Bound states of three fermions forming symmetry-protected topological phases

We propose a simple theoretical construction of certain short-range entangled phases of interacting fermions, by putting the bound states of three fermions (which we refer to as clustons) into topological bands. We give examples in two and three dimensions, and show that they are distinct from any free fermion state. We further argue that these states can be viewed as combinations of certain free fermion topological states and bosonic symmetry-protected topological (SPT) states. This provides a conceptually simple understanding of various SPT phases, and the possibility of realizing them in cold atom systems. New parton constructions of these SPT phases in purely bosonic systems are proposed. We also discuss a related anomaly in two dimensional Dirac theories, which is the gravitational analogue of the parity anomaly.Comment: 4+4 pages, 3 figure

Schr\"odinger Soliton from Lorentzian Manifolds

In this paper, we introduce a new notion named as Schr\"odinger soliton. So-called Schr\"odinger solitons are defined as a class of special solutions to the Schr\"odinger flow equation from a Riemannian manifold or a Lorentzian manifold $M$ into a K\"ahler manifold $N$. If the target manifold $N$ admits a Killing potential, then the Schr\"odinger soliton is just a harmonic map with potential from $M$ into $N$. Especially, if the domain manifold is a Lorentzian manifold, the Schr\"odinger soliton is a wave map with potential into $N$. Then we apply the geometric energy method to this wave map system, and obtain the local well-posedness of the corresponding Cauchy problem as well as global existence in 1+1 dimension. As an application, we obtain the existence of Schr\"odinger soliton of the hyperbolic Ishimori system.Comment: 22 pages, with lower regularity of the initial data required in the revised version

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

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

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

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

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