We discuss phenomena arising from the combined influence of electron correlation and spin-orbit coupling, with an emphasis on emergent quantum phases and transitions in heavy transition metal compounds with 4d and 5d elements. A common theme is the influence of spin-orbital entanglement produced by spin-orbit coupling, which influences the electronic and magnetic structure. In the weak-to-intermediate correlation regime, we show how non-trivial band-like topology leads to a plethora of phases related to topological insulators. We expound these ideas using the example of pyrochlore iridates, showing how many novel phases such as the Weyl semi-metal, axion insulator, topological Mott insulator, and topological insulators may arise in this context. In the strong correlation regime, we argue that spin-orbital entanglement fully or partially removes orbital degeneracy, reducing or avoiding the normally ubiquitous Jahn-Teller effect. As we illustrate for the honeycomb lattice iridates and double perovskites, this leads to enhanced quantum fluctuations of the spin-orbital entangled states and the chance to promote exotic quantum spin liquid and multipolar ordered ground states. Connections to experiments, materials, and future directions are discussed.

We construct a model for interacting electrons with strong spin orbit coupling in the pyrochlore iridates. We establish the importance of the direct hopping process between the Ir atoms and use the relative strength of the direct and indirect hopping as a generic tuning parameter to study the correlation effects across the iridates family. We predict novel quantum phase transitions between conventional and/or topologically non-trivial phases. At weak coupling, we find topological insulator and metallic phases. As one increases the interaction strength, various magnetic orders emerge. The novel topological Weyl semi-metal phase is found to be realized in these different orders, one of them being the all-in/all-out pattern. Our findings establish the possible magnetic ground states for the iridates and suggest the generic presence of the Weyl semi-metal phase in correlated magnetic insulators on the pyrochlore lattice. We discus the implications for existing and future experiments.

We study the contribution to the entanglement entropy of (2+1)-dimensional conformal field theories coming from a sharp corner in the entangling surface. This contribution is encoded in a function a(θ) of the corner opening angle, and was recently proposed as a measure of the degrees of freedom in the underlying CFT. We show that the ratio a(θ)/CT , where CT is the central charge in the stress tensor correlator, is an almost universal quantity for a broad class of theories including various higher-curvature holographic models, free scalars and fermions, and Wilson-Fisher fixed points of the O(N ) models with N = 1, 2, 3. Strikingly, the agreement between these different theories becomes exact in the limit θ → π, where the entangling surface approaches a smooth curve. We thus conjecture that the corresponding ratio is universal for general CFTs in three dimensions.Many interacting gapless quantum systems do not possess simple particle-like excitations, making it difficult to quantify their effective number of degrees of freedom (dof) at low-energy. Conformal field theories (CFTs) constitute an important example. For CFTs in two spacetime dimensions (2d), the Virasoro central charge is a good measure of the dof. It appears in many quantities, such as the thermal free energy and the entanglement entropy (EE), and decreases under renormalization group (RG) flow [1]. In higher dimensions, the concept of quantum entanglement is emerging as a fundamental diagnostic for such measures [2,3]. E.g., it was instrumental in finding an analogous RG monotone for 3d CFTs, with the EE of a disk-shaped region [4]. We shall study another measure of recent interest [5][6][7][8][9][10][11][12][13][14][15][16]: the coefficient capturing the contribution of sharp corners to spatial entanglement.In the context of quantum field theory, the EE is defined for a spatial region V as: S = −Tr (ρ V ln ρ V ), where ρ V is the reduced density matrix produced by integrating out the dof in the complementary region V . In the groundstate of a 3d CFT, the EE takes the form:where δ is a short-distance cutoff, e.g., the lattice spacing, and , a length scale associated with the size of V . The first, 'area law', term depends on the UV regulator and scales with the size of the boundary. The second one appears only when V has a sharp corner with opening angle θ ∈ [0, 2π), Fig. 1. Crucially, a(θ) is a regulator independent coefficient that characterizes the underlying CFT. It is positive and satisfies a(2π − θ) = a(θ) [5], and behaves as follows:in the limits of a nearly smooth entangling surface and a very sharp corner, respectively. It has been studied for a variety of systems: free scalars and fermions [5][6][7], interacting scalar theories via numerical simulations [8][9][10], Lifshitz quantum critical points [11], and holographicFIG. 1: a) An entangling region V of size with a corner; b) The holographic entangling surface γ for a region on the boundary of AdS4 with a corner.models [12]. The results suggest that a(θ) is an effective measure of the do...

