On the existence of KMS states

A necessary and sufficient condition for the existence of KMS states is formulated in terms of a certain left ideal in the algebraA ⁰ A.     

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.     

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.     

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     

Modular Conjugation and the Implementation of Supersymmetry

Any -graded C ^*-dynamical system with a self-adjoint graded-Kubo-Martin-Schwinger (KMS) functional on it can be represented (canonically) as a -graded algebra of bounded operators on a -graded Hilbert space, so that the grading of the latter is compatible with the functional. The modular conjugation operator plays a crucial role in this reconstruction. The results are generalized to the case of an unbounded graded-KMS functional having as dense domain the union of a net of C ^*-subalgebras. It is shown that the modulus of such an unbounded graded-KMS functional is KMS.      

On the possible temperatures of a dynamical system

A simpleC*-algebra and a continuous one-parameter automorphism group are constructed such that the set of inverse temperatures at which there exist equilibrium states (i.e., KMS states, or, for =±, ground or ceiling states) is an arbitrary closed subset of IR{±}.With partial support of the National Science Foundation     

Group duality and the Kubo-Martin-Schwinger condition

We consider clusteringG-invariant states of aC*-algebraU endowed with an action of a locally compact abelian groupG. Denoting as usual byF _AB,G _AB, the corresponding two-point functions, we give criteria for the fulfillment of the KMS condition (w.r.t. some one-parameter subgroup ofG) based upon the existence of a closable mapT such thatTF _AB =G _AB for allA,BU. Closability is either inL (G),B(G), orC (G), according to clustering assumptions. Our criteria originate from the combination of duality results for the groupG (phrased in terms of functions systems), with density results for the two-point functions.Supported in part by the National Science Foundation     

Bose condensation with negative chemical potential

There is widespread prejudice that the existence of Bose-condensed equilibrium states of infinite ideal boson gas requires chemical potential to be strictly zero. This is not true, in general. Using standard techniques of algebraic QFT only, we show that there exists e ^ith invariant extensions T of Schwartz's space D(³) and Bose-condensed KMS states on the CCR algebra (T) for every chemical potential 0 (h=––, the one-particle Hamiltonian). The corresponding condensation fields are, in general, of rapid growth at infinity, with suggested physical implications.     

Approximate ground state in a model with spontaneous symmetry breakdown

The spontaneous symmetry breakdown is treated by means of a variational approach. Use is made of coherent states of Glauber and of pairing states of BCS-type as the translationally invariant vacuum states for the discussion of the real scalar field ⁴ withm ₀ ² 0. The first type of trial states reproduces the usual approach to spontaneous symmetry breakdown (-) in the tree approximation (which is possible only form ₀ ² <0), while the second type of trial states offers the possibility of spontaneous symmetry breakdown even form ₀ ² =0.     

Quasi-stationary states of hydrogen

We consider the hydrogen atom within the context of a theory of relativistic quantum mechanics that allows for a probabilistic interpretation of the wave function. We find the radial equation that determines the energy levels of bound states, represented by quasi-stationary states. We compute the order of magnitude of the shifts from the usual spectrum obtained from the Dirac equation, and we find that the leading terms for these corrections are of the order of ⁶ log fors-states and ⁶ for other states. They are small compared to the Lamb shift, which is of the order of ⁵ log .     

Thermo field dynamics of quantum spin systems

Thermo field dynamics of quantum spin systems is formulated, which gives a new variational principle at finite temperatures. The KMS relation is reformulated as identities among thermal vacuum states. Path integral formulations of the thermal vacuum state are given, which yield a new thermo field Monte Carlo method. Thermo field dynamics of finite-spin systems are studied in detail as simple examples of the present method. Pertubational expansion methods of the thermal state and time-dependent state are also given.     

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.     

Subalgebras of infinite C*-algebras with finite Watatani indices I. Cuntz algebras

Using fusion rules of sectors as a working hypothesis, we construct endomorphisms of the Cuntz algebra whose images have finite Watatani indices. Quasi-free KMS states on appear in a natural way associated with the endomorphisms, and we determine the Murray-von Neumann-Connes types of their GNS representations.Dedicated to Huzihiro Araki     

Unitary equivalence of temperature dynamics for ideal and locally perturbed Fermi-gas

We consider the local perturbation $$V = \varepsilon \sum\limits_{x,y \in \mathbb{Z}^v } {V(x,y)\chi _\Omega (x)\chi _\Omega (y)a * (x)a * (y)a(y)a(x)} $$ of the ideal Fermi-gas on the lattice ? ^v , where Ω is a finite subset of ? ^v and χ_Ω is its indicator. The invertibility of Möller morphisms for small ? is proven. It follows that in the cyclic GNS representation with respect to KMS states the dynamics of ideal and locally perturbed Fermi-gas are unitary equivalent.     

New Coherent States for the BDS-Hamiltonian

In the paper we construct a new set of coherent states for a deformed Hamiltonian of the harmonic oscillator, previously introduced by Beckers, Debergh, and Szafraniec, which we have called the BDS-Hamiltonian. This Hamiltonian depends on the new creation operator a ⁺, i.e. the usual creation operator displaced with the real quantity . In order to construct the coherent states, we use a new measure in the Hilbert space of the Hamiltonian eigenstates, in fact we change the inner product. This ansatz assures that the set of eigenstates be orthonormalized and complete. In the new inner product space the BDS-Hamiltonian is self-adjoint. Using these coherent states, we construct the corresponding density operator and we find the P-distribution function of the unnormalized density operator of the BDS-Hamiltonian. Also, we calculate some thermal averages related to the BDS-oscillators system which obey the quantum canonical distribution conditions.     

Boundary Quantum Field Theory on the Interior of the Lorentz Hyperboloid

We construct local, boost covariant boundary QFT nets of von Neumann algebras on the interior of the Lorentz hyperboloid \({\mathfrak{H}_R}\), x ² ? t ² > R ², x > 0, in the two-dimensional Minkowski spacetime. Our first construction is canonical, starting with a local conformal net on \({\mathbb{R}}\), and is analogous to our previous construction of local boundary CFT nets on the Minkowski half-space. This net is in a thermal state at Hawking temperature. Then, inspired by a recent construction by E. Witten and one of us, we consider a unitary semigroup that we use to build up infinitely many nets. Surprisingly, the one-particle semigroup is again isomorphic to the semigroup of symmetric inner functions of the disk. In particular, by considering the U(1)-current net, we can associate with any given symmetric inner function a local, boundary QFT net on \({\mathfrak{H}_R}\). By considering different states, we shall also have nets in a ground state, rather than in a KMS state.     

Statistical mechanics of quantum spin systems. II

In the first part of this paper we continue the general analysis of quantum spin systems. It is demonstrated, for a large class of interactions, that time-translations form a group of automorphisms of theC*-algebra of quasi-local observables and that the thermodynamic equilibrium states are invariant under this group. Further it is shown that the equilibrium states possess the Kubo-Martin-Schwinger analyticity and boundary condition properties. In the second part of the paper we give a general analysis of states which are invariant under space and time translations and also satisfy the KMS boundary condition. A discussion of these latter conditions and their connection with the decomposition of invariant states into ergodic states is given. Various properties pertinent to this discussion are derived.Supported in part by the Office of Naval Research Contract No. Nonr 1866 (5).     

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.