Stability properties and the KMS condition

First we derive stability properties of KMS states and subsequently we derive the KMS condition from stability properties. New results include a convergent perturbation expansion for perturbed KMS states in terms of appropriate truncated functions and stability properties of ground states. Finally we extend the results of Haag, Kastler, Trych-Pohlmeyer by proving that stable states ofL ¹-asymptotically abelian systems which satisfy a weak three point cluster property are automatically KMS states. This last theorem gives an almost complete characterization of KMS states, ofL ¹-asymptotic abelian systems, by stability and cluster properties (a slight discrepancy can occur for infinite temperature states).Supported during this research by the Norwegian Research Council for Science and Humanities     

Thermal States in Conformal QFT. II

We continue the analysis of the set of locally normal KMS states w.r.t. the translation group for a local conformal net of von Neumann algebras on . In the first part we have proved the uniqueness of the KMS state on every completely rational net. In this second part, we exhibit several (non-rational) conformal nets which admit continuously many primary KMS states. We give a complete classification of the KMS states on the U(1)-current net and on the Virasoro net Vir₁ with the central charge c = 1, whilst for the Virasoro net Vir _c with c > 1 we exhibit a (possibly incomplete) list of continuously many primary KMS states. To this end, we provide a variation of the Araki-Haag-Kastler-Takesaki theorem within the locally normal system framework: if there is an inclusion of split nets and is the fixed point of w.r.t. a compact gauge group, then any locally normal, primary KMS state on extends to a locally normal, primary state on , KMS w.r.t. a perturbed translation. Concerning the non-local case, we show that the free Fermi model admits a unique KMS state.     

On the Structure of KMS States of Disordered Systems

We study the set of KMS states of spin systems with random interactions. This is done in the framework of operator algebras by investigating Connes and Borchers –invariants of W*–dynamical systems. In the case of KMS states exhibiting a property of invariance with respect to the spatial translations, some interesting properties emerge naturally. Such a situation covers KMS states obtained by infinite–volume limits of finite–volume Gibbs states, that is the quenched disorder. This analysis can be considered as a step towards fully understanding the very complicated structure of the set of temperature states of quantum spin glasses, and its connection with the breakdown of the symmetry for replicas.     

Thermal States in Conformal QFT. I

We analyze the set of locally normal KMS states w.r.t. the translation group for a local conformal net A{{\mathcal A}} of von Neumann algebras on \mathbb R{\mathbb R} . In this first part, we focus on the completely rational net A{{\mathcal A}} . Our main result here states that, if A{{\mathcal{A}}} is completely rational, there exists exactly one locally normal KMS state j{\varphi} . Moreover, j{\varphi} is canonically constructed by a geometric procedure. A crucial r?le is played by the analysis of the “thermal completion net” associated with a locally normal KMS state. A similar uniqueness result holds for KMS states of two-dimensional local conformal nets w.r.t. the time-translation one-parameter group.     

A Remark on Formal KMS States in Deformation Quantization

Within the framework of deformation quantization, we define formal KMS states on the deformed algebra of power series of functions with compact support in phase space as C[[]]-linear functionals obeying a formal variant of the usual KMS condition known in the theory of C^*-algebras. We show that for each temperature KMS states always exist and are up to a normalization equal to the trace of the argument multiplied by a formal analogue of the usual Boltzmann factor, a certain formal star exponential.     

The Reeh–Schlieder property for ground states

Recently it has been shown that the Reeh–Schlieder property w.r.t. thermal equilibrium states is a direct consequence of locality, additivity and the relativistic KMS condition. Here we extend this result to ground states.     

On Fermion Grading Symmetry for Quasi-Local Systems

We discuss fermion grading symmetry for quasi-local systems with graded commutation relations. We introduce a criterion of spontaneously symmetry breaking (SSB) for general quasi-local systems. It is formulated based on the idea that each pair of distinct phases (appeared in spontaneous symmetry breaking) should be disjoint not only for the total system but also for every complementary outside system of a local region specified by the given quasi-local structure. Under a completely model independent setting, we show the absence of SSB for fermion grading symmetry in the above sense. We obtain some structural results for equilibrium states of lattice systems. If there would exist an even KMS state for some even dynamics that is decomposed into noneven KMS states, then those noneven states inevitably violate our local thermal stability condition.     

