The Duality Transformation for an n-Species Ising Model Interacting with Z2 Gauge Fields on a Random Surface

The duality transformation is carried out for an n-species Ising spin system interacting with Z₂ gauge fields. For n > 1, we find that the dual model has topological terms when the surface is topologically nontrivial. The plaquette interaction of the gauge fields is dual to an n-spin coupling in the dual model.     

Analytic approximations to Hamiltonian lattice field theories (II). The weak-coupling behavior of U(1) theories

It is shown that at weak coupling physical quantities in hamiltonian U(1) lattive gauge (or global symmetric) theories of arbitrary dimension are provided as expectation values in a d ? 1 dimensional lagrangian Z(2) gauge (or spin) theory with calculable long-range interactions.Confinement and the existence of a magnetic mass gap are equivalent to the existence of infinite-range plaquette-plaquette (or link-link) correlations in the spin field. The existence of infinite range correlations is simply related to the dimension of the lattice and the transformation property of the order parameter. As expected, only the d = 2+1 U(1) gauge theory confines electric charges at all non-vanishing coupling.     

Antisymmetric tensor gauge theories and non-linear σ-models

Antisymmetric tensor fields B_μν subject to the gauge transformation δB_μν = ?_μξ_ν ? ?_νξ_μ can describe spinless particles. We investigate the properties of field theories with a “non-abelian generalization” of this invariance. One class of such theories is equivalent to non-linear principal chiral σ-models, another to massive Yang-Mills theories. A supersymmetric analogue in 2 + 2 superspace is constructed and leads to the supersymmetric σ-model defined on a general riemannian manifold.     

Spectral Duality Between Heisenberg Chain and Gaudin Model

In our recent paper we described relationships between integrable systems inspired by the AGT conjecture. On the gauge theory side an integrable spin chain naturally emerges while on the conformal field theory side one obtains some special reduced Gaudin model. Two types of integrable systems were shown to be related by the spectral duality. In this paper we extend the spectral duality to the case of higher spin chains. It is proved that the N-site GL _k Heisenberg chain is dual to the special reduced k + 2-points gl _N Gaudin model. Moreover, we construct an explicit Poisson map between the models at the classical level by performing the Dirac reduction procedure and applying the AHH duality transformation.     

Operator oriented optimization of block spin transformations

We propose a generally applicable method to optimize block spin transformations. Working in the sector of even operators the renormalization transformation is adjusted such that the resulting renormalized trajectory is close to the trajectory of a simple few-parameter hamiltonian. The idea is tested within the d = 2 Ising model and leads to clear improvements in the determination of the critical exponents.     

Variational approach for the N-state spin and gauge Potts model

A hamiltonian variational treatment is applied both to the spin Potts model and to its gauge version for any number of states N and spatial dimensions d?2. Regarding the former we reproduce the correct critical coupling and latent heat for not too low N and d. For the latter, our approach turns the gauge theory into an equivalent d-dimensional classical spin model, which evaluated for d + 1 = 4 gives results in agreement with 1/N expansions.     

Effects of finite size and shape on scattering and thermodynamic quantities at 2D Ising critical points

Using recent results by Cardy based on the conformal invariance of critical correlation functions we calculate universal results for scattering functions S(k), susceptibilities, correlation lengths and specific heat correction terms for finite Ising systems in two dimensions with circular and rectangular shapes and free boundary conditions. Our results specify the effect of shape on these quantities at the critical point. In particular, the half-width and lineshape of the scattering function is found to be strongly influenced by geometry. For a circle, S(k) follows the infinite system behavior 1/k^2−η, η = 0.25 only for very large k. For a substantial range of intermediate k values it is well represented by 1/k^2−η_app, with an “apparent” exponent η_app. We also discuss the probable influence of end, edge and domain wall effects on the correlation lengths, susceptibilities and specific heat correction terms. The application of our results to experimental systems and other theoretical models is considered.     

