This review summarizes recent developments in the theory of spin glasses, as well as pertinent experimental data. The most characteristic properties of spin glass systems are described, and related phenomena in other glassy systems (dielectric and orientational glasses) are mentioned. The Edwards-Anderson model of spin glasses and its treatment within the replica method and mean-field theory are outlined, and concepts such as "frustration," "broken replica symmetry," "broken ergodicity," etc., are discussed. The dynamic approach to describing the spin glass transition is emphasized. Monte Carlo simulations of spin glasses and the insight gained by them are described. Other topics discussed include site-disorder models, phenomenological theories for the frozen phase and its excitations, phase diagrams in which spin glass order and ferromagnetism or antiferromagnetism compete, the Ne\'el model of superparamagnetism and related approaches, and possible connections between spin glasses and other topics in the theory of disordered condensed-matter systems.

Spin glasses: Experimental facts, theoretical concepts, and open questions

Weyl and Dirac semimetals are three-dimensional phases of matter with gapless electronic excitations that are protected by topology and symmetry. As three-dimensional analogs of graphene, they have generated much recent interest. Deep connections exist with particle physics models of relativistic chiral fermions, and, despite their gaplessness, to solid-state topological and Chern insulators. Their characteristic electronic properties lead to protected surface states and novel responses to applied electric and magnetic fields. The theoretical foundations of these phases, their proposed realizations in solid-state systems, and recent experiments on candidate materials as well as their relation to other states of matter are reviewed.

Weyl and Dirac semimetals in three-dimensional solids

This article reviews the physics of high-temperature superconductors from the point of view of the doping of a Mott insulator. The basic electronic structure of cuprates is reviewed, emphasizing the physics of strong correlation and establishing the model of a doped Mott insulator as a starting point. A variety of experiments are discussed, focusing on the region of the phase diagram close to the Mott insulator (the underdoped region) where the behavior is most anomalous. The normal state in this region exhibits pseudogap phenomenon. In contrast, the quasiparticles in the superconducting state are well defined and behave according to theory. This review introduces Anderson's idea of the resonating valence bond and argues that it gives a qualitative account of the data. The importance of phase fluctuations is discussed, leading to a theory of the transition temperature, which is driven by phase fluctuations and the thermal excitation of quasiparticles. However, an argument is made that phase fluctuations can only explain pseudogap phenomenology over a limited temperature range, and some additional physics is needed to explain the onset of singlet formation at very high temperatures. A description of the numerical method of the projected wave function is presented, which turns out to be a very useful technique for implementing the strong correlation constraint and leads to a number of predictions which are in agreement with experiments. The remainder of the paper deals with an analytic treatment of the $t\text{\ensuremath{-}}J$ model, with the goal of putting the resonating valence bond idea on a more formal footing. The slave boson is introduced to enforce the constraint againt double occupation and it is shown that the implementation of this local constraint leads naturally to gauge theories. This review follows the historical order by first examining the U(1) formulation of the gauge theory. Some inadequacies of this formulation for underdoping are discussed, leading to the SU(2) formulation. Here follows a rather thorough discussion of the role of gauge theory in describing the spin-liquid phase of the undoped Mott insulator. The difference between the high-energy gauge group in the formulation of the problem versus the low-energy gauge group, which is an emergent phenomenon, is emphasized. Several possible routes to deconfinement based on different emergent gauge groups are discussed, which leads to the physics of fractionalization and spin-charge separation. Next the extension of the SU(2) formulation to nonzero doping is described with a focus on a part of the mean-field phase diagram called the staggered flux liquid phase. It will be shown that inclusion of the gauge fluctuation provides a reasonable description of the pseudogap phase. It is emphasized that $d$-wave superconductivity can be considered as evolving from a stable U(1) spin liquid. These ideas are applied to the high-${T}_{c}$ cuprates, and their implications for the vortex structure and the phase diagram are discussed. A possible test of the topological structure of the pseudogap phase is described.

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Doping a Mott insulator: Physics of high-temperature superconductivity

Wetting phenomena are ubiquitous in nature and technology. A solid substrate exposed to the environment is almost invariably covered by a layer of fluid material. In this review, the surface forces that lead to wetting are considered, and the equilibrium surface coverage of a substrate in contact with a drop of liquid. Depending on the nature of the surface forces involved, different scenarios for wetting phase transitions are possible; recent progress allows us to relate the critical exponents directly to the nature of the surface forces which lead to the different wetting scenarios. Thermal fluctuation effects, which can be greatly enhanced for wetting of geometrically or chemically structured substrates, and are much stronger in colloidal suspensions, modify the adsorption singularities. Macroscopic descriptions and microscopic theories have been developed to understand and predict wetting behavior relevant to microfluidics and nanofluidics applications. Then the dynamics of wetting is examined. A drop, placed on a substrate which it wets, spreads out to form a film. Conversely, a nonwetted substrate previously covered by a film dewets upon an appropriate change of system parameters. The hydrodynamics of both wetting and dewetting is influenced by the presence of the three-phase contact line separating "wet" regions from those that are either dry or covered by a microscopic film only. Recent theoretical, experimental, and numerical progress in the description of moving contact line dynamics are reviewed, and its relation to the thermodynamics of wetting is explored. In addition, recent progress on rough surfaces is surveyed. The anchoring of contact lines and contact angle hysteresis are explored resulting from surface inhomogeneities. Further, new ways to mold wetting characteristics according to technological constraints are discussed, for example, the use of patterned surfaces, surfactants, or complex fluids.

