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arXiv:0803.3185 · Physical Review B

Degeneracy and consistency condition for Berry phases: Gap closing under a local gauge twist

Takaaki Hirano, Hosho Katsura, Yasuhiro Hatsugai

Published Aug 21, 2008• 35 citations
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Abstract

We have discussed a consistency condition of Berry phases defined by a local gauge twist and spatial symmetries of the many-body system. It imposes a nontrivial gap closing condition under the gauge twist in both finite- and infinite-size systems. It also implies a necessary condition for the gapped and unique ground state. As for the simplest case, it predicts an inevitable gap closing in the Heisenberg chain of half-integer spins. Its relation to the Lieb-Schultz-Mattis theorem is discussed based on the symmetries of the twisted Hamiltonian. The discussion is also extended to the (approximately) degenerate multiplet and fermion cases. It restricts the number of the states in the low energy cluster of the spectrum by the filling of the fermions. Constraints by the reflection symmetry are also discussed.

DOI
10.1103/physrevb.78.054431

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We have discussed a consistency condition of Berry phases defined by a local gauge twist and spatial symmetries of the many-body system. It imposes a nontrivial gap closing condition under the gauge twist in both finite- and infinite-size systems. It also implies a necessary condition for the gapped and unique ground state. As for the simplest case, it predicts an inevitable gap closing in the Heisenberg chain of half-integer spins. Its relation to the Lieb-Schultz-Mattis theorem is discussed based on the symmetries of the twisted Hamiltonian. The discussion is also extended to the (approximately) degenerate multiplet and fermion cases. It restricts the number of the states in the low energy cluster of the spectrum by the filling of the fermions. Constraints by the reflection symmetry are also discussed.
We review the Lieb–Schultz–Mattis theorem and its variants, which are no-go theorems that state that a quantum many-body system with certain conditions cannot have a locally-unique gapped ground state. We restrict ourselves to one-dimensional quantum spin systems and discuss both the generalized Lieb–Schultz–Mattis theorem for models with U(1) symmetry and the extended Lieb–Schultz–Mattis theorem for models with discrete symmetry. We also discuss the implication of the same arguments to systems on the infinite cylinder, both with the periodic boundary conditions and with the spiral boundary conditions. For models with U(1) symmetry, we here present a rearranged version of the original proof of Lieb, Schultz, and Mattis based on the twist operator. As the title suggests, we take a modern topological point of view and prove the generalized Lieb–Schultz–Mattis theorem by making use of a topological index (which coincides with the filling factor). By a topological index, we mean an index that characterizes a locally-unique gapped ground state and is invariant under continuous (or smooth) modification of the ground state. For models with discrete symmetry, we describe the basic idea of the most general proof based on the topological index introduced in the context of symmetry-protected topological phases. We start from background materials such as the classification of projective representations of the symmetry group. We also review the notion that we call a locally-unique gapped ground state of a quantum spin system on an infinite lattice and present basic theorems. This notion turns out to be natural and useful from the physicists’ point of view. We have tried to make the present chapter readable and almost self-contained. We only assume basic knowledge about quantum spin systems.
We propose a geometric approach to the Lieb-Schultz-Mattis theorem for quantum many-body systems with discrete spin-rotation symmetries and lattice inversion or rotation symmetry, but without translation symmetry assumed. Under symmetry twisting on a $(d\ensuremath{-}1)$-dimensional plane, we find that any ddd-dimensional inversion-symmetric spin system possesses a doubly degenerate spectrum when it hosts a half-integer spin at the inversion-symmetric point. We also show that any rotation-symmetric generalized spin model with a projective representation at the rotation center has a similar degeneracy under symmetry twisting. We argue that these degeneracies imply that a unique symmetric gapped ground state that is smoothly connected to product states is forbidden in the original untwisted systems---generalized inversional or rotational Lieb-Schultz-Mattis theorems without lattice translation symmetry imposed. The traditional Lieb-Schultz-Mattis theorems with translations also fit in the proposed framework.
