### symmetry-breaking, phase transitions and number theory

[abstract:] "This is the first installment of a paper in three parts, where we use noncommutative geometry to study the space of commensurability classes of Q-lattices and we show that the arithmetic properties of KMS states in the corresponding quantum statistical mechanical system, the theory of modular Hecke algebras, and the spectral realization of zeros of L-functions are part of a unique general picture. In this first chapter we give a complete description of the multiple phase transitions and arithmetic spontaneous symmetry breaking in dimension two. The system at zero temperature settles onto a classical Shimura variety, which parameterizes the pure phases of the system. The noncommutative space has an arithmetic structure provided by a rational subalgebra closely related to the modular Hecke algebra. The action of the symmetry group involves the formalism of superselection sectors and the full noncommutative system at positive temperature. It acts on values of the ground states at the rational elements via the Galois group of the modular field."

The following paper contains a useful historical overview of an important body of work which preceded the above:

[excerpt:] "[We] construct a quantum dynamical system with partition function the Riemann zeta function, or the Dedekind zeta function in the general number field case. In order for the quantum dynamical system to reflect the arithmetic of the primes it must capture also some sort of interaction between them. This last feature translates in the statistical mechanical language into the phenomenon of spontaneous symmetry breaking at critical temperature with respect to a natural symmetry group. In the region of high temperature, there is a unique equilibrium state as the system is in disorder and symmetric with respect to a natural symmetry group. In the region of low temperature, a phase transition occurs and the symmetry is broken. This symmetry group acts transitively on a family of possible extremal equilibrium states. The construction of a quantum dynamical system with partition function the Riemann zeta function $\zeta(\beta)$ and spontaneous symmetry breaking or phase transition at its pole $\beta = 1$ with respect to a natural symmetry group was achieved by Bost and Connes in [BC].

A different construction of the basic algebra using crossed products was proposed by Laca and Raeburn and extended to the number field case by them with Arledge in [ALR].

An extension of the work of Bost and Connes to general global fields was done by Harari and Leichtnam in [HL]. The generalisation proposed by Harari and Leichtnam in [HL] fails to capture the Dedekind zeta function as partition function in the case of a number field with class number greater than 1. Their partition function in that case is the Dedekind zeta function with a finite number of non-canonically chosen Euler factors removed. This prompted the author's paper [Coh1] where the full Dedekind zeta function is recovered as partition function. This is achieved by recasting the original construction of Bost and Connes more completely in terms of adeles and ideles."

[BC] J.-B. Bost and A. Connes, "Hecke Algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory", Selecta Math. (New Series), 1 (1995) 411-457.

"In this paper, we construct a natural C*-dynamical system whose partition function is the Riemann zeta function. Our construction is general and associates to an inclusion of rings (under a suitable finiteness assumption) an inclusion of discrete groups (the associated ax + b groups) and the corresponding Hecke algebras of bi-invariant functions. The latter algebra is endowed with a canonical one parameter group of automorphisms measuring the lack of normality of the subgroup. The inclusion of rings Z provides the desired C*- dynamical system, which admits the zeta function as partition function and the Galois group Gal(Q cycl/ Q) of the cyclotomic extension Qcycl of Q as symmetry group. Moreover, it exhibits a phase transition with spontaneous symmetry breaking at inverse temperature beta = 1. The original motivation for these results comes from the work of B. Julia [J] (cf. also [Spe])."

[excerpt from p.413:] "We shall now describe (the precise motivation will be explained below) a C* dynamical system intimately related to the distribution of prime numbers and exhibiting the above behaviour of spontaneous symmetry breaking."

[ALR] J. Arledge, M. Laca, I. Raeburn, "Semigroup crossed products and Hecke algebras arising from number fields", Doc. Mathematica 2 (1997) 115-138.

[HL] D. Harari and E. Leichtnam "Extension du phenomene de brisure spontanee de symetrie de Bost-Connes au cas des corps global quelconques"

[Coh1] P.B. Cohen, "A C*-dynamical system with Dedekind zeta partition function and spontaneous symmetry breaking", soumis aux Actes des Journees Arithmetiques de Limoges, 1997. Preprint de l'IRMA de l'UST de Lille.

