Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0910.1811 · Physical Review B

Entanglement spectrum of a topological phase in one dimension

Frank Pollmann, Ari M. Turner, Erez Berg, Masaki Oshikawa

Published Feb 26, 2010• 1,338 citations
arXivHTMLPDF

Abstract

We show that the Haldane phase of $S=1$ chains is characterized by a double degeneracy of the entanglement spectrum. The degeneracy is protected by a set of symmetries (either the dihedral group of $\ensuremath{\pi}$ rotations about two orthogonal axes, time-reversal symmetry, or bond centered inversion symmetry), and cannot be lifted unless either a phase boundary to another, ``topologically trivial,'' phase is crossed, or the symmetry is broken. More generally, these results offer a scheme to classify gapped phases of one-dimensional systems. Physically, the degeneracy of the entanglement spectrum can be observed by adiabatically weakening a bond to zero, which leaves the two disconnected halves of the system in a finitely entangled state.

DOI
10.1103/physrevb.81.064439

Related papers

Symmetry protection of topological phases in one-dimensional quantum spin systems

Frank Pollmann, Erez Berg, Ari M. Turner +1 more

We discuss the characterization and stability of the Haldane phase in integer spin chains on the basis of simple, physical arguments. We find that an odd-$S$ Haldane phase is a topologically nontrivial phase which is protected by any one of the following three global symmetries: (i) the dihedral group of $\ensuremath{\pi}$ rotations about the $x$, $y$, and $z$ axes, (ii) time-reversal symmetry ${S}^{x,y,z}\ensuremath{\rightarrow}\ensuremath{-}{S}^{x,y,z}$, and (iii) link inversion symmetry (reflection about a bond center), consistent with previous results [Phys. Rev. B 81, 064439 (2010)]. On the other hand, an even-$S$ Haldane phase is not topologically protected (i.e., it is indistinct from a trivial, site-factorizable phase). We show some numerical evidence that supports these claims, using concrete examples.

arXiv:0909.4059•Physical Review B•Feb 22, 2012•849 citations

Relationship between symmetry protected topological phases and boundary conformal field theories via the entanglement spectrum

Gil Young Cho, Ken Shiozaki, Shinsei Ryu +1 more

Abstract Quantum phase transitions out of a symmetry-protected topological (SPT) phase in (1 + 1) dimensions into an adjacent, topologically distinct SPT phase protected by the same symmetry or a trivial gapped phase, are typically described by a conformal field theory (CFT). At the same time, the low-lying entanglement spectrum of a gapped phase close to such a quantum critical point is known (Cho et al arXiv:1603.04016), very generally, to be universal and described by (gapless) boundary conformal field theory. Using this connection we show that symmetry properties of the boundary conditions in boundary CFT can be used to characterize the symmetry-protected degeneracies of the entanglement spectrum, a hallmark of non-trivial symmetry-protected topological phases. Specifically, we show that the relevant boundary CFT is the orbifold of the quantum critical point with respect to the symmetry group defining the SPT, and that the boundary states of this orbifold carry a quantum anomaly that determines the topological class of the SPT. We illustrate this connection using various characteristic examples such as the time-reversal breaking ‘Kitaev chain’ superconductor (symmetry class D), the Haldane phase, and the Z 8 classification of interacting topological superconductors in symmetry class BDI in (1 + 1) dimensions.

arXiv:1606.06402•Journal of Physics A Mathematical and Theoretical•Jun 6, 2017•47 citations

Entanglement properties of the Haldane phases: A finite system-size approach

Shohei Miyakoshi, Satoshi Nishimoto, Y. Ohta

We study the bond-alternating Heisenberg model using the finite-size density-matrix renormalization-group (DMRG) technique and analytical arguments based on the matrix product state, where we pay particular attention to the boundary-condition dependence on the entanglement spectrum of the system. We show that, in the antiperiodic boundary condition (APBC), the parity quantum numbers are equivalent to the topological invariants characterizing the topological phases protected by the bond-centered inversion and $\ensuremath{\pi}$ rotation about the $z$ axis. We also show that the odd parity in the APBC, which characterizes topologically nontrivial phases, can be extracted as a twofold degeneracy in the entanglement spectrum even with finite system size. We then determine the phase diagram of the model with the uniaxial single-ion anisotropy using the level spectroscopy method in the DMRG technique. These results not only suggest the detectability of the symmetry protected topological (SPT) phases via general twisted boundary conditions but also provide a useful and precise numerical tool for discussing the SPT phases in the exact diagonalization and DMRG techniques.

