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:1508.00960 · Physical review. B./Physical review. B

ZN Berry phase and symmetry-protected topological phases of the SU( N ) antiferromagnetic Heisenberg chain

Yuichi Motoyama, Synge Todo

Published Nov 20, 2018• 10 citations
arXivHTMLPDF

Abstract

The local ${Z}_{N}$ quantized Berry phase for the $\mathrm{SU}(N)$ antiferromagnetic Heisenberg spin model is formulated. This quantity, which is a generalization of the local ${Z}_{2}$ Berry phase for SU(2) symmetry, has a direct correspondence to the number of singlet pairs spanning a particular bond, and is effective as a tool to characterize and classify the various symmetry-protected topological phases of one-dimensional $\mathrm{SU}(N)$ spin systems. We extend the path-integral quantum Monte Carlo method for the ${Z}_{2}$ Berry phase in order to calculate the ${Z}_{N}$ Berry phase numerically. We demonstrate our method by calculating the ${Z}_{4}$ Berry phase for the bond-alternating $\mathrm{SU}(4)$ antiferromagnetic Heisenberg chain, which is represented by the Young diagram of four columns.

DOI
10.1103/physrevb.98.195127

Related papers

ZN Berry Phases in Symmetry Protected Topological Phases

Toshikaze Kariyado, Takahiro Morimoto, Yasuhiro Hatsugai

We show that the Z_{N} Berry phase (Berry phase quantized into 2π/N) provides a useful tool to characterize symmetry protected topological phases with correlation that can be directly computed through numerics of a relatively small system size. The Z_{N} Berry phase is defined in a N-1-dimensional parameter space of local gauge twists, which we call the "synthetic Brillouin zone," and an appropriate choice of an integration path consistent with the symmetry of the system ensures exact quantization of the Berry phase. We demonstrate the usefulness of the Z_{N} Berry phase by studying two 1D models of bosons, SU(3) and SU(4) Affleck-Kennedy-Lieb-Tasaki models, where topological phase transitions are captured by Z_{3} and Z_{4} Berry phases, respectively. We find that the exact quantization of the Z_{N} Berry phase at the topological transitions arises from a gapless band structure (e.g., Dirac cones or nodal lines) in the synthetic Brillouin zone.

arXiv:1709.01546•Physical Review Letters•Jun 15, 2018•39 citations

Three-dimensional symmetry-protected topological phase close to antiferromagnetic Néel order

Cenke Xu

It is well known that the Haldane phase of a one-dimensional spin-1 chain is a symmetry-protected topological (SPT) phase, which is described by a nonlinear sigma model (NLSM) with a $\ensuremath{\Theta}$ term at $\ensuremath{\Theta}=2\ensuremath{\pi}$. In this work we study a three-dimensional (3d) SPT phase of a SU(2$N$) antiferromagnetic spin system with a self-conjugate representation on every site. The spin-ordered N\'eel phase of this system has a ground state manifold $\mathcal{M}=\frac{\mathrm{U}(2N)}{\mathrm{U}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(N)}$, and this system is described by a NLSM defined with manifold $\mathcal{M}$. Since the homotopy group ${\ensuremath{\pi}}_{4}[\mathcal{M}]=\mathbb{Z}$ for $N>1$, this NLSM can naturally have a $\ensuremath{\Theta}$ term. We will argue that when $\ensuremath{\Theta}=2\ensuremath{\pi}$ this NLSM describes a SPT phase. This SPT phase is protected by the SU(2$N$) spin symmetry, or its subgroup SU($N$)$\ifmmode\times\else\texttimes\fi{}$SU$(N)\ensuremath{\rtimes}{Z}_{2}$, without assuming any other discrete symmetry. We will also construct a trial SU(2$N$) spin state on a 3d lattice; we argue that the long-wavelength physics of this state is precisely described by the aforementioned NLSM with $\ensuremath{\Theta}=2\ensuremath{\pi}$.

arXiv:1209.4399•Physical Review B•Apr 25, 2013•39 citations

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

Path-integral Monte Carlo method for the local Z2 Berry phase

Yuichi Motoyama, Synge Todo

We present a loop cluster algorithm Monte Carlo method for calculating the local Z(2) Berry phase of the quantum spin models. The Berry connection, which is given as the inner product of two ground states with different local twist angles, is expressed as a Monte Carlo average on the worldlines with fixed spin configurations at the imaginary-time boundaries. The "complex weight problem" caused by the local twist is solved by adopting the meron cluster algorithm. We present the results of simulation on the antiferromagnetic Heisenberg model on an out-of-phase bond-alternating ladder to demonstrate that our method successfully detects the change in the valence bond pattern at the quantum phase transition point. We also propose that the gauge-fixed local Berry connection can be an effective tool to estimate precisely the quantum critical point.

arXiv:1211.3471•Physical Review E•Feb 6, 2013•17 citations

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

R 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 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.

arXiv:1405.3047•Journal of Physics Condensed Matter•Oct 16, 2014•13 citations

Z2×Z2 symmetry and Z4 Berry phase of bosonic ladders

Yoshihito Kuno, Yasuhiro Hatsugai

Bose gas on a two-leg ladder exhibits an interesting topological phase. We show the presence of a bosonic symmetry-protected-topological (SPT) phase protected by ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry. This symmetry leads to ${Z}_{4}$ fractional quantization of the ${Z}_{4}$ Berry phase, which is a topological order parameter to identify the bulk. Using the ${Z}_{4}$ Berry phase, we show that the interacting bosonic system possesses rich topological phases depending on the particle density and strength of interaction. Based on the bulk-edge correspondence, each edge state of the SPT phases is discussed in relation to the ${Z}_{4}$ Berry phases. In particular, we have found an intermediate phase that is not adiabatically connected to a simple adiabatic limit that possesses unconventional edge states, which we numerically demonstrate by employing the density-matrix renormalization-group algorithm.

