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arXiv:1804.10122 · Physical review. B./Physical review. B

Fermionic Lieb-Schultz-Mattis theorems and weak symmetry-protected phases

Meng Cheng

Published Feb 21, 2019• 68 citations
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Abstract

The celebrated Lieb-Schultz-Mattis theorem excludes a nondegenerate gapped ground state of a spin system with an odd number of $s=\frac{1}{2}$ 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.

DOI
10.1103/physrevb.99.075143

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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.
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.
We prove two Lieb-Schultz-Mattis type theorems that apply to any translationally invariant and local fermionic ddd-dimensional lattice Hamiltonian for which fermion-number conservation is broken down to the conservation of fermion parity. We show that when the internal symmetry group Gf{G}_{f}^{\phantom{\rule{0.16em}{0ex}}}Gf​ is realized locally (in a repeat unit cell of the lattice) by a nontrivial projective representation, then the ground state cannot be simultaneously nondegenerate, symmetric (with respect to lattice translations and Gf{G}_{f}^{\phantom{\rule{0.16em}{0ex}}}Gf​), and gapped. We also show that when the repeat unit cell hosts an odd number of Majorana degrees of freedom and the cardinality of the lattice is even, then the ground state cannot be simultaneously nondegenerate, gapped, and translation symmetric.
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.
The Lieb-Schultz-Mattis (LSM) theorem and its higher-dimensional extensions forbid the existence of a unique, symmetric, and gapped ground state at fractional fillings in quantum many-body systems with a conserved particle number (or spin angular momentum) and the conventional translation symmetry of Euclidean lattices. In this work, we propose a generalization of the LSM theorem to quantum many-body systems on hyperbolic lattices, i.e., regular tessellations of two-dimensional negatively curved space. By leveraging concepts from hyperbolic band theory in a many-body setting, we adapt Oshikawa's flux-threading argument to periodic hyperbolic lattices with a non-Euclidean (Fuchsian) translation symmetry and compute a lower-bound to the ground-state degeneracy as a function of filling and lattice geometry. We explore the consequences of LSM constraints for gapped phases of hyperbolic quantum matter and suggest frustrated spin models on hyperbolic analogs of the square and triangular lattices as promising platforms for realizing symmetric spin liquids in hyperbolic space.
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.
Symmetry provides powerful non-perturbative constraints in quantum many-body systems. A prominent example is the Lieb-Schultz-Mattis (LSM) anomaly -- a mixed 't Hooft anomaly between internal and translational symmetries that forbids a trivial symmetric gapped phase. In this work, we investigate lattice translation operators in systems with an LSM anomaly. We construct explicit lattice models in two and three spatial dimensions and show that, after gauging the full internal symmetry, translation becomes non-invertible and fuses into defects of the internal symmetry. The result is supported by the anomaly-inflow in view of topological field theory. Our work extends earlier one-dimensional observations to a unified higher-dimensional framework and clarifies their origin in mixed anomalies and higher-group structures, highlighting a coherent interplay between internal and crystalline symmetries.