Gauge Theory for Quantum Spin Glasses (original) (raw)

Gauge models for spin-glasses

Physical Review B

A Landau-Ginzburg description of a spin-glass which incorporates naturally the concept of "frustration, " or the incompatibility of different local stable spin configurations in neighboring regions, is presented. For a planar spin, the effective Hamiltonian has a form analogous to that of the Landau-Ginzburg functional for a superconductor in a magnetic field, except that the role of the vector potential is taken by a quenched random variable Q(x) which represents the wave vector of the spin-density wave of minimum local free energy. The model is thus a simple transcription to a Landau-Ginzburg picture of the basic notion of a spinglass as a material whose properties are determined by competition between ferromagnetic and antiferromagnetic interactions. The probability distribution of Q(x) is chosen not to depend on Q(x) directly (in order not to favor any particular value of Q), but to be Gaussian in the curl of Q(x). The variance f of this distribution, the mean-square vorticity in Q(z), is a measure of the degree of frustration. [Any longitudinal part of Q(x) is gauged away by rotating the local spin axes appropriately. ] For a classical vector (Heisenberg) spin system, the analogous description is a Hamiltonian of O(3) Yang-Mills form, again with the gauge random variable. Two calculations are presented. The first tests the stability of the f = 0 theory (thermodynamically identical to an ordinary ferromagnet) against the introduction of a small amount of frustration. The result is that the f = 0 fixed point is unstable, and no new fixed point (of order 4d) appears. Thus the spin-glass transition does not appear to be related to any normal sort of critical point with a particular local-spin-density configuration as a "hidden" order parameter. The second is a mean-field analysis of a transition to a state characterized by an Edwards-Anderson order parameter; its qualitative features are similar to those of mean-field theories for other models for spin-glasses. The conditions for the thermodynamic stability of such a state remain unknown,

A remark on gauge symmetries in Ising spin glasses

Probability Theory and Related Fields, 1999

This paper studies a particular line in the parameter space of the FK random interaction random cluster model for spin glasses following Katsura ([K]) and Mazza ([M]). We show that, after averaging over the random couplings, the occupied FK bonds have exactly a Bernoulli distribution. Comparison with explicit calculations on trees confirms the marginal role of FK percolation in determining phase transitions.

On the connection between spin glasses and gauge field theories

Physics Reports, 1980

A simple connection between Ising spin glasses and the Z 2 lattice gauge theory, at negative plaquette temperatures, is presented. It is first shown that annealed models give useful lower bounds on the free energy and ground-state energy of spin glasses. However, they have unphysical low temperature properties (e.g. a negative entropy), which are related to a temperature dependence of the frustration. A restricted annealing scheme is presented which remedies this deficiency through the introduction of a pure gauge coupling counterterm. The possible phase diagrams of the lattice gauge system and their relevance to spin glass transitions are discussed.

On a classical spin glass model

Zeitschrift f�r Physik B Condensed Matter, 1983

A simple, exactly soluble, model of a spin-glass with weakly correlated disorder is presented. It includes both randomness and frustration, but its solution can be obtained without replicas. As the temperature T is lowered, the spin-glass phase is reached via an equilibrium phase transition at T--T I. The spin-glass magnetization exhibits a distinct S-shape character, which is indicative of a field-induced transition to a state of higher magnetization above a certain threshold field. For suitable probability distributions of the exchange interactions. (a) A mixed phase is found where spin-glass and ferromagnetism coexist. (b) The zero-field susceptibility has a flat plateau for 0_<T_< T~ and a Curie-Weiss behaviour for T > T I. (c) At low temperatures the magnetic specific heat is linearly dependent on the temperature. The physical origin of the dependence upon the probability distributions is explained, and a careful analysis of the ground state structure is given.

