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arXiv:1705.00012 · Physical review. B./Physical review. B

Lieb-Schultz-Mattis theorem and its generalizations from the perspective of the symmetry-protected topological phase

Chao‐Ming Jian, Zhen Bi, Cenke Xu

Published Feb 12, 2018• 90 citations
arXivHTMLPDF

Abstract

We ask whether a local Hamiltonian with a featureless (fully gapped and nondegenerate) ground state could exist in certain quantum spin systems. We address this question by mapping the vicinity of certain quantum critical point (or gapless phase) of the $d$-dimensional spin system under study to the boundary of a $(d+1)$-dimensional bulk state, and the lattice symmetry of the spin system acts as an onsite symmetry in the field theory that describes both the selected critical point of the spin system and the corresponding boundary state of the $(d+1)$-dimensional bulk. If the symmetry action of the field theory is nonanomalous, i.e., the corresponding bulk state is a trivial state instead of a bosonic symmetry-protected topological (SPT) state, then a featureless ground state of the spin system is allowed; if the corresponding bulk state is indeed a nontrivial SPT state, then it likely excludes the existence of a featureless ground state of the spin system. From this perspective, we identify the spin systems with $\mathrm{SU}(N$) and $\mathrm{SO}(N$) symmetries on one-, two-, and three-dimensional lattices that permit a featureless ground state. We also verify our conclusions by other methods, including an explicit construction of these featureless spin states.

DOI
10.1103/physrevb.97.054412

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We ask whether a local Hamiltonian with a featureless (fully gapped and nondegenerate) ground state could exist in certain quantum spin systems. We address this question by mapping the vicinity of certain quantum critical point (or gapless phase) of the ddd-dimensional spin system under study to the boundary of a (d+1)(d+1)(d+1)-dimensional bulk state, and the lattice symmetry of the spin system acts as an onsite symmetry in the field theory that describes both the selected critical point of the spin system and the corresponding boundary state of the (d+1)(d+1)(d+1)-dimensional bulk. If the symmetry action of the field theory is nonanomalous, i.e., the corresponding bulk state is a trivial state instead of a bosonic symmetry-protected topological (SPT) state, then a featureless ground state of the spin system is allowed; if the corresponding bulk state is indeed a nontrivial SPT state, then it likely excludes the existence of a featureless ground state of the spin system. From this perspective, we identify the spin systems with SU(N\mathrm{SU}(NSU(N) and SO(N\mathrm{SO}(NSO(N) symmetries on one-, two-, and three-dimensional lattices that permit a featureless ground state. We also verify our conclusions by other methods, including an explicit construction of these featureless spin states.
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.
Lieb-Schultz-Mattis (LSM) theorem and its various generalizations provide powerful guidance towards the search for novel phases of matter. Here, the authors propose and prove a generalized LSM theorem suitable for 2+1D lattice models of interacting bosons or spins, with both magnetic flux and fractional spin in the unit cell. One outcome of this theorem is that, under certain conditions, the gapped ground states preserving all symmetries must be a nontrivial symmetry-protected topological (SPT) phase. Such symmetry-enforced SPTs display a dyonic character in that they associate charge with symmetry flux, which is demonstrated using quasiexactly solvable models constructed by decorating quantum dimer models.
We propose and prove a family of generalized Lieb-Schultz-Mattis~(LSM) theorems for symmetry protected topological~(SPT) phases on boson/spin models in any dimensions. The ``conventional'' LSM theorem, applicable to e.g. any translation invariant system with an odd number of spin-1/2 particles per unit cell, forbids a symmetric short-range-entangled ground state in such a system. Here we focus on systems with no LSM anomaly, where global/crystalline symmetries and fractional spins within the unit cell ensure that any symmetric SRE ground state must be a non-trivial SPT phase with anomalous boundary excitations. Depending on models, they can be either strong or ``higher-order'' crystalline SPT phases, characterized by non-trivial surface/hinge/corner states. Furthermore, given the symmetry group and the spatial assignment of fractional spins, we are able to determine all possible SPT phases for a symmetric ground state, using the real space construction for SPT phases based on the spectral sequence of cohomology theory. We provide examples in one, two and three spatial dimensions, and discuss possible physical realization of these SPT phases based on condensation of topological excitations in fractionalized phases.
The Lieb-Schultz-Mattis (LSM) theorem dictates that emergent low-energy states from a lattice model cannot be a trivial symmetric insulator if the filling per unit cell is not integral and if the lattice translation symmetry and particle number conservation are strictly imposed. In this paper, we compare the one-dimensional gapless states enforced by the LSM theorem and the boundaries of one-higher dimensional strong symmetry-protected topological (SPT) phases from the perspective of quantum anomalies. We first note that they can both be described by the same low-energy effective field theory with the same effective symmetry realizations on low-energy modes, wherein non-on-site lattice translation symmetry is encoded as if it were an internal symmetry. In spite of the identical form of the low-energy effective field theories, we show that the quantum anomalies of the theories play different roles in the two systems. In particular, we find that the chiral anomaly is equivalent to the LSM theorem, whereas there is another anomaly that is not related to the LSM theorem but is intrinsic to the SPT states. As an application, we extend the conventional LSM theorem to multiple-charge multiple-species problems and construct several exotic symmetric insulators. We also find that the (3+1)d chiral anomaly provides only the perturbative stability of the gaplessness local in the parameter space.
In quantum magnetism, peculiar features of quantum spin system sometimes forbid the existence of gapped ``featureless'' paramagnets which are fully symmetric and unfractionalized. The celebrated Lieb-Schultz-Mattis theorem is an example of such a constraint, but it is not known what the most general restriction might be. For given lattice structure and symmetries, it is an interesting question whether a featureless paramagnetic phase exists. In this paper, the authors focus on the existence of such featureless paramagnets on the square and honeycomb lattices and, on the basis of analytical and numerical arguments, propose the corresponding wave functions for several such models. The existence of these nontrivial states not only offers a proof of principle, but also suggests an exotic field theory description when these states are brought to the vicinity of a second-order phase transition.
The celebrated Lieb-Schultz-Mattis theorem excludes a nondegenerate gapped ground state of a spin system with an odd number of s=12s=\frac{1}{2}s=21​ spins per unit cell. Here, the author presents a new family of Lieb-Schultz-Mattis type theorems for fermionic systems that have no analog in spin systems. An example is a particle-hole symmetric system of spinless fermions on a lattice with an odd number of fermionic modes per unit cell. The author shows that a symmetric gapped ground state must be topologically ordered, and the lattice translation symmetries must permute anyon types, resulting in non-Abelian zero modes bound to lattice dislocations. These results can be interpreted as quantum anomalies on the surface of three-dimensional fermionic symmetry-protected phases.