, we should be

able to find almost all the statistical quantities including the zeros of the partition

function of the underlying system. That is the exact reason that we spent the

whole Chapter 2 in the discussion of the density of states. However more often

than usual, we can only get an approximate form of the density of states which

is subject to errors of different kinds. In this case, using an analytical function

to approximate the numerical density of states is necessary. These functions can

provide a starting point to explore the complex region based on the information

collected on the real axis.

64

3.3.1 LargeNp limit with a fixed f (x)

It has been speculated [21] that the condition Re f ′′(x) = 0 served as the

boundary to separate the actual zeros from the region where the Gaussian dis-

tribution dominates and is free of any zeros. In this section, we should further

examine the relation between the roots of f ′′(x) = 0 and the actual zeros. By

”actual zeros”, we mean the actual roots of the partition function at that partic-

ular volume. We will show that with some proper conditions being satisfied,

there exists a simple scaling relation between the actual zeros and the roots of the

f ′′(x) = 0.

We know the entropy density function f (x) has a volume dependence in the

weak coupling regime which has been discussed in Section 2.3.2. However it is

still interesting to know that if we scale f (x) − βx by an arbitrary Np where f (x)

is obtained at a finite volume, how will the Fisher zeros respond to such change.

We will show in the following that in the limit Np → ∞, the produced zeros will

approach to the points that correspond to the roots of f ′′(x) = 0 mapped to the

complex-βplane. The importance of this property is two-folded. First, it guides us

where to search for the actual zeros because empirically these points have almost

identical real parts as the actual zeros; second, at very high volumes, these points

are very close to the actual zeros and therefore can be good approximations.

We will only demonstrate it in a simplified case with several assumptions.

Let Z(β) =∫

dx exp[Np( f (x) − βx)]. Suppose that f (x) is analytic and free of

singularities in the complex x-plane except the bounds of the integration. Then

Z(β) should also be an analytic function. Assume that both Z(β) and f ′′(x) have

only zeros of order no more than one and in all cases the integration path can be

replaced by a simple path through a single saddle point, i.e., Eq. (2.12) is satisfied.

Then in the limit ofNp →∞, β∗ ≡ f ′(x∗) are the roots of Z(β)/(Z′(β)/Np), where x∗

65

are the roots of f ′′(x), i.e., f ′′(x∗) = 0.

In general, Z(β) scales with Np. To make the discussion unambiguous, we

should consider the scale insensitive version Z(β)/(Z′(β)/Np) which should have

the same zeros as Z(β) by assumption.

To prove that this is true, we will start with the saddle point approximation

which we assume valid all the time. We have

Z(β) = eNp( f (x0)−βx0)

√2π

−Np f ′′(x0), (3.9)

it is easy to see that

Z(β)Z′(β)/Np

= − 1

x0 + (1/2) f ′′′(x0)Np f ′′(x0)2

,

= − f ′′(x0)2

x0 f ′′(x0)2 + 12Np

f ′′′(x0), (3.10)

where we have used the saddle point condition f ′(x0) = β and the fact that

dx0/dβ = 1/ f ′′(x0).

We now consider the roots of f ′′(x) = 1/Nαp where α < 1 to ensure the

validity of the saddle point approximation. We will label them as xNp . For large

Np, we can show that for a x∗ which satisfies f ′′(x∗) = 0, there is a xNp close to

it. To see that, since f ′′′(x∗) , 0 by assumption, the inverses mapping of f ′′(x),

which is between the domain of f ′′ and the sheet of the Riemann surfaces that x∗

is located, is well defined. The inverse mapping ( f ′′)−1x∗ exists, then ( f ′′)−1

x∗ (1/Nαp )

gives one of the xNp and

xNp − x∗ ∝ ([( f ′′)x∗]−1)′(0)(1/Nαp − 0)

∝ 1f ′′′(x∗)

N−αp . (3.11)

We now let α > 1/2, so that the expression |NpxNp f ′′(xNp)2| ≪ | f ′′′(xNp)|

holds always true for large enough Np. Then we map them to the β-plane by

66

βNp = f ′(xNp). Using Eq. (3.10), we will have

Z(βNp)

Z′(βNp)/Np∼ −

f ′′(x′Np)2

f ′′′(x′Np)∝ N−2α

p . (3.12)

Since f ′ is a continuous mapping, both (βNp − β∗) and Z(βNp)/[Z′(βNp)/Np] are

arbitrarily small for largeNp, which means β∗ is the root of Z(βNp)/[Z′(βNp)/Np] in

the largeNp limit.

Note:

• At x∗, the mapping f ′(x) is singular and the inverse is ill-defined. But the

conclusion will not be affected.

• The saddle point approximation 3.10 cannot be applied to a integration path

with multiple saddle points, taking the U(1) gauge model as an example.

• The points from f ′′(x) = 0 are not corresponding to the zeros in infinite

volume limit due to the volume dependence of f (x) in practice. The zeros

of both the actual zeros and these roots move with volumes. However there

exists a clear scaling of the distance of the zeros calculated with aNp to the

f ′′(x) = 0 zeros. Since the volume effect decreases with the volumes, it is

interesting to know whether such a scaling approximate the actual scaling

of the system.

Let’s examine this phenomenon through the following toy model. We as-

sume there is such a system whose entropy density function can be described in

the following simple form

f (x) = 1 + 2(x − 511

) − 12

(x − 511

)2 − (x − 511

)4. (3.13)

f ′′(x) has two roots: x = 0.454545±0.288675i which can be mapped to the β-plane.

We know that for such a system, Z(β) as well as Z′(β) should have no singularities

at any volume. So these correspond to the zeros of Z(β) asNp →∞, and they are

67

at β = 2. ± 0.19245i.

Now we can plug f (x) back to the expression of the partition function and

useNp as the scale factor, then we can actually search for the zeros of the partition

function using the method we will endeavor to describe in the next subsection.

To better see the scaling of the zeros with the ”volume”, we add some unrealistic

”volumes” which admit half of the lattice spacing. In the following, we will call

these zeros obtained from the variation ofNp theNp-scaled zeros. The results are

summarized in Table. (3.1) and Fig. (3.7).

”L” ”Volume” Reβ Imβ

2.5 235 2.000000000 0.283832564

3 486 2.000000000 0.248845504

3.5 900 2.000000000 0.229913015

4 1536 2.000000000 0.218722786

4.5 2460 2.000000000 0.211668011

5 3750 2.001 0.2071

∞ ∞ 2 0.19245

Table 3.1: The results of the zeros with the increasing ”volumes”.

