Davida Kollmar

Department of Physics, Yeshiva University, New York, NY

Collaborators:

Lea F. Santos (Yeshiva University, New York, NY)

Outline

Introduction to QPTs

Introduction to ICE

Examine specific spin-½ systems

Introduction to QPTs

Quantum Phase Transitions

Competition between terms

𝐻 = 𝐻0 + 𝑔𝐻1

Opening/closing energy gap

Orders of Transitions

○ Berezinskii-Kosterlitz-Thouless transition

Examine the ground state and first

excited state

Detecting Transitions

Have been different ways to detect these transitions, borrowing tools from quantum information

Entanglement

○ Concurrence

○ Entanglement entropy

○ Quantum Discord

Fidelity

𝐹 = Ψ λ Ψ λ + 𝛿

ICE

Invariant Correlational Entropy

This quantity is given by

𝑆 = −Tr {𝜌 ln(𝜌 )}

Use average energy density matrix:

𝜌 =𝜌𝑔 + 𝜌𝑔+𝛿 +⋯+ 𝜌𝑔+(𝑁−1)𝛿

𝑁

Check graph for peaks or inflection points

1D Spin-½

Typical many body system

Good model for real materials

XXZ Model-Heisenberg Model

The Hamiltonian for this model is given

by:

𝐻 = 𝐽 𝑆𝑗𝑥𝑆𝑗+1𝑥 + 𝑆𝑗

𝑦𝑆𝑗+1𝑦+ ∆𝑆𝑗

𝑧𝑆𝑗+1𝑧

𝐿

𝑗=1

Parameter to vary: Δ

antiferromagnetic ferromagnetic xy

-1 1

XXZ Model-Heisenberg Model

-2 -1 0 1 2D

-3.5

-3

-2.5

-2

-1.5

-1

-0.5E

ner

gy

Sz=-3,3 Sz=0

Sz=-1,1

Sz=0

Sz=-1,1

XXZ Model-Heisenberg Model

0 0.5 1 1.5 2D

0

0.2

0.4

0.6

0.8

1

Fid

elit

y

First Excited State

L=10 𝐹 = Ψ λ Ψ λ + 𝛿

Using ICE on the XXZ Model

First excited state: peak at transition

Ground state: inflection point at

transition

0 0.5 1 1.5 2D

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Inv

aria

nt

Co

rrel

atio

nal

En

tro

py

First Excited

State

L=10

0 0.5 1 1.5D

0.02

0.04

0.06

0.08

Inv

ari

an

t C

orr

ela

tio

nal

En

tro

py

L=12

L=10

L=8

L=6

Ground

State

NN+NNN Model

The Hamiltonian for this model is:

𝐻 = 𝑆𝑗𝑆𝑗+1 + 𝜆𝑆𝑗𝑆𝑗+2

𝐿

𝑗=1

Parameter to vary: λ

The transition is found numerically,

where an energy gap appears

fluid dimer

0.241

NN+NNN Model

First excited state, not ground state

Sz=0 subspace

0 0.1 0.2 0.3 0.4 0.5D

-4.4

-4.2

-4

-3.8

-3.6

-3.4

-3.2

En

erg

y

Sz=-1,0,1

Sz=-1,0,1

Sz=0

Sz=0

λ

Chen et al

PRE 76, 061108 (2007)

NN+NNN Model: ICE

First excited state: peak at transition

L=8

L=6

L=10

Future Directions

Examine different models

Ising Model in the Transverse Field

Bose Einstein Condensate

Study larger systems

Scaling analysis

Acknowledgements

Dr. Berliner, Dr. Kressel, and the Kressel

Research Scholarship committee

Deans at Stern College for Women