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:1512.05229 · Physical review. B./Physical review. B

Chiral Haldane phases of SU(N) quantum spin chains

Abhishek Roy, Thomas Quella

Published Apr 23, 2018• 33 citations
arXivHTMLPDF

Abstract

Gapped quantum spin chains with symmetry $\text{PSU}(N)=\text{SU}(N)/{\mathbb{Z}}_{N}$ are known to possess $N$ distinct symmetry protected topological phases. Besides the trivial phase, there are $N\ensuremath{-}1$ Haldane phases which are distinguished by the occurrence of massless boundary spins. We give an expression for a universal AKLT Hamiltonian for an arbitrary symmetry group, including the case where inversion symmetry is broken. Using a diagrammatic method, we construct Hamiltonians for two of the topological classes in $\text{SU}(N)$ that arise when the physical spin is in the adjoint representation. Since our results are valid for all $N$, we also discuss the large-$N$ limit.

DOI
10.1103/physrevb.97.155148

Related papers

Symmetry-protected topological phases in a two-leg SU(N) spin ladder with unequal spins

Sylvain Capponi, Pierre Fromholz, P. Lecheminant +1 more

Chiral Haldane phases are examples of one-dimensional topological states of matter which are protected by the projective $\mathrm{SU}(N)$ group (or its subgroup ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N})$ with $N>2$. The unique feature of these symmetry-protected topological (SPT) phases is that they are accompanied by inversion-symmetry breaking and the emergence of different left and right edge states which transform, for instance, respectively in the fundamental $(\mathbit{N})$ and antifundamental $(\overline{\mathbit{N}})$ representations of $\mathrm{SU}(N)$. We show, by means of complementary analytical and numerical approaches, that these chiral SPT phases as well as the nonchiral ones are realized as the ground states of a generalized two-leg $\mathrm{SU}(N)$ spin ladder in which the spins in the first chain transform in $\mathbit{N}$ and the second in $\overline{\mathbit{N}}$. In particular, we map out the phase diagram for $N=3$ and 4 to show that all the possible symmetry-protected topological phases with projective $\mathrm{SU}(N)$ symmetry appear in this simple ladder model.

arXiv:1909.13553•Physical review. B./Physical review. B•May 12, 2020•13 citations

Z3 symmetry-protected topological phases in the SU(3) AKLT model

Takahiro Morimoto, Hiroshi Ueda, Tsutomu Momoi +1 more

We study ${\mathbb{Z}}_{3}$ symmetry-protected topological (SPT) phases in one-dimensional spin systems with ${Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3}$ symmetry. We construct ground-state wave functions of the matrix product form for nontrivial ${\mathbb{Z}}_{3}$ phases and their parent Hamiltonian from a cocycle of the group cohomology ${H}^{2}({Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3},U(1))$. The Hamiltonian is an SU(3) version of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, consisting of bilinear and biquadratic terms of su(3) generators in the adjoint representation. A generalization to the $\mathrm{SU}(N)$ case, the $\mathrm{SU}(N)$ AKLT Hamiltonian, is also presented, which realizes nontrivial ${\mathbb{Z}}_{N}$ SPT phases. We use the infinite-size variant of the density matrix renormalization group (iDMRG) method to determine the ground-state phase diagram of the SU(3) bilinear-biquadratic model as a function of the parameter $\ensuremath{\theta}$ controlling the ratio of the bilinear and biquadratic coupling constants. The nontrivial ${\mathbb{Z}}_{3}$ SPT phase is found for a range of the parameter $\ensuremath{\theta}$ including the point of vanishing biquadratic term $(\ensuremath{\theta}=0)$ as well as the SU(3) AKLT point [$\ensuremath{\theta}=arctan(2/9)$]. A continuous phase transition to the SU(3) dimer phase takes place at $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.027\ensuremath{\pi}$, with a central charge $c\ensuremath{\approx}3.2$. For SU(3) symmetric cases, we define string order parameters for the ${\mathbb{Z}}_{3}$ SPT phases in a similar way to the conventional Haldane phase. We propose simple spin models that effectively realize the SU(3) and SU(4) AKLT models.