Migdal renormalization group approach to deconfinement phase transition

The Migdal renormalization group approach is applied to a finite temperature lattice gauge theory. Imposing the periodic boundary condition in the timelike orientation, the phase structure of the finite temperature lattice gauge system with a gauge groupG in (d+1)-dimensional space is determined by two kinds of recursion equations, describing spacelike and timelike correlations, respectively. One is the recursion equation for ad-dimensional gauge system with the gauge groupG, and the other corresponds to ad-dimensional spin system for which the effective theory is described by the nearest neighbor interaction of the Wilson lines. Detailed phase structure is investigated for theSU(2) gauge theory in (3+1)-dimensional space. Deconfinement phase transition is obtained. Using the recursion equation for the three dimensional spin system of the Wilson lines, it is shown that the flow of the renormalization group trajectories leads to a phase transition of the three dimensional Ising model.     

Renormalization and scaling of a two-dimensional Ising system in a transverse field at Tc>0

A quantum generalization of the Niemeijer and van Leeuwen (N-vL) renormalization group transformation is constructed which allows to study the critical properties (at T_c>0) of a two-dimensional Ising system with a transverse field on a triangular lattice. Explicit calculations are performed in a second order cumulant expansion. Only one fixed point is found corresponding to the same Ising-like hamiltonian given by N-vl. The linearized transformation has a zero eigenvalue associated with Γ, the transverse field strength. The critical properties of the system are briefly discussed; in particular we show that the singular behaviour of the transverse susceptibility at Γ = 0, which turns out to be of the same kind as that of the energy density, is explained by the existence of such a zero eigenvalue. Our arguments suggest a natural extension of this result to three dimensions.     

Inverse freezing in the Hopfield fermionic Ising spin glass with a transverse magnetic field

The Hopfield fermionic Ising spin glass (HFISG) model in the presence of a magnetic transverse field Γ is used to study the inverse freezing transition. The mean field solution of this model allows introducing a parameter a that controls the frustration level. Particularly, in the present fermionic formalism, the chemical potential μ and the Γ provide a magnetic dilution and quantum spin flip mechanism, respectively. Within the one step replica symmetry solution and the static approximation, the results show that the reentrant transition between the spin glass and the paramagnetic phases, which is related to the inverse freezing for a certain range of μ, is gradually suppressed when the level of frustration a is decreased. Nevertheless, the quantum fluctuations caused by Γ can destroy this inverse freezing for any value of a.     

Generalized roughening transition and its effect on the string tension

The well-understood roughening transition of an interface in the d = 3 Ising model implies an essential singularity in the string tension of the dual Z₂ gauge model. The roughening transition corresponds to the delocalization of the string due to strong long-wavelength fluctuations, and this reformulation can be generalized to other gauge groups and to d = 4 also. It is not a deconfining transition - it is expected to occur deep in the confining region - but its presence would raise serious questions about the continuation of strong coupling expansions of the tension beyond this point. In this paper predictions on the roughening transition are confronted with the available information on the string tension for different gauge groups in three dimensions.     

Spectral self-similarity,one-dimensional Ising chains and random matrices

Partition functions of one-dimensional Ising chains with specific long distance exchange between N spins are connected to the N-soliton τ-functions of the Korteweg-de Vries (KdV) and B-type Kadomtsev-Petviashvili (BKP) integrable equations. The condition of translational invariance of the spin lattice selects infinite-soliton solutions with soliton amplitudes forming a number of geometric progressions. The KdV equation generates a spin chain with exponentially decaying antiferromagnetic exchange. The BKP case is richer. It comprises both ferromagnets and anti ferromagnets and, as a special case, includes an exchange decaying as 1/(i − j)² for large |i − j|. The corresponding partition functions are calculated exactly for a homogeneous magnetic field and some fixed values of the temperature. The connection between these Ising chains and random matrix models is considered as well. A short account of the basic ideas underlying the present work has been published in JETP Lett. 66 (1997) 789.     

Finite-size crossover in systems with slab geometry

A renormalization group study of the finite-size (dimensional) crossover is carried out with the help pf ε = 4 − d and ε₀ = 3 − d expansion techniques. The finite-size crossover and the invariance relation for the length scale transformation are proven up to the two-loop approximation. The formal equivalence between the finite-size crossover in classical systems and the quantum-to-classical dimensional crossover in certain quantum statistical models is emphasized and exploited. The finite-size corrections to the fluctuation shift of the critical temperature and the width of the critical region are investigated. It is shown that the shift exponent λ describing the fractional rounding of the critical temperature obeys the relation λ = D − 2, where D is the dimensionality of the system.     

