Abstract:
We study geometrically frustrated antiferromagnets with magnetoelastic coupling. Frustration in these systems may be relieved by a structural transition to a low temperature phase with reduced lattice symmetry. We examine the statistical mechanics of this transition and the effects on it of quenched disorder, using Monte Carlo simulations of the classical Heisenberg model on the pyrochlore lattice with coupling to uniform lattice distortions. The model has a transition between a cubic, paramagnetic high-temperature phase and a tetragonal, Neel ordered low-temperature phase. It does not support the spin-Peierls phase, which is predicted as an additional possibility within Landau theory, and the transition is first-order for reasons unconnected with the symmetry analysis of Landau theory. Quenched disorder stabilises the cubic phase, and we find a phase diagram as a function of temperature and disorder strength similar to that observed in ZnCdCrO.

Abstract:
We develop a novel renormalization approach for frustrated quantum antiferromagnets. It is designed to consistently treat spin-wave interactions all over the magnetic Brillouin zone, including high-energy modes in outer regions as well as low energy modes in the center. Focusing on the paradigmatic $J_1$-$J_2$ model, we find a unifying description of the second-order transition between the N\'eel phase and the paramagnetic phase and the first-order transition between the N\'eel phase and the columnar phase. Our approach provides explicit results for the renormalized values of the spin stiffness and spin-wave velocity characterizing the low-energy magnons in the N\'eel phase.

Abstract:
We study the ground-state phase transitions of quasi-one-dimensional quantum Heisenberg antiferromagnets by the quantum Monte Carlo method with the continuous-time loop algorithm and finite-size scaling. For a model which consists of S=1 chains with bond alternation coupled on a square lattice, we determine the ground state phase diagram and the universality class of the quantum phase transitions.

Abstract:
We study the critical properties of three dimensional frustrated magnets, diluted with non-magnetic impurities. We show that these systems exhibit a second order phase transition, corresponding to a new universality class. In the pure case, the phase transition is expected to be weakly of first order. We therefore argue that these frustrated systems can be used to study experimentally the rounding effect of disorder on discontinuous phase transitions. We give first estimates of the critical exponents associated with this universality class, by using the method of the effective average action.

Abstract:
Confinement effects on the phase transitions in antiferromagnets are studied as a function of the surface coupling v and the surface field h for bcc(110) films. Unusual topologies for the phase diagram are attained for particular combinations of v and h. It is shown that some of the characteristics of the finite-temperature behavior of the system are driven by its low-temperature properties and consequently can be explained in terms of a ground-state analysis. Cluster variation free energies are used for the investigation of the finite temperature behavior.

Abstract:
I begin with a proposed global phase diagram of the cuprate superconductors as a function of carrier concentration, magnetic field, and temperature, and highlight its connection to numerous recent experiments. The phase diagram is then used as a point of departure for a pedagogical review of various quantum phases and phase transitions of insulators, superconductors, and metals. The bond operator method is used to describe the transition of dimerized antiferromagnetic insulators between magnetically ordered states and spin-gap states. The Schwinger boson method is applied to frustrated square lattice antiferromagnets: phase diagrams containing collinear and spirally ordered magnetic states, Z_2 spin liquids, and valence bond solids are presented, and described by an effective gauge theory of spinons. Insights from these theories of insulators are then applied to a variety of symmetry breaking transitions in d-wave superconductors. The latter systems also contain fermionic quasiparticles with a massless Dirac spectrum, and their influence on the order parameter fluctuations and quantum criticality is carefully discussed. I conclude with an introduction to strong coupling problems associated with symmetry breaking transitions in two-dimensional metals, where the order parameter fluctuations couple to a gapless line of fermionic excitations along the Fermi surface.

Abstract:
We consider the thermal phase transition from a paramagnetic to stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions J1<0 (nearest neighbor, ferromagnetic) and J2 >0 (second neighbor, antiferromagnetic). The striped phase breaks a Z4 symmetry and is stabilized at low temperatures for g=J2/|J1|>1/2. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [PRL 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The J1-J2 model has a line of very weak first-order phase transitions in the whole region 1/21/2. Most of our results are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological bench-marks for weakly first-order behaviors and Ashkin-Teller criticality.

Abstract:
In this paper, we study phase structure of $Z_2$ lattice gauge theories that appear as an effective field theory describing low-energy properties of frustrated antiferromagnets in two dimensions. Spin operators are expressed in terms of Schwinger bosons, and an emergent U(1) gauge symmetry reduces to a $Z_2$ gauge symmetry as a result of condensation of a bilinear operator of the Schwinger boson describing a short-range spiral order. We investigated the phase structure of the gauge theories by means of the Monte-Carlo simulations, and found that there exist three phases, phase with a long-range spiral order, a dimer state, and a spin liquid with deconfined spinons. Detailed phase structure and properties of phase transitions depend on details of the models.

Abstract:
We present a general introduction to the non-zero temperature dynamic and transport properties of low-dimensional systems near a quantum phase transition. Basic results are reviewed in the context of experiments on the spin-ladder compounds, insulating two-dimensional antiferromagnets, and double-layer quantum Hall systems. Recent large N computations on an extended t-J model (cond-mat/9906104) motivate a global scenario of the quantum phases and transitions in the high temperature superconductors, and connections are made to numerous experiments.

Abstract:
We identify two "universality" classes in the Coulomb frustrated phase separation phenomenon. They correspond to two different kind of electronic compressibility anomalies often encountered in strongly correlated electronic systems. We discuss differences and similarities of their corresponding phase diagrams in two- and three-dimensional systems.