Understanding the real time dynamics of quantum systems without quasiparticles constitutes an important yet challenging problem. We study the superfluid-insulator quantum-critical point of bosons on a two-dimensional lattice, a system whose excitations cannot be described in a quasiparticle basis. We present detailed quantum Monte Carlo results for two separate lattice realizations: their low-frequency conductivities are found to have the same universal dependence on imaginary frequency and temperature. We then use the structure of the real time dynamics of conformal field theories described by the holographic gauge/gravity duality to make progress on the difficult problem of analytically continuing the Monte Carlo data to real time. Our method yields quantitative and experimentally testable results on the frequency-dependent conductivity near the quantum critical point, and on the spectrum of quasinormal modes in the vicinity of the superfluid-insulator quantum phase transition. Extensions to other observables and universality classes are discussed.1 arXiv:1309.2941v2 [cond-mat.str-el]

We study the finite temperature and magnetic field phase diagram of electrons on the pyrochlore lattice subject to a local repulsion as a model for the pyrochlore iridates. We provide the most general symmetry-allowed Hamiltonian, including next-nearest neighbour hopping, and relate it to a Slater-Koster based Hamiltonian for the iridates. It captures Lifshitz and/or thermal transitions between several phases such as metals, semimetals, topological insulators and Weyl semimetals, and gapped antiferromagnets with different orders. Our results on the charge conductivity, both DC and optical, Hall coefficient, magnetization and susceptibility show good agreement with recent experiments and provide new predictions. As such, our effective model sheds light on the pyrochlore iridates in a unified way.

We compute the non-zero temperature conductivity of conserved flavor currents in conformal field theories (CFTs) in 2+1 spacetime dimensions. At frequencies much greater than the temperature, ω k B T , the ω dependence can be computed from the operator product expansion (OPE) between the currents and operators which acquire a non-zero expectation value at T > 0. Such results are found to be in excellent agreement with quantum Monte Carlo studies of the O(2) Wilson-Fisher CFT. Results for the conductivity and other observables are also obtained in vector 1/N expansions. We match these large ω results to the corresponding correlators of holographic representations of the CFT: the holographic approach then allows us to extrapolate to small ω/(k B T ). Other holographic studies implicitly only used the OPE between the currents and the energy-momentum tensor, and this yields the correct leading large ω behavior for a large class of CFTs. However, for the Wilson-Fisher CFT a relevant "thermal" operator must also be considered, and then consistency with the Monte Carlo results is obtained without a previously needed ad hoc rescaling of the T value. We also establish sum rules obeyed by the conductivity of a wide class of CFTs.

The entanglement entropy in three-dimensional conformal field theories (CFTs) receives a logarithmic contribution characterized by a regulator-independent function $a(\theta)$ when the entangling surface contains a sharp corner with opening angle $\theta$. In the limit of a smooth surface ($\theta\rightarrow\pi$), this corner contribution vanishes as $a(\theta)=\sigma\,(\theta-\pi)^2$. In arXiv:1505.04804, we provided evidence for the conjecture that for any $d=3$ CFT, this corner coefficient $\sigma$ is determined by $C_T$, the coefficient appearing in the two-point function of the stress tensor. Here, we argue that this is a particular instance of a much more general relation connecting the analogous corner coefficient $\sigma_n$ appearing in the $n$th R\'enyi entropy and the scaling dimension $h_n$ of the corresponding twist operator. In particular, we find the simple relation $h_n/\sigma_n=(n-1)\pi$. We show how it reduces to our previous result as $n\rightarrow 1$, and explicitly check its validity for free scalars and fermions. With this new relation, we show that as $n\rightarrow 0$, $\sigma_n$ yields the coefficient of the thermal entropy, $c_S$. We also reveal a surprising duality relating the corner coefficients of the scalar and the fermion. Further, we use our result to predict $\sigma_n$ for holographic CFTs dual to four-dimensional Einstein gravity. Our findings generalize to other dimensions, and we emphasize the connection to the interval R\'enyi entropies of $d=2$ CFTs.Comment: 26 + 15 pages, 6 + 1 figures, 4 + 1 tables; v2: minor modifications to match published version, references adde

Topological phases of quantum matter defy characterization by conventional order parameters but can exhibit a quantized electromagnetic response and/or protected surface states. We examine such phenomena in a model for three-dimensional correlated complex oxides, the pyrochlore iridates. The model realizes interacting topological insulators, with and without time-reversal symmetry, and topological Weyl semimetals. We use cellular dynamical mean-field theory, a method that incorporates quantum many-body effects and allows us to evaluate the magnetoelectric topological response coefficient in correlated systems. This invariant is used to unravel the presence of an interacting axion insulator absent within a simple mean-field study. We corroborate our bulk results by studying the evolution of the topological boundary states in the presence of interactions. Consequences for experiments and for the search for correlated materials with symmetry-protected topological order are given.