Application of the renormalization group method to the s = 12XY model on the triangular lattice

The two-dimensional spin one half XY model may undergo an unusual type of phase transition. A formulation of the renormalization group approach of Niemeijer and van Leeuwen suitable for quantum-mechanical systems is presented. The method is applied to the spin one half XY model on the triangular lattice using a second order cumulant expansion based on division of the triangular lattice into seven spin cells. The resulting free energy curve is in excellent agreement with that obtained from the high temperature series expansion. An unstable fixed point is found but the corresponding critical temperature is not in good agreement with the series expansion value. Pathological values are found for the critical exponents.     

The anisotropic quantum antiferromagnet on the Sierpiński gasket: Ground state and thermodynamics

We investigate an antiferromagnetic s = 1/2 quantum spin system with anisotropic spin exchange on a fractal lattice, the Sierpiski gasket. We introduce a novel approximative numerical method, the configuration selective diagonalization (CSD) and apply this method to a the Sierpiski gasket with N = 42. Using this and other methods we calculate ground state energies, spin gap, spin-spin correlations and specific heat data and conclude that the s = 1/2 quantum antiferromagnet on the Sierpinski gasket shows a disordered magnetic ground state with on the Sierpinski gasket shows a disordered magnetic ground state with a very short correlation length of and an, albeit very small, spin gap. This conclusion holds for Heisenberg as well a for XY exchange.Received: 18 February 2004, Published online: 20 April 2004PACS: 75.10.-b General theory and models of magnetic ordering - 05.45.Df Fractals - 75.40.Mg Numerical simulation studies     

Quantum corrections for lattice gases

Classical lattice gases consisting of structureless particles (with spin) have been quantized by introducing a kinetic energy operator that produces nearest-neighbor hops. Systematic quantum corrections for the partition function and the particle distribution functions appear naturally as power series inX = ²/2ml ² ( ^–1 =k _B T,m is the mass,l is a distance related to lattice spacing). These corrections require knowledge of certain particle displacement probabilities in the corresponding classical lattice gases. Leading-order corrections have been derived in forms that should facilitate their use in computer simulation studies of lattice gases by the standard Monte Carlo method.     

Extremity of the disordered phase in the Ising model on the Bethe lattice

We prove that the disordered Gibbs distribution in the ferromagnetic Ising model on the Bethe lattice is extreme forTT _c ^SG , whereT _c ^SG is the critical temperature of the spin glass model on the Bethe lattice, and it is not extreme forT _c ^SG .     

Finite-size-scaling analyses of the chiral ordert in the Josephson-junction ladder with half a flux quantum per plaquette

Chiral order of the Josephson-junction ladder with half a flux quantum per plaquette is studied by means of the exact diagonalization method. We consider an extreme quantum limit where each superconductor grain (order parameter) is represented by S=1/2 spin. So far, the semi-classical case, where each spin reduces to a plane rotator, has been considered extensively. We found that in the case of S=1/2, owing to the strong quantum fluctuations, the chiral (vortex lattice) order becomes dissolved except in a region, where attractive intrachain and, to our surprise, repulsive interchain interactions both exist. On the contrary, for considerably wide range of parameters, the superconductor (XY) order is kept critical. The present results are regarded as a demonstration of the critical phase accompanying chiral-symmetry breaking predicted for frustrated XXZ chain field-theoretically. Received 20 February 2000     

On the critical dynamics of disordered spin systems

The dynamic critical behaviour of spin systems with quenched impurities, and of amorphous spin systems as characterized by the additional presence of random anisotropy directions, is studied by renormalization group methods to second order in=4–d. For the Halperin-Hohenberg-Ma model with purely relaxational dynamics it is concluded that in three dimensions (d=3) the critical slowing down should be enhanced by impurities for systems with Ising type statics, whereas there is no change forXY- and Heisenberg systems. For amorphous systems, however, the critical dynamics should change also in theXY- and Heisenberg cases. Furthermore, it is concluded that additional conserved, but noncritical modes become always statically decoupled from the order parameter for systems with impurities, but not for amorphous systems. Thus, for the impure system, the energy density mode and the asymmetric models of Halperin, Hohenberg and Siggia are ruled out. But the effects of dynamic coupling remain: Especially, the relationz=d/2 for the dynamic exponent of planar and isotropic antiferromagnets is modified for impure or amorphous systems.     

