Emergent Phase Space Description of Unitary Matrix...

Emergent Phase Space Description of Unitary MatrixModel

Suvankar DuttaIndian Institute of Science Education and Research Bhopal

Strings AttachedIIT Kanpur

February, 2017

JHEP 1604, 104 (2016) and arXiv:1610.07743 [hep-th](with Parikshit Dutta).

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Unitary matrix model plays an important role in physics and mathematics.

The field was initiated almost a century ago by statisticians and introducedin physics in the 50s-60s by Wigner and Dyson.

Partition functions of different super-symmetric gauge theories, inparticular Chern-Simons theories on certain manifolds boil down to UMMs.

These models also have applications in a broad class of condensed mattersystems.

There exists a strong similarity between non-trivial zeros of ζ(s) functionand eigenvalues of unitary matrix.

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Unitary matrix model

An unitary matrix model is a statistical ensemble of unitary matrices,defined by the partition function

Z =∫

[DU] exp [S(U)] ,

S(U) : Action, DU→ Haar measure.

Both DU and S(U) are invariant under unitary transformation.

One can go to a diagonal basis

U → eiθi δij , θi ’s are eigenvalues of U.

In this basis the Haar measure and the action can be written as∫[DU] =

N∏i=1

∫ π−π

dθi∏i

Unitary matrix model : the action

Gauge theories on compact manifolds can be written as a unitary matrixmodel.For example, the thermal partition function of a gauge theory at zerocoupling, with adjoint matter and gauge group SU(N) on a compactmanifold S3 can be written as unitary matrix model with

S(U) =∞∑

n=0

an(β)n TrU

nTrU†n

A particular class of unitary matrix model known as plaquette model isgiven by

Z =∫DU exp

[N∞∑

n=0

βnn (Tr [U

n] + Tr [U†n])].

This model is equivalent to the previous one for βn = an〈TrUn〉.444444444444444444444444444444

Phase space description in nut-shell

Solution of UMMs in large N limit (N being dimension of matrix) renders adistribution of eigenvalues ρ({θi}) of unitary matrices for different phases.

EVs of unitary matrices behave like positions of free fermions (FFs).

UMM also has an interpretation in the language of FFs with number ofboxes in Young diagram, corresponding to different representations ofU(N), being like momentum. u({hi}) → momentum distribution.

Thus, a relation between them defines a Fermi surface (phase spacedistribution/droplet) in two dimensions for different phases of the UMMunder consideration.

Different phases are distinguished by different topologies of fermisurfaces/droplets.

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Why phase space description ?

First we need to understand why there exists a phase space description (interms of non-interacting fermions) for large N saddle points of aninteracting gauge theory.

Secondly, phase space description of gauge theories can be used to providea dual quantum mechanical description of string degrees of freedom, sincedifferent large N phases of gauge theory are dual to classical solutions ofstring theory in AdS space.

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Eigenvalues and Fermions

The partition function is given by (for plaquette model)

Z =∫DU exp

[N∞∑

n=0

βnn (Tr [U

n] + Tr [U†n])]

=∫ ∏

i[dθi ] e−NSeff ({θi}),

where Seff ({θi}) = −N∑

i=1

∞∑n=1

2βnn cos nθi −

2N∑i 6=j

ln | sin θi − θj2 |

It describes a system of N particles interacting by the two-dimensionalrepulsive potential ln | sin(θi − θj)/2| in the common potential

V (θi ) = −∞∑

n=1

2βnn cos nθi , ∀ i .

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Eigenvalues and Fermions

Because of the strong repulsive potential ln | sin(θi − θj)/2| two particles donot come close to each other.

If we neglected the Coulomb force, all eigenvalues would sit at the minimaof the potential V (θ).

Due to the Coulomb repulsion theyare spread around these minima andfill some finite intervals.

N Fermions moving in potential V (θ).

In large N limit, partition function isdominated by minimum energyconfiguration.

V (θ)

−π π0

−θ1 θ1

Eigenvalue distribution. 888888888888888888888888888888

Solution of unitary matrix model : Eigenvalue analysis

Distribution of eigenvalues is characterized by the density function

ρ(θ) = 1N

N∑i=1

δ(θ − θi ).

In large N limit we define continuous variables :

θ(x) = θi , x =iN , x ∈ [0, 1], ρ(θ) =

∂x∂θ.

In this limit we solve the matrix model : to find which configurationsdominate the partition function.

Different configurations means different distributions of EVs. As we varythe parameters of the theory, system goes from one configuration to other- phase transition.

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Solution of unitary matrix model : How to find ρ(θ)

We introduce resolvent R(z) : R(z) = N−1〈Tr[(1− z U)−1]〉.

R(z) → analytic function inside the unit disk in complex z plane.

R(z) is also analytic for |z | > 1.

R(z →∞)→ 0, R(1/z) = 1.Given a matrix model R(z) satisfies an algebraic equation :Dyson-Schwinger equation.