Electric-magnetic duality in non-abelian gauge theories

The duality transformation of the vacuum expectation value of the operator which creates magnetic vortices (the 't Hooft loop operator in the Higgs phase), is performed in the radial gauge (x_uA_u^a(x) = 0). It is found that in the weak coupling region (small g) of a pure Yang-Mills theory the dual operator creates electric vortices whose strength is $\text{1}\text{g}$. The theory is self-dual in this region, and the effective coupling of the dual Lagrangian is $\text{1}\text{g}$. (It is self-dual also in the extreme strong coupling region.) Thus the above duality transformation reduces to electric-magnetic duality where the electric field in the 't Hooft loop operators transforms into a magnetic field in the dual operator. In a spontaneously broken gauge theory these results are valid only within the region where the vortices (or the monopoles) are concentrated, or in directions of the algebra space of unbroken symmetry, as self-duality holds only for this subset of fields. Noting that the 't Hooft loop operator project into the subspace of these field configurations we find that it is an electric-magnetic duality for the spontaneously broken theory as well. In the strong coupling region a strong coupling expansion in powers $\text{1}\text{g}$ is suggested.     

Hard pions and axial meson exchange current effects in negative muon capture in deuterium

The contribution of the axial meson exchange current effects to the doublet transition rate in the reaction μ^? +d → 2n+ ν_μ is calculated by using the minimal, chiral and approximately gauge invariant Lagrangian model for the A₁ρπ system. The contribution from the ρ-π weak decay process current usually considered is found to be nearly cancelled by that from the A₁ pole graph which is prescribed by the underlying invariance principles. Correct treatment of the N¹ propagator in the N¹ excitation current of the pion range leads to ≈ 30 % suppression of the N¹ effect.     

Replica symmetry breaking in a fermionic cluster spin glass model in a transverse field

We analyze a fermionic Ising spin glass model in the presence of a transverse magnetic field Γ within a cluster mean field theory. The model considers a Sherrington-Kirkpatrick type interaction between magnetic moments of clusters with a ferromagnetic intra-cluster coupling J₀. The spin site operators are written as a bilinear combination of fermionic operators. In these quantum spin glass model, the inter-cluster disorder is treated by using a framework of one-step replica symmetry breaking within the static approximation. The effective intra-cluster interaction is then computed by means of an exact diagonalization method. Results for several cluster size n_s, values of Γ and J₀ are presented. For instance, the specific heat shows a broad maximum (for n_s>1) at a temperature above the freezing temperature T_f, which is characterized by the inter-cluster replica symmetry breaking. Phase diagrams T versus Γ show that the critical temperature T_f(Γ) decreases for any value of n_s when Γ increases until it reaches a quantum critical point at some value of Γ_c.     

On the connection between spin glasses and gauge field theories

A simple connection between Ising spin glasses and the Z₂ 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.     

Sufficient condition for confinement of static quarks by a vortex condensation mechanism

We derive a sufficient condition for confinement of static quarks by a vortex condensation mechanism. It admits vortices that are thick at all times at the cost of constraining them to a finite volume Γ_i whose complement is not simply connected. The confining potential V(L) is estimated in terms of the change of free energy of a system enclosed in Γ_i which is induced by a change in vorticity (= singular gauge transformation applied to boundary conditions on ?Γ_i.) For Abelian gauge theories in 3 dimensions the confining Coulomb potential is reproduced as a lower bound.     

Dirac dynamics on stochastic phase spaces for spin 1/2 particles

The Foldy-Wouthuysen representation of the dynamics of a free spin $\text{1}\text{2}$ particle is formulated in a Hilbert space $\text{H}$(Γ) of spinor-valued functions over Γ-space. $\text{H}$(Γ) carries a reducible Wigner-type representation of the Poincaré group. The transition to the Dirac representation in a new bispinor Hilbert space $\text{K}$(Γ) is effected by means of a generalized inverse Foldy-Wouthuysen transformation. Explicit expressions are derived for the resolution generators η of invariant subspaces $\text{K}$±(Γ_η) carrying irreducible representations of the resulting representations of the Poincaré group. The formalism is recast in a manifestly covariant form and the Dirac equation on $\text{H}$(Γ_s) with minimal coupling to a four-potential is examined. It is shown that the resulting external field theory is gauge-invariant and relativistically covariant.