Fate of fractional quantum Hall states in open quantum systems:
characterization of correlated topological states for the full Liouvillian
- URL: http://arxiv.org/abs/2005.12635v2
- Date: Tue, 8 Sep 2020 09:59:23 GMT
- Title: Fate of fractional quantum Hall states in open quantum systems:
characterization of correlated topological states for the full Liouvillian
- Authors: Tsuneya Yoshida, Koji Kudo, Hosho Katsura, and Yasuhiro Hatsugai
- Abstract summary: We introduce the pseudo-spin Chern number of the Liouvillian which is computed by twisting the boundary conditions only for one of the subspaces of the doubled Hilbert space.
The existence of such a topological invariant elucidates that the topological properties remain unchanged even in the presence of the jump term.
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- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Despite previous extensive analysis of open quantum systems described by the
Lindblad equation, it is unclear whether correlated topological states, such as
fractional quantum Hall states, are maintained even in the presence of the jump
term. In this paper, we introduce the pseudo-spin Chern number of the
Liouvillian which is computed by twisting the boundary conditions only for one
of the subspaces of the doubled Hilbert space. The existence of such a
topological invariant elucidates that the topological properties remain
unchanged even in the presence of the jump term which does not close the gap of
the effective non-Hermitian Hamiltonian (obtained by neglecting the jump term).
In other words, the topological properties are encoded into an effective
non-Hermitian Hamiltonian rather than the full Liouvillian. This is
particularly useful when the jump term can be written as a strictly block-upper
(-lower) triangular matrix in the doubled Hilbert space, in which case the
presence or absence of the jump term does not affect the spectrum of the
Liouvillian. With the pseudo-spin Chern number, we address the characterization
of fractional quantum Hall states with two-body loss but without gain,
elucidating that the topology of the non-Hermitian fractional quantum Hall
states is preserved even in the presence of the jump term. This numerical
result also supports the use of the non-Hermitian Hamiltonian which
significantly reduces the numerical cost. Similar topological invariants can be
extended to treat correlated topological states for other spatial dimensions
and symmetry (e.g., one-dimensional open quantum systems with inversion
symmetry), indicating the high versatility of our approach.
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