Eigenvalue density

ρ(θ) = 12π[2

Unitary matrix model : example

Consider plaquette model with β1, β2 6= 0 and βn>2 = 0. Jurkiewicz andZalewski

Action is given by

S(U) = N[β1(TrU + TrU†) +

β22 (TrU

2 + TrU†2)]

= NN∑

i=1(2β1 cos θi + β2 cos 2θi )

= N2∫

dx(2β1 cos θ(x) + β2 cos 2θ(x))

= −N2∫

dθρ(θ)V (θ).

The potential V (θ) is given by

V (θ) = −2β1 cos θ − β2 cos 2θ. 111111111111111111111111111111111111111111111111111111111111


We categorize different classes according to gaps in eigenvaluedistributions.

No-gap phase : Two possibilities

Potential V (θ) = 0

All the eigenvalues are uniformlydistributed on circle due to coulombrepulsion.

ρ(θ) = 12π .

Potential V (θ) 6= 0, but depth is less -can not accommodate N fermions.

ρ(θ) = 12π (1 + 2β1 cos θ + 2β2 cos 2θ)

for θ ∈ [−π, π].

V (θ)

−π π0

−π π0

ρ(θ)

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One-gap phase

Potential depth is well enough toaccommodate N states.

Eigenvalue density is given by

ρ(θ) = 2π

√sin2 θ12 − sin

2 θ2

cos θ2 [β1 + β2(2 cos θ + cos θ1 − 1)] .

θ1 is determined in terms of β1 andβ2.

V (θ)

−π π0

−θ1 θ1

Highest level

ρ(θ)

−θ1 θ1−ππ

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Two-gap phase

Potential looks like a double-welltype.In large N limit, lowest energyconfiguration : lowest energy statesare symmetrically filled in both thewells.


ρ(θ) = −β2π| sin θ|√

(cos θ2 − cos θ)(cos θ − cos θ1)

β2 < 0 and θ1, θ2 are determined interms of β1 and β2.

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Two-gap phase

Potential has minima at θ = 0 andθ = π.In large N limit, lowest energyconfiguration : lowest energy statesare symmetrically filled in both thewells.


ρ(θ) = 1π

√(cos θ2 − cos θ)(cos θ1 − cos θ)

[β1 + β2(2 cos θ + cos θ1 + cos θ2)]

β2 > 0 and θ1, θ2 are determined interms of β1 and β2. 151515151515151515151515151515151515151515151515151515151515

Unitary matrix model : Phase space

In large N (classical) limit, energy levels become continuous. Particles arecharacterized by their coordinates in the phase space.

In this case phase space is two- dimensional. Each particle occupies thearea 2π/N.

Ground state of the system correspond to a configuration where all thelevels are filled up to �F .

Therefore single particle Hamiltonian

Hs(p, θ) =p22m + V (θ) = �F

defines a Fermi surface in (h, θ) phase space. p being momentum of afermion.

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Unitary matrix model : Phase space

Since total number of states below �F is N we have

N =∮ dθ

2π/N

∫dp Θ(�F − Hs(p, θ)).

Total energy of the lowest energy configuration

E0 =∮ dθ

2π/N

∫dp Hs(p, θ)Θ(�F − Hs(p, θ)).

Phase space density

ω(p, θ) = 12πΘ(�F − Hs(p, θ)).

In large N limit, classical phase space density satisfies[−∂H∂θ

∂

∂p +∂H∂p

∂

∂θ

]ω(p, θ) = 0. 171717171717171717171717171717171717171717171717171717171717

Goal :

To understand how free fermi phase picture emerges from a generic class ofmatrix model ?

What is the underlying quantum mechanics ?

What does momentum distribution correspond to ?

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Partition function in momentum space

Partition function

Z =∫

[DU] exp[N∞∑

n=1

βnn (TrU

n + TrU†n)]

Expanding the exponential

Z ∼∑

R

[−−−

]χ~kRχ

~lR .

χ~kR : character of conjugacy class

~k of permu. group SK ,K =

∑α αkα

R denotes representation of U(N).

Sum over R can be decomposedas

∑R→

∞∑K=1

∑{λi}

δ

( N∑i=1

λi − K).

Hence, partition function

Z =∑~λ

[−−−

]χ~k~λχ~l~λ

In large N limit one can extremizepartition function w.r.t. λi ’s andfind dominant representations.χ~k~λ

are difficult to find.

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Partition function

Z =∫

[DU] exp[N∞∑

n=1

βnn (TrU

n + TrU†n)]

Expanding the exponential

Z ∼∑

R

[−−−

]χ~kRχ

~lR .

χ~kR : character of conjugacy class

~k of permu. group SK ,K =

∑α αkα

R denotes representation of U(N).

Sum over R can be decomposedas

∑R→

∞∑K=1

∑{λi}

δ

( N∑i=1

λi − K).

