Composite fermi liquids in the lowest Landau level

We study composite fermi liquid (CFL) states in the lowest Landau level (LLL) limit at a generic filling $\nu = \frac{1}{n}$. We begin with the old observation that, in compressible states, the composite fermion in the lowest Landau level should be viewed as a charge-neutral particle carrying vorticity. This leads to the absence of a Chern-Simons term in the effective theory of the CFL. We argue here that instead a Berry curvature should be enclosed by the fermi surface of composite fermions, with the total Berry phase fixed by the filling fraction $\phi_B=-2\pi\nu$. We illustrate this point with the CFL of fermions at filling fractions $\nu=1/2q$ and (single and two-component) bosons at $\nu=1/(2q+1)$. The Berry phase leads to sharp consequences in the transport properties including thermal and spin Hall conductances, which in the RPA approximation are distinct from the standard Halperin-Lee-Read predictions. We emphasize that these results only rely on the LLL limit, and do not require particle-hole symmetry, which is present microscopically only for fermions at $\nu=1/2$. Nevertheless, we show that the existing LLL theory of the composite fermi liquid for bosons at $\nu=1$ does have an emergent particle-hole symmetry. We interpret this particle-hole symmetry as a transformation between the empty state at $\nu=0$ and the boson integer quantum hall state at $\nu=2$. This understanding enables us to define particle-hole conjugates of various bosonic quantum Hall states which we illustrate with the bosonic Jain and Pfaffian states. The bosonic particle-hole symmetry can be realized exactly on the surface of a three-dimensional boson topological insulator. We also show that with the particle-hole and spin $SU(2)$ rotation symmetries, there is no gapped topological phase for bosons at $\nu=1$.Comment: 16 pages, 1 figure, new version with minor change

Dual Dirac liquid on the surface of the electron topological insulator

We discuss a non-fermi liquid gapless metallic surface state of the topological band insulator. It has an odd number of gapless Dirac fermions coupled to a non-compact U(1) gauge field. This can be viewed as a vortex dual to the conventional Dirac fermion surface state. This surface duality is a reflection of a bulk dual description discussed recently for the gauged topological insulator. All the other known surface states can be conveniently accessed from the dual Dirac liquid, including the surface quantum hall state, the Fu-Kane superconductor, the gapped symmetric topological order and the "composite Dirac liquid". We also discuss the physical properties of the dual Dirac liquid, and its connection to the half-filled Landau level.Comment: 5+2 page

Time-reversal symmetric U(1) quantum spin liquids

We study possible quantum $U(1)$ spin liquids in three dimensions with time-reversal symmetry. We find a total of 7 families of such $U(1)$ spin liquids, distinguished by the properties of their emergent electric/magnetic charges. We show how these spin liquids are related to each other. Two of these classes admit nontrivial protected surface states which we describe. We show how to access all of the 7 spin liquids through slave particle (parton) constructions. We also provide intuitive loop gas descriptions of their ground state wave functions. One of these phases is the `topological Mott insulator' conventionally described as a topological insulator of an emergent fermionic `spinon'. We show that this phase admits a remarkable dual description as a topological insulator of emergent fermionic magnetic monopoles. This results in a new (possibly natural) surface phase for the topological Mott insulator and a new slave particle construction. We describe some of the continuous quantum phase transitions between the different $U(1)$ spin liquids. Each of these seven families of states admits a finer distinction in terms of their surface properties which we determine by combining these spin liquids with symmetry protected topological phases. We discuss lessons for materials such as pyrochlore quantum spin ices which may harbor a $U(1)$ spin liquid. We suggest the topological Mott insulator as a possible ground state in some range of parameters for the quantum spin ice Hamiltonian.Comment: 25 pages, 11 figures, 1 tabl

Half-filled Landau level, topological insulator surfaces, and three dimensional quantum spin liquids

We synthesize and partly review recent developments relating the physics of the half-filled Landau level in two dimensions to correlated surface states of topological insulators in three dimensions. The latter are in turn related to the physics of certain three dimensional quantum spin liquid states. The resulting insights provide an interesting answer to the old question of how particle-hole symmetry is realized in composite fermion liquids. Specifically the metallic state at filling $\nu = \frac{1}{2}$ - described originally in pioneering work by Halperin , Lee, and Read as a liquid of composite fermions - was proposed recently by Son to be described by a particle-hole symmetric effective field theory distinct from that in the prior literature. We show how the relation to topological insulator surface states leads to a physical understanding of the correctness of this proposal. We develop a simple picture of the particle-hole symmetric composite fermion through a modification of older pictures as electrically neutral "dipolar" particles. We revisit the phenomenology of composite fermi liquids (with or without particle-hole symmetry), and show that their heat/electrical transport dramatically violates the conventional Wiedemann-Franz law but satisfies a modified one. We also discuss the implications of these insights for finding physical realizations of correlated topological insulator surfaces.Comment: 22 pages, 7 figures; (v2) Added some clarifications and corrected typo