arXiv:1612.04686•Physical review. B./Physical review. B•Dec 27, 2016•12 citations

Disentangling topological degeneracy in the entanglement spectrum of one-dimensional symmetry-protected topological phases

Wen-Jia Rao, Guang-Ming Zhang, Kun Yang

One-dimensional valence bond solid (VBS) states represent the simplest symmetry-protected topological phases. We show that their ground state entanglement spectrum contains both topological and nontopological structures. For the $\text{SO}(3)$ symmetric VBS states with odd-integer spins, the twofold topological degeneracy is associated with an underlying ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry that protects the corresponding topological phase. In general, for the $\text{SO}(2S+1)$ symmetric VBS states with integer spins $S$, the corresponding protecting symmetry is identified as the ${({Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2})}^{S}$ symmetry, yielding the ${2}^{S}$-fold topological degeneracy. The topological degeneracy and associated protecting symmetry can be identified by a nonlocal unitary transformation, which changes the topological order of the VBS states into conventional ferromagnetic order.

arXiv:1403.0660•Physical Review B•Mar 17, 2014•8 citations

Universal Entanglement Spectrum in One-Dimensional Gapless Symmetry Protected Topological States

Xue-Jia Yu, Sheng Yang, Hai‐Qing Lin +1 more

Quantum entanglement marks a definitive feature of topological states. However, the entanglement spectrum remains insufficiently explored for topological states without a bulk energy gap. Using a combination of field theory and numerical techniques, we accurately calculate and analyze the entanglement spectrum of gapless symmetry protected topological states in one dimension. We highlight that the universal entanglement spectrum not only encodes the nontrivial edge degeneracy, generalizing the Li-Haldane conjecture to gapless topological states, but also contains the operator content of the underlying boundary conformal field theory. This implies that the bulk wave function can act as a fingerprint of both quantum criticality and topology in gapless symmetry protected topological states. We also identify a symmetry enriched conformal boundary condition that goes beyond the conventional conformal boundary condition.

arXiv:2402.04042•Physical Review Letters•Jul 12, 2024•39 citations

Entanglement and inversion symmetry in topological insulators

Ari M. Turner, Yi Zhang, Ashvin Vishwanath

Topological band insulators are usually characterized by symmetry-protected surface modes or quantized linear-response functions (like Hall conductance). Here we present a way to characterize them based on certain bulk properties of just the ground-state wave function, specifically, the properties of its entanglement spectrum. We prove that whenever protected surface states exist, a corresponding protected ``mode'' exists in the entanglement spectrum as well. Besides this, the entanglement spectrum sometimes succeeds better at indicating topological phases than surface states. We discuss specifically the example of insulators with inversion symmetry which is found to act as an antiunitary symmetry on the entanglement spectrum. A Kramers degeneracy can then arise even when time-reversal symmetry is absent. This degeneracy persists for interacting systems. The entanglement spectrum is therefore a promising tool to characterize topological band insulators and superconductors beyond the free-particle approximation.

