Stability and equilibrium states of infinite classical systems

We prove that any stationary state describing an infinite classical system which is stable under local perturbations (and possesses some strong time clustering properties) must satisfy the classical KMS condition. (This in turn implies, quite generally, that it is a Gibbs state.) Similar results have been proven previously for quantum systems by Haag et al. and for finite classical systems by Lebowitz et al.Supported by N.S.F. Grant MPS 71-03375 A03. Part of this work carried out at the Courant Institute where it was supported by N.S.F. Grant GP-37069X.Supported in part by AFOSR Grant #73-2430 and N.S.F. Grant MP S75-20638.Supported by N.S.F. Grant # GP33136X-2. Part of this work was carried out at the Institute for Advanced Study.     

Uniqueness of gibbs fields via cluster expansions

For the unbounded spin systems one cannot get cluster expansion if there exist large enough boundary values. A simple idea to avoid these difficulties is to prove that with probabilityp 1 when ^v there is a large subvolume of such that on all spin values do not exceed some fixed number. This gives a new method to prove uniqueness results for the unbounded spin systems generalizing some results of Refs. 1 and 2. The formulations of these results are in Section 1; the proofs are in Section 2.     

Self-avoiding walk in 5 or more dimensions

Using an expansion based on the renormalization group philosophy we prove that for aT step weakly self-avoiding random walk in five or more dimensions the variance of the endpoint is of orderT and the scaling limit is gaussian, asT.Dedicated to the memory of Kurt Symanzik whose profound contributions have guided and inspired usWork partially supported by N.S.F. Grant DMR 81-00417A. P. Sloan Foundation Fellow. Work partially supported by N.S.F. Grant MCS 82-02115     

The quantum ground state of a Heisenberg ferromagnet with an “easy-plane” type anisotropy

For the Heisenberg ferromagnet with the easy-plane type anisotropy the ground state energy and the magnetization are found with the help of perturbation theory supposing that the anisotropy energy is less than the exchange one. The study is carried out exactly without using any spin operators representation. Therefore, it is valid for a spin of any magnitude.     

Stability of motions near resonances in quasi-integrable Hamiltonian systems

Nekhoroshev's theorem on the stability of motions in quasi-integrable Hamiltonian systems is revisited. At variance with the proofs already available in the literature, we explicitly consider the case of weakly perturbed harmonic oscillators; furthermore we prove the confinement of orbits in resonant regions, in the general case of nonisochronous systems, by using the elementary idea of energy conservation instead of more complicated mechanisms. An application of Nekhoroshev's theorem to the study of perturbed motions inside resonances is also provided.Partially supported by Ministere della Pubblica Istruzione.Partially supported by Grant N.S.F. DMS 85-03333 and by Ministero della Pubblica Istruzione.     

Notes on recentU 4 theories of gravitation

Some of the proposed Lagrangians and their corresponding field equations for a gravitational theory based on a Riemann-Cartan space with metric-compatible connection (U ₄, theory) are compared and a new one is suggested. This Lagiangian, and that of P. von der Heyde and F. W. Hehl et al. are examined applying the Gordon-decomposition argument. Finally, Einstein's field equations with cosmological term are shown to be included in some sense, but the cosmological constant naturally has microphysical origin. To simplify notation, Cartan's calculus is used throughout.Work supported by Fonds zur Förderung der wissenschaftlichen Forschung in österreich, project No. 3666.Most of this work was carried out during the stay at the 6th Course of the International School of Cosmology and Gravitation at Erice, 1979.     