Spin and density fluctuations in disordered narrow-band superconductors

A tight-binding theory of strong-coupling superconductivity is formulated on the basis of a random lattice model with local electron-electron and contact-type electronphonon interactions. Functional-derivative technique yields a systematic deduction of the electron self-energy in terms of Coulomb repulsion, density and spin fluctuations. A novel procedure is proposed to average disordered Berk-Schrieffer-type equations. The phonon ( ^ph)-, Coulomb ^C-, and paramagnon ( ^m)-mediated coupling parameters entering the superconductingT _c are calculated for diffusive behaviour in dirty systems or alloys. The critical reduction ofT _c due to ^m differs from the logarithmic singularity for a clean sample.     

Natural Nonequilibrium States in Quantum Statistical Mechanics

A quantum spin system is discussed where a heat flow between infinite reservoirs takes place in a finite region. A time-dependent force may also be acting. Our analysis is based on a simple technical assumption concerning the time evolution of infinite quantum spin systems. This assumption, physically natural but currently proved for few specific systems only, says that quantum information diffuses in space-time in such a way that the time integral of the commutator of local observables converges: ⁰ _– dt [B, ^t A]<. In this setup one can define a natural nonequilibrium state. In the time-independent case, this nonequilibrium state retains some of the analyticity which characterizes KMS equilibrium states. A linear response formula is also obtained which remains true far from equilibrium. The formalism presented here does not cover situations where (for time-independent forces) the time-translation invariance and uniqueness of the natural nonequilibrium state are broken.     

Crossover from Coulomb-blockade to ohmic conduction in small tunnel junctions

We discuss the suppression of Coulomb effects in the low temperature conductanceg(T) of small tunnel junctions with increasing dissipation or bare conductanceg. The conductance is expressed in terms of the spin correlation fuction of a classical one dimensionalXY-model with ferromagnetic nearest neighbor plus inverse square interaction. It is shown that at low temperatures the conductance vanishes asymptotically likeT ² instead of exponentially. A Coulomb gap in the sense of a thermally activated contribution tog(T) is present only for bare conductances smaller thang _c 1. A simple model for the spin correlation functions is compared with experiments.     

Paramagnetic Proton Nuclear Spin Relaxation Theory of Low-Symmetry Complexes for Electron Spin Quantum NumberS=

A generalization of the modified Solomon–Bloembergen–Morgan (MSBM) equations has been derived in order to describe paramagnetic relaxation enhancement (PRE) of paramagnetic complexes characterized by both a transient (Δ^ZFS_t) and a static (Δ^ZFS_s) zero-field splitting (ZFS) interaction. The new theory includes the effects of static ZFS, hyperfine coupling, and angular dependence and is presented for the case of electron spin quantum numberS= , for example, Mn(II) and Fe(III) complexes. The model gives the difference from MSBM theory in terms of a correction term δ which is given in closed analytical form. The theory may be important in analyzing the PRE of proton spin–lattice relaxation dispersion measurements (NMRD profiles) of low-symmetry aqua–metal complexes which are likely to be formed upon transition metal ions associated with charged molecular surfaces of biomacromolecules. The theory has been implemented with a computer program which calculates solvent water protonT₁NMRD profiles using both MSBM and the new theory.     

Phase transitions in two-dimensional uniformly frustratedXY models. I. Antiferromagnetic model on a triangular lattice

A most popular model in the family of two-dimensional uniformly-frustratedXY models is the antiferromagnetic model on a triangular lattice [AFXY(t) model]. Its ground state is both continuously and twofold discretely degenerated. Different phase transitions possible in such systems are investigated. Relevant topological excitations are analyzed and a new class of such (vortices with a fractional number of circulation quanta) is discovered. Their role in determining the properties of the system proves itself essential. The characteristics of phase transitions related to breaking of discrete and continuous symmetries change. The phase diagram of the generalized AFXY(t) model is constructed. The results obtained are rederived in the representation of the Coulomb gas with half-integer charges, equivalent to the AFXY(t) model with the Berezinskii-Villain interaction.     

The random walk representation of classical spin systems and correlation inequalities

Ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion. By using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities. As an application of these inequalities we obtain a simple proof of the mass gap for the (⁴)₂ quantum field model. We also establish new upper bounds on critical temperatures.Partially supported by N.S.F. Grant No. 79-02490On leave from the University of VirginiaPartially supported by N.S.F. Grant No. DMR 81-00417     

New solvable lattice models in three dimensions

In this paper we establish a remarkable connection between two seemingly unrelated topics in the area of solvable lattice models. The first is the Zamolodchikov model, which is the only nontrivial model on a three-dimen-sional lattice so far solved. The second is the chiral Potts model on the square lattice and its generalization associated with theU _q(sl(n)) algebra, which is of current interest due to its connections with high-genus algebraic curves and with representations of quantum groups at roots of unity. We show that this last sl(n)-generalized chiral Potts model can be interpreted as a model on a threedimensional simple cubic lattice consisting ofn square-lattice layers with anN- valued (N2) spin at each site. Further, in theN=2 case this three-dimen-sional model reduces (after a modification of the boundary conditions) to the Zamolodchikov model we mentioned above.     