Interacting fermionic topological insulators/superconductors in three dimensions

Symmetry Protected Topological (SPT) phases are a minimal generalization of the concept of topological insulators to interacting systems. In this paper we describe the classification and properties of such phases for three dimensional(3D) electronic systems with a number of different symmetries. For symmetries representative of all classes in the famous 10-fold way of free fermion topological insulators/superconductors, we determine the stability to interactions. By combining with results on bosonic SPT phases we obtain a classification of electronic 3D SPT phases for these symmetries. In cases with a normal U(1) subgroup we show that this classification is complete. We describe the non-trivial surface and bulk properties of these states. In particular we discuss interesting correlated surface states that are not captured in a free fermion description. We show that in many, but not all cases, the surface can be gapped while preserving symmetry if it develops intrinsic topological order.Comment: 14+1 pages, an erratum is added at the end, the original paper is unchange

Topological Paramagnetism in Frustrated Spin-One Mott Insulators

Time reversal protected three dimensional (3D) topological paramagnets are magnetic analogs of the celebrated 3D topological insulators. Such paramagnets have a bulk gap, no exotic bulk excitations, but non-trivial surface states protected by symmetry. We propose that frustrated spin-1 quantum magnets are a natural setting for realising such states in 3D. We describe a physical picture of the ground state wavefunction for such a spin-1 topological paramagnet in terms of loops of fluctuating Haldane chains with non-trivial linking phases. We illustrate some aspects of such loop gases with simple exactly solvable models. We also show how 3D topological paramagnets can be very naturally accessed within a slave particle description of a spin-1 magnet. Specifically we construct slave particle mean field states which are naturally driven into the topological paramagnet upon including fluctuations. We propose bulk projected wave functions for the topological paramagnet based on this slave particle description. An alternate slave particle construction leads to a stable U(1) quantum spin liquid from which a topological paramagnet may be accessed by condensing the emergent magnetic monopole excitation of the spin liquid.Comment: 16 pages, 5 figure

Symmetry enriched U(1) quantum spin liquids

We classify and characterize three dimensional $U(1)$ quantum spin liquids (deconfined $U(1)$ gauge theories) with global symmetries. These spin liquids have an emergent gapless photon and emergent electric/magnetic excitations (which we assume are gapped). We first discuss in great detail the case with time reversal and $SO(3)$ spin rotational symmetries. We find there are 15 distinct such quantum spin liquids based on the properties of bulk excitations. We show how to interpret them as gauged symmetry-protected topological states (SPTs). Some of these states possess fractional response to an external $SO(3)$ gauge field, due to which we dub them "fractional topological paramagnets". We identify 11 other anomalous states that can be grouped into 3 anomaly classes. The classification is further refined by weakly coupling these quantum spin liquids to bosonic Symmetry Protected Topological (SPT) phases with the same symmetry. This refinement does not modify the bulk excitation structure but modifies universal surface properties. Taking this refinement into account, we find there are 168 distinct such $U(1)$ quantum spin liquids. After this warm-up we provide a general framework to classify symmetry enriched $U(1)$ quantum spin liquids for a large class of symmetries. As a more complex example, we discuss $U(1)$ quantum spin liquids with time reversal and $Z_2$ symmetries in detail. Based on the properties of the bulk excitations, we find there are 38 distinct such spin liquids that are anomaly-free. There are also 37 anomalous $U(1)$ quantum spin liquids with this symmetry. Finally, we briefly discuss the classification of $U(1)$ quantum spin liquids enriched by some other symmetries.Comment: 24 pages + appendices + reference

Classification of interacting electronic topological insulators in three dimensions

A fundamental open problem in condensed matter physics is how the dichotomy between conventional and topological band insulators is modified in the presence of strong electron interactions. We show that there are 6 new electronic topological insulators that have no non-interacting counterpart. Combined with the previously known band-insulators, these produce a total of 8 topologically distinct phases. Two of the new topological insulators have a simple physical description as Mott insulators in which the electron spins form spin analogs of the familiar topological band-insulator. The remaining are obtained as combinations of these two `topological paramagnets' and the topological band insulator. We prove that these 8 phases form a complete list of all possible interacting topological insulators, and are classified by a Z_2^3 group-structure. Experimental signatures are also discussed for these phases.Comment: New version contains more results on experimental signatures and a more rigorous proof of a key statement (see Appendix D,E), with references reorganize

Gapped Symmetry Preserving Surface-State for the Electron Topological Insulator

It is well known that the 3D electronic topological insulator (TI) with charge-conservation and time-reversal symmetry cannot have a trivial insulating surface that preserves symmetry. It is often implicitly assumed that if the TI surface preserves both symmetries then it must be gapless. Here we show that it is possible for the TI surface to be both gapped and symmetry-preserving, at the expense of having surface-topological order. In contrast to analogous bosonic topological insulators, this symmetric surface topological order is intrinsically non-Abelian. We show that the surface-topological order provides a complete non-perturbative definition of the electron TI that transcends a free-particle band-structure picture, and could provide a useful perspective for studying strongly correlated topological Mott insulators.Comment: 12 pages, 2 figures, (published version