Metastable electronic states induced by nuclear decay and light

Radioactive atoms incorporated in insulating solid-state compounds create various kinds of chemical and physical after-effects upon nuclear disintegration. Mössbauer emission spectroscopy of⁵⁷Co-labelled coordination compounds has undoubtedly become the most informative technique to detect such after-effects like aliovalent charge and spin states of the nucleogenic iron atom resulting from the⁵⁷Co(EC)⁵⁷Fe decay, low energy excitations of crystal field and Zeeman states, linkage isomerism, radical formation with subsequent redox reactions, and others. We have extensively studied⁵⁷Co-labelled complexes with [Co^IIN₆] cores employing time-integral and time-resolved Mössbauer emission spectroscopy. In complexes of strong ligands fields (where the corresponding synthesized iron(II) complexes show low spin behaviour with¹A₁ ground slate) and in complexes of intermediate ligand field strength (where the corresponding iron(II) compounds exhibit thermal spin-crossover¹A₁ ⁵T₂) we have observed the population of metastable high spin states, which is strongly time- and temperature-dependent in the case of the strong-field complexes. Irradiation of iron(II) spin-crossover complexes with light also induces the formation of metastable high spin states, which are the same in nature as those resulting from decaying nuclei as an intramolecular light source. The mechanisms for light-induced excited spin state trapping (LIESST) and nuclear decay induced excited spin state trapping (NIESST) have been elucidated; they are strongly related to each other. In⁵⁷Co-labelled LiNbO₃ and other matrices, we have observed nonthermalized populations of the Zeeman sublevels of the ligand field ground state. These low energy excitations and the metastable ligand field states constitute the last stages of the slowing down process after the nuclear decay.     

Hidden symmetry breaking and the Haldane phase inS=1 quantum spin chains

We study the phase diagram ofS=1 antiferromagnetic chains with particular emphasis on the Haldane phase. The hidden symmetry breaking measured by the string order parameter of den Nijs and Rommelse can be transformed into an explicit breaking of aZ ₂×Z ₂ symmetry by a nonlocal unitary transformation of the chain. For a particular class of Hamiltonians which includes the usual Heisenberg Hamiltonian, we prove that the usual Néel order parameter is always less than or equal to the string order parameter. We give a general treatment of rigorous perturbation theory for the ground state of quantum spin systems which are small perturbations of diagonal Hamiltonians. We then extend this rigorous perturbation theory to a class of diagonally dominant Hamiltonians. Using this theory we prove the existence of the Haldane phase in an open subset of the parameter space of a particular class of Hamiltonians by showing that the string order parameter does not vanish and the hiddenZ ₂×Z ₂ symmetry is completely broken. While this open subset does not include the usual Heisenberg Hamiltonian, it does include models other than VBS models.     

Local Ward identities and the decay of correlations in ferromagnets

Using local Ward identities we prove a number of correlation inequalities forN-component, isotropically coupled, pair interacting ferromagnets; some for allN2 and some forN=2, 3, 4. These are used to prove a mass gap above the mean field temperature, for allN2. ForN=2, 3, 4 we prove an upper bound on a critical exponent, and a lower bound on the susceptability which diverges asm0.Research partially supported by US National Science Foundation under Grant PHY-7825390 A01.Research partially supported by US National Science Foundation under Grant MCS-78-01885.     

The classical limit of quantum spin systems

We derive a classical integral representation for the partition function,Z ^Q , of a quantum spin system. With it we can obtain upper and lower bounds to the quantum free energy (or ground state energy) in terms of two classical free energies (or ground state energies). These bounds permit us to prove that when the spin angular momentumJ (but after the thermodynamic limit) the quantum free energy (or ground state energy) is equal to the classical value. In normal cases, our inequality isZ ^C (J)Z ^Q (J)Z ^C (J+1).On leave from the Department of Mathematics, M.I.T., Cambridge, Mass. 02139, USA. Work partially supported by National Science Foundation Grant GP-31674X and by a Guggenheim Memorial Foundation Fellowship.     

Further extension of the Gauss-Hertz principle

The continuum form of the Gauss-Hertz principle is extended to include the time domain as well as space. The Schrödinger equation and general relativity are derived by this method. The equivalence of the principle is shown to that of the Hamiltonian method where the energy is the expression –[ ² +A·² A], with being the difference between the acceleration potential and potential energy density, andA being the difference between the vector potentials of the acceleration field and the force field. The goal of Hertz to demonstrate a third arrangement of the principles of mechanics...which starts with... time, space and mass has apparently been achieved for relativity and for quantum mechanics, in addition to those classical equations previously found.     

Classical and quantum mechanical systems of Toda-lattice type

Solutions to the classical periodic and non-periodic Toda lattice type Hamiltonian systems are expressed in terms of an Iwasawa-type factorization of a large Lie group. The scattering of these systems is determined in the non-periodic case. For the generalized periodic Toda lattices a generalization of Kostant's formula is obtained using standard representations of affine Lie groups.Research partially supported by NSF Grant MCS 83-01582Research partially supported by NSF Grant MCS 79-03153     

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.     

