Magnets with random uniaxial anisotropy: Thermodynamic properties in the large-N limit

We show that the random-axis model lends itself to a systematic large-N calculation. The model shows different behavior below and above four dimensions. The equation of state is derived and discussed in terms of “Arrott” plots. Higher-order terms in the disorder, when summed, have a crucial effect on the susceptibility which is found to be finite below four dimensions (and above four dimensions for strong disorder). A spin-glass to paramagnetic phase transition is characterized by the vanishing of the Edwards-Anderson order parameter, which differs from zero in the spin-glass phase. A cusp in the specific-heat and susceptibility is seen across the transition. The cross-over exponent and other exponents of interest are calculated. Above four dimensions a third phase appears for weak disorder and low-temperature ferromagnetic in nature. The transverse and longitudinal susceptibilities are discussed. Whereas the ferromagnetic transition is characterized by mean-field exponents, the ferromagnetic to spin-glass exponents are equal to their counterparts in the non-random system in d ? 2 dimensions. This is shown to originate from an effective random field proportional to the EA order parameter. The flow equations in the large-N limit are also discussed.     

A phase cell cluster expansion for Euclidean field theories

We adapt the cluster expansion first used to treat infrared problems for lattice models (a mass zero cluster expansion) to the usual field theory situation. The field is expanded in terms of special block spin functions and the cluster expansion given in terms of the expansion coefficients (phase cell variables); the cluster expansion expresses correlation functions in terms of contributions from finite coupled subsets of these variables. Most of the present work is carried through in d space time dimensions (for φ₂⁴ the details of the cluster expansion are pursued and convergence is proven). Thus most of the results in the present work will apply to a treatment of φ₃⁴ to which we hope to return in a succeeding paper. Of particular interest in this paper is a substitute for the stability of the vacuum bound appropriate to this cluster expansion (for d = 2 and d = 3), and a new method for performing estimates with tree graphs. The phase cell cluster expansions have the renormalization group incorporated intimately into their structure. We hope they will be useful ultimately in treating four dimensional field theories.     

Infrared Bound and Mean-Field Behaviour in the Quantum Ising Model

We prove an infrared bound for the transverse field Ising model. This bound is stronger than the previously known infrared bound for the model, and allows us to investigate mean-field behaviour. As an application we show that the critical exponent γ for the susceptibility attains its mean-field value γ = 1 in dimension at least 4 (positive temperature), respectively 3 (ground state), with logarithmic corrections in the boundary cases.     

Mean-field renormalization group for the q-state clock spin-glass model

The mean-field renormalization group is used to study the phase diagrams of a d-dimensional q-state clock spin-glass model. We found, for q = 3 clock, the transition from paramagnet to spin glass is an isotropic spin-glass phase, but for q = 4 clock, the transition from paramagnet to spin glass is an anisotropic spin-glass phase. However, for q ⩾ 5 clock, the result of anisotropic spin-glass phase depends on the temperature and the distribution of random coupling. While the coordinate number approaches infinity, the critical temperature evaluated by the mean-field renormalization group method is equal to that by the replica method.     

On the renormalization of the Kardar-Parisi-Zhang equation

The Kardar-Parisi-Zhang (KPZ) equation of nonlinear stochastic growth in d dimensions is studied using the mapping onto a system of directed polymers in a quenched random medium. The polymer problem is renormalized exactly in a minimally subtracted perturbation expansion about d = 2. For the KPZ roughening transition in dimensions d > 2, this renormalization group yields the dynamic exponent z = 2 and the roughness exponent χ = 0, which are exact to all orders in ε ≡ (2 − d)/2. The expansion becomes singular in d = 4. If this singularity persists in the strong-coupling phase, it indicates that d = 4 is the upper critical dimension of the KPZ equation. Further implications of this perturbation theory for the strong-coupling phase are discussed. In particular, it is shown that the correlation functions and the coupling constant defined in minimal subtraction develop an essential singularity at the strong-coupling fixed point.     

Quantum effects in the critical properties of the transverse Ising model for 2 < d ⩽ 3

The crossover behaviour of a d-dimensional (2<d?3) Ising model in a transverse field Г is investigated near the multicritical point $\text{[Г, T] = [Г}{}_{\mathrm{c}}\text{(0), 0].}$ A renormalization scheme which removes divergences in the zero-temperature limit is presented. The crossover exponent and scaling function for the longitudinal susceptibility are found.     

Localization in the ground state of the ising model with a random transverse field

We study the zero-temperature behavior of the Ising model in the presence of a random transverse field. The Hamiltonian is given by $$H = - J\sum\limits_{\left\langle {x,y} \right\rangle } {\sigma _3 (x)\sigma _3 (y) - \sum\limits_x {h(x)\sigma _1 (x)} } $$ whereJ>0,x,y∈Z ^d, σ₁, σ₃ are the usual Pauli spin 1/2 matrices, andh={h(x),x∈Z ^d} are independent identically distributed random variables. We consider the ground state correlation function 〈σ₃(x)σ₃(y)〉 and prove:

1. Letd be arbitrary. For anym>0 andJ sufficiently small we have, for almost every choice of the random transverse fieldh and everyx∈Z ^d, that $$\left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle \leqq C_{x,h} e^{ - m\left| {x - y} \right|} $$ for ally∈Z ^d withC _{x h} <∞.
2. Letd≧2. IfJ is sufficiently large, then, for almost every choice of the random transverse fieldh, the model exhibits long range order, i.e., $$\mathop {\overline {\lim } }\limits_{\left| y \right| \to \infty } \left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle > 0$$ for anyx∈Z ^d.