arXiv:1409.1659•Physical Review B•Dec 4, 2014•54 citations

Anyonic quantum spin chains: Spin-1 generalizations and topological stability

Charlotte Gils, Eddy Ardonne, Simon Trebst +4 more

There are many interesting parallels between systems of interacting non-Abelian anyons and quantum magnetism occurring in ordinary SU(2) quantum magnets. Here we consider theories of so-called su(2)${}_{k}$ anyons, well-known deformations of SU(2), in which only the first $k+1$ angular momenta of SU(2) occur. In this paper, we discuss in particular anyonic generalizations of ordinary SU(2) spin chains with an emphasis on anyonic spin $S=1$ chains. We find that the overall phase diagrams for these anyonic spin-1 chains closely mirror the phase diagram of the ordinary bilinear-biquadratic spin-1 chain including anyonic generalizations of the Haldane phase, the AKLT construction, and supersymmetric quantum critical points. A novel feature of the anyonic spin-1 chains is an additional topological symmetry that protects the gapless phases. Distinctions further arise in the form of an even/odd effect in the deformation parameter $k$ when considering su(2)${}_{k}$ anyonic theories with $k\ensuremath{\ge}5$, as well as for the special case of the su(2)${}_{4}$ theory for which the spin-1 representation plays a special role. We also address anyonic generalizations of spin-$\frac{1}{2}$ chains with a focus on the topological protection provided for their gapless ground states. Finally, we put our results into the context of earlier generalizations of SU(2) quantum spin chains, in particular so-called (fused) Temperley-Lieb chains.

arXiv:1303.4290•Physical Review B•Jun 17, 2013•85 citations

Non-Landau quantum phase transition in modulated SU( N ) Heisenberg spin chains

Sylvain Capponi, Lukas Devos, P. Lecheminant +2 more

We investigate the nature of the quantum phase transition in modulated $\mathrm{SU}(N)$ Heisenberg spin chains. In the odd-$N$ case, the transition separates a trivial nondegenerate phase to a doubly degenerate gapped chiral $\mathrm{PSU}(N)$ symmetry-protected topological (SPT) phase which breaks spontaneously the inversion symmetry. The transition is not an Ising transition associated to the breaking of the ${\mathbb{Z}}_{2}$ inversion symmetry, but is governed by the delocalization of the edge states of the SPT phase. In this respect, a modulated $\mathrm{SU}(N)$ Heisenberg spin chain provides a simple example in one dimension of a non-Landau phase transition which is described by the $\mathrm{SU}{(N)}_{1}$ conformal field theory. We show that the chiral SPT phase exhibits fractionalized spinon excitations, which can be confined by slightly changing the model parameters.