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Wetting and Spreading

The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly become the method of choice for numerical studies of such systems. Its applications to the calculation of static, dynamic, and thermodynamic quantities in these systems are reviewed here. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and nonequilibrium statistical physics, and time-dependent phenomena is also discussed. This review additionally considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by the DMRG.

http://www.physics.udel.edu/~bnikolic/QTTG/NOTES/DMRG/SCHOLLWOCK=RMP_dmrg.pdf

The density-matrix renormalization group

A phenomenological theory of the ordered phase of short-range quantum Ising spin glass is developed in terms of droplet excitations, and presented in detail. These excitations have free energies that provene from an interplay between the classical excitation energies ϵL, with a broad distribution whose characteristic magnitude grows with length scale L as Lθ, and quantum tunneling rates ΓL, decreasing faster than exponentially with L. At temperature T = 0, the equal-time spatial correlations due to quantum fluctuations do not show the power-law decay that occurs for T > 0 due to thermal fluctuations, but ather an exponential decay of Ornstein-Zernicke type. At finite, but still very small T, there exists a crossover length scale L ∗ (T), set by ħΓ L∗(T)⋍ kBT, below which the droplets behave quantum-mechanically and above which the behaviour is essentially classical. As for the classical spin glass, only a small fraction of droplets are thermally active at low T; it is shown that many of the low-T static properties are dominated by the thermally active droplets at the crossover length L ∗ (T) . The uniform static linear susceptibility is found to diverge at T = 0 below the lower critical dimension, d l , and to be finite above d l . The static nonlinear susceptibility diverges in all dimensions, d. The zero temperature linear ac susceptibility χ(ω) is dominated by droplets at a length scale such that ω is of order the characteristic frequency of the quantum system. The behaviour near to the quantum critical point is discussed within a conventional scaling framework if it is approached at strictly zero temperature as well as from finite T. Implications of the existence of Griffiths singularities at the critical point and the disordered phase are pointed out: In the disordered phase, Griffiths singularities dominate the low-T specific heat and the long-time correlations.

Equilibrium behaviour of quantum Ising spin glass

We present a simplified strong-randomness renormalization group (RG) that captures some aspects of the many-body localization (MBL) phase transition in generic disordered one-dimensional systems. This RG can be formulated analytically and is mathematically equivalent to a domain coarsening model that has been previously solved. The critical fixed-point distribution and critical exponents (that satisfy the Chayes inequality) are thus obtained analytically or to numerical precision. This reproduces some, but not all, of the qualitative features of the MBL phase transition that are indicated by previous numerical work and approximate RG studies: our RG might serve as a ``zeroth-order'' approximation for future RG studies. One interesting feature that we highlight is that the rare Griffiths regions are fractal. For thermal Griffiths regions within the MBL phase, this feature might be qualitatively correctly captured by our RG. If this is correct beyond our approximations, then these Griffiths effects are stronger than has been previously assumed.

Many-body localization phase transition: A simplified strong-randomness approximate renormalization group

MAGNETISM and superconductivity are manifestations of two different ordered states into which metals can condense at low temperatures In general these states are mutually exclusive1; they do not coexist at the same place in a sample The study of the interplay between these properties has recently been revitalized by the discovery2,3 of a class of compounds with formula RNi2B2C (where R is a rare-earth element) which are both antiferromagnetic and superconducting at sufficiently low temperature4 It has been suggested5 that magnetic and superconducting order can coexist in these materials on an atomic scale Here we use small-angle neutron scattering to study the structure of the superconducting vortex lattice in ErNi2B2C Our results show that the development of magnetic order causes the vortex lines to disorder and rotate away from the direction of the applied magnetic field This coupling of superconductivity and magnetism provides clear evidence for microscopic coexistence of magnetic and superconducting order, and indicates that magnetic superconductors may exhibit a range of unusual phenomena not observed in conventional superconductors

Microscopic coexistence of magnetism and superconductivity in ErNi2B2C

We examine the quantum phase transition at zero temperature between paramagnetic and spin-glass-ordered phases as the strength of a uniform transverse field, \ensuremath{\Gamma}, is varied. At the critical point, ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$, the spin autocorrelation function decays with time, t, as ${\mathit{t}}^{\mathrm{\ensuremath{-}}2}$. As the transition is approached from the paramagnetic phase the nonlinear susceptibility diverges as {\ensuremath{\Vert}ln(\ensuremath{\Gamma}-${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$)\ensuremath{\Vert}/(\ensuremath{\Gamma}-${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$)${\mathrm{}}}^{1/2}$.

Zero-temperature critical behavior of the infinite-range quantum Ising spin glass.

It is the conventional wisdom that the correlation length for the XY model with linear damping should asymptotically grow diffusively as the square root of time after a quench into the ordered phase. This implies that the defect density \ensuremath{\rho} should decay with time as \ensuremath{\rho}\ensuremath{\propto}${\mathit{t}}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$ with the scaling exponent \ensuremath{\nu}=1. We present evidence, by numerically integrating the equations of motion for a two-dimensional XY model, for a logarithmic correction to this scaling which makes it difficult to reach the asymptotic regime \ensuremath{\nu}=-d(ln\ensuremath{\rho})/d(lnt)=1. Even after the defect density has decayed by three orders of magnitude \ensuremath{\nu}=0.91, which still deviates by 10% from the asymptotic value.

David A. Huse

Papers

Equilibrium behaviour of quantum Ising spin glass

Many-body localization phase transition: A simplified strong-randomness approximate renormalization group

Microscopic coexistence of magnetism and superconductivity in ErNi2B2C

Zero-temperature critical behavior of the infinite-range quantum Ising spin glass.

Coarsening dynamics of the XY model