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arXiv:cond-mat/0306732 · Physical Review Letters

Chiral Spin Liquid Wave Function and the Lieb-Schultz-Mattis Theorem

Sandro Sorella, Luca Capriotti, Federico Becca, Alberto Parola

Published Dec 17, 2003• 18 citations
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Abstract

We study a chiral spin liquid wave function defined as a Gutzwiller projected BCS state with a complex pairing function. After projection, spontaneous dimerization is found for any odd but finite number of chains, thus satisfying the Lieb-Schultz-Mattis theorem, whereas for an even number of chains there is no dimerization. The two-dimensional thermodynamic limit is consistently reached for a large number of chains since the dimer order parameter vanishes in this limit. This property clearly supports the possibility of a spin liquid ground state in two dimensions with a gap to all physical excitations and with no broken translation symmetry.

DOI
10.1103/physrevlett.91.257005

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We study a chiral spin liquid wave function defined as a Gutzwiller projected BCS state with a complex pairing function. After projection, spontaneous dimerization is found for any odd but finite number of chains, thus satisfying the Lieb-Schultz-Mattis theorem, whereas for an even number of chains there is no dimerization. The two-dimensional thermodynamic limit is consistently reached for a large number of chains since the dimer order parameter vanishes in this limit. This property clearly supports the possibility of a spin liquid ground state in two dimensions with a gap to all physical excitations and with no broken translation symmetry.
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.
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 consider the Gutzwiller-projected Hofstadter-Slater determinant as a resonating-valence-bond wave function in a (2+1)-dimensional antiferromagnet. A strong case is made that spinons in the spin liquid state obey (1/2 statistics when a mass gap is present. We arrive at this conclusion from an effective continuum limit gauge theory. When a mass gap opens up, the parity symmetry is spontaneously broken resulting in twofold degeneracy of the ground state. The picture is consistent with that of the fractional quantum Hall state advocated by Laughlin.
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.
In this work we study the crystal symmetry fractionalization in chiral spin liquids with the chiral-semion topological order. We show that if such a chiral spin liquid is realized in a two-dimensional lattice model with odd number of spin-12\frac1221​ per unit cell and the state preserves spin rotation symmetry and translation symmetries, the semion excitation must carry both half-integer spin and fractional crystal quantum numbers. As a result, only a unique symmetry enriched topological phase can be realized in chiral spin liquids in a spin-12\frac1221​ kagome lattice model. These fractional symmetry quantum numbers are confirmed numerically using ground state wave functions obtained with the density matrix renormalization group method.