3.3.2 Chebyshev Approximation

An analytic form of f (x) will enable us to understand the properties that

are associated with the correspondence between the derivatives of the partition

function and the density of states. The other benefit of having an analytic form

68

0.192

0.2

0.22

0.24

0.26

0.28

0.3

2 3 4 5 6 8 ∞

Im

β

L

Imaginary Part of Lowest Zero vs Size

ç

ç

ç

ç

ç

ç

Slope = -2.6457

5.02.0 3.00.010

0.100

0.050

0.020

0.030

0.015

0.070

L

ÈImHΒ

L-Β¥LÈ

Figure 3.7: The left are the imaginary parts of the zeros in the toy model describedin the text in different ”volumes” plotted against the f ′′(x) = 0 zeros whichcorresponds to the zeros in infinite ”volume” limit. The right is the log-log plotof the difference of the zeros to the f ′′ = 0 zero vs the volume. We found thatIm(βL − β∞) ∝ L−2.6457.

is on the evaluation of Z(β). The original grids of the density of states sometimes

may not be sufficient to do precise numerical integrations (which is how we define

the partition function). It is especially true when the imaginary component of β

is large and, as a consequence, the partition function oscillates more frequently

than what the original bin width can resolve.

We will use the Chebyshev Interpolation for such purposes [14, 75, 65] . It

has been shown that, for the Chebyshev interpolation of numerical data, the Least

Square Fit method is more efficient and robust than the discrete and integration

methods [26, 7, 13]. In this paper, we will primarily follow this approach.

Given a range of interest [a, b], it can be mapped to [−1, 1] in which we can

expand the target function by

f (y) =Nc∑

n=0

cnTn(y) (3.14)

where Tn(y) = cos[n arccos(y)] are the Chebyshev polynomials of the first kind.

We then minimize the distance of the function to a data set or multiple data sets,

69

which will uniquely determine the coefficients cn by a set of linear equations.

3.3.3 The Ellipse of Convergence

When doing a complex continuation of a series function, we should keep in

mind that, like other polynomial approximations, Chebyshev interpolation with

a finite order and a mapped range may introduce artifacts such as fake zeros,

especially when we are taking the derivatives of the approximating function. We

should be careful that we stay in the valid region of the approximation which can

be quantified by the ellipse of convergence [65].

The Chebyshev approximation is really a relabeled Fourier series expansion

[13], differed by a further mapping arccos(x). To work in the complex plane,

the following relation is helpful: Tn(z) = (ωn + ω−n)/2, where z = (ω + ω−1)/2.

The connection is straightforward with the aid of intermediate parameter y,

cosh ny = Tn(z) and cosh y = z.

The convergence of a Chebyshev series can be analyzed through the variable

ω. We turn Eq. (3.14) intoNc∑

n=0

cn (ωn + ω−n) =Nc∑

n=0

cnωn +

Nc∑n=0

cnω−n. (3.15)

A sufficient but not necessary condition for the expression to converge is

that both of the series on the r.h.s of Eq. (3.15) are convergent. If there is a

singularity z0 which is the closest to the expansion point (the origin) in the z-

plane, then it corresponds to two in the ω-plane: ω0(±) = z0 ±√

z20 − 1 which

are conjugate about the unit circle. Both of the Lauren expansions in the r.h.s

of Eq. (3.15) will converge uniformly in the area defined by the two concentric

circles which are determined by ω0(±). Interestingly, these two concentric circles

correspond to the same ellipse in the z-plane, with the semi-major and semi-minor

to be |ω0| ± 1/|ω0|[65, 68]. The region inside the ellipse corresponds to the region

70

enclosed by the two concentric circles in the ω-plane.

The determination of the ellipse of convergence of a Chebyshev expansion

is not always straightforward. Fig. (3.8) shows the logarithm of the coefficients

in the Chebyshev series approximating the numerical f (x) on a 44 lattice using a

maximum order 40 and the range (0.3, 0.6). There is an obvious cut at order 15.

For order n > 15, the small slope indicates a very small radius of convergence. The

tail of the orders up to 15 can be conservatively fitted. A fit for order 4 ∼ 15 gives

a slope of −0.609, which corresponds to a radius of ωR = exp(−0.609) = 0.5435.

So the ellipse in the z-plane has a semi-major 1.1917 and focus length 1. Mapped

back to the fitting range [0.3, 0.6], the ellipse’s semi-major and semi-minor are

0.1788 and 0.0973 and it is centered at 0.45. The artifact of the approximation can

also be indicated by the roots of f ′(x) = β. The outer ring of roots of f ′(x) = β

appear to be independent of the values of β. Fig. (3.9) shows the roots of f ′(x) = β

where β = 2.2 + 0.02i, 2.2 + 0.04i, 2.2 + 0.06i and 2.2 + 0.08i. We can see that the

outer ring of roots of f ′(x) = β seem to have no dependence on β’s and they are

coincident with the ellipse of convergence defined above. However there are

three non-trivial solutions inside the ellipse. Two of them tend to merge as the β

approach the value f ′(x∗) where x∗ is the root of f ′′(x) = 0 (the cross in the graph).

As we can see from Fig. (3.10), the roots of f ′′(x) = 0 are well inside the ellipse

and the two non-trivial saddle points collapse right at x∗.

71

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á á

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á á

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á

á áá

á

á

á

á

á

á á

á

á

á á á á áá

á

á

0 5 10 15 20 25 30

-20

-15

-10

-5

0

ordern

lnÈc

nÈ

Figure 3.8: The logarithm of the absolute coefficients of the Chebyshev approxi-mation discussed in the text versus the order number n.

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á

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ç

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çà

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à

à

à

à

à

à

à

à

0.3 0.4 0.5 0.6-0.10

-0.05

0.00

0.05

0.10

Re x

Imx

Figure 3.9: The ellipse of convergence of a Chebyshev series ( described in thetext) and the roots of f ′(x) = β with the values given in the text. The two saddlepoints which are obviously inside the ellipse merge to the root of f ′′(x) = 0 (redcross) as the value of β approaches to the value corresponding to the root off ′′(x) = 0.

72

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+

0.3 0.4 0.5 0.6-0.10

-0.05

0.00

0.05

0.10

Re x

Imx

Figure 3.10: The ellipse of convergence of a Chebyshev series and the roots off ′′(x) = 0(empty circles). The empty squares are the roots of f ′(x) = β where β isgiven by f ′(rootsof f ′′ = 0) ( the cross ).

3.3.4 The Moment Tests

To estimate the goodness of the approximation, we can look at the moments

from both the data and the approximations. We can define the following scale-

invariant moments,

M1 = ⟨x⟩,

M2 = Np⟨(x − ⟨x⟩)2⟩,

M3 = N2p ⟨(x − ⟨x⟩)3⟩,

M4 = N3p (⟨(x − ⟨x⟩)4⟩. − 3M2

2) (3.16)

Due to the fact that these variables are not all independent, to estimate the

errors on these moments, we can use either Jackknife or bootstrap method [64] to

address the correlations between them. We found out that the results from these

two method are similar. There is also a method introduced by Wolff [78] which is

73

to estimate the error bars more precisely. The results are shown in Fig. (3.11) and

Fig. (3.12), which indicate that the Chebyshev approximation is consistent with

the numerical data.

0.24

0.28

0.32

0.36

0.4

2 2.1 2.2 2.3 2.4

2nd Moment

β

4x4x4x4

MC3CH40CH34CH30

Figure 3.11: The second moment of the SU(2) model at volume 44. The numericalresult is plotted against various orders of Chebyshev approximation.

-0.5

0

0.5

1

1.5

2 2.1 2.2 2.3 2.4

3rd Moment

β

4x4x4x4

MCCH40CH34CH30

Figure 3.12: The third moment. Similar to Fig. (3.11).