Hence, partition function

Z =∑~λ

[−−−

]χ~k~λχ~l~λ

In large N limit one can extremizepartition function w.r.t. λi ’s andfind dominant representations.χ~k~λ

are difficult to find. 191919191919191919191919191919191919191919191919191919191919


For simpler model β1 6= 0 and βn>1 = 0, character is easy to find. It isequal to the dimension of the representation. [SD-Gopakumar; Dutta-SD]

Introduce hi = λi + N − i and define Young tableaux density in large N limit

u(h) = −∂x∂h , h(x) =

hiN , x =

iN .

In large N limit the saddle point equation is given by

−∫ hU

hLdh′ u(h

′)h − h′ = ln

[hξ

], ξ2 = β1

∫ hUhL

dh′u(h′)h′.

It is easy to find u(h) for the most dominant representation.202020202020202020202020202020202020202020202020202020202020


We observed that [SD-Gopakumar, 2007] and [Dutta-SD, 2015]

h↔ ρ(θ), and u(h) = θ/π.

This allows one to write eigenvalue distribution and Young tableauxdistribution in terms of a single constant phase space distribution functionω(h, θ) such that,

ρ(θ) =∫ ∞

0ω(h, θ)dh, u(h) =

∫ π−πω(h, θ)dθ

where ω(h, θ) is a distribution in a two dimensions.

ω(h, θ) = 12π ; (h, θ) ∈ R

= 0 ; otherwise.212121212121212121212121212121212121212121212121212121212121

Partition function in momentum space : A generic case

Frobenius formula

χ~k~h =

∆(x).∏j

(Pj(x))kj

(h1,··· ,hN )

; Pj(x) =N∑

i=1x ji , ∆(x) =

∏i


In large N limit we defineziN = z(x), kn = N

2k ′n.

Saddle point equation for z(x)

∑n

nk ′nzn(x)Zn

+−∫ 1

0

z(x)dyz(x)− z(y) = h(x), Zn =

∫ 10

dxzn(x).

Original contour was about the origin, we take z(x) = eiθ(x)

∑n

nk ′nZn

einθ +−∫ π−π

dθ′ eiθρ(θ′)

eiθ − eiθ′ = h(θ)

ρ(θ) = ∂x∂θ

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Imaginary part of this equation,

12−∫ π−π

dθ′ρ(θ′) cot(θ − θ′

2

)=∑

n

nk ′nZn

sin nθ.

It turns out thatnk ′n/Zn = βn

and the above equation is exactly same as eigenvalue equation.

Auxiliary variables which appear in Frobenius formula are eigenvalues ofunitary matrices under consideration.

This allows us to write a generalized partition function

Z =∫ ∏

idθi

∏i

dhi Ω(~h, ~θ).242424242424242424242424242424242424242424242424242424242424


The real part of equation is an algebraic equation, which relates number ofboxes in a particular Young diagram with eigenvalues of unitary matrices,

h(θ) = 12 +∑

n

nk ′nZn

cos nθ = 12 +∑

nβn cos nθ

One can show that value of character remains unchanged if we replace

h→ h + f (θ), f (θ) is an even function.

It turns out to be [Dutta-SD, 2016]

f (θ) = πρ(θ).

Hence final relation between h and θ

h±(θ) =12 +

∑nβn cos nθ ± πρ(θ).

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Fermi surface in (h, θ) plane

h±(θ) = 12 +∑

n βn cos nθ ± πρ(θ) defines a closed region in (h, θ) plane.

All the states inside this region are completely filled.

All the states outside this region are empty.

Thus boundary defines a fermi surface in (h, θ) plane.

Topology of fermi region is different for different phases of the model.

Two regions are topologically different if : i) one can not continuouslydeform (without cutting or glueing) one region to other and (ii) boundaryof the region should not cross the axis.Phase transition corresponds to change of topology of fermi region.

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No-gap phase

In this case

h+ = ρ(θ) and h− = 0.

One-gap phase

In this case h+ and h− are non-zero

This region is topologically differentthan that of no-gap phase becauseorigin is outside the region.

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Two-gap Solution - 1In this case

h+ = ρ(θ) and h− = 0.

Two-gap Solution - 1

In this case h+ and h− are non-zero

This region is topologically differentthan that of no-gap phase becauseorigin is outside the region.

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Underlying quantum mechanics

Phase space density is given by

ω(h, θ) = 12πΘ ((h − h−)(h+ − h)) .

ω(h, θ) satisfies Liouville equation[−∂H∂θ

∂

∂h +∂H∂h

∂

∂θ

]ω(p, θ) = 0

Solving this one cam solve for the Hamiltonian

H(h, θ) = A[h2 − (h+ + h−)h + h+h−

]+ C.

This Hamiltonian is not in canonical form.292929292929292929292929292929292929292929292929292929292929

Work in progress

Does it related to usual canonical Hamiltonian by some canonicaltransformation ?

What is the underlying quantum mechanics for ABJM and CS theory ?

Partition function of ABJM can be calculated exactly for all values ofcoupling constant.

There exists an interpolating function between weak and strong coupling.

Thus emergent phase space description can be used to explore a dualquantum mechanical description of string theory.

Thank you

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