On uniqueness of KMS states of one-dimensional quantum lattice systems

We present a proof of the theorem on the uniqueness of KMS states of one-dimensional quantum lattice systems, which is based on some equicontinuity.     

Existence of ground states and KMS states for approximately inner dynamics

A strongly continuous one parameter group of *-automorphisms of aC*-algebra with unit is said to be approximately inner if it can be approximated strongly by inner one parameter groups of *-automorphisms. It is shown that an approximately inner one parameter group of *-automorphisms has a ground state and, if there exists a trace state, a KMS state for all inverse temperatures. It follows that quantum lattice systems have ground states and KMS states. Conditions that a strongly continuous one parameter group of *-automorphisms of a UHF algebra be approximately inner are given in terms of the unbounded derivation which generates the automorphism group.     

KMS States, Entropy and the Variational Principle¶in Full C*-Dynamical Systems

To any periodic and full C ^*-dynamical system , an invertible operator s acting on the Banach space of trace functionals of the fixed point algebra is canonically associated. KMS states correspond to positive eigenvectors of s. A Perron–Frobenius type theorem asserts the existence of KMS states at inverse temperatures equals the logarithms of the inner and outer spectral radii of s (extremal KMS states). Examples arising from subshifts in symbolic dynamics, self-similar sets in fractal geometry and noncommutative metric spaces are discussed. Certain subshifts are naturally associated to the system, and criteria for the equality of their topological entropy and inverse temperatures of extremal KMS states are given. Unital completely positive maps implemented by partitions of unity {x _j } of grade 1 are considered, resembling the “canonical endomorphism” of the Cuntz algebras. The relationship between the Voiculescu topological entropy of and the topological entropy of the associated subshift is studied. Examples where the equality holds are discussed among Matsumoto algebras associated to non finite type subshifts. In the general case is bounded by the sum of the entropy of the subshift and a suitable entropic quantity of the homogeneous subalgebra. Both summands are necessary. The measure-theoretic entropy of , in the sense of Connes–Narnhofer–Thirring, is compared to the classical measure-theoretic entropy of the subshift. A noncommutative analogue of the classical variational principle for the entropy is obtained for the “canonical endomorphism” of certain Matsumoto algebras. More generally, a necessary condition is discussed. In the case of Cuntz–Krieger algebras an explicit construction of the state with maximal entropy from the unique KMS state is done. Received: 1 February 2000 / Accepted: 23 February 2000     

Stability,equilibrium and metastability in statistical mechanics

We survey a body of work, containing some new material, concerning the characterisation of equilibrium and metastable states of large assemblies of particles in terms of a variety of stability conditions. The theory is formulated in the thermodynamic limit and is based on the premise that the former states are those that are stable against all dynamical and thermodynamical perturbations, whereas the latter ones are endowed with only limited stability, sufficing to guarantee their long lifetimes and good thermodynamical behaviour. The Kubo-Martin-Schwinger (KMS) fluctuation-dissipation conditions play a central role in the developments stemming from this viewpoint, since it turns out that these conditions represent stability against localised disturbances of both the dynamical and thermo-dynamical kinds. Consequently, the stability arguments invoked here lead us to the following principal conclusions: (1) The equilibrium states are those that minimise the free energy density of the system and also satisfy the KMS conditions. This substantiates Gibbs's hypothesis that these states correspond to the standard ensembles. (2) Metastable states are of two kinds, that we term “ideal” and “normal”. Those of the former type satisfy the KMS conditions but minimise only the restriction of the free energy density to some reduced state space: those of the latter type are characterised by a still lower grade of stability. (3) The conditions on the forces under which ideal metastable states can exist are very restrictive, and thus the normal ones generally correspond to those observed in nature.     

KMS conditions and local thermodynamical stability of quantum lattice systems

We formulate local thermodynamical stability conditions for states of quantum lattice systems, and show that these conditions are implied by, and in the case of translationally invariant states equivalent to, those of Kubo-Martin-Schwinger (KMS).     

Ergodic Properties for a Quantum Nonlinear Dynamics

We present a quantum system composed of infinitely many particles, subject to a nonquadratic Hamiltonian, for which it is possible to investigate the long time behavior of the dynamics and its ergodic properties. We do so both for the KMS states and for a large class of locally normal invariant states, whose very existence is already of some interest.     