74

3.3.5 Evaluation of the Partition Function with Nu-merical Integration

We now get to the detail on how to evaluate the partition function once we

have the approximate form of the density of states. The partition function Z(β)

is often used as a normalization and its value bares no direct physical meaning.

What we are evaluating here is actually Z′(β)/Z(β).

From the definition, the difficulty of the numerical calculation of Z(β) is that

when the imaginary part of β is introduced, it will scale withNp which makes the

integrand highly oscillating and it is typically hard to integrate numerically. In

the following, we will compare two different methods to resolve this problem.

3.3.5.1 Path of the Steepest Descent

The first method is to make use of the saddle points and the path of steepest

descent. Constructing the path of steepest descent can be numerically consuming

and we found that approximating the path by a line which is tangent to the

steepest descent path at the saddle point is very reliable. Let x0 be a saddle point,

f ′(x0) = β. Let f ′′(x0) = | f ′′(x0)|eiθ0 . Then we can deform the integration to the

path

x

Figure 3.13: Path of Steepest Descent.

75

x = x0 + t exp(i2

(θ0 − π)), (3.17)

where t is the new integration variable and is real. This method works very

effectively and accurately. However there is a limitation when applying to the

actual problem. It arises from the locations of the saddle points. We can see

from Fig. (3.9) that as the imaginary part of β increases, the two non-trivial

saddle points tend to approach to each other and at a certain value of β which

is close enough to the f ′′(x) = 0 zero (the cross), the saddle point approximation

Eq. (3.10) will break down because the integrating path now has to include two

saddle points.

3.3.5.2 The Improved Trapezoidal Integration

Ironically, simple integration algorithms like the Trapezoidal rule works

also very well in the special case of SU(2). The fundamental reason is that in

general, the probability density associated with the SU(2) at a real β is very close

to a Gaussian distribution. For an exact Gaussian distribution, let’s consider

I =∫ a

−adxe−

12 Ax2

cos(x). (3.18)

We can use the simple Trapezoidal integration with 2N points,

IN =

N∑n=−N

exp(−12

An2h2) cos(nh), (3.19)

where h = a/N.

By using the Jacobi Imaginary Transformation ( See Appendix for a proof),

we can show that the error of the integration goes like

∆II≡ I − IN

I∝ exp(− π

2

2Aa2 N2). (3.20)

It shows that the integration improved with the order of exponential of negative

N2 which makes the convergence of the integration very fast.

76

For a quasi-Gaussian distribution with small perturbations, empirical re-

sults show that it also exhibits similar rate of convergence.

3.3.6 Localization of Zeros Using Cauchy’s Theorem

There is a general algorithm to find the zeros of an analytic function by using

the Cauchy’s Integral Theorem [54, 39]. For simplicity, we will only consider the

special case when all the zeros are of order 1 which applies to our problem.

Suppose that an analytic function Z(β) has K simple zeros enclosed by a closed

contour C, then there are the following relations,

12πi

∮c(ln Z)′ βn dβ =

K∑i=1

(βi)n, n = 0, 1, 2, ... (3.21)

where βi are all the zeros in contour C. When n = 0, the summation on the right

hand side is just the number of zeros.

The partition function we are considering is an analytic function of β, which

is obtained through either naive summation or extended trapezoidal rule. In

principle, we could construct a large contour and perform the integral with n = 0

to determine the number of zeros enclosed. If there are K zeros, then we do

K integrations with n = 1, ...,K and solve the resulting K-order equations for

the zeros. However it turns out that higher order nonlinear equations are hard

to solve and are extremely sensitive to the error on the coefficients. The slight

changes in the coefficients can result in drastic shifts of the roots or even cause

a totally different root structure. So practically, we will only work with contours

that include at most two zeros. We scan the complex plane with rectangular

contours which enclose zeros of two or less. We found the method quite robust

and reliable.

77

Noticing that P(β) = ⟨x⟩β = (1/Np)(ln Z)′, we can rewrite the equation into

Np

2πi

∮cP(β) βn dβ =

K∑i=1

(βi)n, n = 0, 1, 2, ... (3.22)

What we need for the loop integrals is P(β), but we have to evaluate P(β)

through ratio of Z′(β) and Z(β) which we will obtain using the numerical integra-

tion technique developed in the last subsection. There is a common constant in

both of Z(β) and Z′(β) which will be canceled in the ratio. This constant needs

to be known in advance to prevent overflow or underflow in the evaluation of Z

and Z′. Unfortunately, this constant varies with β. As a result, since the difference

will be scaled by Np in the exponential, different points in the loop may require

canceling constants that are very different. However, it is remarkable that the

constants have a simple connection to P(β) itself. We know for large enough

volume, approximately, P(β) ≈ x0 at the saddle point x0 and the dominant term

in the exponential is close to the function evaluated at the saddle point x0, i.e.,

f (x0) − βx0. So the constant needed to calculate P(β) can be well approximated

by f (P(β)) − βP(β). Considering that we are calculating P(β) in a loop, if the step

is taken small enough, the canceling constant of the preceding point can be good

enough for the succeeding one.

There is a further reason to look carefully at the relation between P(β) and

β. If the integration over β is along a simple loop, what is the resulting loop of

P(β) in the complex x-plane? In general it follows the inverse mapping of the

relation f ′(x0) = β. If f ′(x0) is a simple polynomial, the inverses of the mapping

may possess several branch cuts.

When the loop is too close to a zero, a small step in β-plane may result in a

different branch in x-plane, which will cause a drastic change in the value of P(β).

The integration will fail, for example, the loop integral with n = 0 will deviate

obviously from an integer. So a new loop is needed to insure the reliability of the

78

A B

CD

2.10 2.15 2.200.13

0.14

0.15

0.16

0.17

0.18

0.19

0.20

ReΒ

ImΒ

AB

CD

0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50-0.10

-0.05

0.00

0.05

0.10

ReHxL

ImHxL

Figure 3.14: The plot is showing the correspondence of a loop in the β-plane tothe loop in the x-plane.

integration. We often apply a method called rerouting to go around the zero. An

illustrating graph is shown in Fig. (3.15). The basic idea is that when a path is too

close to a zero, the P(β) will appear to diverge. So we can set a reasonable zone (

for the case involving the Chebyshev Polynomials, this is set to be the ellipse of

convergence). Whenever the value P exceeds this zone, it will trigger a reroute

which is typically to construct a rectangular path to go around the region that is

too close to the zero.

3.4 SU(2) Zeros

We have now gathered enough tools to analyze the partition function zeros

of the SU(2) model. We know there is no phase transition for the SU(2) model in

the infinite volume limit. But it is interesting to know how its zero structure is

like. As we have mentioned in the introduction, the non-vanishing height of the

lowest zeros of the partition function may admit a confinement theory extending

to the weak coupling domain.

79

zero 1

zero 2

Figure 3.15: An actual loop in which two zeros are presented. The path is detouredto get a better integration.

3.4.1 Zeros From Single-Point Reweighting

We can first look at the result of using single-point reweighting method. Fig.

