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:1405.3047 · Journal of Physics Condensed Matter

Symmetry fractionalization: symmetry-protected topological phases of the bond-alternating spin-1/2 Heisenberg chain

R Haghshenas, A Langari, A T Rezakhani

Published Oct 16, 2014• 13 citations
arXivHTMLPDF

Abstract

We study different phases of the one-dimensional bond-alternating spin-1/2 Heisenberg model by using the symmetry fractionalization mechanism. We employ the infinite matrix-product state representation of the ground state (through the infinite-size density matrix renormalization group algorithm) to obtain inequivalent projective representations and commutation relations of the (unbroken) symmetry groups of the model, which are used to identify the different phases. We find that the model exhibits trivial as well as symmetry-protected topological phases. The symmetry-protected topological phases are Haldane phases on even/odd bonds, which are protected by the time-reversal (acting on the spin as σ → -σ), parity (permutation of the chain about a specific bond), and dihedral (π-rotations about a pair of orthogonal axes) symmetries. Additionally, we investigate the phases of the most general two-body bond-alternating spin-1/2 model, which respects the time-reversal, parity, and dihedral symmetries, and obtain its corresponding twelve different types of the symmetry-protected topological phases.

DOI
10.1088/0953-8984/26/45/456001

Related papers

Symmetry fractionalization: symmetry-protected topological phases of the bond-alternating spin-1/2 Heisenberg chain

Reza Haghshenas, A. Langari, A. T. Rezakhani

We study different phases of the one-dimensional bond-alternating spin-$1/2$ Heisenberg model by using the symmetry fractionalization mechanism. We employ the infinite matrix-product state representation of the ground state (through the infinite-size density matrix renormalization group algorithm) to obtain inequivalent projective representations of the (unbroken) symmetry groups of the model, which are used to identify the different phases. We find that the model exhibits trivial as well as symmetry-protected topological phases. The symmetry-protected topological phases are Haldane phases on even/odd bonds, which are protected by the time-reversal (acting on the spin as $\sigma\rightarrow-\sigma$), parity (permutation of the chain about a specific bond), and dihedral ($\pi$-rotations about a pair of orthogonal axes) symmetries. Additionally, we investigate the phases of the most general two-body bond-alternating spin-$1/2$ model, which respects the time-reversal, parity, and dihedral symmetries, and obtain its corresponding twelve different types of the symmetry-protected topological phases.

•Nov 12, 2014•10 citations

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

Symmetry fractionalization in the topological phase of the spin- 1 2 J 1 − J 2 triangular Heisenberg model

S. N. Saadatmand, Ian P. McCulloch

Using density-matrix renormalization-group calculations for infinite cylinders, we elucidate the properties of the spin-liquid phase of the spin-$\frac{1}{2}\phantom{\rule{4pt}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the triangular lattice. We find four distinct ground states characteristic of a nonchiral, ${Z}_{2}$ topologically ordered state with vison and spinon excitations. We shed light on the interplay of topological ordering and global symmetries in the model by detecting fractionalization of time-reversal and space-group dihedral symmetries in the anyonic sectors, which leads to the coexistence of symmetry protected and intrinsic topological order. The anyonic sectors, and information on the particle statistics, can be characterized by degeneracy patterns and symmetries of the entanglement spectrum. We demonstrate the ground states on finite-width cylinders are short-range correlated and gapped; however, some features in the entanglement spectrum suggest that the system develops gapless spinonlike edge excitations in the large-width limit.

arXiv:1606.00334•Physical review. B./Physical review. B•Sep 13, 2016•71 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

All spin-1 topological phases in a single spin-2 chain

Augustine Kshetrimayum, Hong-Hao Tu, Román Orús

Here we study the emergence of different symmetry-protected topological (SPT) phases in a spin-2 quantum chain. We consider a Heisenberg-like model with bilinear, biquadratic, bicubic, and biquartic nearest-neighbor interactions, as well as uniaxial anisotropy. We show that this model contains four different effective spin-1 SPT phases, corresponding to different representations of the $({\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2})+T$ symmetry group, where ${\mathbb{Z}}_{2}$ is some $\ensuremath{\pi}$ rotation in the spin internal space and $T$ is time reversal. One of these phases is equivalent to the usual spin-1 Haldane phase, while the other three are different but also typical of spin-1 systems. The model also exhibits an $SO(5)$-Haldane phase. Moreover, we also find that the transitions between the different effective spin-1 SPT phases are continuous and can be described by a $c=2$ conformal field theory. At such transitions, indirect evidence suggests a possible effective field theory of four massless Majorana fermions. The results are obtained by approximating the ground state of the system in the thermodynamic limit using matrix product states via the infinite time-evolving block decimation method, as well as by effective field theory considerations. Our findings show that different large effective spin-1 SPT phases separated by continuous quantum phase transitions can be stabilized in a simple quantum spin chain.

arXiv:1412.3370•Physical Review B•May 19, 2015•14 citations

Symmetry-protected topological phases in the SU(N) Heisenberg spin chain: A Majorana fermion approach