Non-equilibrium Relations for Spin Glasses with Gauge Symmetry

2010

We study the applications of non-equilibrium relations such as the Jarzynski equality and fluctuation theorem to spin glasses with gauge symmetry. It is shown that the exponentiated free-energy difference appearing in the Jarzynski equality reduces to a simple analytic function written explicitly in terms of the initial and final temperatures if the temperature satisfies a certain condition related to gauge symmetry. This result is used to derive a lower bound on the work done during the non-equilibrium process of temperature change. We also prove identities relating equilibrium and non-equilibrium quantities. These identities suggest a method to evaluate equilibrium quantities from non-equilibrium computations, which may be useful to avoid the problem of slow relaxation in spin glasses.

A probabilistic approach to the models of spin glasses

Journal of Statistical Physics, 1983

Introducing the notions of quenched and annealed probability measures, a systematic study of some problems in the description of spin glasses is attempted. Inequalities and variational principles for the free energies are derived. The absence of spontaneous breakdown of the gauge symmetry is discussed and some high-temperature properties are studied. Examples of annealed models with more than one phase transition are shown.

Some considerations of finite-dimensional spin glasses

Journal of Physics A: Mathematical and Theoretical, 2008

In talk I will review the theoretical results that have been obtained for spin glasses, paying a particular attention to finite dimensional spin glasses. I will concentrate my attention on the formulation of the mean field approach and on its numerical and experimental verifications. I will mainly considered equilibrium properties at zero magnetic field, where the situation is clear and it should be not controversial. I will present the various hypothesis at the basis of the theory and I will discuss their physical status.

Quantum field theory of metallic spin glasses

Physical Review B, 1995

We introduce an effective field theory for the vicinity of a zero temperature quantum transition between a metallic spin glass ("spin density glass") and a metallic quantum paramagnet. Following a mean field analysis, we perform a perturbative renormalization-group study and find that the critical properties are dominated by static disorder-induced fluctuations, and that dynamic quantum-mechanical effects are dangerously irrelevant. A Gaussian fixed point is stable for a finite range of couplings for spatial dimensionality d > 8, but disorder effects always lead to runaway flows to strong coupling for d ≤ 8. Scaling hypotheses for a static strong-coupling critical field theory are proposed. The non-linear susceptibility has an anomalously weak singularity at such a critical point. Although motivated by a perturbative study of metallic spin glasses, the scaling hypotheses are more general, and could apply to other quantum spin glass to paramagnet transitions.

Dynamical Response of Quantum Spin-Glass Models at T=0

Physical Review Letters, 2001

We study the behavior of two archetypal quantum spin glasses at T 0 by exact diagonalization techniques: the random Ising model in a transverse field and the random Heisenberg model. The behavior of the dynamical spin response is obtained in the spin-glass ordered phase. In both models it is gapless and has the general form x 00 ͑v͒ qd͑v͒ 1 x 00 reg ͑v͒, with x 00 reg ͑v͒ ϳ v for the Ising and x 00 reg ͑v͒ ϳ const for the Heisenberg, at low frequencies. The method provides new insight to the physical nature of the low-lying excitations.

Frustration and ground-state degeneracy in spin glasses

Physical Review B, 1977

The problem of an Ising model with random nearest-neighbor interactions is reformulated to make manifest Toulouse's recent suggestion that a broken "lattice gauge symmetry" is responsible for the unusual properties of spin glasses. Exact upper and lower bounds on the ground-state energy for models in which the interactions are of constant magnitude but fluctuating sign are obtained, and used to place restrictions on possible geometries of the unsatisfied interactions which must be present in the ground state. Proposed analogies between the ferromagnetspin-glass phase boundary at zero temperature and a percolation threshold for the "strings" of unsatisfied bonds are reviewed in the light of this analysis. Monte Carlo simulations show that the upper bound resulting from a "one-dimensional approximation" to the spin-glass ground-state energy is reasonably close to the actual result. The transition between spin glass and ferromagnet at 0 K appears to be weakly first order in these models. The entropy of the ground state is obtained from the temperature dependence of the internal energy, and compared with the density of free spins at very low temperatures. For a two-dimensional spin glass in which half the bonds are antiferromagnetic, S(0)-0.099 k~; for the analogous three-dimensional spin glass the result is S(0)-0.062 k~. Monte Carlo kinetic simulations are reported which demonstrate the existence and stability of a fieldcooled moment in the spin-glass ground state.