(3.16) [23] shows a contour plot at β = 2.18 with 1, 000, 000 configurations. Two

different methods are compared here. The single-point reweighting gives zeros

with imaginary part higher than 0.18. However these zeros are all outside the

circle of confidence. We also include here the quasi-Gaussian fitting result. We

fit the perturbation parameters which are discussed in Section 3.2 and it spotted

two zeros at β = 2.18 + i0.13 and β = 2.22 + i0.12 which are right on the border of

the circle of confidence. However they are excluded by the level of confidence.

So we can conclude that near Reβ ∼ 2.18, there is no zero below Imβ = 0.13.

We did a thorough test using the existing data, in order to estimate the

statistical fluctuations in the location of these complex zeros. We generated

200 bootstraps (each contains 40,000 configurations) at β = 2.225. For each of

the 200 sets, we calculated the zero contours of the real part of the partition

function. Using this procedure, 383895 zeros of the real part were located. The

distribution of these zeros are plotted using a 200 by 200 grid in the complex

80

Figure 3.16: The contour plot shows the zeros of a SU(2) gauge model on a 44

lattice. Blue and green lines are the zero curves of the imaginary and real part ofthe partition function from the Monte Carlo simulation at β = 2.18. The crossesand circles are the counterparts from a quasi-Gaussian approximation whichoverlaps the data in the bottom part. Both show isolated zeros of the partitionfunction. However they are all lying outside the radius of confidence, or morestrictly, they are all above the level of confidence ( the squares). Therefore thesezeros are artificial. The Monte Carlo data of a neighboring β doesn’t indicate theconsistent locations of these zeros.

β-plane. The results are shown in Fig. (3.17) [21]. The zeros of the real part

accumulate in several bands. Each band contains zeros of number from 60 to

more than 100. The circle of confidence in the Gaussian approximation for 40,000

independent configurations as well as level of confidence are also shown in this

graph for references. It is clear that as we get closer to the boundary of the region

of confidence, the fluctuation becomes larger. When it is beyond the circle of

confidence, it becomes merely random noise.

3.4.2 Histograms Analysis

We have seen that the Fisher zeros arise from the deviation of its distribution

from the Gaussian distribution. To determine the region where the zeros might

81

Figure 3.17: Distribution of eros of the real part of the partition function in thecomplex β plane and regions of confidence described in the text.

appear, we can examine its signal of such deviation. By looking at the coherent

pattern of the distribution with the major Gaussian part subtracted, we can tell

whether a zero is close-by. We use the histogram residue method. We first sort

the data into n number of bins. Define the residues

ri = (Ni −NPi)/√

(NPi) , (3.23)

where Ni is the number of configurations in i-th bin and Pi is the probability in the

i-th bin (here the probability is not the probability of the original distribution but

that of the Gaussian distribution determined by the average and variance of the

original distribution). N is the total number of configurations. Fig. (3.18,3.19,3.20)

[23] shows such a pattern. For the volume 44 at β = 2.18, we can see the strong

pattern departing from the pure Gaussian distribution which implies zeros pos-

sibly exist in the vicinity. In contrast, for the volume 64, the difference from the

Gaussian looks purely random which suggests that the Gaussian distribution

dominates here. However the coherent pattern appears for the volume 64 at

β = 2.34.

82

Figure 3.18: The residue distribution after subtracting the Gaussian part withβ = 2.18 on a 44 lattice.

Figure 3.19: The residue distribution after subtracting the Gaussian with β = 2.18on a 64 lattice.

3.4.3 Zeros using Analytic Approximation

We now use the analytic approximation of the entropy density function f (x)

and the series of tools we have developed to find the Fisher’s zeros. We have

83

Figure 3.20: The residue distribution after subtracting the Gaussian with β = 2.348on a 64 lattice.

discussed the structure and property of the roots of f ′′(x) = 0, which correspond

to the zeros in Large-Np limit. Now we can look at the actual location of the zeros

at these finite volumes. We use the numerical integration to evaluate ⟨x⟩β (which

is −Z′(β)/NpZ(β)), and then do loop integrations in the β-plane. We typically

break the region into a collection of boxes and search for zeros in each one. We

use Eq. (3.22) with n = 0 to determine the number of zeros in each box and decide

whether to do a refined integration. If the box contains no zeros, we skip; if the

box encloses more than two zeros, then we break it into two smaller boxes. If the

box contains only one zero, then the integration in Eq. (3.22) with n = 1 gives

directly the location of the zero; if the box contains two zeros, the we have the

following two equations

β1 + β2 = I1,

β21 + β

22 = I2,

where I1,2 are the loop integration described in Eq. (3.22) with n = 1, 2. The

solutions of these equations will give the locations of the two zeros. Fig. (3.21)

84

0

0.1

0.2

0.3

0.4

0.5

0.6

1.5 2 2.5 3

Im(

β)

Re(β)

SU(2) zeros

L=4 f’’=0 L=4 res.L=6 f’’=0 L=6 res.

Figure 3.21: The plot shows a typical pattern of the SU(2) zeros. The zeros oftwo different volumes, 44(filled squares) and 64 (filled circles), are plotted againsttheir respective f ′′(x) = 0 zeros (empty squares for the 44 and empty circles forthe 64). The density of states are approximated using Chebyshev Polynomial oforder 44. The fitting range is over x ∈ [0, 2].

shows a plot of all the zeros in the box of (1.5,3) (0,0.55) at the two different

volumes 44 and 64. The roots of f ′′(x) = 0 zeros (empty squares and circles) are

right beneath the strands of the actual zeros.

We are interested in both the actual zeros and the roots of f ′′(x) = 0. How-

ever the latter needs to be examined carefully with the ellipse of convergence

which is important for the analytic continuation to the complex plane. We have

shown that the effective orders of approximation can be determined through the

plot of the logarithm of the absolute value of the Chebyshev coefficients versus

the order. The ellipse determined through the slope of the plot should well en-

close the roots of f ′′(x) = 0 to make them reliable, as has been illustrated in Section

3.3.3.

To ensure the zeros we got are not artifacts from the approximation, we

85

should make sure of two aspects: first, the zeros should not vary with the ranges

of the approximation; second, the zeros should be stable with the variation of the

orders of approximation.

Table (3.2) shows the Fisher zeros obtained at the volume 44 with three

different ranges: [0.1, 0.8], [0.2, 0.7] and [0.3, 0.6]. The order is determined through

the ln |cn| ∼ n plots. The points corresponding to the roots of f ′′(x) are inside the

ellipse of convergence. The actual zeros are obtained through the Cauchy loop

integration method. All the error bars are estimated by the replica of numerical

f (x) computed from 20 seeds of Monte Carlo data (Table (3.2)). The similar results

for the volume 64 are also given in Table (3.3). We can see that the approximation

is rather stable with the change of fitting ranges (see Fig. (3.22)).

Range order Ell’s Major f ′′ - Reβ f ′′- Imβ Actual Reβ Actual Imβ

[0.1,0.8] 38 0.356453 2.2045(45) 0.0766(56) 2.1970(25) 0.1516(35)

[0.2,0.7] 24 0.261115 2.2014(30) 0.0787(47) 2.1989(44) 0.1488(54)

[0.3,0.6] 14 0.178755 2.2032(48) 0.0771(50) 2.1972(34) 0.1496(45)

Table 3.2: Both of the actual zeros and the f ′′(x) zeros are shown with differentranges of fitting for the volume 44. The semi-major of the corresponding ellipsesare also given ( the focus length is half the range).

