Davida Kollmar Department of Physics, Yeshiva University ... · Detecting Transitions Have been...
Transcript of Davida Kollmar Department of Physics, Yeshiva University ... · Detecting Transitions Have been...
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