     

The quantum transverse spin-2 Ising model with a bimodal random-field in the pair approximation

In this paper, we have investigated the bimodal random-field spin-2 Ising system in a transverse field by combining the pair approximation with the discretized path-integral representation. The exact equations for the second-order phase transition lines and tricritical points are obtained in terms of the random field H, the transverse field G and the coordination number z. It is found that there are some critical values for H and G where the tricritical points disappear for given z. We have also observed that the system presents reentrant behavior which may be caused by the quantum effects and randomness. The phase diagram with respect to the random field and the second-order phase transition temperature are studied extensively for given values of the transverse field and the coordination number.     

The large-N phase transition in the U(N) lattice gauge theory

We present an outline for a proof (the precise details of which will be presented in a follow-up paper) of a large-N phase transition in dimensions greater than two. The critical couplings are calculated in d=3 and d=4 and are found to be β=0.44 and β=0.40, respectively.     

Theory of Anderson localization in weak magnetic fields

We derive self-consistency equations determining the transverse dynamical conductivity for the case of Wegner's local gauge invariant model in a weak magnetic field B. The solution in the critical regime connected with Anderson localization is given for dimensionalities d = 2, 3. In d = 2 the self-consistency equations generate a logarithmic singularity in second order in the coupling constant. This is shown to be in agreement with the loop expansion and yields localization for arbitrarily weak coupling. In d = 3 there is a metal-insulator transition. In its vicinity the self-consistency equations reduce to a two-parameter scaling law, which is consistent with the results of Khmelnitskii and Larkin.     

Nontrivial critical behaviour in a lattice model of random surfaces

A model of “planar random surfaces without spikes” on hypercubical lattices was introduced some years ago as a discretization of quantum string theory. We review some general properties of this model and present results from a Monte Carlo study of its critical behaviour in d = 4, 8 and 10 dimensions. In d = 4 dimensions we find a Hausdorff dimension d_H ≈ 4 and an anomalous dimensions η ≈ 1. These critical exponents imply a deviation from mean field theory in contrast to other lattice random surface models. Furthermore, we find evidence for mean field behaviour in 8 and 10 dimensions, indicating an upper critical dimension d_c^u ⩽ 8.     

Linear nonreciprocal dichroism in boracite Co3B7O13I

Linear nonreciprocal dichroism that is odd in magnetic field B was observed in the transverse geometry and studied for cubic (symmetry class T _d ) phase of boracite Co₃B₇O₁₃I. Nonreciprocal dichroism was observed in the range of absorption bands corresponding to the transitions of the Co²⁺ ion in the energy range ΔE=1.2–3.2 eV. The sign and magnitude of nonreciprocal dichroism depend on the mutual orientation of the magnetic field and crystallographic axes. Nonreciprocal dichroism refers to the phenomena of magnetically induced spatial dispersion, and its anisotropy is typical for the manifestation of the second-order magnetoelectric susceptibility in the optical range.     

Measuring the interface tension when the electroweak phase transition becomes weak

We measure the interface tension near the phase transition endpoint of the 3d SU(2)-Higgs model. The tunnel correlation length method is used and compared to other approaches. A modified scaling behaviour for the mass gap as function of the transverse area is proposed.     

Diffusion in a random medium: A renormalization group approach

We investigate through a continuous random diffusion equation the long-distance properties of the general non-symmetric hopping model. The lower and upper critical dimensionalities are d = 1 and d = 2 respectively. A renormalization group analysis shows that the velocity and the diffusion constant obey scaling laws with non-classical exponents, which are computed to first order in ε = 2 ? d. Similar scaling laws, based on heuristic arguments, are conjectured for the AC conductivity.     

Disorder-induced quantum-Griffiths-like features from a non-conventional renormalization group analysis near four dimensions

We investigate the role played by symmetry conserving quenched disorder on quantum criticality of a variety of d-dimensional systems with a continuous symmetry order parameter. We employ a non-standard procedure which combines a preliminary reduction to an effective classical random problem and a successive conventional renormalization group treatment. Solving the effective flow equations to first order in ε=4−d and then restoring the original coupling parameters, for d<4 we find a quantum critical point scenario exhibiting unusual features, which remind us of some predictions of the quantum Griffiths phase model.     

Instability of Heisenberg-spin-glass-phase transitions

It is pointed out that there should be no stable phase transitions for XY- and Heisenberg spin glasses with d ? 4 dimensions, and for Ising spin glasses with d ? 2, in the presence of arbitrarily small random magnetic fields. In the absence of such fields the critical dimensions are 2 and 1, respectively.     

Dynamics of the random field Ising model

Dynamics of the kinetic Ising model in the presence of static random fields is investigated using a self-consistent method. It is shown that if the interface fluctuations of the low temperature phase are small the system at low temperatures stays in a state without long range order. For this state the spin correlation function 〈S_q(t)S_?q(O)> averaged over all configurations of random fields decays exponentially in time with a single wavevector dependent relaxation time which is finite at the transition temperature T₀ and remains very long below T₀. In the mean field approximation the correlation time at the magnetic Bragg peak and at T₀ scales with the magnitude of the random field as τ∞h^?z_h with z_h = 1 for d = 2 and $\text{z}{}_{\mathrm{h}}\text{=}\text{4}\text{3}$ for d = 3, respectively.     

On the susceptibility exponent of random anisotropy magnets

The temperature dependence of the non-linear susceptibility ≈₂(T) of random anisotropy magnets in the Ising limit (speromagnets) is calculated for temperatures above the freezing temperature T_f within the framework of the correlated molecular field theory. For the effective susceptibility exponent λ_s(T) = (T?T_f)≈₂d^-1_≈2/dT a non-monotonic temperature dependence is found as for the case of spin glasses. This must be taken into account in order to obtain reliable values for the critical susceptibility exponent from experimental data.