Quantum Statistical Mechanical Systems Research Articles

Quantum statistical mechanics and the boundary of modular curves

The theory of limiting modular symbols provides a noncommutative geometric model of the boundary of modular curves that includes irrational points in addition to cusps. A noncommutative space associated to this boundary is constructed, as part of a family of noncommutative spaces associated to different continued fractions algorithms, endowed with the structure of a quantum statistical mechanical system. Two special cases of this family of quantum systems can be interpreted as a boundary of the system associated to the Shimura variety of GL2 and an analog for SL2. The structure of equilibrium states for this family of systems is discussed. In the geometric cases, the ground states evaluated on boundary arithmetic elements are given by pairings of cusp forms and limiting modular symbols.

Noncommutative Geometry of Groups Like Γ0(N)

We show that the Connes-Marcolli GL2-system can be represented on the Big Picture, a combinatorial gadget introduced by Conway in order to understand various results about congruence subgroups pictorially. In this representation the time evolution of the GL2-system is implemented by Conway’s distance between projective classes of commensurable lattices. We exploit these results in order to associate quantum statistical mechanical systems to congruence subgroups. This work is motivated by the study of congruence subgroups and their principal moduli in connection with monstrous moonshine.

Q-Deformations of statistical mechanical systems and motives over finite fields

We consider q-deformations of Witt rings, based on geometric operations on zeta functions of motives over finite fields, and we use these deformations to construct q-analogs of the Bost-Connes quantum statistical mechanical system. We show that the q-deformations obtained in this way can be related to Habiro ring constructions of analytic functions over F_1 and to categorifications of Bost-Connes systems.

Bost–Connes systems, categorification, quantum statistical mechanics, and Weil numbers

In this article we develop a broad generalization of the classical Bost–Connes system, where roots of unity are replaced by an algebraic datum consisting of an abelian group and a semi-group of endomorphisms. Examples include roots of unity, Weil restriction, algebraic numbers,Weil numbers, CM fields, germs, completion ofWeil numbers, etc. Making use of the Tannakian formalism, we categorify these algebraic data. For example, the categorification of roots of unity is given by a limit of orbit categories of Tate motives while the categorification of Weil numbers is given by Grothendieck’s category of numerical motives over a finite field. To some of these algebraic data (e.g. roots of unity, algebraic numbers, Weil numbers, etc), we associate also a quantum statistical mechanical system with several remarkable properties, which generalize those of the classical Bost–Connes system. The associated partition function, low temperature Gibbs states, and Galois action on zero-temperature states are then studied in detail. For example, we show that in the particular case of the Weil numbers the partition function and the low temperature Gibbs states can be described as series of polylogarithms.

Quantum statistical mechanics in arithmetic topology

This paper provides a construction of a quantum statistical mechanical system associated to knots in the 3-sphere and cyclic branched coverings of the 3-sphere, which is an analog, in the sense of arithmetic topology, of the Bost-Connes system, with knots replacing primes, and cyclic branched coverings of the 3-sphere replacing abelian extensions of the field of rational numbers. The operator algebraic properties of this system differ significantly from the Bost-Connes case, due to the properties of the action of the semigroup of knots on a direct limit of knot groups. The resulting algebra of observables is a noncommutative Bernoulli crossed product. We describe the main properties of the associated quantum statistical mechanical system and of the relevant partition functions, which are obtained from simple knot invariants like genus and crossing number.

Twisted spectral triples and quantum statistical mechanical systems

Spectral triples and quantum statistical mechanical systems are two important constructions in noncommutative geometry. In particular, both lead to interesting reconstruction theorems for a broad range of geometric objects, including number fields, spin manifolds, graphs. There are similarities between the two structures, and we show that the notion of twisted spectral triple, introduced recently by Connes and Moscovici, provides a natural bridge between them. We investigate explicit examples, related to the Bost-Connes quantum statistical mechanical system and to Riemann surfaces and graphs.

Multifractals, Mumford curves and eternal inflation

We relate the Eternal Symmetree model of Harlow, Shenker, Stanford, and Susskind to constructions of stochastic processes related to quantum statistical mechanical systems on Cuntz-Krieger algebras. We extend the eternal inflation model from the Bruhat-Tits tree to quotients by p-adic Schottky groups, again using quantum statistical mechanics on graph algebras.

Endomotives of toric varieties

We construct endomotives associated to toric varieties, in terms of the decomposition of a toric variety into torus orbits and the action of a semigroup of toric morphisms. We show that the endomotives can be endowed with time evolutions and we discuss the resulting quantum statistical mechanical systems. We show that in particular, one can construct a time evolution related to the logarithmic height function. We discuss relations to F1-geometry.

Graph reconstruction and quantum statistical mechanics

We study how far it is possible to reconstruct a graph from various Banach algebras associated to its universal covering, and extensions thereof to quantum statistical mechanical systems. It turns out that most the boundary operator algebras reconstruct only topological information, but the statistical mechanical point of view allows for complete reconstruction of multigraphs with minimal degree three.