Pierre Fromholz, P. Lecheminant

The nature of symmetry-protected topological phases of Heisenberg spin chains in totally symmetric representations of rank $N$ of the $\mathrm{SU}(N$) group is investigated through a Majorana fermion study starting from an integrable point. The latter approach generalizes the one pioneered by Tsvelik [Phys. Rev. B 42, 10499 (1990)] to describe the low-energy properties of the Haldane phase of the spin-1 Heisenberg chain from three massive Majorana fermions. We find, for all $N$'s, the emergence of a nondegenerate gapped phase with edge states whose topological protection depends on the parity of $N$. Whereas for $N$ odd, there is no such protection, the phase with even $N$ is shown to be topologically protected. We find that the phase belongs to the same topological class as the phase with edge states living in self-conjugate fully antisymmetric representation of the $\mathrm{SU}(N)$ group.

arXiv:2006.13553•Physical review. B./Physical review. B•Sep 8, 2020•8 citations
We study different phases of the one-dimensional bond-alternating spin-1/2 Heisenberg model by using the symmetry fractionalization mechanism. We employ the infinite matrix-product state representation of the ground state (through the infinite-size density matrix renormalization group algorithm) to obtain inequivalent projective representations and commutation relations of the (unbroken) symmetry groups of the model, which are used to identify the different phases. We find that the model exhibits trivial as well as symmetry-protected topological phases. The symmetry-protected topological phases are Haldane phases on even/odd bonds, which are protected by the time-reversal (acting on the spin as σ → -σ), parity (permutation of the chain about a specific bond), and dihedral (π-rotations about a pair of orthogonal axes) symmetries. Additionally, we investigate the phases of the most general two-body bond-alternating spin-1/2 model, which respects the time-reversal, parity, and dihedral symmetries, and obtain its corresponding twelve different types of the symmetry-protected topological phases.
We study different phases of the one-dimensional bond-alternating spin-1/21/21/2 Heisenberg model by using the symmetry fractionalization mechanism. We employ the infinite matrix-product state representation of the ground state (through the infinite-size density matrix renormalization group algorithm) to obtain inequivalent projective representations of the (unbroken) symmetry groups of the model, which are used to identify the different phases. We find that the model exhibits trivial as well as symmetry-protected topological phases. The symmetry-protected topological phases are Haldane phases on even/odd bonds, which are protected by the time-reversal (acting on the spin as σ→−σ\sigma\rightarrow-\sigmaσ→−σ), parity (permutation of the chain about a specific bond), and dihedral (π\piπ-rotations about a pair of orthogonal axes) symmetries. Additionally, we investigate the phases of the most general two-body bond-alternating spin-1/21/21/2 model, which respects the time-reversal, parity, and dihedral symmetries, and obtain its corresponding twelve different types of the symmetry-protected topological phases.
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.
Using density-matrix renormalization-group calculations for infinite cylinders, we elucidate the properties of the spin-liquid phase of the spin-$\frac{1}{2}\phantom{\rule{4pt}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the triangular lattice. We find four distinct ground states characteristic of a nonchiral, Z2{Z}_{2}Z2​ topologically ordered state with vison and spinon excitations. We shed light on the interplay of topological ordering and global symmetries in the model by detecting fractionalization of time-reversal and space-group dihedral symmetries in the anyonic sectors, which leads to the coexistence of symmetry protected and intrinsic topological order. The anyonic sectors, and information on the particle statistics, can be characterized by degeneracy patterns and symmetries of the entanglement spectrum. We demonstrate the ground states on finite-width cylinders are short-range correlated and gapped; however, some features in the entanglement spectrum suggest that the system develops gapless spinonlike edge excitations in the large-width limit.
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.
Here we study the emergence of different symmetry-protected topological (SPT) phases in a spin-2 quantum chain. We consider a Heisenberg-like model with bilinear, biquadratic, bicubic, and biquartic nearest-neighbor interactions, as well as uniaxial anisotropy. We show that this model contains four different effective spin-1 SPT phases, corresponding to different representations of the $({\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2})+T$ symmetry group, where Z2{\mathbb{Z}}_{2}Z2​ is some $\ensuremath{\pi}$ rotation in the spin internal space and TTT is time reversal. One of these phases is equivalent to the usual spin-1 Haldane phase, while the other three are different but also typical of spin-1 systems. The model also exhibits an SO(5)SO(5)SO(5)-Haldane phase. Moreover, we also find that the transitions between the different effective spin-1 SPT phases are continuous and can be described by a c=2c=2c=2 conformal field theory. At such transitions, indirect evidence suggests a possible effective field theory of four massless Majorana fermions. The results are obtained by approximating the ground state of the system in the thermodynamic limit using matrix product states via the infinite time-evolving block decimation method, as well as by effective field theory considerations. Our findings show that different large effective spin-1 SPT phases separated by continuous quantum phase transitions can be stabilized in a simple quantum spin chain.
The nature of symmetry-protected topological phases of Heisenberg spin chains in totally symmetric representations of rank NNN of the SU(N\mathrm{SU}(NSU(N) group is investigated through a Majorana fermion study starting from an integrable point. The latter approach generalizes the one pioneered by Tsvelik [Phys. Rev. B 42, 10499 (1990)] to describe the low-energy properties of the Haldane phase of the spin-1 Heisenberg chain from three massive Majorana fermions. We find, for all NNN's, the emergence of a nondegenerate gapped phase with edge states whose topological protection depends on the parity of NNN. Whereas for NNN odd, there is no such protection, the phase with even NNN is shown to be topologically protected. We find that the phase belongs to the same topological class as the phase with edge states living in self-conjugate fully antisymmetric representation of the SU(N)\mathrm{SU}(N)SU(N) group.