The approximation should also be stationary with respect to the variation

of the orders. Table (3.4) shows the Fisher zeros obtained using various orders of

approximation for the two volumes 44 and 64. We see that variation is within the

statistical fluctuation (less than one sigma), which is shown in Fig. (3.23).

86

Range order Ell’s Major f ′′ - Reβ f ′′- Imβ Actual Reβ Actual Imβ

[0.2,0.7] 38 0.256916 2.3179(23) 0.0904(37) 2.3142(22) 0.1223(33)

[0.3,0.6] 24 0.156440 2.3158(19) 0.0924(35) 2.3135(17) 0.1224(37)

Table 3.3: Both of the actual zeros and the f ′′(x) zeros are shown with differentranges of fitting for the volume 64.

0.04

0.06

0.08

0.1

0.12

0.14

0.16

2.16 2.2 2.24 2.28 2.32

Im

β

Re β

Zeros with diff. ranges

Actual Zerosf’’(x)=0 Zeros

Figure 3.22: The plot is the zeros (empty cirdles) of two volumes: 44 and 64. Theirf ′′(x) = 0 zeros (filled circles) are also plotted for comparison. Both of the actualzeros and the f ′′(x) = 0 zeros tend to approach to the same limit which is distantaway from the real axis.

We can notice that the imaginary part of the actual zeros decreases slightly

with volume. However, the imaginary part of the f ′′(x) = 0 zeros increases

with volume which shows the actual zeros will stay away from the real axis.

This corresponds to a crossover rather than a real phase transition in the infinite

volume limit [56, 16]. This fits in what is widely accepted on the critical behavior

about the SU(2) pure gauge model at the zero temperature.

87

Volume order Actual Reβ Actual Imβ

4 14 2.1970(25) 0.1516(35)

15 2.1974(35) 0.1516(39)

16 2.1976(37) 0.1502(51)

6 24 2.3135(17) 0.1224(37)

25 2.3159(28) 0.1203(41)

26 2.3161(46) 0.1194(32)

Table 3.4: The actual zeros of various orders of approximationfor the two volumes 44 and 64 discussed in the text.

0.08

0.1

0.12

0.14

0.16

2.16 2.2 2.24 2.28 2.32

Im

β

Re β

Zeros with diff. orders

44 order 1444 order 1544 order 1664 order 2464 order 2564 order 26

Figure 3.23: The lowest zeros obtained using various orders of approximation attwo different volumes: 44 and 64.

Due to the rapidly increasingNp and the noise level of the data, the searching

for the actual zeros of the SU(2) model at the volume 84 and higher becomes

difficult. However as the toy model in Section 3.3.1 hints, we might be able to

reveal the scaling property by scaling a finite-volume entropy density function

88

with an arbitraryNp. We can look at the distances of theseNp-scaled zeros to the

roots of f ′′(x) = 0 on the complex β-plane.

We use f (x) obtained from the volume 44. We change the value of Np and

use the Cauchy loop integral method to calculate the corresponding zeros. The

results are given in Table.(3.5). The difference between the Np = 6 × 44 zero and

theNp →∞ zero is about 0.076. The difference between theNp = 6× 64 zero and

theNp →∞ zero is about 0.027 which is close to what is actually calculated using

the f (x) at volume 64. The series of zeros are plotted against the f ′′(x) = 0 zero in

Fig. (3.24). Fig. (3.25) shows the log-log plot of the distances of the zeros to the

f ′′(x) = 0 zero using f (x) obtained from two different volumes: 44, 64. The 44 case

shows a power law of order −2.6207 while the 64 shows a power law of order

−2.62175. The two scalings are remarkably close which means that the volume

has rather small effect on this type of scaling.

L Np f4, Reβ f4, Imβ f6, Reβ f6, Imβ

4 1536 2.198301129 0.138674394 2.312452261 0.180337608

5 3750 2.200263410 0.104969858 2.319579161 0.137947670

6 7776 2.201087484 0.088794723 2.322613344 0.117721356

7 14406 2.201508553 0.079996177 2.324184327 0.106733300

8 24576 2.201750744 0.074768590 2.325094886 0.100204463

9 39366 2.201901660 0.071452152 2.325666611 0.096061124

10 60000 2.201999080 0.069231458 - -

∞ ∞ 2.202309682 0.062313622 2.327234710 0.08462087

Table 3.5: The zeros with the increasing ”volumes” using the density of states f4

and f6 of the 44 and 64 lattices.

89

L = 4

5

6

78

9 10¥

2.197 2.198 2.199 2.200 2.201 2.202 2.203

0.06

0.08

0.10

0.12

0.14

Re Β

ImΒ

Figure 3.24: The locations of the zeros calculated using the entropy density func-tion f at volume 44 scale withNp = 6×L4. The red point corresponds to the f ′′ = 0zero.

0.006

0.012

0.024

0.048

0.096

4 5 6 7 8 9 10

|β L -

β ∞|

L

Scale of the zero distances: 44,64

Using 44 fUsing 64 f

Figure 3.25: The zeros βL obtained using various Np = 6 × L4 using the entropydensity function from two different volumes: 44,64. The plot shows the distancefrom βL to the f ′′ = 0 zero β∞ versus L in the logarithmic scale.

90

3.5 U(1) Zeros

We now move on to the discussion of the Fisher’s zeros of the U(1) model.

Unlike the case of the SU(2), the discrete reweighting method works very reliably

here in finding the zeros, especially for higher volumes, due to the fact that the

zeros tend to approach the real axis with a power law of the volume. We applied

both of the discrete reweighting and the analytic approximation methods. We

found that in general they agree with a very high accuracy. The results from the

U(1) model provide a contrasting example for case of SU(2) because these two

systems belong to different universalities.

3.5.1 Volume Dependence of the Double Peak

The plaquette distribution of the U(1) model appears to have a double peak

structure near the βcrit which is around 1. If the double peak persists at large

volume, we should expect a first-order transition. So we should first look at the

change of the double peak over the volumes.

To determine the separation of the double peaks, we can visually check the

distribution f (s) − βs (here s ≡ x which is the average action) for the value of

β such that the two peaks have equal heights. In Fig. (3.26) [9], we show that

f (s)− βs is slightly tilted to the left for β = 1.00175 and to the right for β = 1.00179.

So the two peaks make a tie around βS = 1.00177(2). With the same graphs, we

can also tell approximate values s1 and s2 which are the two maximum values of

s. The numerical results for the three volume are summarized in Table 3.6.

A plot of the double peak distribution is displayed in Fig. (3.27). Fig. (3.27)

makes clear that the dip between the peaks deepens and the peak separation

slightly decreases as the volume increases. The peak separation s2 − s1 scales

91

-0.7474

-0.74739

-0.74738

-0.74737

0.34 0.36 0.38 0.4 0.42

f(s)-

βs

s

U(1) 64

β=1.00175β=1.00177β=1.00179

Figure 3.26: f (s) − βs for β = 1.00175, 100177 and 1.00179 on a 64 lattice. Thehorizontal lines is drawn to indicate the asymmetry of the heights. The error barsare provided with the same scale as f (s) − βs.