High-frequency sum rules for the quasi-one-dimensional quantum plasma dielectric tensor. II. Spin effect

We derive a high-frequency expansion for all elements of the quasi-one-dimensional quantum plasma dielectric tensor atT=0 K for quantum particles with spins. In addition to the known results for spinless case, we find that _S.5 ^Emphasis>/12 (k) and _S.5 ²³ (k) are the only frequency moments of the dielectric tensor with spin terms. Further, we find that there is no spin effect on quantum plasma dispersion for both ordinary and transverse modes propagating either along or across the external field.     

Low temperature phase diagrams for quantum perturbations of classical spin systems

We consider a quantum spin system with Hamiltonian $$H = H^{(0)} + \lambda V,$$ whereH ⁽⁰⁾ is diagonal in a basis ∣s〉=? _x ∣s _x 〉 which may be labeled by the configurationss={s_x} of a suitable classical spin system on ? ^d , $$H^{(0)} |s\rangle = H^{(0)} (s)|s\rangle .$$ We assume thatH ⁽⁰⁾(s) is a finite range Hamiltonian with finitely many ground states and a suitable Peierls condition for excitation, whileV is a finite range or exponentially decaying quantum perturbation. Mapping thed dimensional quantum system onto aclassical contour system on ad+1 dimensional lattice, we use standard Pirogov-Sinai theory to show that the low temperature phase diagram of the quantum spin system is a small perturbation of the zero temperature phase diagram of the classical HamiltonianH ⁽⁰⁾, provided λ is sufficiently small. Our method can be applied to bosonic systems without substantial change. The extension to fermionic systems will be discussed in a subsequent paper.     

The exact phase diagram of the quantumXY spin glass model in a transverse field

The infinite rangeXY spin glass model in a transverse field is investigated by means of the Trotter-Suzuki approach. The exact phase diagram is obtained showing that a spin glass transition takes place for non-zero values of the transverse field up to a critical value _c =1.440.01. The present numerically exact calculations are in good agreement with our previous approximate results and they clear remaining discrepancies from previous work.     

Generalized quantum spins,coherent states,and lieb inequalities

A mathematical generalization of the concept of quantum spin is constructed in which the role of the symmetry groupO ₃ is replaced byO _v (=2,3,4, ...). The notion of spin direction is replaced by a point on the manifold of oriented planes in ^v . The theory of coherent states is developed, and it is shown that the natural generalizations of Lieb's formulae connecting quantum spins and classical configuration space hold true. This leads to the Lieb inequalities [1] and with it to the limit theorems as the quantum spinl approaches infinity. The critical step in the proofs is the validity of the appropriate generalization of the Wigner-Eckart theorem.This paper is based largely on the Indiana University Ph. D. thesis of the first named author     

Quantum XY spin glass model with planar Dzyaloshinskii-Moriya interactions in longitudinal field

In the replica symmetric approximation and static limit in Matsubara “imaginary time”, the quantum XY spin glass model with planar Dzyaloshinskii-Moriya interaction in longitudinal field is investigated. Several thermodynamic quantities are calculated numerically as well as spin self-interaction and spin glass order parameter for spin S=1/2. It is shown that the entropy is not independent of the field. A crossover behavior of the specific heat depending on temperature is found. There is a deviation from the parabolic approximation, C/T=A+Bh ² . Received 11 March 1998     

Effect of the block-spin configuration on the location of βc in two-dimensional Ising modelsin two-dimensional Ising models

We consider the nearest neighbor Ising model on the 2D square lattice and divide the lattice into 2 by 2 blocks. Each block is assigned one spin value (1 or –1) and these block spin values are kept fixed. We then impose the majority rule and look at the effect on the phase transition that was present in the original unconstrained spin system. We find that for the checkerboard block-spin configuration, Monte Carlo simulations show that _c is close to 1, which, compared to the original nearest neighbor Ising _c = 0.44..., shows that the critical temperature has been reduced by more than one half. For none of the other 11 block-spin configurations that we have considered is there any indication of a phase transition in the constrained system of original spins.