[J] B.L. Julia "Statistical theory of numbers", from Number Theory and Physics, M. Waldschmidt, et. al. (eds.), Springer Proceedings in Physics 47 (Springer, 1989) [outline]

E. Ha and F. Paugam, "Bost-Connes-Marcolli systems for Shimura varieties" (preprint 03/05)

[abstract:] "We construct a Quantum Statistical Mechanical system $(A,\sigma_t)$ analogous to the Bost-Connes-Marcolli system...in the case of Shimura varieties. Along the way, we define a new Bost-Connes system for number fields which has the "correct" symmetries and "correct" partition function. We give a formalism that applies to general Shimura data (G,X). The object of this series of papers is to show that these systems have phase transitions and spontaneous symmetry breaking, and to classify their KMS states, at least for low temperature."   [additional background information]

B.L. Julia, "Theories statistique et thermodynamique des nombres", in: Conference de Strasbourg en l'honneur de P. Cartier, Proc. IRMA-RCP25, Vol. 44 (1993).

"We shall in fact bring a large chunk of theoretical physics technology to bear on important mathematical problems and conversely one hopes to learn from a century of analytic number theory to understand better several issues of modern physics like the quark-gluon plasma transition and the Hagedorn critical temperature."

D. Merlini, "The Riemann magneton of the primes" (preprint 04/04)

[abstract:] "We present a calculation involving a function related to the Riemann Zeta function and suggested by two recent works concerning the Riemann Hypothesis: one by Balazard, Saias and Yor and the other by Volchkov. We define an integral m(r) involving the Zeta function in the complex variable s = r + it and find a particularly interesting expression for m(r) which is rigorous at least in some range of r. In such a range we find that there are two discontinuities of the derivative m'(r) at r = 1 and r = 0, which we calculate exactly. The jump at r = 1 is given by 4*Pi. The validity of the expression for m(r) up to r = 1/2 is equivalent to the truth of the Riemann Hypothesis (RH). Assuming RH the expression for m (r) gives m = 0 at r = 1/2 and the slope m'(r) = Pi*(1 + gamma) = 4.95 at r = 1/2 (where gamma = 0.577215... is the Euler constant). As a consequence, if the expression for m(r) can be continued up to r = 1/2, then if we interpret m(r) as a magnetization in the presence of a magnetic field h = r - 1/2 (or as a "free energy" at inverse temperature beta proportional to r), there is a first order phase transition at r = 1/2 (h = 0) with a jump of m'(r) given by 2*Pi times the first Lin coefficient lambda_1 = [1+gamma/2-(1/2)ln(4*Pi)] = 0.0230957. Independently of the RH, by looking at the behavior of the convergent Taylor expansion of m(r) at r = 1-, m(r = 1/2+) as well as the first Lin coefficient may be evaluated using the Euler product formula, in terms of the primes. This gives further evidence for the possible truth of the Riemann Hypothesis."

"A so-called Renormalization Group (RG) analysis is performed in order to shed some light on why the density of prime numbers in N* decreases like the single power of the inverse naperian logarithm."

"...in this note, our aim is to look for the deep reason why the density of primes decreases with the single power of the natural logarithm. We hope that we have been able to shed some light on this fact: the breaking of a symmetry, namely that of scale invariance...is the very factor repsonsible for this specific decrease.

The coincidence of the results obtained is striking when compared to the formulas of the first non-trivial approximation of Quantum ChromoDynamics...But a main common feature emerges: in both cases the two fields are afflicted by the same broken symmetry, that of scale invariance."

fascinating speculative excerpt from "On Fourier and Zeta(s)" by J.-F. Burnol

P. Kleban, A. E. Özlük, "A Farey fraction spin chain"

"We introduce a new number-theoretic spin chain and explore its thermodynamics and connections with number theory. The energy of each spin configuration is defined in a translation-invariant manner in terms of the Farey fractions, and is also expressed using Pauli matrices...The number of states of energy E = log(n) summed over chain length is expressed in terms of a restricted divisor problem. We conjecture that its asymptotic form is (n log n), consistent with the phase transition at beta = 2, and suggesting a possible connection with the Riemann zeta function.."

P. Contucci and A. Knauf, "The phase transition of the number-theoretical spin chain", Forum Mathematicum 9 (1997) 547-567.

[abstract:] "The number-theoretical spin chain has exactly one phase transition, which is located at inverse temperature $\beta_{cr} = 2$. There the magnetization jumps from one to zero. The energy density, being zero in the low temperature phase, grows at least linearly in $\beta_{cr} - \beta$.