Natural Energy Bounds in Quantum Thermodynamics

Given a stationary state for a noncommutative flow, we study a boundedness condition, depending on a parameter β>0, which is weaker than the KMS equilibrium condition at inverse temperature β. This condition is equivalent to a holomorphic property closely related to the one recently considered by Ruelle and D'Antoni–Zsido and shared by a natural class of non-equilibrium steady states. Our holomorphic property is stronger than Ruelle's one and thus selects a restricted class of non-equilibrium steady states. We also introduce the complete boundedness condition and show this notion to be equivalent to the Pusz–Woronowicz complete passivity property, hence to the KMS condition. In Quantum Field Theory, the β-boundedness condition can be interpreted as the property that localized state vectors have energy density levels increasing β-subexponentially, a property which is similar in the form and weaker in the spirit than the modular compactness-nuclearity condition. In particular, for a Poincaré covariant net of C^*-algebras on Minkowski spacetime, the β-boundedness property,β≥ 2π, for the boosts is shown to be equivalent to the Bisognano–Wichmann property. The Hawking temperature is thus minimal for a thermodynamical system in the background of a Rindler black hole within the class of β-holomorphic states. More generally, concerning the Killing evolution associated with a class of stationary quantum black holes, we characterize KMS thermal equilibrium states at Hawking temperature in terms of the boundedness property and the existence of a translation symmetry on the horizon. Received: 2 October 2000 / Accepted: 5 December 2000     

The Onsager relations for quantum systems weakly coupled to KMS reservoirs

We show that the Onsager relations for energy and matter flows through small quantum systems are generically direct consequences of the KMS (Kubo-Martin-Schwinger) property of the reservoir (Gibbs) equilibrium states in the (van Hove) weak coupling limit.     

Global C*-Dynamics and Its KMS States of Weakly Inhomogeneous Bipolaronic Superconductors

We establish the limiting dynamics of a class of inhomogeneous bipolaronic models for superconductivity which incorporate deviations from the homogeneous models strong enough to require disjoint representations. The models are of the Hubbard type and the thermodynamics of their homogeneous part has been already elaborated by the authors. Now the dynamics of the systems is evaluated in terms of a generalized perturbation theory and leads to a C*-dynamical system over a classically extended algebra of observables. The classical part of the dynamical system, expressed by a set of 15 nonlinear differential equations, is observed to be independent from the perturbations. The KMS states of the C*-dynamical system are determined on the state space of the extended algebra of observables. The subsimplices of KMS states with unbroken symmetries are investigated and used to define the type of a phase. The KMS phase diagrams are worked out explicitly and compared with the thermodynamic phase structures obtained in the preceding works.     

KMS states for the Weyl algebra

We consider the possible automorphism groups for the Weyl algebra overR, resp.T, and classify those for which KMS states, unique or not unique, exist.     

Thermodynamic Properties of Non-equilibrium States in Quantum Field Theory

Within the framework of relativistic quantum field theory, a novel method is established which allows for distinguishing non-equilibrium states admitting locally a thermodynamic interpretation. The basic idea is to compare these states with global equilibrium states (KMS states) by means of local thermal observables. With the help of such observables, the states can be ordered into classes of increasing local thermal stability. Moreover, it is possible to identify states exhibiting certain specific thermal properties of interest, such as a definite local temperature or entropy density. The method is illustrated in a simple model describing the spatio-temporal evolution of a “big heat bang.”     

Equilibrium states for a classical lattice gas

Various definitions of thermodynamic equilibrium states for a classical lattice gas are given and are proved to be equivalent. In all cases, a set of equations is given, the solutions of which are by definition equilibrium states. Examples are the condition of Lanford and Ruelle, and the KMS boundary condition. In connection with this, it is shown that the time translation for classical interactions exists as an automorphism of the quantum algebra of observables, under conditions which are weaker than those found for quantum interactions.     

On the equivalence between KMS-states and equilibrium states for classical systems

It is shown that for any KMS-state of a classical system of non-coincident particles, the distribution functions are absolutely continuous with respect to Lebesgue measure; the equivalence between KMS states and Canonical Gibbs States is then established.Supported in part by NSF Grant MCS 75-21684Supported in part by NSF Grant MPS 72-04534Supported in part by NSF Grant MPS 75-20638