Majorana spin liquids and projective realization of SU(2) spin symmetry

We revisit the fermionic parton approach to S = 1/2 quantum spin liquids with SU(2) spin rotation symmetry, and the associated projective symmetry group (PSG) classification. We point out that the existing PSG classification is incomplete; upon completing it, we find spin liquid states with S=1 and S=0 Majorana fermion excitations coupled to a deconfined Z2 gauge field. The crucial observation leading us to this result is that, like space group and time reversal symmetries, spin rotations can act projectively on the fermionic partons; that is, a spin rotation may be realized by simultaneous SU(2) spin and gauge rotations. We show that there are only two realizations of spin rotations acting on fermionic partons: the familiar naive realization where spin rotation is not accompanied by any gauge transformation, and a single type of projective realization. We discuss the PSG classification for states with projective spin rotations. To illustrate these results, we show that there are four such PSGs on the two-dimensional square lattice. We study the properties of the corresponding states, finding that one -- with gapless Fermi points -- is a stable phase beyond mean-field theory. In this phase, depending on parameters, a small Zeeman magnetic field can open a partial gap for the Majorana fermion excitations. Moreover, there are nearby gapped phases supporting Z2 vortex excitations obeying non-Abelian statistics. We conclude with a discussion of various open issues, including the challenging question of where such S=1 Majorana spin liquids may occur in models and in real systems.Comment: 19 pages, 8 figures. Typos corrected, references adde

Topological Entanglement Entropy of Fracton Stabilizer Codes

Entanglement entropy provides a powerful characterization of two-dimensional gapped topological phases of quantum matter, intimately tied to their description by topological quantum field theories (TQFTs). Fracton topological orders are three-dimensional gapped topologically ordered states of matter, but the existence of a TQFT description for these phases remains an open question. We show that three-dimensional fracton phases are nevertheless characterized, at least partially, by universal structure in the entanglement entropy of their ground state wave functions. We explicitly compute the entanglement entropy for two archetypal fracton models --- the `X-cube model' and `Haah's code' --- and demonstrate the existence of a topological contribution that scales linearly in subsystem size. We show via Schrieffer-Wolff transformations that the topological entanglement of fracton models is robust against arbitrary local perturbations of the Hamiltonian. Finally, we argue that these results may be extended to characterize localization-protected fracton topological order in excited states of disordered fracton models.Comment: published versio

Odd Fracton Theories, Proximate Orders, and Parton Constructions

The Lieb-Schultz-Mattis (LSM) theorem implies that gapped phases of matter must satisfy non-trivial conditions on their low-energy properties when a combination of lattice translation and $U(1)$ symmetry are imposed. We describe a framework to characterize the action of symmetry on fractons and other sub-dimensional fractional excitations, and use this together with the LSM theorem to establish that X-cube fracton order can occur only at integer or half-odd-integer filling. Using explicit parton constructions, we demonstrate that "odd" versions of X-cube fracton order can occur in systems at half-odd-integer filling, generalizing the notion of odd $Z_2$ gauge theory to the fracton setting. At half-odd-integer filling, exiting the X-cube phase by condensing fractional quasiparticles leads to symmetry-breaking, thereby allowing us to identify a class of conventional ordered phases proximate to phases with fracton order. We leverage a dual description of one of these ordered phases to show that its topological defects naturally have restricted mobility. Condensing pairs of these defects then leads to a fracton phase, whose excitations inherit these mobility restrictions

Entanglement Entropy of 3-d Conformal Gauge Theories with Many Flavors

Three-dimensional conformal field theories (CFTs) of deconfined gauge fields coupled to gapless flavors of fermionic and bosonic matter describe quantum critical points of condensed matter systems in two spatial dimensions. An important characteristic of these CFTs is the finite part of the entanglement entropy across a circle. The negative of this quantity is equal to the finite part of the free energy of the Euclidean CFT on the three-sphere, and it has been proposed to satisfy the so called F-theorem, which states that it decreases under RG flow and is stationary at RG fixed points. We calculate the three-sphere free energy of non-supersymmetric gauge theory with a large number N_F of bosonic and/or fermionic flavors to the first subleading order in 1/N_F. We also calculate the exact free energies of the analogous chiral and non-chiral {\cal N} = 2 supersymmetric theories using localization, and find agreement with the 1/N_F expansion. We analyze some RG flows of supersymmetric theories, providing further evidence for the F-theorem.Comment: 31 pages, 2 figures; v2 refs added, minor change

An SU(N) Mott insulator of an atomic Fermi gas realized by large-spin Pomeranchuk cooling