L βS s1 s2

4 0.9793(1) 0.370(5) 0.445(5)

6 1.00177(2) 0.353(2) 0.411(2)

8 1.00734(1) 0.349(1) 0.395(1)

Table 3.6: βS, s1 and s2 defined in the textfor L = 4, 6 and 8.

92

0

5

10

15

20

0.3 0.35 0.4 0.45 0.5

P(s)

s

U(1) plaquette distr. at βS

44

64

84

Figure 3.27: The double peak distribution of three volumes: 44, 64 and 84.

approximately like L−0.7. Given that we have only a small set of volumes, this

statement should be used to estimate the range of β that is necessary to calculate

the density of states at larger volumes. The general scaling may receive con-

siderable correction from higher volumes, which will be revisited later in this

chapter.

3.5.2 Zeros using the Discrete Reweighting Method

With the numerical expression of the density of states, we can do a simple

summation to evaluate the partition function without doing an analytic approxi-

mation,

Z(β) ≃ ∆s∑

s

eNp( f (s)−βs) . (3.24)

Then we can find the zero contours of the real part and the imaginary part of the

partition function. These plots are shown, for an example, in Fig. (3.28). The

lowest two zeros can be resolved by this method rather precisely. The results are

given in Table.(3.7) and Table.(3.8). Out of density of states from 20 seeds of data,

the zeros can be located as precisely as down to 10−5.

93

0

0.02

0.04

0.06

0.08

0.1

0.97 0.975 0.98 0.985 0.99

Im

β

Reβ

U(1) 44

ReZ=0ImZ=0

0

0.01

0.02

0.03

0.04

0.05

0.99 0.995 1 1.005 1.01

Im

β

Reβ

U(1) 64 ReZ=0ImZ=0

Figure 3.28: Zeros of the real (point +) and imaginary (point x) part of Z for U(1)using the density of states for 44 and 64 lattices.

L first second third

4 0.9791(1) 0.9780(4) -

6 1.00180(5) 1.0007(1) 0.9993(5)

8 1.00744(2) 1.0068(2) 1.0061(4)

Table 3.7: Real part of the first three zerosfor L = 4, 6 and 8.

We now look at how the cuts of ranges that we use to evaluate the partition

function will affect the locations of the zeros. We first fix the right integration

bound to be at s = 0.95 which is close to the edge of the domain, and we change

the left bound incrementally from a low value. We monitor the change of the

location of the first zero ( typically the imaginary part of the zero because it is

more sensitive to the change). We can do the same to fix the left bound at s = 0.05.

Fig. (3.29) shows an example of how the zeros respond to the variation of the cuts

in the volume of 64. Typically, we will expect a drastic change of the locations of

94

L first second third

4 0.0259(1) 0.057(1) -

6 0.00758(2) 0.018(1) 0.027(2)

8 0.00306(2) 0.008(1) 0.012(1)

Table 3.8: Imaginary part of the firstthree zeros for L = 4, 6 and 8.

the zeros on the cuts, which indicates that the distribution decays rapidly with

the cuts and beyond certain values, the bounds of the integration have no effect

to the locations of the zeros.

ç ç ç ç ç ç ç ççç

ç

ç

ç

ç

ç

ç

ç

ç

ç

ç

ç

ççç

0.27 0.28 0.29 0.30 0.31 0.32

- 5.5

- 5.0

- 4.5

- 4.0

- 3.5

s

logHÈImΒ-

ImΒ0ÈL

ççççççççççççççç

ç

ç

ç

ç

ç

ç

ç

ç

ç

ç

ç

0.41 0.42 0.43 0.44 0.45 0.46

- 5.5

- 5.0

- 4.5

- 4.0

- 3.5

s

logHÈImΒ-

ImΒ0ÈL

Figure 3.29: The shift in the imaginary part of the zeros due to the cuts of theintegration described in the text.

95

L Reβ σs σc Imβ σs σc

0.9791235 3.6e-5 5.3e-8 0.0260065 3.7e-5 3.9e-9

4 0.9777314 3.5e-4 7.1e-6 0.0572764 1.4e-4 3.3e-6

0.9752954 1.1e-3 2.9e-4 0.0831705 1.3e-3 3.2e-4

1.0017969 1.7e-5 1.7e-6 0.0075821 8.7e-6 1.4e-6

6 1.0007433 6.0e-5 2.3e-5 0.0182044 2.8e-5 4.0e-6

0.9988964 1.4e-4 2.7e-4 0.0271866 4.5e-4 1.5e-4

1.0074380 1.1e-5 7.7e-8 0.0030653 3.6e-6 6.8e-8

8 1.0068296 2.3e-5 2.1e-6 0.0077673 2.4e-5 3.3e-7

1.0060410 1.1e-4 1.2e-5 0.0115079 1.0e-4 8.5e-5

Table 3.9: The lowest three zeros in the three volumes 44,64 and 84. Col-umn 1-4 are, the real parts of the zeros, the estimate error σs from differentseeds of Monte Carlo runs and the error σc due to the orders of Cheby-shev interpolation( we used three different orders 40,44 and 50 for allthree volumes). Same for the imaginary part.

3.5.3 Zeros from analytic approximation

The other method we used is to approximate f (x) using analytic functions

and use Cauchy’s loop integral method as we did for the SU(2) model. We found

that the zeros obtained in the way are almost identical to those obtained using

discrete reweighting. To estimate the error due to the polynomial approximation,

we calculate them using three different orders of Chebyshev approximation.

The error received from the order of the approximation is very small compared

with the statistical error (typically about one magnitude off). The results are

summarized in Table.(3.9).

96

0

0.02

0.04

0.06

0.08

0.1

0.96 0.98 1 1.02

Im(

β)

Re(β)

U(1) zeros

Figure 3.30: The lowest zeros from three volumes 44,64 and 84(from left to right).The error bars have taken account of both of the Monte Carlo statistical er-ror(seeds) and the Chebyshev interpolation error(orders). The three guidelinesare the fits for the first, second and third lowest zeros, using only the zeros of 64

and 84. They intersect the real axis approximately at the same point β = 1.01134.The diamonds on the real axes(Imβ=0) are the double-peak β’s from Table.(3.6).

Fig. (3.30) shows how these zeros are distributed on the complex plane. The

precision of the zeros increase as the imaginary part of the zeros decreases. We

can see that the first, second and the third zeros of the three volumes 44, 64 and 84

crosses roughly at the same point βc ∼ 1.01134 on the real axis, which corresponds

to the transition point in the infinite volume limit.

97

0.01

4 8

Im

β 1

L

The imaginary part vs lattice size L: log-log

fitdata

Figure 3.31: The log-log plot of the imaginary part of the lowest zero vs the latticesize L = 4, 6, 8.

