We explore the ground states and quantum phase transitions of two-dimensional, spin S = 1/2, antiferromagnets by generalizing lattice models and duality transforms introduced by Sachdev and Jalabert (Mod. Phys. Lett. B 4, 1043Lett. B 4, (1990). The 'minimal' model for square lattice antiferromagnets is a lattice discretization of the quantum non-linear sigma model, along with Berry phases which impose quantization of spin. With full SU(2) spin rotation invariance, we find a magnetically ordered ground state with Néel order at weak coupling, and a confining paramagnetic ground state with bond charge (e.g. spin Peierls) order at strong coupling. We study the mechanisms by which these two states are connected in intermediate coupling. We extend the minimal model to study different routes to fractionalization and deconfinement in the ground state, and also generalize it to cases with a uniaxial anisotropy (the spin symmetry group is then U(1)). For the latter systems, fractionalization can appear by the pairing of vortices in the staggered spin order in the easy-plane; however, we argue that this route does not survive the restoration of SU (2) spin symmetry. For SU(2) invariant systems we study a separate route to fractionalization associated with the Higgs phase of a complex boson measuring non-collinear, spiral spin correlations: we present phase diagrams displaying competition between magnetic order, bond charge order, and fractionalization, and discuss the nature of the quantum transitions between the various states. A strong check on our methods is provided by their application to S = 1/2 frustrated antiferromagnets in one dimension: here, our results are in complete accord with those obtained by bosonization and by the solution of integrable models.Contents

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.

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We show that the solid phase between the 1/5 and 2/9 fractional quantum Hall states arises from an extremely delicate interplay between type-1 and type-2 composite fermion crystals, clearly demonstrating its nontrivial, strongly correlated character. We also compute the phase diagram of various crystals occurring over a wide range of filling factors and demonstrate that the elastic constants exhibit nonmonotonic behavior as a function of the filling factor, possibly leading to distinctive experimental signatures that can help mark the phase boundaries separating different kinds of crystals.

The ground states and excitations of two-dimensional insulating and doped Mott insulators are described by a bond operator formalism. While the method represents the degrees of freedom of an arbitrary antiferromagnet exactly, it is especially suited to systems in which there is a natural pairing of sites into bonds, as in states with spontaneous or explicit spin-Peierls order (or bond-centered charge order). In the undoped insulator, as discussed previously, we obtain both paramagnetic and magnetically-ordered states. We describe the evolution of superconducting order in the ground state with increasing doping-at low doping, the superconductivity is weak, can co-exist with magnetic order, and there are no gapless spin 1/2 fermionic excitations; at high doping, the magnetic order is absent and we obtain a BCS d-wave superconductor with gapless spin 1/2, nodal fermions. We present the critical theory describing the onset of these nodal fermionic excitations. We discuss the evolution of the spin spectrum, and obtain regimes where a spin 1 exciton contributes a sharp resonance in the dynamic spin susceptiblity. We also discuss the experimental consequences of low-energy, dynamically fluctuating, spin-Peierls order in an isotropic CuO2 plane-we compute consequences for the damping and dispersion of an optical phonon involving primarily the O ions, and compare the results with recent neutron scattering measurements of phonon spectra.

When confined to two dimensions and exposed to a strong magnetic field, electrons screen the Coulomb interaction in a topological fashion; they capture an even number of quantum vortices and transform into particles called 'composite fermions' (refs 1-3). The fractional quantum Hall effect occurs in such a system when the ratio (or 'filling factor, nu) of the number of electrons and the degeneracy of their spin-split energy states (the Landau levels) takes on particular values. The Landau level filling nu = 1/2 corresponds to a metallic state in which the composite fermions form a gapless Fermi sea. But for nu = 5/2, a fractional quantum Hall effect is observed instead; this unexpected result is the subject of considerable debate and controversy. Here we investigate the difference between these states by considering the theoretical problem of two composite fermions on top of a fully polarized Fermi sea of composite fermions. We find that they undergo Cooper pairing to form a p-wave bound state at nu = 5/2, but not at nu = 1/2. In effect, the repulsive Coulomb interaction between electrons is overscreened in the nu = 5/2 state by the formation of composite fermions, resulting in a weak, attractive interaction.