Error-Correcting Codes and Phase Transitions

The theory of error-correcting codes is concerned with constructing codes that optimize simultaneously transmission rate and relative minimum distance. These conflicting requirements determine an asymptotic bound, which is a continuous curve in the space of parameters. The main goal of this paper is to relate the asymptotic bound to phase diagrams of quantum statistical mechanical systems. We first identify the code parameters with Hausdorff and von Neumann dimensions, by considering fractals consisting of infinite sequences of code words. We then construct operator algebras associated to individual codes. These are Toeplitz algebras with a time evolution for which the KMS state at critical temperature gives the Hausdorff measure on the corresponding fractal. We extend this construction to algebras associated to limit points of codes, with non-uniform multi-fractal measures, and to tensor products over varying parameters.

Cyclotomy and endomotives

We compare two different models of noncommutative geometry of the cyclotomic tower, both based on an arithmetic algebra of functions of roots of unity and an action by endomorphisms, the first based on the Bost-Connes (BC) quantum statistical mechanical system and the second on the Habiro ring, where the Habiro functions have, in addition to evaluations at roots of unity, also full Taylor expansions. Both have compatible endomorphisms actions of the multiplicative semigroup of positive integers. As a higher dimensional generalization, we consider a crossed product ring obtained using Manin’s multivariable generalizations of the Habiro functions and an action by endomorphisms of the semigroup of integer matrices with positive determinant. We then construct a corresponding class of multivariable BC endomotives, which are obtained geometrically from self maps of higher dimensional algebraic tori, and we discuss some of their quantum statistical mechanical properties. These multivariable BC endomotives are universal for (torsion free) Λ-rings, compatibly with the Frobenius action. Finally, we discuss briefly how Habiro’s universal Witten-Reshetikhin-Turaev invariant of integral homology 3-spheres may relate invariants of 3-manifolds to gadgets over \( \mathbb{F}_1 \) and semigroup actions on homology 3-spheres to endomotives.

Fun with [formula omitted

Text We show that the algebra and the endomotive of the quantum statistical mechanical system of Bost–Connes naturally arises by extension of scalars from the “field with one element” to rational numbers. The inductive structure of the abelian part of the endomotive corresponds to the tower of finite extensions of that “field,” while the endomorphisms reflect the Frobenius correspondences. This gives in particular an explicit model over the integers for this endomotive, which is related to the original Hecke algebra description. We study the reduction at a prime of the endomotive and of the corresponding noncommutative crossed product algebra. Video For a video summary of this paper, please visit http://www.youtube.com/watch?v=az_0pxm1jrI.

KMS states and complex multiplication

We construct a quantum statistical mechanical system which generalizes the Bost–Connes system to imaginary quadratic fields K of arbitrary class number and fully incorporates the explicit class field theory for such fields. This system admits the Dedekind zeta function as partition function and the idele class group as group of symmetries. The extremal KMS states at zero temperature intertwine this symmetry with the Galois action on the values of the states on the arithmetic subalgebra. The geometric notion underlying the construction is that of commensurability of K-lattices.

The cluster expansion for potentials with exponential fall-off

Continuing the work of a previous paper, the Glimm-Jaffe-Spencer cluster expansion from constructive quantum field theory is adapted to treat quantum statistical mechanical systems of particles interacting by potentials that fall off exponentially at large distance. The HamiltonianH 0+V need be stable in the extended sense thatH 0+4V+BN≧0 for someB. In this situation, with a mild technical condition on the potentials, the cluster expansion converges and the infinite volume limit of the correlation functions exists, at low enough density. These infinite volume correlation functions cluster exponentially. A natural system included in the present treatment is that of matter with ther −1 potential replaced bye −ar/r. The Hamiltonian is stable, but the system would collapse in the absence of the exclusion principle—the potential is unstable. Therefore this system cannot be handled by the classic work of Ginibre, which requires stable potentials.

The cluster expansion in statistical mechanics

The Glimm-Jaffe-Spencer cluster expansion from constructive quantum field theory is adapted to treat quantum statistical mechanical systems of particles interacting by finite range potentials. The HamiltonianH0+V need be stable in the extended sense thatH0+4V+BN≧0 for someB. In this situation, with a mild technical condition on the potentials, the cluster expansion converges and the infinite volume limit of the correlation functions exists, at low enough density. These infinite volume correlation functions cluster exponentially. We define a class of interacting boson and fermion particle theories with a matter-like potential, 1/r suitably truncated at large distance. This system would collapse in the absence of the exclusion principle—the potential is unstable—but the Hamiltonian is stable. This provides an example of a system for which our method proves existence of the infinite volume limit, that is not covered by the classic work of Ginibre, which requires stable potentials.