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 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 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.
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.
In this study, we consider one-dimensional (1D) quantum spin systems with translation and discrete symmetries (spin reversal, space inversion, and time reversal symmetries). By combining the continuous U(1) symmetry with the discrete symmetries and using the extended Lieb–Schultz–Mattis (LSM) theorem [E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. 16, 407 (1961); K. Nomura, J. Morishige, and T. Isoyama, J. Phys. A 48, 375001 (2015)], we investigate the relation between the ground states, energy spectra, and symmetries. For half-integer spin cases, we generalize the dimer and Néel concepts using the discrete symmetries, and we can reconcile the LSM theorem with the dimer or Néel states, since there was a subtle dilemma. Furthermore, a part of discrete symmetries is enough to classify possible phases. Thus we can deepen our understanding of the relation between the LSM theorem and discrete symmetries.
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 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-1/2 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.
The qualitative difference in low-energy properties of spin S quantum antiferromagnetic chains with integer S and half-odd-integer S discovered by Haldane [F. D. M. Haldane, arXiv:1612.00076 (1981); Phys. Lett. A 93, 464–468 (1983); Phys. Rev. Lett. 50, 1153–1156 (1983)] and Tasaki [Tasaki, Graduate Texts in Physics (Springer, 2020)] can be intuitively understood in terms of the valence-bond picture proposed by Affleck et al. [I. Affleck, Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477–528 (1988)]. Here, we develop a similarly intuitive diagrammatic explanation of the qualitative difference between chains with odd S and even S, which is at the heart of the theory of symmetry-protected topological (SPT) phases. (There is a 24 min video in which the essence of the present work is discussed: https://youtu.be/URsf9e_PLlc.) More precisely, we define one-parameter families of states, which we call the asymmetric valence-bond solid (VBS) states, that continuously interpolate between the Affleck–Kennedy–Lieb–Tasaki (AKLT) state and the trivial zero state in quantum spin chains with S = 1 and 2. The asymmetric VBS state is obtained by systematically modifying the AKLT state. It always has exponentially decaying truncated correlation functions and is a unique gapped ground state of a short-ranged Hamiltonian. We also observe that the asymmetric VBS state possesses the time-reversal, the Z2×Z2, and the bond-centered inversion symmetries for S = 2 but not for S = 1. This is consistent with the known fact that the AKLT model belongs to the trivial SPT phase if S = 2 and to a nontrivial SPT phase if S = 1. Although such interpolating families of disordered states were already known, our construction is unified and is based on a simple physical picture. It also extends to spin chains with general integer S and provides us with an intuitive explanation of the essential difference between models with odd and even spins.