arXiv:2304.09533•Physical Review A•Jul 27, 2023•2 citations
The local ZN{Z}_{N}ZN​ quantized Berry phase for the SU(N)\mathrm{SU}(N)SU(N) antiferromagnetic Heisenberg spin model is formulated. This quantity, which is a generalization of the local Z2{Z}_{2}Z2​ Berry phase for SU(2) symmetry, has a direct correspondence to the number of singlet pairs spanning a particular bond, and is effective as a tool to characterize and classify the various symmetry-protected topological phases of one-dimensional SU(N)\mathrm{SU}(N)SU(N) spin systems. We extend the path-integral quantum Monte Carlo method for the Z2{Z}_{2}Z2​ Berry phase in order to calculate the ZN{Z}_{N}ZN​ Berry phase numerically. We demonstrate our method by calculating the Z4{Z}_{4}Z4​ Berry phase for the bond-alternating SU(4)\mathrm{SU}(4)SU(4) antiferromagnetic Heisenberg chain, which is represented by the Young diagram of four columns.
We show that the Z_{N} Berry phase (Berry phase quantized into 2π/N) provides a useful tool to characterize symmetry protected topological phases with correlation that can be directly computed through numerics of a relatively small system size. The Z_{N} Berry phase is defined in a N-1-dimensional parameter space of local gauge twists, which we call the "synthetic Brillouin zone," and an appropriate choice of an integration path consistent with the symmetry of the system ensures exact quantization of the Berry phase. We demonstrate the usefulness of the Z_{N} Berry phase by studying two 1D models of bosons, SU(3) and SU(4) Affleck-Kennedy-Lieb-Tasaki models, where topological phase transitions are captured by Z_{3} and Z_{4} Berry phases, respectively. We find that the exact quantization of the Z_{N} Berry phase at the topological transitions arises from a gapless band structure (e.g., Dirac cones or nodal lines) in the synthetic Brillouin zone.
It is well known that the Haldane phase of a one-dimensional spin-1 chain is a symmetry-protected topological (SPT) phase, which is described by a nonlinear sigma model (NLSM) with a $\ensuremath{\Theta}$ term at $\ensuremath{\Theta}=2\ensuremath{\pi}$. In this work we study a three-dimensional (3d) SPT phase of a SU(2NNN) antiferromagnetic spin system with a self-conjugate representation on every site. The spin-ordered N\'eel phase of this system has a ground state manifold $\mathcal{M}=\frac{\mathrm{U}(2N)}{\mathrm{U}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(N)}$, and this system is described by a NLSM defined with manifold M\mathcal{M}M. Since the homotopy group ${\ensuremath{\pi}}_{4}[\mathcal{M}]=\mathbb{Z}$ for N>1N>1N>1, this NLSM can naturally have a $\ensuremath{\Theta}$ term. We will argue that when $\ensuremath{\Theta}=2\ensuremath{\pi}$ this NLSM describes a SPT phase. This SPT phase is protected by the SU(2NNN) spin symmetry, or its subgroup SU(NNN)$\ifmmode\times\else\texttimes\fi{}$SU$(N)\ensuremath{\rtimes}{Z}_{2}$, without assuming any other discrete symmetry. We will also construct a trial SU(2NNN) spin state on a 3d lattice; we argue that the long-wavelength physics of this state is precisely described by the aforementioned NLSM with $\ensuremath{\Theta}=2\ensuremath{\pi}$.
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 present a loop cluster algorithm Monte Carlo method for calculating the local Z(2) Berry phase of the quantum spin models. The Berry connection, which is given as the inner product of two ground states with different local twist angles, is expressed as a Monte Carlo average on the worldlines with fixed spin configurations at the imaginary-time boundaries. The "complex weight problem" caused by the local twist is solved by adopting the meron cluster algorithm. We present the results of simulation on the antiferromagnetic Heisenberg model on an out-of-phase bond-alternating ladder to demonstrate that our method successfully detects the change in the valence bond pattern at the quantum phase transition point. We also propose that the gauge-fixed local Berry connection can be an effective tool to estimate precisely the quantum critical point.
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.
Bose gas on a two-leg ladder exhibits an interesting topological phase. We show the presence of a bosonic symmetry-protected-topological (SPT) phase protected by ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry. This symmetry leads to Z4{Z}_{4}Z4​ fractional quantization of the Z4{Z}_{4}Z4​ Berry phase, which is a topological order parameter to identify the bulk. Using the Z4{Z}_{4}Z4​ Berry phase, we show that the interacting bosonic system possesses rich topological phases depending on the particle density and strength of interaction. Based on the bulk-edge correspondence, each edge state of the SPT phases is discussed in relation to the Z4{Z}_{4}Z4​ Berry phases. In particular, we have found an intermediate phase that is not adiabatically connected to a simple adiabatic limit that possesses unconventional edge states, which we numerically demonstrate by employing the density-matrix renormalization-group algorithm.