arXiv:2409.01019•Physical review. B./Physical review. B•Jan 10, 2025•2 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-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
Gapped quantum spin chains with symmetry PSU(N)=SU(N)/ZN\text{PSU}(N)=\text{SU}(N)/{\mathbb{Z}}_{N}PSU(N)=SU(N)/ZN​ are known to possess NNN distinct symmetry protected topological phases. Besides the trivial phase, there are N\ensuremath−1N\ensuremath{-}1N\ensuremath−1 Haldane phases which are distinguished by the occurrence of massless boundary spins. We give an expression for a universal AKLT Hamiltonian for an arbitrary symmetry group, including the case where inversion symmetry is broken. Using a diagrammatic method, we construct Hamiltonians for two of the topological classes in SU(N)\text{SU}(N)SU(N) that arise when the physical spin is in the adjoint representation. Since our results are valid for all NNN, we also discuss the large-NNN limit.
Chiral Haldane phases are examples of one-dimensional topological states of matter which are protected by the projective SU(N)\mathrm{SU}(N)SU(N) group (or its subgroup ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N})$ with N>2N>2N>2. The unique feature of these symmetry-protected topological (SPT) phases is that they are accompanied by inversion-symmetry breaking and the emergence of different left and right edge states which transform, for instance, respectively in the fundamental $(\mathbit{N})$ and antifundamental $(\overline{\mathbit{N}})$ representations of SU(N)\mathrm{SU}(N)SU(N). We show, by means of complementary analytical and numerical approaches, that these chiral SPT phases as well as the nonchiral ones are realized as the ground states of a generalized two-leg SU(N)\mathrm{SU}(N)SU(N) spin ladder in which the spins in the first chain transform in $\mathbit{N}$ and the second in $\overline{\mathbit{N}}$. In particular, we map out the phase diagram for N=3N=3N=3 and 4 to show that all the possible symmetry-protected topological phases with projective SU(N)\mathrm{SU}(N)SU(N) symmetry appear in this simple ladder model.
We study Z3{\mathbb{Z}}_{3}Z3​ symmetry-protected topological (SPT) phases in one-dimensional spin systems with ${Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3}$ symmetry. We construct ground-state wave functions of the matrix product form for nontrivial Z3{\mathbb{Z}}_{3}Z3​ phases and their parent Hamiltonian from a cocycle of the group cohomology ${H}^{2}({Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3},U(1))$. The Hamiltonian is an SU(3) version of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, consisting of bilinear and biquadratic terms of su(3) generators in the adjoint representation. A generalization to the SU(N)\mathrm{SU}(N)SU(N) case, the SU(N)\mathrm{SU}(N)SU(N) AKLT Hamiltonian, is also presented, which realizes nontrivial ZN{\mathbb{Z}}_{N}ZN​ SPT phases. We use the infinite-size variant of the density matrix renormalization group (iDMRG) method to determine the ground-state phase diagram of the SU(3) bilinear-biquadratic model as a function of the parameter $\ensuremath{\theta}$ controlling the ratio of the bilinear and biquadratic coupling constants. The nontrivial Z3{\mathbb{Z}}_{3}Z3​ SPT phase is found for a range of the parameter $\ensuremath{\theta}$ including the point of vanishing biquadratic term $(\ensuremath{\theta}=0)$ as well as the SU(3) AKLT point [$\ensuremath{\theta}=arctan(2/9)$]. A continuous phase transition to the SU(3) dimer phase takes place at $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.027\ensuremath{\pi}$, with a central charge $c\ensuremath{\approx}3.2$. For SU(3) symmetric cases, we define string order parameters for the Z3{\mathbb{Z}}_{3}Z3​ SPT phases in a similar way to the conventional Haldane phase. We propose simple spin models that effectively realize the SU(3) and SU(4) AKLT models.
There are many interesting parallels between systems of interacting non-Abelian anyons and quantum magnetism occurring in ordinary SU(2) quantum magnets. Here we consider theories of so-called su(2)k{}_{k}k​ anyons, well-known deformations of SU(2), in which only the first k+1k+1k+1 angular momenta of SU(2) occur. In this paper, we discuss in particular anyonic generalizations of ordinary SU(2) spin chains with an emphasis on anyonic spin S=1S=1S=1 chains. We find that the overall phase diagrams for these anyonic spin-1 chains closely mirror the phase diagram of the ordinary bilinear-biquadratic spin-1 chain including anyonic generalizations of the Haldane phase, the AKLT construction, and supersymmetric quantum critical points. A novel feature of the anyonic spin-1 chains is an additional topological symmetry that protects the gapless phases. Distinctions further arise in the form of an even/odd effect in the deformation parameter kkk when considering su(2)k{}_{k}k​ anyonic theories with $k\ensuremath{\ge}5$, as well as for the special case of the su(2)4{}_{4}4​ theory for which the spin-1 representation plays a special role. We also address anyonic generalizations of spin-12\frac{1}{2}21​ chains with a focus on the topological protection provided for their gapless ground states. Finally, we put our results into the context of earlier generalizations of SU(2) quantum spin chains, in particular so-called (fused) Temperley-Lieb chains.
We investigate the nature of the quantum phase transition in modulated SU(N)\mathrm{SU}(N)SU(N) Heisenberg spin chains. In the odd-NNN case, the transition separates a trivial nondegenerate phase to a doubly degenerate gapped chiral PSU(N)\mathrm{PSU}(N)PSU(N) symmetry-protected topological (SPT) phase which breaks spontaneously the inversion symmetry. The transition is not an Ising transition associated to the breaking of the Z2{\mathbb{Z}}_{2}Z2​ inversion symmetry, but is governed by the delocalization of the edge states of the SPT phase. In this respect, a modulated SU(N)\mathrm{SU}(N)SU(N) Heisenberg spin chain provides a simple example in one dimension of a non-Landau phase transition which is described by the SU(N)1\mathrm{SU}{(N)}_{1}SU(N)1​ conformal field theory. We show that the chiral SPT phase exhibits fractionalized spinon excitations, which can be confined by slightly changing the model parameters.
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 $\ensuremath{\Theta}=2\ensuremath{\pi}$.
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.