3.5.4 The Scaling of Zeros

It was suggested in [40] that the jth Fisher zero has an dependence on the

volume which can be expressed using the critical exponent ν by

β j(L) ∼(

jLd

)1/νd

, (3.25)

where d is the dimension. For the lowest zero, the scaling is Imβ1 ∼ L1/ν. Fig. (3.31)

shows a log-log plot of the imaginary part of the lowest zeros vs the volumes.

It gives a scaling of L−3.07 which corresponds to ν = 0.325. The result is in good

agreement with the result in [50] which was obtained using high-statistics finite

size scaling of cumulants for their smaller volumes.

Now let’s look at theNp-scaled zeros similar to those discussed in the SU(2)

98

case. It was suggested from the SU(2)Np-scaled zeros that this scaling has small

volume dependence. We use the entropy density function from the 84 lattice. We

change the value of Np = 6 × L4 with L varying from 8 up to 21 and search for

the zeros. The lowest zeros are plotted in Fig. (3.32). Unlike the case of SU(2),

the roots of f ′′ = 0 are not corresponding to theNp →∞ limit because the saddle

point approximation Eq. (2.12) breaks down due to the presence of three saddle

points on the real axis for β near the value of βS given in Table (3.6). There are

two real solutions of f ′′ = 0, but neither is the saddle point for the βS . Since the

zeros are close enough to the real axis, so we can just analyze the scaling of the

imaginary part of the zeros. We fit ln(Im(βL)) ∼ ln L with a linear function for the

scaling power. We analyze how the power changes with datasets included in the

fit by throwing out the smallest lattice size L one at a time until the last two. The

fit result of the power scaling versus the lowest L is plotted in Fig. (3.33). Clearly

the scaling is approaching to L−4 which corresponds to ν = 0.25 and indicates a

strong sign of a first order phase transition in the infinite volume limit.

3.5.5 The Derivatives of f (x)

We can look at the problem from a different angle. We have shown the

double peaks of the probability distribution of the U(1) model. If the width of the

double peaks is not vanishing in the V → ∞ limit, then we should expect a first-

order transition. However as the volume increases, it becomes harder to precisely

find the peaks and there is ambiguity on the definition of double peaks (equal

area or equal height). However we found that the real roots of f ′′(x) = 0 can be

good alternatives to the double peaks. These roots correspond to the locations

99

0

0.0006

0.0012

0.0018

0.0024

0.003

1.00736 1.0074 1.00744

Im

β

Re β

Np scaled zeros using 84 f(x): U(1)

Np scaled zerosIntersect 1.007354

Figure 3.32: TheNp scaled zeros discussed in the text.

-4.12

-4.1

-4.08

-4.06

-4.04

-4.02

-4

8 10 12 14 16 18 20

fit of -1/

ν

L of fit lower bound

Np scaled zeros using 84 f(x): U(1)

Figure 3.33: The orders of power law from the fits are plotted versus the lowestsize L that is included in the fit.

where the distribution function exp[Np( f (x)− βx)] changes convexity. Obviously

the distance between the roots is always smaller than the width of the double

peaks. If the roots’ separation survives the V →∞ limit, so does the width of the

100

double peaks.

We have the preliminary data for larger lattice sizes: L = 10, 12, 14, 20. Fig.

(3.34) shows the numerical derivative of f (x) with respective to x. The depth

between the extrema is flattening as volume increases. However the width of the

extrema will survive the V →∞ limit which is shown in Fig. (3.35). The two real

roots of f ′′(x) (the extrema of f ′(x)) are plotted versus L. The preliminary data

from higher volumes show a persisting width of about 0.014, which indicates a

scenario of first-order transition.

0.96

0.98

1

1.02

1.04

1.06

0.3 0.35 0.4 0.45

f’(x)

x

Numerical Derivatives of f(x)

L=4L=6L=8L=10L=12L=14L=20

Figure 3.34: The first derivative of f (x) from different volumes.

101

0.34

0.36

0.38

0.4

0.42

4 8 12 16 20

x

L

Real roots of f’’(x)=0

left rootright root

Figure 3.35: The locations of the two real roots of f ′′(x) = 0 are plotted versus thelattice size.

102

CHAPTER 4CONCLUSION

In this work, we studied the Fisher’s zero in the complex coupling plane

which is important to identify the type of phase transition in lattice gauge theory.

We used an general approach to find the zeros by analyzing the density of states

of the models. We used both perturbative and non-perturbative methods to con-

struct the density of states. With the numerical calculation of the entropy density

function, we were able to locate the zeros by using discrete reweighting method

and the Cauchy’s loop integral method with the Chebyshev approximation of

the entropy density function. We applied these methods on the SU(2) and the

U(1) lattice gauge theory with a fundamental Wilson action on the lattices. The

locations of the zeros were presented and the scaling of the zeros was discussed.

In Chapter 2, we calculated the density of states perturbatively. By using

the saddle point approximation, we constructed series expansions of the density

of states in both of the strong and weak coupling regimes by using the strong

and weak coupling expansions of the average plaquettes. We calculated the

series of the SU(2) and U(1) lattice gauge models and found that they exhibited

finite radii of convergence and suggested the complex singularities. We also

constructed the density of states numerically using Monte Carlo simulations

and the Ferrenberg-Swendsen’s histogram reweighting method. We analyzed

the volume dependence of the density of states and gave the low order volume

corrections for the U(1) lattice gauge model.

In Chapter 3, we calculated the Fisher’s zeros using both of the discrete

reweighting and analytic approximation method. We carefully examined our

numerical calculations by analyzing the artifacts which may arise from either the

statistical fluctuations or the approximation method. With the quasi-Gaussian

103

models, we showed that the artificial zeros can be excluded effectively by circle

of confidence [2] and concluded that for the SU(2) model at the volume 44, the

zeros should be located above Imβ = 0.13. By approximating the entropy density

function using Chebyshev polynomials, we calculated the locations of the lowest

zeros: β = 2.197(3) + i0.150(5) (for 44) and β = 2.319(5) + i0.119(6) (for 64). The

imaginary part of the zeros which correspond to the root of f ′′ = 0 tend to

increases with volumes, which shows that the zeros of the SU(2) will stabilize at a

distance from the real axis. TheNp scaled zeros showed that these zeros approach

to the f ′′ = 0 roots with a scaling of L−2.62 with respective to the lattice size L.

For the U(1) lattice gauge theory, we used both of the method to calculated the

locations of th lowest three zeros and found that they were in precise agreement.

By doing an extrapolation to the real axis, we found that the transition point in the

infinite limit is located at βc = 1.01134 and the imaginary parts of the zeros show

a scaling of L−3.07 for the three volumes 44, 64 and 84, which is consistent with the

result obtained using the finite size scaling of the cumulants [50]. Preliminary

results at larger volumes show a first order transition in the infinite volume limit,

which is also indicated by the scaling of theNp-scaled zeros.