We discuss quantum many-body systems with lattice translation and discrete on-site symmetries. We point out that, under a boundary condition twisted by a symmetry operation, there is an exact degeneracy of ground states if the unit cell forms a projective representation of the on-site discrete symmetry. Based on the quantum transfer matrix formalism, we show that, if the system is gapped, the ground-state degeneracy under the twisted boundary condition also implies a ground-state (quasi)degeneracy under the periodic boundary conditions. This gives a compelling evidence for the recently proposed Lieb-Schultz-Mattis-type ingappability due to the on-site discrete symmetry in two and higher dimensions.
We construct a Z2×Z2\mathbb{Z}_2 \times \mathbb{Z}_2Z2​×Z2​ gauge theory coupled to matter on a one-dimensional chain, aiming to study the ground-state physics in the Gauss law subspace. We show that the theory in the Gauss law subspace has a U(1)(1)(1) symmetry whose generator commutes with lattice translations, but anticommutes with the lattice reflection operator. This leads to a Lieb-Schultz-Mattis (LSM) theorem that always rules out a trivial gapped ground state in the Gauss law subspace, if the hamiltonian is invariant under translations and reflection. Any point in the parameter space must realize either a spontaneously symmetry broken (SSB) ground state, or a gapless ground state. Imposing the Gauss law is pivotal for the existence of the U(1)(1)(1) symmetry, and hence of the LSM theorem. We thus demonstrate a novel mechanism to obtain an LSM-type theorem, wherein the symmetry responsible for the theorem originates from the kinematic constraints of a gauge theory. We identify a point in the parameter space at which the system is gapless. At the gapless point, the excitations admit a description in terms of free Dirac fermions with a constraint on the total fermion number. The asymptotic behavior of the two-point correlation function of the simplest local gauge-invariant quantity at the gapless point is found to be $ \propto \cos{(πr)}\,r^{-2/9}$, where rrr is the lattice separation between the two points. This model is also a natural platform to study phase diagram topological defects residing in families of SSB phases.
The theorem of Lieb, Schultz and Mattis (LSM), which states that the S=1/2 XXZ spin chain has gapless or degenerate ground states, can be applied to broader models. Independently, Kolb considered the relation between the wave number qqq and the twisting boundary condition, and he obtained a similar result as LSM. However, in frustrating cases it is known that there exist several exceptions for the assumption of the unique lowest state for the finite size, which is important in the traditional LSM theorem. In our previous paper, without the assumption of the uniqueness, we have extended the LSMK theorem for frustrating and non-symmetric cases. However, there remains a complexity in the proof of continuity. In this paper, we will simplify the proof than the previous work.
Twisting-operator approach to identifying gaplessness in SU(NNN) fermionic systems
We propose a general necessary condition for a spinful fermion chain with SU(2) spin-rotation symmetry to be gapped. Specifically, we prove that the expectation value of a properly defined fermionic twisting operator asymptotically approaches unity in any gapped phase with finite ground-state degeneracy, with finite-size corrections bounded by O(1/L)\mathscr{O}(1/L)O(1/L). Consequently, a non-unity value in the thermodynamic limit provides a sufficient criterion for identifying gapless fermion chains. We confirm this criterion using the sss-wave Bardeen-Cooper-Schrieffer (BCS) Hamiltonian and determinant quantum Monte Carlo (DQMC) simulations of interacting Hubbard models. We further extend the twisting-operator approach to SU(NNN)-symmetric fermionic systems, where the gapped ground-state sector must be SU(NNN)-singlet and ⟨U^M⟩=1+O(1/L)\langle \hat{\mathcal{U}}^M\rangle=1+\mathscr{O}(1/L)⟨U^M⟩=1+O(1/L) by a fermionic twisting operator U^\hat{\mathcal{U}}U^ with a suitably chosen integer MMM.