P. Contucci, P. Kleban, and A. Knauf, "A fully magnetizing phase transition", Journal of Statistical Physics (1999)

J. Fiala, P. Kleban, A. Özlük, "The phase transition in statistical models defined on Farey fractions" (accepted for publication, J. Stat. Physics)

[abstract:] "We consider several statistical models defined on the Farey fractions. Two of these models may be regarded as "spin chains", with long-range interactions, while another arises in the study of multifractals associated with chaotic maps exhibiting intermittency. We prove that these models all have the same free energy. Their thermodynamic behavior is determined by the spectrum of the transfer operator (Ruelle-Perron-Frobenius operator), which is defined using the maps (presentation functions) generating the Farey "tree". The spectrum of this operator was completely determined by Prellberg. It follows that these models have a second-order phase transition with a specific heat divergence of the form [t (ln t)2]-1. The spin chain models are also rigorously known to have a discontinuity in the magnetization at the phase transition."

J. Fiala and P. Kleban, "Thermodynamics of the Farey Fraction Spin Chain", J. Stat. Physics 116 (2004) 1471-1490

[abstract:] "We consider the Farey fraction spin chain, a one-dimensional model defined on (the matrices generating) the Farey fractions. We extend previous work on the thermodynamics of this model by introducing an external field h. From rigorous and renormalization group arguments, we determine the phase diagram and phase transition behavior of the extended model. Our results are fully consistent with scaling theory (for the case when a "marginal" field is present) despite the unusual nature of the transition for h=0."

J. Fiala and P. Kleban, "Generalized number theoretic spin chain-connections to dynamical systems and expectation values", J. of Stat. Physics 121 (2005) 553-577

[abstract:] "We generalize the number theoretic spin chain, a one-dimensional statistical model based on the Farey fractions, by introducing a new parameter x > 0. This allows us to write recursion relations in the length of the chain. These relations are closely related to the Lewis three-term equation, which is useful in the study of the Selberg zeta-function. We then make use of these relations and spin orientation transformations. We find a simple connection with the transfer operator of a model of intermittency in dynamical systems. In addition, we are able to calculate certain spin expectation values explicitly in terms of the free energy or correlation length. Some of these expectation values appear to be directly connected with the mechanism of the phase transition."

"The purpose of this paper is to present a quantum statistical theory of 2-dimensional vortex gas based on the generalized Hamiltonian dynamics recently developed...A remarkable consequence is that the partition function and related quantities are given in terms of the generalized Riemann zeta function. The topological phase transition is naturally understood as the pole structure of the zeta function."

"In the recent publication (Journal of Geometry and Physics, 33 (2000) 23-102) we demonstrated that dynamics of 2+1 gravity can be described in terms of train tracks. Train tracks were introduced by Thurston in connection with description of dynamics of surface automorphisms. In this work we provide an example of utilization of general formalism developed earlier. The complete exact solution of the model problem describing equilibrium dynamics of train tracks on the punctured torus is obtained. Being guided by similarities between the dynamics of 2d liquid crystals and 2+1 gravity the partition function for gravity is mapped into that for the Farey spin chain. The Farey spin chain partition function, fortunately, is known exactly and has been thoroughly investigated recently. Accordingly, the transition between the pseudo-Anosov and the periodic dynamic regime (in Thurston's terminology) in the case of gravity is being reinterpreted in terms of phase transitions in the Farey spin chain whose partition function is just a ratio of two Riemann zeta functions. The mapping into the spin chain is facilitated by recognition of a special role of the Alexander polynomial for knots/links in study of dynamics of self homeomorphisms of surfaces. At the end of paper, using some facts from the theory of arithmetic hyperbolic 3-manifolds (initiated by Bianchi in 1892), we develop systematic extension of the obtained results to noncompact Riemannian surfaces of higher genus. Some of the obtained results are also useful for 3+1 gravity. In particular, using the theorem of Margulis, we provide new reasons for the black hole existence in the Universe: black holes make our Universe arithmetic. That is the discrete Lie groups of motion are arithmetic."

A.P.C. Malbouisson and J.M.C. Malbouisson, "Boundary dependence of the coupling constant and the mass in the vector N-component $(\lambda \phi^{4})_{D}$ theory", Journal of Physics A 35 (2002) 2263-2273.