The Hubbard model, containing only the minimum ingredients of nearest neighbor hopping and on-site interaction for correlated electrons, has succeeded in accounting for diverse phenomena observed in solid-state materials. One of the interesting extensions is to enlarge its spin symmetry to SU(N>2), which is closely related to systems with orbital degeneracy. Here we report a successful formation of the SU(6) symmetric Mott insulator state with an atomic Fermi gas of ytterbium (173Yb) in a three-dimensional optical lattice. Besides the suppression of compressibility and the existence of charge excitation gap which characterize a Mott insulating phase, we reveal an important difference between the cases of SU(6) and SU(2) in the achievable temperature as the consequence of different entropy carried by an isolated spin. This is analogous to Pomeranchuk cooling in solid 3He and will be helpful for investigating exotic quantum phases of SU(N) Hubbard system at extremely low temperatures.Comment: 20 pages, 6 figures, to appear in Nature Physic

Three-body interactions with cold polar molecules

We show that polar molecules driven by microwave fields give naturally rise to strong three-body interactions, while the two-particle interaction can be independently controlled and even switched off. The derivation of these effective interaction potentials is based on a microscopic understanding of the underlying molecular physics, and follows from a well controlled and systematic expansion into many-body interaction terms. For molecules trapped in an optical lattice, we show that these interaction potentials give rise to Hubbard models with strong nearest-neighbor two-body and three-body interaction. As an illustration, we study the one-dimensional Bose-Hubbard model with dominant three-body interaction and derive its phase diagram.Comment: 8 pages, 4 figure

The Spin Liquid State of the Tb2Ti2O7 Pyrochlore Antiferromagnet: A Puzzling State of Affairs

The pyrochlore antiferromagnet Tb2Ti2O7 has proven to be an enigma to experimentalists and theorists working on frustrated magnetic systems. The experimentally determined energy level structure suggests a local Ising antiferromagnet at low temperatures, T < 10 K. An appropriate model then predicts a long-range ordered Q = 0 state below approximately 2 K. However, muon spin resonance experiments reveal a paramagnetic structure down to tens of milli-Kelvin. The importance of fluctuations out of the ground state effective Ising doublet has been recently understood, for the measured paramagnetic correlations can not be described without including the higher crystal field states. However, these fluctuations treated within the random phase approximation (RPA) fail to account for the lack of ordering in this system below 2 K. In this work, we briefly review the experimental evidence for the collective paramagnetic state of Tb2Ti2O7. The basic theoretical picture for this system is discussed, where results from classical spin models are used to motivate the investigation of quantum effects to lowest order via the RPA. Avenues for future experimental and theoretical work on Tb2Ti2O7 are presented.Comment: Latex2e,6 pages, IOP format, introduction shortened and other minor corrections, replaced with published version in the Proceedings of the Highly Frustrated Magnetism 2003 Conference, Grenobl

Reversing non-local transport through a superconductor by electromagnetic excitations

Superconductors connected to normal metallic electrodes at the nanoscale provide a potential source of non-locally entangled electron pairs. Such states would arise from Cooper pairs splitting into two electrons with opposite spins tunnelling into different leads. In an actual system the detection of these processes is hindered by the elastic transmission of individual electrons between the leads, yielding an opposite contribution to the non-local conductance. Here we show that electromagnetic excitations on the superconductor can play an important role in altering the balance between these two processes, leading to a dominance of one upon the other depending on the spatial symmetry of these excitations. These findings allow to understand some intriguing recent experimental results and open the possibility to control non-local transport through a superconductor by an appropriate design of the experimental geometry.Comment: 6 pages, 3 figure

Gauge fields for ultracold atoms in optical superlattices

We present a scheme that produces a strong U(1)-like gauge field on cold atoms confined in a two-dimensional square optical lattice. Our proposal relies on two essential features, a long-lived metastable excited state that exists for alkaline-earth or Ytterbium atoms, and an optical superlattice. As in the proposal by Jaksch and Zoller [New Journal of Physics 5, 56 (2003)], laser-assisted tunneling between adjacent sites creates an effective magnetic field. In the tight-binding approximation, the atomic motion is described by the Harper Hamiltonian, with a flux across each lattice plaquette that can realistically take any value between 0 and $\pi$. We show how to take advantage of the superlattice to ensure that each plaquette acquires the same phase, thus simulating a uniform magnetic field. We discuss the observable consequences of the artificial gauge field on non-interacting bosonic and fermionic gases. We also outline how the scheme can be generalized to non-Abelian gauge fields