非エルミート系における分数量子ホール状態

T. Yoshida (U. Tsukuba)

TY-Kudo-Hatsugai Sci. Rep. 9 16895 (2019)
Workshop: 熱場の量子論
2020/08/26 @Kyoto U

TY-Kudo-Katsura-Hatsugai arXiv: 2005.12635

Motivation

Short-range entangled

Long-range entangled(topo. + U )・ Topological insulators
・ Weyl semimetals

・ Fractional quantum Hall・ Z2 spin liquid・nH topo. ins.
・ exceptional points???

Question

Topological ordered phases for non-Hermitian systems?

Yes !
FQH states with 1/3 filling can emerge
for non-Hermitian systems.

Hamiltonian~Spinless fermions in magnetic flux~

nearest neighbor interactions:non-Hermitian

～Relevance with cold atoms～
Inelastic collision (Feshbach resonance)

Y. Ashida et al., PRA (2016)
M. Nakagawa et al., PRL (2018)
K. Yamamoto et al., PRL (2019)

Atoms may release their internal energy as kinetic energy

SystemEnvironment
Two-body loss:
Time-evolution of open quantum systems:density matrix of the system

～Relevance with cold atoms～
Time-evolution of open quantum systems

Suppress jump term by measurement

SystemObserve
particle loss
Environment

～Relevance with cold atoms～
Time-evolution of open quantum systems

Suppress jump term by measurement

SystemObserve
particle loss
Environment

～Relevance with cold atoms～
Time-evolution of open quantum systems

Suppress jump term by measurement

SystemObserve
particle loss
Environment

～Warm up: Hermitian case～

・Presence of the bulk gap

・3-fold degeneracy for ground states
Haldane PRL (1983)

・Ctot=1 for the ground state triplet

～non-Hermitian case～For non-Hermitian systems, eigenvalues take complex numbers…
Q:definition of ground states?
・minimumState with??

・ longest lifetime??
A:We define ground states with minimum∵) it also has longest lifetime

We define energy gap as

We conclude emergence of non-Hermitian FQH states (1/3 filling)based on・3-fold degeneracy for PBC (topological degeneracy)
・Ctot=1 for 3-fold degenerated ground states

3-fold degeneracy for ground statesn : label of states

Many-body translation symmetry

three-fold degeneracy for the PBC
Hermitian case: Haldane PRL (1983)

We conclude emergence of non-Hermitian FQH states (1/3 filling)based on・3-fold degeneracy for PBC (topological degeneracy)✔

・Ctot=1 for 3-fold degenerated ground statestotal Chern number (Ctot=1) for ground states triplet?

Left and Right eigenvectors

Ctot=1

Non-Hermitian FQH states emerge.

But,What are unique phenomena induced by non-Hermiticity?

Energy gaps opens for V=i・3-fold degeneracy for PBC
・Ctot=1 for ground state triplet〇: V=iWhy purely dissipative interaction
opens the gap?

Energyfor V=0

Why purely dissipative interaction
opens the gap?

Energyfor V=0

Repulsive interaction
due to Landau level mixingWhy purely dissipative interaction
opens the gap?

Energyfor V=0~ i2=-1~ -ℏω0

Repulsive interaction
due to Landau level mixing

Repulsive interactions induced by dissipations: reminiscentof quantum Zeno effects

Summary
Non-Hermitian FQH states (1/3 filling)
・3-fold degeneracy for PBC (topological degeneracy)
・Ctot=1 for 3-fold degenerated ground states
Dissipation induces nHFQH
reminiscent of the Zeno effectWe also find that topology of FQH states is maintainedeven in the presence of the jump termby introducing the pseudo-spin Chern number

TY-Kudo-Katsura-Yoshida arXiv:2005.12635

Thank you!