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arXiv:2409.01019 · Physical review. B./Physical review. B

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

Sylvain Capponi, Lukas Devos, P. Lecheminant, Keisuke Totsuka, Laurens Vanderstraeten

Published Jan 10, 2025• 2 citations
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Abstract

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.

DOI
10.1103/physrevb.111.l020404

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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.
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.
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 a large-N generalization of J1J_1J1​-J2J_2J2​ Heisenberg model on square lattice -- an Sp(2N)Sp(2N)Sp(2N) spin model. The possible quantum spin liquid phases of the Sp(2N)Sp(2N)Sp(2N) model are studied using the SU(2) projective construction. We find several spin liquid states at least in large NNN limit, which include SU(2) πππ-flux state, SU(2) chiral spin state and Z2Z_2Z2​ spin liquid states. All those spin liquid states have non-trivial quantum orders. We show how projective symmetry group, which characterizes quantum order, protects the stability of even gapless spin liquids. We also study the continuous quantum phase transition from the SU(2) πππ-flux state to the SU(2) chiral spin liquid state, and from the SU(2) πππ-flux state to the Z2Z_2Z2​ spin liquid state. We show that those phase transitions are beyond the paradigm of Landau symmetry-breaking theory. The first phase transition, although a Z2Z_2Z2​ symmetry breaking transition, contains critical exponents that are different from those obtained from the Ginzburg-Landau theory of Ising universality class. The second transition does not even involve a change of symmetry and has no symmetry breaking order parameter.
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 $N\ensuremath{-}1$ 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.
We study different quantum phases in integer spin systems with on-site ${D}_{2h}={D}_{2}\ensuremath{\bigotimes}{Z}_{2}$ and translation symmetry. We find four distinct nontrivial phases in S=1S=1S=1 spin chains despite the fact that they all have the same symmetry. All four phases have gapped bulk excitations, doubly degenerate end states, and the doubly degenerate entanglement spectrum. These nontrivial phases are examples of symmetry-protected topological (SPT) phases introduced by Gu and Wen [Phys. Rev. B 80, 155131 (2009)]. One of the SPT phases corresponds to the Haldane phase and the other three are new. These four SPT phases can be distinguished experimentally by their different responses of the end states to weak external magnetic fields. According to the Chen--Gu--Wen classification, the D2h{D}_{2h}D2h​ symmetric spin chain can have a total of 64 SPT phases that do not break the symmetry. Here we constructed seven nontrivial phases from the seven classes of nontrivial projective representations of the D2h{D}_{2h}D2h​ group. Four of these are found in S=1S=1S=1 spin chains and studied in this paper.
We propose a series of simple two-dimensional (2D) lattice interacting fermion models that we demonstrate at low energy describe bosonic symmetry-protected topological (SPT) states and quantum phase transitions between them. This is because due to interaction, the fermions are gapped both at the boundary of the SPT states and at the bulk quantum phase transition, thus these models at low energy can be described completely by bosonic degrees of freedom. We show that the bulk of these models is described by a Sp(N\mathrm{Sp}(NSp(N) principal chiral model with a topological $\mathrm{\ensuremath{\Theta}}$ term, whose boundary is described by a Sp(N\mathrm{Sp}(NSp(N) principal chiral model with a Wess-Zumino-Witten term at level 1. The quantum phase transition between SPT states in the bulk is tuned by a particular interaction term, which corresponds to tuning $\mathrm{\ensuremath{\Theta}}$ in the field theory, and the phase transition occurs at $\mathrm{\ensuremath{\Theta}}=\ensuremath{\pi}$. The simplest version of these models with N=1N=1N=1 is equivalent to the familiar O(4) nonlinear sigma model (NLSM) with a topological term, whose boundary is a (1+1)D(1+1)\text{D}(1+1)D conformal field theory with central charge c=1c=1c=1. After breaking the O(4) symmetry to its subgroups, this model can be viewed as bosonic SPT states with U(1), or Z2{Z}_{2}Z2​ symmetries, etc. All of these fermion models, including the bulk quantum phase transitions, can be simulated with the determinant quantum Monte Carlo method without the sign problem. Recent numerical results strongly suggest that the quantum disordered phase of the O(4) NLSM with precisely $\mathrm{\ensuremath{\Theta}}=\ensuremath{\pi}$ is a stable (2+1)D(2+1)\text{D}(2+1)D conformal field theory with gapless bosonic modes.