The essence of the ν = 5/2 fractional quantum Hall effect is believed to be captured by the Moore-Read Pfaffian (or anti-Pfaffian) description. However, a mystery regarding the formation of the Pfaffian state is the role of the three-body interaction Hamiltonian H3 that produces it as an exact ground state and the concomitant particle-hole symmetry breaking. We show that a twobody interaction Hamiltonian H2 constructed via particle-hole symmetrization of H3 produces a ground state nearly exactly approximating the Pfaffian and anti-Pfaffian states, respectively, in the spherical geometry. Importantly, the ground state energy of H2 exhibits a "Mexican-hat" structure as a function of particle number in the vicinity of half filling for a given flux indicating spontaneous particle-hole symmetry breaking. This signature is absent for the second Landau level Coulomb interaction at 5/2. The fractional quantum Hall effect (FQHE) [1] at orbital Landau level (LL) filling factor ν = 5/2 [2] (1/2 filling of the second LL (SLL)) is the subject of recent theoretical and experimental research. This is partly due to the Moore-Read Pfaffian state [3], the leading theoretical description of the ν = 5/2 FQHE, possessing non-Abelian quasiparticle excitations with potential applications towards fault-tolerant topological quantum computation [4]. Recent theoretical results [5,6] along with previous work [7,8] provide compelling evidence that this non-abelian description is essentially correct.However, a question remains regarding the MooreRead Pfaffian (Pf) description best illustrated by contrasting it to the celebrated Laughlin state [9] for the FQHE at ν = 1/q (q odd). When confined to a single LL, two-body interaction Hamiltonians can be parameterized by Haldane pseudopotentials V m -the energy for a pair of electrons in a state of relative angular momentum m [10] where only odd m enters for spin-polarized electron systems [11]. The Laughlin state is the exact ground state of a two-body Hamiltonian with only V 1 non-zero (the interaction is hard-core). Thus, through the pseudopotential description, the Laughlin state is shown to be continuously connected to the exact ground state of the Coulomb Hamiltonian at ν = 1/q. The Pf wave function, by contrast, is an exact ground state of a repulsive three-body Hamiltonian H 3 for even number of electrons N e in a half-filled LL [12]. There is no two-body Hamiltonian, and hence no exact pseudopotential description, for which the Pf is an exact eigenstate. So, as good as the physical description for the ν = 5/2 FQHE state provided by the Pf may be, it is not continuously connected to the exact Coulomb ground state or, in fact, the ground state of any two-body Hamiltonian. This notion has been discussed in the literature [13,14,15] for over ten years, and recently questions have been raised [16,17] about the applicability of the Pf for the physical 5/2-state.However, evidently some two-body Hamiltonians produce ground states that have nearly unity overlap (≈ 0.99) with the Pf. For example, ...

In this work, we present an analytical theory of strongly correlated Wigner crystals (WCs) in the lowest Landau level (LLL) by constructing an approximate, but accurate effective two-body interaction for composite fermions (CFs) participating in the WCs. This requires integrating out the degrees of freedom of all surrounding CFs, which we accomplish analytically by approximating their wave functions by delta functions. This method produces energies of various strongly correlated WCs that are in excellent agreement with those obtained from the Monte Carlo simulation of the full CF crystal wave functions. We compute the compressibility of the strongly correlated WCs in the LLL and predict discontinuous changes at the phase boundaries separating different crystal phases. [33] to provide an accurate description at low filling factors (ν ≤ 1/5). Despite these extensive theoretical works, the calculation of a precise phase diagram of quantum Hall liquids versus CF crystals (CFCs) remained stalled for many years due to difficulties in obtaining the energy of CFCs in the thermodynamic limit accurately. This issue was resolved in a recent work [38] inspired by the Thomson problem [39]. Here, the CFC wave functions are constructed in the spherical geometry by placing the WC wave packet centers at the locations that minimize the Coulomb energy of N charged point particles on the surface of a sphere. Locally, these minimum energy positions resemble the hexagonal lattice, which is the minimum energy symmetry for a classical 2D electron crystal [40]. This allows a precise investigation of the CFC wave functions up to a fairly large system size (N ∼ 100) [38].The Monte Carlo (MC) simulation of the CFC wave functions is computationally quite expensive and rather difficult to implement. Furthermore, it turns out that even though the energy obtained from this method enables a determination of the phase diagram, it is not sufficiently accurate to allow an evaluation of quantities such as compressibility, which is related to the second derivative of the energy. We develop in this work an analytical theory of the CFCs by constructing an accurate effective two-body interaction, which is based on the two-body wave function of CFs participating in the CFCs. This requires integrating out the degrees of freedom of all surrounding CFs, which we accomplish analytically by approximating their wave functions as delta functions. We call this approach the "renormalized two-body formalism," to be contrasted with the "isolated two-body formalism" where the effects of all surrounding CFs are neglected. The CFC energies obtained from the renormalized two-body formalism are in excellent agreement with those obtained from the MC simulation of the full CFC wave functions. With these analytical results, we obtain the compressibility and predict that its measurements, such as those carried out in GaAs heterostructures or in graphene [41][42][43][44][45][46], can detect the phase diagram of the CFCs.We begin by constructing the wave function for the CFC...