Spin-glass dynamics determined from muon spin relaxation and neutron spin echo measurements

Muon spin relaxation (SR) and neutron spin echo (NSE) measurements of magnetic ion correlation times and correlation functions in the spin glass systemsCuMn,AgMn, andAuFe are compared. It is found that theSR and NSE measurements are in excellent agreement both above and below the spin-glass freezing temperatures. The experimental results are compared to recent theories of spin-glass dynamics.We are grateful to D.L. Huber and R.E. Walstedt for stimulating discussions. This work was performed under the auspices of the U.S. Department of Energy, and was also supported by the U.S. National Science Foundation, Grant No. DMR-8115543.     

On alternative forms of the variational principle implying the generalized Thomas-Fermi theory

A gas of fermions in a static potential is investigated within the framework of the March-Murray perturbation scheme equivalent to a generalization of the Thomas-Fermi theory. The corresponding variational principle, due to Stoddart and March, is complemented by an alternative form. In contrast with the Stoddart-March formulation whose nature is Hamiltonian and which requires variations of the particle density in the total energy functional, the author's formulation, found to be of Lagrangian nature, does variations with respect to the potential function. Both the (alternative) quantum variational (many-body) principles are formulated self-consistently, with due attention to the Poisson equation of electrodynamics. They are considered first for zero temperature and afterwards for arbitrary temperatures. The functional considerations involved in the present paper are related with thermodynamics of quantum microcontinuum. An Appendix is added, to clarify the functional concepts by demonstrating them on some simple analogy. Therefore, a variational calculation of the ground state energy of ^– mesoatoms or mesoions is given. It leads mathematically to the well-known problem of calculating the ground 1 S (parastate) energy of two electrons in the helium atom.     

A phase cell approach to Yang-Mills theory

For the abelian Yang-Mills theory, a one-to-one correspondence is established between continuum gauge potentials and compatible lattice configurations on an infinite sequence of finer and finer lattices. The compatibility is given by a block spin transformation determining the configuration on a lattice in terms of the configuration on any finer lattice. Thus the configuration on any single lattice is not an approximation to the continuum field, but rather a subset of the variables describing the field.It is proven that the Wilson actions on the lattices monotonically increase to the continuum action as one passes to finer and finer lattices. Configurations that minimize the continuum action, subject to having the variables fixed on some lattice, are studied.This work was supported in part by the National Science Foundation under Grant No. PHY-85-02074     

Non-translation invariant Gibbs states with coexisting phases III: Analyticity properties

We consider the spatially inhomogeneous Gibbs states for the three dimensional Ising and Widom-Rowlinson models. We prove the analyticity inz=exp(–2J) for small |z| of the spin correlation functions of these Gibbs states and of the surface tension.Supported in part by N.S.F. Grant PHY 77-22302Supported by the Swiss National Foundation for Scientific Research     

Monte Carlo study of the critical behavior of pure and site-diluted Ising ferro-and ferrimagnets

Monte Carlo simulations are performed for pure and site-diluted Ising ferro- and ferrimagnets on a simple cubic lattice with up to 40³ sites and with impurity concentrationx. For the diluted ferromagnet (x=0.2) the exponent= 0.392±0.03 is definitely larger than the pure model value of=0.304±0.03. In contrast, for ferrimagnetic systems (x=0, 0.1, 0.2) the values appear to be independent ofx and within the error limits consistent with the value for the pure ferromagnet, possibly because the width of the asymptotic random critical regime (or of the crossover regime) is even smaller than in the case of ferromagnets.     

Exponential decay near resonance,without analyticity

We consider a quantum mechanical model which displays the behaviour associated with having a resonance or metastable state. The Hamiltonian depends on a parameter . When =0, there is an eigenstate ₀; when 0, ₀ dissolves into the continuous spectrum, showing approximate exponential decay. We prove this result without using dilatation analyticity. The model describes a two-state atom coupled to the quantized radiation field. The state space of the field is truncated, so that only the vacuum and one-photon states are included.This work was partially supported by NSF Grant DMS-8922941