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     

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.     

States and automorphism groups of operator algebras

Suppose that a group of automorphisms of a von Neumann algebraM, fixes the center elementwise. We show that if this group commutes with the modular (KMS) automorphism group associated with a normal faithful state onM, then this state is left invariant by the group of automorphisms. As a result we obtain a “noncommutative” ergodic theorem. The discrete spectrum of an abelian unitary group acting as automorphisms ofM is completely characterized by elements inM. We discuss the KMS condition on the CAR algebra with respect to quasi-free automorphisms and gauge invariant generalized free states. We also obtain a necessary and sufficient condition for the CAR algebra and a quasi-free automorphism group to be η-abelian.     

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.     

Correlation inequalities and equilibrium states

For an infinite dynamical system, idealized as a von Neumann algebra acted upon by a time translation implemented by a HamiltonianH, we characterize equilibrium states (KMS) by stationarity, a Bogoliubov-type inequality and continuous spectrum ofH, except at zero.Aangesteld Navorser NFWO, Belgium     

Extended Rearrangement Inequalities and Applications to Some Quantitative Stability Results

In this paper, we prove a new functional inequality of Hardy–Littlewood type for generalized rearrangements of functions. We then show how this inequality provides quantitative stability results of steady states to evolution systems that essentially preserve the rearrangements and some suitable energy functional, under minimal regularity assumptions on the perturbations. In particular, this inequality yields a quantitative stability result of a large class of steady state solutions to the Vlasov–Poisson systems, and more precisely we derive a quantitative control of the L¹ norm of the perturbation by the relative Hamiltonian (the energy functional) and rearrangements. A general non linear stability result has been obtained by Lemou et al. (Invent Math 187:145–194, 2012) in the gravitational context, however the proof relied in a crucial way on compactness arguments which by construction provides no quantitative control of the perturbation. Our functional inequality is also applied to the context of 2D-Euler systems and also provides quantitative stability results of a large class of steady-states to this system in a natural energy space.     

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     

Field operators and their spectral properties in finite-dimensional quantum field theory

In Ref. 1 we have considered the finite-dimensional quantum mechanics. There the quantum mechanical space of states wasV=C ^r. It is known that the second quantization of this space is the space of square-summable functions of finite number of variables(L ²(R^r,dx)) (Segal isomorphism). Creation and annihilation operators were introduced in Ref. 1, and the former coincided with the usual position and momentum operators in the conventional quantum mechanics. In this paper we shall investigate the spectral properties of field operators. We shall show that the isomorphism between the exponential ofV andL ²(R^r,dx) can be understood as the decomposition by generalized eigenvectors of field operators (Fourier transform).     

Non-abelian spin-singlet quantum Hall states: wave functions and quasihole state counting

We investigate a class of non-abelian spin-singlet (NASS) quantum Hall phases, proposed previously. The trial ground and quasihole excited states are exact eigenstates of certain (k+1)-body interaction Hamiltonians. The k=1 cases are the familiar Halperin abelian spin-singlet states. We present closed-form expressions for the many-body wave functions of the ground states, which for k>1 were previously defined only in terms of correlators in specific conformal field theories. The states contain clusters of k electrons, each cluster having either all spins up, or all spins down. The ground states are non-degenerate, while the quasihole excitations over these states show characteristic degeneracies, which give rise to non-abelian braid statistics. Using conformal field theory methods, we derive counting rules that determine the degeneracies in a spherical geometry. The results are checked against explicit numerical diagonalization studies for small numbers of particles on the sphere.     

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.     

Disordered Fermions on Lattices and Their Spectral Properties

We study Fermionic systems on a lattice with random interactions through their dynamics and the associated KMS states. These systems require a more complex approach compared with the standard spin systems on a lattice, on account of the difference in commutation rules for the local algebras for disjoint regions, between these two systems. It is for this reason that some of the known formulations and proofs in the case of the spin lattice systems with random interactions do not automatically go over to the case of disordered Fermion lattice systems. We extend to the disordered CAR algebra some standard results concerning the spectral properties exhibited by temperature states of disordered quantum spin systems. We investigate the Arveson spectrum, known to physicists as the set of the Bohr frequencies. We also establish its connection with the Connes and Borchers spectra, and with the associated invariants for such W ^∗-dynamical systems which determine the type of von Neumann algebras generated by a temperature state. We prove that all such spectra are independent of the disorder. Such results cover infinite-volume limits of finite-volume Gibbs states, that is the quenched disorder for Fermions living on a standard lattice ℤ ^d , including models exhibiting some standard spin-glass-like behavior. As a natural application, we show that a temperature state can generate only a type III\mathop {\rm {III}} von Neumann algebra (with the type III₀\mathop {\rm {III_{0}}} component excluded). In the case of the pure thermodynamic phase, the associated von Neumann algebra is of type III_l\mathop {\rm {III_{\lambda }}} for some λ∈(0,1], independent of the disorder. All such results are in accordance with the principle of self-averaging which affirms that the physically relevant quantities do not depend on the disorder. The approach pursued in the present paper can be viewed as a further 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 the replicas.     

Ground state(s) of the spin-boson hamiltonian

We establish that the finite temperature KMS states of the spin-boson hamiltonian have a limit as the temperature drops to zero. Using recent advances on the one-dimensional Ising model with long range, 1/r ², interactions, we prove qualitative properties of the ground state(s) in the ohmic case. We show (i) the asymptotics of the critical coupling as the bare energy gap goes to zero and to infinity, (ii) a jump in the order parameter, and (iii) that the number of bosons is finite below and infinite at and above the critical coupling strength.     

