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arXiv:1507.00361 · Physical review. B./Physical review. B

Existence of featureless paramagnets on the square and the honeycomb lattices in 2+1 dimensions

Chao‐Ming Jian, Michael P. Zaletel

Published Jan 13, 2016• 30 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.93.035114

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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.
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.
Lieb-Schultz-Mattis (LSM) theorems provide powerful constraints on the problem, i.e. whether a quantum phase or phase transition can emerge in a many-body system. We derive the topological partition functions that characterize the LSM constraints in spin systems with G_s\times G_{int} G s × G i n t symmetry, where G_s G s is an arbitrary space group in one or two spatial dimensions, and G_{int} G i n t is any internal symmetry whose projective representations are classified by \mathbb{Z}_2^k ℤ 2 k with k k an integer. We then apply these results to study the emergibility of a class of exotic quantum critical states, including the well-known deconfined quantum critical point (DQCP), U(1) U ( 1 ) Dirac spin liquid (DSL), and the recently proposed non-Lagrangian Stiefel liquid. These states can emerge as a consequence of the competition between a magnetic state and a non-magnetic state. We identify all possible realizations of these states on systems with SO(3)\times \mathbb{Z}_2^T S O ( 3 ) × ℤ 2 T internal symmetry and either p6m p 6 m or p4m p 4 m lattice symmetry. Many interesting examples are discovered, including a DQCP adjacent to a ferromagnet, stable DSLs on square and honeycomb lattices, and a class of quantum critical spin-quadrupolar liquids of which the most relevant spinful fluctuations carry spin- 2 2 . In particular, there is a realization of spin-quadrupolar DSL that is beyond the usual parton construction. We further use our formalism to analyze the stability of these states under symmetry-breaking perturbations, such as spin-orbit coupling. As a concrete example, we find that a DSL can be stable in a recently proposed candidate material, NaYbO _2 2 .
We show how to construct fully symmetric states without topological order on a honeycomb lattice for S=12S=\frac{1}{2}S=21​ spins using the language of projected entangled pair states. An explicit example is given for the virtual bond dimension D=4D=4D=4. Four distinct classes differing by lattice quantum numbers are found by applying the systematic classification scheme introduced by two of the authors [S. Jiang and Y. Ran, Phys. Rev. B 92, 104414 (2015)]. Lack of topological degeneracy or other conventional forms of symmetry breaking in the proposed wave functions are checked by numerical calculations of the entanglement entropy and various correlation functions. Exponential decay of all correlation functions measured are strongly indicative of the energy gap for the putative parent Hamiltonian of the state. Our work provides the first explicit realization of a featureless quantum state for spin-12\frac{1}{2}21​ particles on a honeycomb lattice.
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.
The Lieb-Schultz-Mattis (LSM) theorem and its extensions forbid trivial phases from arising in certain quantum magnets. Constraining infrared behavior with the ultraviolet data encoded in the microscopic lattice of spins, these theorems tie the absence of spontaneous symmetry breaking to the emergence of exotic phases like quantum spin liquids. In this work, we take a new topological perspective on these theorems, by arguing they originate from an obstruction to "trivializing" the lattice under smooth, symmetric deformations, which we call the "lattice homotopy problem." We conjecture that all LSM-like theorems for quantum magnets (many previously unknown) can be understood from lattice homotopy, which automatically incorporates the full spatial symmetry group of the lattice, including all its point-group symmetries. One consequence is that any spin-symmetric magnet with a half-integer moment on a site with even-order rotational symmetry must be a spin liquid. To substantiate the claim, we prove the conjecture in two dimensions for some physically relevant settings.
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.