1. Local convertibility in a quantum quench
Speaker: Li Dai 戴笠 (postdoc in NCHU)
Supervisor：Prof Ming-Chiang Chung 張明強教授
國立中興大學物理系 Department of Physics,
National Chung Hsing University, Taichung, Taiwan
PRA 91, (2015)

2. Outline Introduction Quantum quench in 1D Kitaev chain Conclusion
1. Topological phases and entanglement spectrum;
2. Local convertibility.
Quantum quench in 1D Kitaev chain
1. Stationary state and correlation length;
2. Relation to the Majorana modes.
Conclusion

3. 1. Introduction
1.1 Topological phases cannot be described by local order parameter.
1.2 Novel properties: topologically protected ground state degeneracy;
quantum hall effects (integer, fractional, spin, anomalous)…
1.3 Entanglement spectrum
Reduced state of subsystem A A B
Entanglement Hamiltonian
(1) Classify quantum phases;
(2) Detect edge modes.
1.1 Topological phase is beyond the Landau Paradigm of condensed matter physics.
1.2 QHE: 2D electron gas in a strong magnetic field and low temperature, quantized Hall voltage.
QSHE: mercury telluride, 2D quantum well, spin-orbit coupling, no magnetic field, spin-hall conductance, no charge-hall conductance.
QAHE: ferromagnetic material (or paramagnetic material in a magnetic field), multi-bands.
“Classify quantum phases” means (two-fold) degeneracy of ES can characterize topological phases (even in the absence of edge states); trivial (non-topological) phases have no such degeneracy.
F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa,
Phys. Rev. B 81, (2010);
M.-C. Chung, Y.-H. Jhu, P. Chen, and S. Yip,
Europhys. Lett. 95, (2011).

4. ' ' ' LOCC 1.4 Local convertibility Rényi entropy for all LOCC
Local Operations and Classical Communication
1.4 Local convertibility '
Rényi entropy ' '
for all
Reduced state of subsystem A
d: effective rank of
Von Neumann entanglement entropy (EE)
Single-Copy entanglement
LOCC
Higher EE
Same or Lower EE
Not always true!
S. Turgut, J. Phys. A: Math. Theor (2007).

5. 1.5 Differential Local Convertibility (DLC)
LOCC
(1) Analytical property, mathematically elegant;
(2) Connection to the adiabatic evolution.
Topological phase with symmetry
Ferromagnetic phase breaking symmetry
e.g. transverse Ising model A B A B
symmetry:
For the transverse Ising model with 0<=g<1, both OBC and PBC have (nearly) doubly degenerate ground states. Namely, boundary conditions are not important here. A B A B
Doubly degenerate ground states
Quantum channel
Classical channel
J. Cui, M. Gu, L. C. Kwek, M. F. Santos, H. Fan, and
V. Vedral, Nat. Commun. 3, 812 (2012).

6. g
t =1
Zero-energy edge mode
Sign of g -1
The presence of Majorana fermions at the two ends with some overlap leads to the breakdown of DLC. 1 g
1. Large alpha Renyi entropy is dominated by the edge-state behavior. Low alpha entropy increases because more states are needed for Schmidt decomposition.
2. Esp., etg of bulk states increases toward QPT, while etg of edge states does the opposite.
3. When the correlation length increases to be comparable to the size of the subsystem, MFs start recombining. As a result, DLC breaks down.
F. Franchini, J. Cui, L. Amico, H. Fan, M. Gu, V. Korepin,
L. C. Kwek, and V. Vedral, Phys. Rev. X 4, (2014).

7. (esp. topological property)
However, for other models such as XXZ spin chain, DLC is not necessarily related to quantum phase transitions.
H. Bragança, et. al., Phys. Rev. B 89, (2014).
In this work, we study DLC in the 1D Kitaev chain with a quantum quench.
Motivation
The quantum states with time evolution involves excited states/levels, so that DLC may agree with the quantum phases.
DLC of
quenched states
Quantum phases
(esp. topological property)
For spin ½, spin 1, XXZ chains, DLC is a good indicator of SU(2) symmetry of the model. Change of DLC can occur within the same phase.
The point 1 indicates that DLC is more related to the properties of the Hamiltonian than its ground state.
Entanglement
property
Majorana fermionic
quantum computation
S. B. Bravyi and A. Y. Kitaev, Ann. Phys. 298, 210 (2002).

8. . . 2. Quantum quench in 1D Kitaev chain 2.1 The model
Pseudo-magnetic field . .
M.-C. Chung, Y.-H. Jhu, P. Chen, C.-Y. Mou, and X. Wan, arXiv: v2
S.-Q. Shen, W.-Y. Shan, and H.-Z. Lu, SPIN 01, 33 (2011).

9. ? 2.2 Quantum quench Question: LOCC
Answer: Yes, if the sign of is uniform up to 0, for all
2.3 Stationary state: for quenched :
Thermalization
M. Fagotti and F. H. L. Essler, Phys. Rev. B 87, (2013).

10. How to calculate the Rényi entropy of stationary state
Correlation function matrix
Quantum quench
Pseudo-magnetic field
P. Calabrese and J. Cardy, J. Stat. Mech.: Theor. Exp. 2005, P04010 (2005).
M.-C. Chung, Y.-H. Jhu, P. Chen, C.-Y. Mou, and X. Wan, arXiv: v2

11. Sign of

12. Critical length for breakdown of DLC
edge mode
Critical length for breakdown of DLC
Ceiling function for f*L0
Maximum deviation for the best fitting is 7 sites.
Correlation length of Majorana fermions
A. Y. Kitaev, Phys.-Usp. 44, 131 (2001).

13. Physical interpretation
Steady part in
(1)
(2)

14. Subsystem A
Subsystem B
When increases,
(1) The Majorana wave function within one subsystem extends inward.
insusceptible to
(2) The interaction between the Majorana fermions in A and those in B decreases.

15. Non-topological regime
The von Neumann entanglement entropy of the quenched state decreases for both regimes.

16. Conclusion
We have studied the local convertibility of the quantum state in the Kitaev chain with a quantum quench.
The quenched state is locally inconvertible in the topological regime (edge modes and Majorana fermions).
The many-body quantum state may have rich structure that cannot
be well characterized by the entanglement entropy.
Our result should help to better understand many-body phenomena in topological systems, and the Majorana fermionic quantum computation.

17. Thank you for your attention!