Invariant states in statistical mechanics

Mathematically we consider aC*-algebra , acted upon by the groupT of space-translations, which has an asymptotic abelian property. We analyse invariant states over . Physically this programme can be considered as a kinematical study of equilibrium states in statistical mechanics. Each invariant state can be uniquely decomposed into elementary invariant states (E-states). These elementary states have, amongst other characteristics, the physical property that space-averages of local observables are constants in the corresponding representations. In anE-state the discrete spectrum S _D of space-translations is additive which gives rise to the classificationE _I,E _II, andE _III corresponding to the three possibilities that S _D contains one point, a lattice of points, or a set with accumulation points. AnE _II-state can be uniquely decomposed into states (L-states) having a symmetry with respect to a closed subgroupT _L of (S _D and T _L are reciprocal lattices).L-states have properties with respect toT _L analogous to the properties ofE _I-states with respect toT. The decomposition intoL-states is the inverse process of homogenizing a lattice state by smearing it over a lattice distance. The mathematical methods which we employ have more general applications.     

Valence and conduction bands in MPS3 layered compounds studied by synchrotron radiation

Summary With the resonant photomeission technique we investigated the valence bands of FePS₃ and NiPS₃. The experimental results, support the ionic picture of the compounds and our previous identification of the valence band structures. The structures rapidly varying in intensity when the excitation energy is scanned across the Fe and NiM _2,3 absorption edge are associated to the transition metal 3d states; the nonresonating features are ascribed to the (P₂P₆)⁴⁻ cluster states. With the yield technique we measured the high-resolution absorption spectra of the phosphorus and sulphur inner-core levels in Mn, Fe and Ni thiophosphates. TheL _2,3(P) andL _2,3(S) spectra are similar to each other in all the compounds and are interpreted in terms of the projected density of states of the conduction bands derived from the (P₂S₆)⁴⁻ cluster states. To speed up publication, the authors of this paper have agreed to not receive the proofs for correction.     

Uniqueness and clustering properties of Gibbs states for classical and quantum unbounded spin systems

We consider quantum unbounded spin systems (lattice boson systems) in -dimensional lattice space Z. Under appropriate conditions on the interactions we prove that in a region of high temperatures the Gibbs state is unique, is translationally invariant, and has clustering properties. The main methods we use are the Wiener integral representation, the cluster expansions for zero boundary conditions and for general Gibbs state, and explicitly -dependent probability estimates. For one-dimensional systems we show the uniqueness of Gibbs states for any value of temperature by using the method of perturbed states. We also consider classical unbounded spin systems. We derive necessary estimates so that all of the results for the quantum systems hold for the classical systems by straightforward applications of the methods used in the quantum case.     

Noncommutativity of quantum observables

Given a quantum logic (L,L), a measure of noncommutativity for the elements ofL was introduced by Román and Rumbos. For the special case whenL is the lattice of closed subspaces of a Hilbert space, the noncommutativity between two atoms ofL was related to the transition probability between their corresponding pure states. Here we generalize this result to the case where one of the elements ofL is not necessarily an atom.     

Dynamics of fluctuations for quantum lattice systems

For short range interactions and forL ¹-space clustering states it is proved that there exists a bonafide time evolution on the set of normal fluctuations. This dynamics is applied to derive the notion of equilibrium state of the algebra of fluctuations.     

Constructive Approximations of Markov Operators

We construct piecewise linear Markov finite approximations of Markov operators defined on L ¹([0, 1] ^N ) and we study various properties, such as consistency, stability, and convergence, for the purpose of numerical analysis of Markov operators.     

Alpha-particle and triton cluster states in 19F

We calculate the energies and other properties of three- and four-particle cluster states in the nuclei ¹⁹F and ¹⁹Ne, treated as eigenstat es of a local cluster-core potential. The potential we consider is a symmetrized Saxon-Woods well which has the advantage that a single value of the potential depth can generate rotational spectra of levels with different values of orbital angular momentum L; in this respect, it is very similar to the folding potentials which have been used in previous cluster calculations. We discuss the cluster states in the light of recent three- and four-particle heavy ion transfer reactions and studies of the electromagnetic decay properties of bound and continuum states. Most of the states observed to be strongly populated in the transfer experiments can be unambiguously assigned to cluster bands based on ¹⁶O + t and ¹⁵N + α configurations with various angular momenta and node numbers. Some evidence for mixing between triton and alpha configurations exists and is discussed. We also calculate the electromagnetic properties of the cluster states and find that, with few exceptions, they are in good agreement with the experimental data. Some El transitions are predicted to be very large, contrary to existing experiments, and new experiments are proposed to investigate this discrepancy.     

The BGK Model with External Confining Potential: Existence,Long-Time Behaviour and Time-Periodic Maxwellian Equilibria

We study global existence and long time behaviour for the inhomogeneous nonlinear BGK model for the Boltzmann equation with an external confining potential. For an initial datum f ₀≥0 with bounded mass, entropy and total energy we prove existence and strong convergence in L ¹ to a Maxwellian equilibrium state, by compactness arguments and multipliers techniques. Of particular interest is the case with an isotropic harmonic potential, in which Boltzmann himself found infinitely many time-periodic Maxwellian steady states. This behaviour is shared with the Boltzmann equation and other kinetic models. For all these systems we study the multistability of the time-periodic Maxwellians and provide necessary conditions on f ₀ to identify the equilibrium state, both in L ¹ and in Lyapunov sense. Under further assumptions on f, these conditions become also sufficient for the identification of the equilibrium in L ¹.