symmetry-breaking, phase transitions and number theory

A. Connes
and M.
Marcolli, "From Physics to Number
Theory via Noncommutative Geometry. Part I: Quantum Statistical
Mechanics of **Q**-lattices" (preprint 04/04)

[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."

M. Marcolli and A. Connes, "From
physics to number theory via noncommutative geometry. Part II: Renormalization, the Riemann-Hilbert correspondence,
and motivic Galois theory", from *Frontiers in Number Theory,
Physics, and Geometry: On Random Matrices, Zeta Functions, and Dynamical Systems* (Springer, 2006)

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

P. Cohen, "Dedekind zeta functions and quantum statistical
mechanics"

[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
**Q**^{cycl} 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."

Notes on Lee-Yang type theorems in relation to
number theory

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. Petermann,
"The so-called renormalization group method applied to the specific
prime number logarithmic decrease"

"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

A.O. Lopes, "The zeta function, non-differentiability
of pressure, and the critical exponent of transition"

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."

H. Ono and H. Kuratsuji,
"Statistical theory of 2-dimensional quantum vortex gas: non-canonical
effect and generalized zeta function"

"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."

A.L. Kholodenko,
"Statistical mechanics of 2+1 gravity from Riemann zeta function and
Alexander polynomial: Exact results"

"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."

J. Hilgert and D. Mayer,
"The dynamical zeta function and transfer
operators for the Kac-Baker model"

"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).