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arXiv:1410.4211 · Physical review. B./Physical review. B

Effective field theory for one-dimensional valence-bond-solid phases and their symmetry protection

Yohei Fuji

Published Mar 31, 2016• 32 citations
arXivHTMLPDF

Abstract

We investigate valence-bond-solid (VBS) phases in one-dimensional spin systems by an effective field theory developed by Schulz [Phys. Rev. B 34, 6372 (1986)]. While the distinction among the VBS phases is often understood in terms of different entanglement structures protected by certain symmetries, we adopt a different but more fundamental point of view, that is, different VBS phases are separated by a gap closing under certain symmetries. In this way, the effective field theory reproduces the known three symmetries: time reversal, bond-centered inversion, and dihedral group of $\ensuremath{\pi}$ spin rotations. It also predicts that there exists another symmetry: site-centered inversion combined with a spin rotation by $\ensuremath{\pi}$. We demonstrate that the last symmetry gives distinct trivial phases, which cannot be characterized by their entanglement structure, in terms of a simple perturbative analysis in a spin chain. We also discuss several applications of the effective field theory to the phase transitions among VBS phases in microscopic models and an extension of the Lieb-Schultz-Mattis theorem to non-translational-invariant systems.

DOI
10.1103/physrevb.93.104425

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We investigate valence-bond-solid (VBS) phases in one-dimensional spin systems by an effective field theory developed by Schulz [Phys. Rev. B 34, 6372 (1986)]. While the distinction among the VBS phases is often understood in terms of different entanglement structures protected by certain symmetries, we adopt a different but more fundamental point of view, that is, different VBS phases are separated by a gap closing under certain symmetries. In this way, the effective field theory reproduces the known three symmetries: time reversal, bond-centered inversion, and dihedral group of \ensuremathπ\ensuremath{\pi}\ensuremathπ spin rotations. It also predicts that there exists another symmetry: site-centered inversion combined with a spin rotation by \ensuremathπ\ensuremath{\pi}\ensuremathπ. We demonstrate that the last symmetry gives distinct trivial phases, which cannot be characterized by their entanglement structure, in terms of a simple perturbative analysis in a spin chain. We also discuss several applications of the effective field theory to the phase transitions among VBS phases in microscopic models and an extension of the Lieb-Schultz-Mattis theorem to non-translational-invariant systems.
We investigate disordered phases realized in one-dimensional quantum spin systems.We revisit an effective field theory proposed by Schulz [Phys.Rev. B 34, 6372 (1986)], which encapsulates several essential properties in low energy, from the point of view of symmetry.Although this effective theory was originally derived by perturbation theory, we find that it is fully consistent with the symmetry of the Hamiltonian and thus uniquely identified in the non-perturbative sense.This theory explains many possible phase transitions known to occur among valence-bond-solid phases, which are typical disordered phases in one-dimensional spin systems.Furthermore, this theory indicates that the phase transitions only occur under one of four symmetries: time reversal, bond-centered inversion, dihedral group of spin rotations, and site-centered inversion combined with a spin rotation.Surprisingly, the first three symmetries are consistent with those protecting the Haldane phase in spin-1 chains, which have already been found by a completely different approach using the matrix-product state (MPS).On the other hand, the last one is not known so far.To understand it, we explicitly construct a microscopic model that has two distinct disordered phases protected by the combined inversion symmetry.We examine this model and confirm the existence of those disordered phases by perturbation theory and numerical simulations.For completeness, we further give a more rigorous proof of their existence, based on the non-translation-invariant MPS.We also propose an extension of the Lieb-Schultz-Mattis theorem for the absence of a unique gapped ground state in half-odd-integer spin systems with site-centered inversion symmetry.
We study a one-dimensional (1D) system that shows many analogies to proposed two-dimensional (2D) deconfined quantum critical points (DQCP). Our system is a translationally invariant spin-12\frac{1}{2}21​ chain with onsite ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry and time-reversal symmetry. It undergoes a direct continuous transition from a ferromagnet (FM), where one of the Z2{Z}_{2}Z2​ symmetries and the time reversal are broken, to a valence bond solid (VBS), where all onsite symmetries are restored while the translation symmetry is broken. The other Z2{Z}_{2}Z2​ symmetry remains unbroken throughout, but its presence is crucial for both the direct transition (via specific Berry phase effect on topological defects, also related to a Lieb-Schultz-Mattis--type theorem) and the precise characterization of the VBS phase (which has crystalline-symmetry-protected-topological--like property). The transition has a description in terms of either two domain-wall species that ``fractionalize'' the VBS order parameter or in terms of two partons that ``fractionalize'' the FM order parameter, with each picture having its own Z2{Z}_{2}Z2​ gauge theory structure. The two descriptions are dual to each other and, at long wavelengths, take the form of a self-dual gauged Ashkin-Teller model, reminiscent of the self-dual easy-plane noncompact CP1{\mathrm{CP}}^{1}CP1 model that arises in the description of the 2D easy-plane DQCP. We also find an exact reformulation of the transition that leads to a simple field-theory description that explicitly unifies the FM and VBS order parameters; this reformulation can be interpreted as a new parton approach that does not attempt to fractionalize either of the FM and VBS order parameters but instead encodes them in instanton operators. Aside from providing explicit realizations of many ideas proposed in the context of the 2D DQCP, here in the simpler and fully tractable 1D setting with continuous transition, our study also suggests a possible line of approach to the 2D DQCP.
We continue recent efforts to discover examples of deconfined quantum criticality in one-dimensional models. In this work we investigate the transition between a Z3{\mathbb{Z}}_{3}Z3​ ferromagnet and a phase with valence bond solid (VBS) order in a spin chain with ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}$ global symmetry. We study a model with alternating projective representations on the sites of the two sublattices, allowing the Hamiltonian to connect to an exactly solvable point having VBS order with the character of SU(3)SU(3)SU(3)-invariant singlets. Such a model does not admit a Lieb-Schultz-Mattis theorem typical of systems realizing deconfined critical points. Nevertheless, we find evidence for a direct transition from the VBS phase to a Z3{\mathbb{Z}}_{3}Z3​ ferromagnet. Finite-entanglement scaling data are consistent with a second-order or weakly first-order transition. We find in our parameter space an integrable lattice model apparently describing the phase transition, with a very long, finite, correlation length of 190878 lattice spacings. Based on exact results for this model, we propose that the transition is extremely weakly first order and is part of a family of deconfined quantum critical points described by walking of renormalization group flows.
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.
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.
We propose an order parameter to characterize valence-bond-solid (VBS) states in quantum spin systems, given by the ground-state expectation value of a unitary operator appearing in the Lieb-Schultz-Mattis argument. We show that the order parameter changes the sign according to the configuration of the valence bonds. This allows us to determine the phase transition point in between different VBS states accurately. We demonstrate this theory in the bond-alternating Heisenberg chain and in the frustrated spin ladder.