arXiv:2207.10558 · Physical review. B./Physical review. B

Entanglement and particle fluctuations of one-dimensional chiral topological insulators

Kyle Monkman, Jesko Sirker

Published Sep 11, 2023• 5 citations

Abstract

We consider the topological protection of entanglement and particle fluctuations for a general one-dimensional chiral topological insulator with winding number $\mathcal{I}$. We prove, in particular, that when the periodic system is divided spatially into two equal halves, the single-particle entanglement spectrum has $2|\mathcal{I}|$ protected eigenvalues at $1/2$. Therefore the number fluctuations are bounded from below by $\mathrm{\ensuremath{\Delta}}{N}^{2}\ensuremath{\ge}|\mathcal{I}|/2$ and the entanglement entropy by $S\ensuremath{\ge}2|\mathcal{I}|ln2$. We note that our results are obtained by applying directly an index theorem to the microscopic model and do not rely on an equivalence to a continuum model or a bulk-boundary correspondence for a slowly varying boundary.

DOI: 10.1103/physrevb.108.125116

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arXiv:2207.10558 · Physical review. B./Physical review. B

Entanglement and particle fluctuations of one-dimensional chiral topological insulators

Kyle Monkman, Jesko Sirker

Published Sep 11, 2023• 5 citations

arXiv HTML PDF

Abstract

We consider the topological protection of entanglement and particle fluctuations for a general one-dimensional chiral topological insulator with winding number

\mathcal{I}

. We prove, in particular, that when the periodic system is divided spatially into two equal halves, the single-particle entanglement spectrum has

2|\mathcal{I}|

protected eigenvalues at

1/2

. Therefore the number fluctuations are bounded from below by

\mathrm{\ensuremath{\Delta}}{N}^{2}\ensuremath{\ge}|\mathcal{I}|/2

and the entanglement entropy by

S\ensuremath{\ge}2|\mathcal{I}|ln2

. We note that our results are obtained by applying directly an index theorem to the microscopic model and do not rely on an equivalence to a continuum model or a bulk-boundary correspondence for a slowly varying boundary.

DOI: 10.1103/physrevb.108.125116

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Entanglement spectrum classification of C n -invariant noninteracting topological insulators in two dimensions

Chen Fang, Matthew J. Gilbert, Bogdan Andrei Bernevig

We study the single-particle entanglement spectrum in 2D topological insulators which possess $n$ -fold rotation symmetry. By defining a series of special choices of subsystems on which the entanglement is calculated, or real space cuts, we find that the number of protected in-gap states for each type of these real space cuts is a quantum number indexing (if any) nontrivial topology in these insulators. We explicitly show that the number of protected in-gap states is determined by a ${Z}^{n}$ index $({z}_{1},...,{z}_{n})$ , where ${z}_{m}$ is the number of occupied states that transform according to $m$ th one-dimensional representation of the ${C}_{n}$ point group. We find that for a space cut separating $1/p$ th of the system, the entanglement spectrum contains in-gap states pinned in an interval of entanglement eigenvalues $[1/p,1\ensuremath{-}1/p]$. We determine the number of such in-gap states for an exhaustive variety of cuts, in terms of the ${Z}^{n}$ index. Furthermore, we show that in a homogeneous system, the ${Z}^{n}$ index can be determined through an evaluation of the eigenvalues of point-group symmetry operators at all high-symmetry points in the Brillouin zone. When disordered $n$ -fold rotationally symmetric systems are considered, we find that the number of protected in-gap states is identical to that in the clean limit as long as the disorder preserves the underlying point-group symmetry and does not close the bulk insulating gap.We study the single-particle entanglement spectrum in 2D topological insulators which possess $n$-fold rotation symmetry. By defining a series of special choices of subsystems on which the entanglement is calculated, or real space cuts, we find that the number of protected in-gap states for each type of these real space cuts is a quantum number indexing (if any) nontrivial topology in these insulators. We explicitly show that the number of protected in-gap states is determined by a ${Z}^{n}$ index $({z}_{1},...,{z}_{n})$, where ${z}_{m}$ is the number of occupied states that transform according to $m$th one-dimensional representation of the ${C}_{n}$ point group. We find that for a space cut separating $1/p$th of the system, the entanglement spectrum contains in-gap states pinned in an interval of entanglement eigenvalues $[1/p,1\ensuremath{-}1/p]$. We determine the number of such in-gap states for an exhaustive variety of cuts, in terms of the ${Z}^{n}$ index. Furthermore, we show that in a homogeneous system, the ${Z}^{n}$ index can be determined through an evaluation of the eigenvalues of point-group symmetry operators at all high-symmetry points in the Brillouin zone. When disordered $n$-fold rotationally symmetric systems are considered, we find that the number of protected in-gap states is identical to that in the clean limit as long as the disorder preserves the underlying point-group symmetry and does not close the bulk insulating gap.