104

APPENDIX ATWO-DATA-POINT REWEIGHTING

We will use an over line to denote sample average and a pair of brackets to

denote large data limit ( statistical average). Let

δ∆A = ∆A − ∆A, (A.1)

where ∆A = A2 − A1 and we define ∆A ≡ f (Z1,Z2). It is obvious that ⟨δ∆A⟩ = 0,

but the variance is not

(δ∆A)2 =

(∂ f∂Z0

)2

(δZ0)2 +

(∂ f∂Z1

)2

(δZ1)2 +

(∂ f∂Z0

) (∂ f∂Z1

)(δZ0)(δZ1). (A.2)

The third term is zero in statistical limit, assuming that the two Monte Carlo runs

are independent. And note that σ2X= σ2

X/N, let n0, n1 be the data size of ensemble

Z0,Z1, so we have

(δ∆A)2 =1n0

(δZ0)2

Z20

+1n0

(δZ1)2

Z21

=1n0

⟨W2e−2U0⟩0 − ⟨W2e−U0⟩20⟨W2e−U0⟩0

+1n1

⟨W2e−2U1⟩1 − ⟨W2e−U1⟩21⟨W2e−U1⟩1

=

∫T(q)W2e−U0−U1dq(∫

We−U0−U1dq)2 −

1n0− 1

n1, (A.3)

where T(q) = Z0n0

e−U0 + Z1n1

e−U1 . Now make it stationary with W(q), we have

∂(δ∆A)2

∂W=

∫dqdq′e−U0(q)−U1(q)−U0(q′)−U1(q′) {T(q)W(q) − T(q′)W(q′)

}W(q′)δW(q)(∫

We−U0−U1dq)3 .

(A.4)

The only solution for ∂(δ∆A)2

∂W = 0 is

T(q)W(q) = const. (A.5)

for any q. The we have

W(q) =const.

Z0n0

e−U0 + Z1n1

e−U1. (A.6)

105

APPENDIX BFERRENBERG-SWENDSEN’S FORMULA FOR THE WEIGHT

We assume the errors are mainly from the histograms, which approach to

the large statistical limit like

⟨δH(x))2⟩ = (1 + 2τc)⟨H(x)⟩ ≡ g⟨H(x)⟩. (B.1)

The error on the density of states is then

⟨(δn)2⟩ =⟨ R∑α=1

∂n(x)∂H̄α(x)

δHα(x)

2⟩

=

R∑α,γ

∂n(x)∂H̄α(x)

∂n(x)∂H̄γ(x)

⟨δHαδHγ⟩.

Since individual Monte Carlo runs are independent to each other, we can assume

the correlation to be zero for distinct data points,

⟨δHαδHγ⟩ = δαγ⟨(δHα)2⟩ = δαγg⟨Hα⟩. (B.2)

Notice that∂n(x)∂H̄α(x)

=Wα

(eβαx−Fα

Nα∆x

)(B.3)

and the fact that in the limit of infinite data

n(x) =eβαx−Fα

Nα∆x⟨H(x)⟩. (B.4)

So we have

⟨(δn)2⟩ =∑α

W2α

eβαx−Fα

Nα∆xgαn(x) (B.5)

Then we can use Lagrange’s Multiplier to minimize this error with respect

to Wα(x) using the constraint∑αWα(x) = 1. And we can get readily

Wα(x) =Nα/gα eFα−βαx∑′αNα′/gα′ eFα′−βα′x

. (B.6)

106

APPENDIX CTRAPEZOIDAL INTEGRATION FOR QUASI-GAUSSIAN FUNCTIONS

The problem is associated with the numerical integration of

I(A) =∫ ∞

−∞e−

12 Ax2

cos xdx (C.1)

using the simple trapezoidal rules which can be described by∫ b

af (x)dx ≈

N−1∑n=1

f (xi)h +f (x0) + f (xN)

2h . (C.2)

It has an error of order∼ 1/N2. However in the following we should prove that for

the integration in Eq.(C.1), the error of the trapezoidal approximation is of order

exp(−kN2) where k is a constant. As a result, trapezoidal rule works exceptionally

well in this type of integral.

To be precise, let us work with the integral with a proper cut

I =∫ ym

−ym

e−12 Ax2

cos xdx. (C.3)

Using 2N trapezes, we can write

IN = hN∑

n=−N

e−12 An2h2

cos(nh), (C.4)

where h = ym/N. It is actually a truncated form of the elliptic theta function

θ3(ν, τ) =∞∑

n=−∞qn2

cos(2nπν), (C.5)

where q = eiπτ. Use the Jacobi Imaginary transformation,

θ3(ν, τ) =√

(i/τ)e−πiν2/τθ3(ντ,−1τ

). (C.6)

107

Let us take τ = −Ah2/(2iπ), ν = h/(2π), we then have

IN→∞ ≈ h

√iτ

e−iπν2/τ∞∑

m=−∞

(e−iπ/τ

)m2

cos(2πmν)

=

√2πA

e−1

2A

(1 + 2e−

2π2

Ah2 cosh(2πAh

) + ...)

=

√2πA

e−1

2A

(1 + 2e−

2π2

Ah2(e

2πAh + e−

2πAh

)+ ...

)=

√2πA

e−1

2A

(1 + e−

2π2

Ah2 +2πAh + ....

). (C.7)

The exact result of the integration is I =√

2πA e−

12A , so now we have

∆I/I = exp(−η(h)2A

), (C.8)

where η(h) = π2

h2 − πh . Since h = ym/N, so ∆I/I ∝ exp(−kN2) where k = π2/(2y2mA).

In the following, we will do a numerical verification. We will use the fact

log(∆I/I) = − 2π2

Ay2m

(N − ym

2π)2 +

12A

(C.9)

and log-log plot of log(∆I/I) − 12A vs (N − ym

2π ) should follow a power law of two.

C.0.5.1 Generalization

1. Shifting

∫ ∞

−∞e−ax2/2 cos(x + δ)dx = cos(δ)

∫ ∞

−∞e−ax2/2 cos(x)dx. (C.10)

2. Sandwiched Polynomials.

In =

∫ ∞

−∞e−ax2/2xn cos(x)dx (C.11)

It is obvious that I2k−1 = 0 where k is an integer. For even orders,

I2n =

(−2∂∂A

)n

I0. (C.12)

108

ç

ç

ç

ç

ç

ççççççççççççççç

3.0 3.5 4.0 4.5 5.0 5.52

3

4

5

6

7

LogHN - ym�2ΠL

LogHLogHDI�IL-1�2AL

Figure C.1: The log-log plot of log(∆I/I) − 12A as a function of the number of

grids used in the Trapezoidal integration. We use A = 0.004069 and cut theintegral range beyond the precision goal at ym = 582.692 The fit gives a linear re-lation of −4.248355148205+ 2.0000000000000x, which agrees with ln((2π2)/(Ay2

m))= −4.2483551482054.

Let ξ(A) = 1/A2 − 1/A, the lowest several orders read

I2 = −ξI0 ,

I4 = −ξI2 + 2ξ′I0 ,

I6 = −ξI4 + 4ξ′I2 − 4ξ′′I0 ,

I8 = −ξI6 + 6ξ′I4 − 12ξ′′I2 + 8ξ′′′I0 .

For A << 1, the errors can be written as

∆I2 ∼ I2η(h)

1 − Ae−

12Aη(h) ,

∆I4 ∼(I4η(h) + I2

η2(h)A2

)e−

12Aη(h).

109

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