Physical Review B•Dec 13, 2010•269 citations
We show that the Haldane phase of S=1S=1S=1 chains is characterized by a double degeneracy of the entanglement spectrum. The degeneracy is protected by a set of symmetries (either the dihedral group of \ensuremathπ\ensuremath{\pi}\ensuremathπ rotations about two orthogonal axes, time-reversal symmetry, or bond centered inversion symmetry), and cannot be lifted unless either a phase boundary to another, ``topologically trivial,'' phase is crossed, or the symmetry is broken. More generally, these results offer a scheme to classify gapped phases of one-dimensional systems. Physically, the degeneracy of the entanglement spectrum can be observed by adiabatically weakening a bond to zero, which leaves the two disconnected halves of the system in a finitely entangled state.
We discuss the characterization and stability of the Haldane phase in integer spin chains on the basis of simple, physical arguments. We find that an odd-SSS Haldane phase is a topologically nontrivial phase which is protected by any one of the following three global symmetries: (i) the dihedral group of $\ensuremath{\pi}$ rotations about the xxx, yyy, and zzz axes, (ii) time-reversal symmetry ${S}^{x,y,z}\ensuremath{\rightarrow}\ensuremath{-}{S}^{x,y,z}$, and (iii) link inversion symmetry (reflection about a bond center), consistent with previous results [Phys. Rev. B 81, 064439 (2010)]. On the other hand, an even-SSS Haldane phase is not topologically protected (i.e., it is indistinct from a trivial, site-factorizable phase). We show some numerical evidence that supports these claims, using concrete examples.
Abstract Quantum phase transitions out of a symmetry-protected topological (SPT) phase in (1 + 1) dimensions into an adjacent, topologically distinct SPT phase protected by the same symmetry or a trivial gapped phase, are typically described by a conformal field theory (CFT). At the same time, the low-lying entanglement spectrum of a gapped phase close to such a quantum critical point is known (Cho et al arXiv:1603.04016), very generally, to be universal and described by (gapless) boundary conformal field theory. Using this connection we show that symmetry properties of the boundary conditions in boundary CFT can be used to characterize the symmetry-protected degeneracies of the entanglement spectrum, a hallmark of non-trivial symmetry-protected topological phases. Specifically, we show that the relevant boundary CFT is the orbifold of the quantum critical point with respect to the symmetry group defining the SPT, and that the boundary states of this orbifold carry a quantum anomaly that determines the topological class of the SPT. We illustrate this connection using various characteristic examples such as the time-reversal breaking ‘Kitaev chain’ superconductor (symmetry class D), the Haldane phase, and the Z 8 classification of interacting topological superconductors in symmetry class BDI in (1 + 1) dimensions.
We study the bond-alternating Heisenberg model using the finite-size density-matrix renormalization-group (DMRG) technique and analytical arguments based on the matrix product state, where we pay particular attention to the boundary-condition dependence on the entanglement spectrum of the system. We show that, in the antiperiodic boundary condition (APBC), the parity quantum numbers are equivalent to the topological invariants characterizing the topological phases protected by the bond-centered inversion and $\ensuremath{\pi}$ rotation about the zzz axis. We also show that the odd parity in the APBC, which characterizes topologically nontrivial phases, can be extracted as a twofold degeneracy in the entanglement spectrum even with finite system size. We then determine the phase diagram of the model with the uniaxial single-ion anisotropy using the level spectroscopy method in the DMRG technique. These results not only suggest the detectability of the symmetry protected topological (SPT) phases via general twisted boundary conditions but also provide a useful and precise numerical tool for discussing the SPT phases in the exact diagonalization and DMRG techniques.
One-dimensional valence bond solid (VBS) states represent the simplest symmetry-protected topological phases. We show that their ground state entanglement spectrum contains both topological and nontopological structures. For the SO(3)\text{SO}(3)SO(3) symmetric VBS states with odd-integer spins, the twofold topological degeneracy is associated with an underlying ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry that protects the corresponding topological phase. In general, for the SO(2S+1)\text{SO}(2S+1)SO(2S+1) symmetric VBS states with integer spins SSS, the corresponding protecting symmetry is identified as the ${({Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2})}^{S}$ symmetry, yielding the 2S{2}^{S}2S-fold topological degeneracy. The topological degeneracy and associated protecting symmetry can be identified by a nonlocal unitary transformation, which changes the topological order of the VBS states into conventional ferromagnetic order.
Quantum entanglement marks a definitive feature of topological states. However, the entanglement spectrum remains insufficiently explored for topological states without a bulk energy gap. Using a combination of field theory and numerical techniques, we accurately calculate and analyze the entanglement spectrum of gapless symmetry protected topological states in one dimension. We highlight that the universal entanglement spectrum not only encodes the nontrivial edge degeneracy, generalizing the Li-Haldane conjecture to gapless topological states, but also contains the operator content of the underlying boundary conformal field theory. This implies that the bulk wave function can act as a fingerprint of both quantum criticality and topology in gapless symmetry protected topological states. We also identify a symmetry enriched conformal boundary condition that goes beyond the conventional conformal boundary condition.
Topological band insulators are usually characterized by symmetry-protected surface modes or quantized linear-response functions (like Hall conductance). Here we present a way to characterize them based on certain bulk properties of just the ground-state wave function, specifically, the properties of its entanglement spectrum. We prove that whenever protected surface states exist, a corresponding protected ``mode'' exists in the entanglement spectrum as well. Besides this, the entanglement spectrum sometimes succeeds better at indicating topological phases than surface states. We discuss specifically the example of insulators with inversion symmetry which is found to act as an antiunitary symmetry on the entanglement spectrum. A Kramers degeneracy can then arise even when time-reversal symmetry is absent. This degeneracy persists for interacting systems. The entanglement spectrum is therefore a promising tool to characterize topological band insulators and superconductors beyond the free-particle approximation.