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Kiran Babasaheb Estake, Dibyendu Roy

We establish a quantitative bulk-boundary correspondence in interacting topological insulators by relating many-body topological invariants to characteristic degeneracy structures in the entanglement spectrum. Focusing on generalized Su-Schrieffer-Heeger chains with higher winding number, we construct a gauge-invariant many-body winding invariant based on Pancharatnam geometric phases that remains well defined in the presence of interactions. We show that this invariant uniquely determines the low-lying entanglement-spectrum degeneracy, which exhibits a universal $4^ν$ scaling with the winding number $ν$ , providing a concrete formulation of bulk-boundary correspondence beyond single-particle topology. Using exact diagonalization, we demonstrate the robustness of this correspondence under interactions and symmetry-preserving disorder, and identify inversion symmetry as a minimal protecting symmetry that stabilizes both the quantization of the invariant and the associated entanglement degeneracies. Our results unify geometric-phase invariants and entanglement diagnostics within a many-body framework and provide a route to identifying interacting topological phases beyond band theory.We establish a quantitative bulk-boundary correspondence in interacting topological insulators by relating many-body topological invariants to characteristic degeneracy structures in the entanglement spectrum. Focusing on generalized Su-Schrieffer-Heeger chains with higher winding number, we construct a gauge-invariant many-body winding invariant based on Pancharatnam geometric phases that remains well defined in the presence of interactions. We show that this invariant uniquely determines the low-lying entanglement-spectrum degeneracy, which exhibits a universal $4^ν$ scaling with the winding number $ν$, providing a concrete formulation of bulk-boundary correspondence beyond single-particle topology. Using exact diagonalization, we demonstrate the robustness of this correspondence under interactions and symmetry-preserving disorder, and identify inversion symmetry as a minimal protecting symmetry that stabilizes both the quantization of the invariant and the associated entanglement degeneracies. Our results unify geometric-phase invariants and entanglement diagnostics within a many-body framework and provide a route to identifying interacting topological phases beyond band theory.

arXiv:2604.09801Apr 10, 20260 citations

We consider the topological protection of entanglement and particle fluctuations for a general one-dimensional chiral topological insulator with winding number

\mathcal{I}

. We prove, in particular, that when the periodic system is divided spatially into two equal halves, the single-particle entanglement spectrum has

2|\mathcal{I}|

protected eigenvalues at

1/2

. Therefore the number fluctuations are bounded from below by

\mathrm{\ensuremath{\Delta}}{N}^{2}\ensuremath{\ge}|\mathcal{I}|/2

and the entanglement entropy by

S\ensuremath{\ge}2|\mathcal{I}|ln2

We study the single-particle entanglement spectrum in 2D topological insulators which possess

n

-fold rotation symmetry. By defining a series of special choices of subsystems on which the entanglement is calculated, or real space cuts, we find that the number of protected in-gap states for each type of these real space cuts is a quantum number indexing (if any) nontrivial topology in these insulators. We explicitly show that the number of protected in-gap states is determined by a

{Z}^{n}

index

({z}_{1},...,{z}_{n})

, where

{z}_{m}

is the number of occupied states that transform according to

m

th one-dimensional representation of the

{C}_{n}

point group. We find that for a space cut separating

1/p

th of the system, the entanglement spectrum contains in-gap states pinned in an interval of entanglement eigenvalues $[1/p,1\ensuremath{-}1/p]$. We determine the number of such in-gap states for an exhaustive variety of cuts, in terms of the

{Z}^{n}

index. Furthermore, we show that in a homogeneous system, the

{Z}^{n}

index can be determined through an evaluation of the eigenvalues of point-group symmetry operators at all high-symmetry points in the Brillouin zone. When disordered

n

-fold rotationally symmetric systems are considered, we find that the number of protected in-gap states is identical to that in the clean limit as long as the disorder preserves the underlying point-group symmetry and does not close the bulk insulating gap.

Higher-order topological insulators are materials with a bulk energy gap featuring gapless modes at

(d-n)

-dimensional edges, where

n > 1

. Despite this modified bulk-boundary correspondence, their nontrivial topology can nevertheless be observed through the bipartite entanglement spectrum. Inspired by works suggesting fluctuations in conserved quantities could be used to study topology in one-dimensional systems, which likewise feature zero-dimensional edge modes, we extend this treatment to a two-dimensional second-order topological insulator where corner states play an analogous role. We show that the standard bipartite entanglement entropy and particle number fluctuation lack striking signals of topological phase transitions, both in the static case and in the time evolution after a quench, obscured by an area-law term from the subsystem edges. By introducing a quadripartite construction of the subsystem, we show that it is possible to isolate the topological contributions, producing sharp transition peaks in both quantities, observable by studying subsystems much smaller than the full system. As particle number fluctuations are accessible in existing experiments, these results establish a concrete and scalable path to experimental detection of the entanglement in higher-order topological insulators.

For a many-body system of arbitrary dimension, we consider fermionic ground states of noninteracting Hamiltonians invariant under a

{C}_{2}

cyclic group. The absolute difference $\mathrm{\ensuremath{\Delta}}$ between the number of occupied symmetric and antisymmetric single-particle states is a topological invariant in this case. We prove lower bounds on the configurational and the number entropy based on this invariant. Entanglement entropies are particularly important for identifying topological phases protected by spatial symmetries because they lack gapless boundary modes in general. In topological crystalline insulators, the topological invariant $\mathrm{\ensuremath{\Delta}}$ and the entropy bounds can be directly determined from high-symmetry points in the Brillouin zone.

4^ν

scaling with the winding number

ν

, providing a concrete formulation of bulk-boundary correspondence beyond single-particle topology. Using exact diagonalization, we demonstrate the robustness of this correspondence under interactions and symmetry-preserving disorder, and identify inversion symmetry as a minimal protecting symmetry that stabilizes both the quantization of the invariant and the associated entanglement degeneracies. Our results unify geometric-phase invariants and entanglement diagnostics within a many-body framework and provide a route to identifying interacting topological phases beyond band theory.