[Abstract:] "Using the Matsubara formalism, we consider the massive $(\lambda \phi^{4})_{D}$ vector N component model in the large N limit, the system being confined between two infinite parallel planes. We investigate the behavior of the coupling constant as a function of the separation L between the planes. For the Wick-ordered model in D = 3 we are able to give an exact formula to the L-dependence of the coupling constant. For the non-Wick-ordered model we indicate how expressions for the coupling constant and the mass can be obtained for arbitrary dimension D in the small-L regime. Closed exact formulas for the L-dependent renormalized coupling constant and mass are obtained in D = 3 and their behaviors as functions of L are displayed. We are also able to obtainn in generic dimension D, an equation for the critical value of L corresponding to a second order phase transition in terms of the Riemann zeta-function. In D = 3 a renormalization is done and an explicit formula for the critical L is given."

G. Chalmers, "Comment on the Riemann hypothesis" (preprint 03/05)

[abstract:] "The Riemann hypothesis is identified with zeros of ${\cal N}=4$ supersymmetric gauge theory four-point amplitude. The zeros of the $\zeta(s)$ function are identified with the complex dimension of the spacetime, or the dimension of the toroidal compactification. A sequence of dimensions are identified in order to map the zeros of the amplitude to the Riemann hypothesis."

E. Elizalde, Ten Physical Applications of Spectral Zeta Functions (Lecture Notes in Physics. New Series M, Monographs, M35) (Springer-Verlag, 1995)

"Zeta-function regularization is a powerful method in perturbation theory. This book is meant as a guide for the student of this subject. Everything is explained in detail, in particular the mathematical difficulties and tricky points, and several applications are given to show how the procedure works in practice (e.g. Casimir effect, gravity and string theory, high-temperature phase transition, topological symmetry breaking). The formulas some of which are new can be used for accurate numerical calculations. The book is to be considered as a basic introduction and a collection of exercises for those who want to apply this regularization procedure in practice."

"The Kac-Baker model describes a 1-dim. classical lattice spin system with exponentially fast decaying two body interaction. The model was introduced by M. Kac and G. Baker to investigate the phenomenon of phase transition in systems with weak long-range interactions like van der Waals gas. Ruelle's dynamical zeta function for this model can be expressed in terms of Fredholm determinants of two transfer operators and hence is a meromorphic function. One of the two operators, found by M. Kac, is an integral operator with symmetric kernel acting in the Hilbert space of square integrable functions on the line. The other one is Ruelle's transfer operator acting in some Banach space of holomorphic observables of the system. In this paper we show how the Kac operator can be explicitly related basically through the Segal-Bargmann transform to the Ruelle operator restricted to a certain Segal-Bargmann space of entire functions in the complex plane. This allows us to show that Ruelle's zeta function for the Kac-Baker model has infinitely many "non-trivial" zeros on the real axis. In a special case we can show the existence of also infinitely many "trivial" zeros on the line Re s = ln 2 in the complex s-plane. Hence some kind of Riemann hypothesis seems to hold for this dynamical zeta function."

V. Di Clemente, S. F. King and D.A.J. Rayner, "Supersymmetry and electroweak breaking with large and small extra dimensions", Nucl. Phys. B 617 (2001) 71-100

[abstract:] "We consider the problem of supersymmetry and electroweak breaking in a 5d theory compactified on an $S^{1}/Z_{2}$ orbifold, where the extra dimension may be large or small. We consider the case of a supersymmetry breaking 4d brane located at one of the orbifold fixed points with the Standard Model gauge sector, third family and Higgs fields in the 5d bulk, and the first two families on a parallel 4d matter brane located at the other fixed point. We compute the Kaluza-Klein mass spectrum in this theory using a matrix technique which allows us to interpolate between large and small extra dimensions. We also consider the problem of electroweak symmetry breaking in this theory and localize the Yukawa couplings on the 4d matter brane spatially separated from the brane where supersymmetry is broken. We calculate the 1-loop effective potential using a zeta-function regularization technique, and find that the dominant top and stop contributions are separately finite. Using this result we find consistent electroweak symmetry breaking for a compactification scale {$1/R \approx 830$ GeV} and a lightest Higgs boson mass $m_{h} \approx 170$ GeV."

The following is indirectly related to number theoretical concerns via random matrix theory:

C. Hughes, J. Keating, and N. O'Connell, "On the Characteristic Polynomial of a Random Unitary Matrix", Communications in Mathematical Physics 220 (2001) 429-451

[abstract:] "We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial $Z$ of a random $N \times N$ unitary matrix, as $N \rightarrow \infty$. First we show that $\ln Z/\sqrt{\frac{1}{2}\ln N}$, evaluated at a finite set of distinct points, is asymptotically a collection of i.i.d. complex normal random variables. This leads to a refinement of a recent central limit theorem due to Keating and Snaith, and also explains the covariance structure of the eigenvalue counting function. Next we obtain a central limit theorem for $\ln Z$ in a Sobolev space of generalised functions on the unit circle. In this limiting regime, lower-order terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting Gaussian measure. Large deviations results for $\ln Z/A$, evaluated at a finite set of distinct points, can be obtained for $\sqrt{\ln N} \ll A \ll \ln N$. For higher-order scalings we obtain large deviations results for $\ln Z/A$ evaluated at a single point. There is a phase transition at $A = \ln N$ (which only applies to negative deviations of the real part) reflecting a switch from global to local conspiracy."

S. Mertens, "Phase transition in the number partitioning problem", Phys. Rev. Lett. 81 (1998) 4281-4284

[abstract:] "Number partitioning is an NP-complete problem of combinatorial optimization. A statistical mechanics analysis reveals the existence of a phase transition that separates the easy from the hard to solve instances and that reflects the pseudo-polynomiality of number partitioning. The phase diagram and the value of the typical ground state energy are calculated."

S. Mertens, "The easiest hard problem: number partitioning", to appear in A.G. Percus, G. Istrate and C. Moore, eds., Computational Complexity and Statistical Physics (Oxford University Press, 2004)

[abstract:] "Number partitioning is one of the classical NP-hard problems of combinatorial optimization. It has applications in areas like public key encryption and task scheduling. The random version of number partitioning has an "easy-hard" phase transition similar to the phase transitions observed in other combinatorial problems like k-SAT. In contrast to most other problems, number partitioning is simple enough to obtain detailled and rigorous results on the "hard" and "easy" phase and the transition that separates them. We review the known results on random integer partitioning, give a very simple derivation of the phase transition and discuss the algorithmic implications of both phases."

L. Lacasa, B. Luque, O. Miramontes, "Phase transition and computational complexity in a stochastic prime number generator" (preprint 12/2007)

[abstract:] "We introduce a prime number generator in the form of a stochastic algorithm. The character of such algorithm gives rise to a continuous phase transition which distinguishes a phase where the algorithm is able to reduce the whole system of numbers into primes and a phase where the system reaches a frozen state with low prime density. In this paper we firstly pretend to give a broad characterization of this phase transition, both in terms of analytical and numerical analysis. Critical exponents are calculated, and data collapse is provided. Further on we redefine the model as a search problem, fitting it in the hallmark of computational complexity theory. We suggest that the system belongs to the class NP. The computational cost is maximal around the threshold, as common in many algorithmic phase transitions, revealing the presence of an easy-hard-easy pattern. We finally relate the nature of the phase transition to an average-case classification of the problem."

I. Vardi, "Deterministic percolation", Communications in Mathematical Physics 207 (1999) 43-66

[excerpt from introduction:] "...percolation theory has been of great interest in physics, as it is one of the simplest models to exhibit phase transitions. In this paper, I will examine how questions of percolation theory can be posed in a deterministic setting. Thus deterministic percolation is the study of unbounded walks on a single subset of a graph, e.g., defined by number theoretic conditions. This might be of interest in physics and probability theory as it studies percolation in a deterministic setting and in number theory where it can be interpreted as studying the disorder inherent in the natural numbers."

I. Vardi, "Prime percolation", Experimental Mathematics 7 (1998) 275-288

[abstract:] "This paper examines the question of whether there is an unbounded walk of bounded step size along Gaussian primes. Percolation theory predicts that for a low enough density of random Gaussian integers no walk exists, which suggests that no such walk exists along prime numbers, since they have arbitrarily small density over large enough regions. In analogy with the Cramer conjecture, I construct a random model of Gaussian primes and show that an unbounded walk of step size $k\sqrt{\log|z|}$ at $z$ exists with probability 1 if $k \gt \sqrt{2\pi\lambda_{c}}$ and does not exist with probability 1 if $k \lt \sqrt{2\pi\lambda_{c}}$ where $\lambda_{c}$ ~ 0.35 is a constant in continuum percolation, and so conjecture that the critical step size for Gaussian primes is also $\sqrt{2\pi\lambda_{c}\log|z|}$.

C.Borgs, J.Chayes, and B.Pittel, "Phase transition and finite-size scaling for the optimum partitioning problem", Technical report, Microsoft Research http://www.research.microsoft.com/pubs/ (2000).

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