work on spin-chains by A. Knauf,
*et. al.*
A. Knauf,
"The
number theoretical spin chain and the Riemann zeroes", *Comm. Math.
Phys.* **196** (1998), no. 3, 703-731
"It is an empirical observation that the Riemann zeta function can be
well approximated in its critical strip using the Number-Theoretical Spin
Chain. A proof of this would imply the Riemann Hypothesis.
Here we relate that question to the one of spectral radii of a family of
Markov chains...The general idea is to explain the pseudorandom features
of certain number theoretical functions by considering them as observables
of a spin chain of statistical mechanics."
P. Kleban,
A. E. Özlük,
"A Farey fraction
spin chain", *Commun. Math. Phys.* **203** (1999) 635-647
"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.."
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 $x \geq 0$. 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."
T. Prellberg, J. Fiala and P. Kleban, "Cluster approximation for the Farey fraction spin chain" (prepring 07/05)
[abstract:] "We consider the Farey fraction spin chain in an external field *h*. Utilising ideas
from dynamical systems, the free energy of the model is derived by means of an effective
cluster energy approximation. This approximation is valid for divergent cluster sizes,
and hence appropriate for the discussion of the magnetizing transition. We calculate the
phase boundaries and the scaling of the free energy. At $h = 0$ we reproduce the rigorously known asymptotic temperature dependence of the free energy. For $h \neq 0$, our results are largely consistent with those found previously using mean field theory and
renormalization group arguments."
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 and A. Knauf, "The low activity phase of some Dirichlet
series", *Journal of Mathematical Physics ***37**, (1996)
5458-5475.
P. Contucci, P. Kleban, and
A. Knauf, "A fully magnetizing phase transition", *Journal of
Statistical Physics* (1999)
F. Guerra and A. Knauf, "Free energy and correlations of the number theoretical spin
chain", *Journal of Mathematical Physics* **39** (1998) 3188-3202.
A. Knauf, "On a ferromagnetic spin chain", *Communications in
Mathematical Physics* **153** (1993) 77-115.
A. Knauf, "On a ferromagnetic spin chain. Part II: Thermodynamic
limit", *Journal of Mathematical Physics* **35** (1994) 228-236.
A. Knauf, "Phases of the number-theoretical spin chain", *Journal of
Statistical Physics* **73** (1993) 423-431.
A. Knauf, "Irregular scattering, number theory, and statistical
mechanics" , from *Stochasticity and Quantum Chaos* (eds. Z. Haba,
et. al.) Dordrecht (Kluwer,1995)
H.E. Boos and
V.E. Korepin,
"Quantum spin chains and Riemann zeta function with odd arguments",
*J. Phys.* A **34** (2001) 5311-5316
"Riemann zeta function is an important object of number theory. We
argue that it is related to Heisenberg spin 1/2 anti-ferromagnet. In
the XXX spin chain we study the probability of formation of a
ferromagnetic string in the anti-ferromagnetic ground state. We prove
that for short strings the probability can be expressed in terms of
Riemann zeta function with odd arguments."
H.E. Boos and V.E. Korepin,
"Evaluation of integrals representing correlations in XXX Heisenberg
spin chain" (preprint 05/01)
"We study XXX Heisenberg spin 1/2 anti-ferromagnet. We evaluate
a probability of formation of a ferromagnetic string in the anti-ferromagnetic
ground state in thermodynamics limit. We prove that for short strings
the probability can be expressed in terms of Riemann zeta function with
odd arguments."
H. E. Boos, V. E. Korepin, Y. Nishiyama and M. Shiroishi,
"Quantum correlations
and number theory", *J. Phys.* A **35** (2002) 4443-4452
[abstract:] "We study spin-1/2 Heisenberg XXX antiferromagnet. The
spectrum of the Hamiltonian was found by Hans Bethe in 1931. We study
the probability of formation of ferromagnetic string in the
antiferromagnetic ground state, which we call emptiness formation
probability *P*(*n*). This is the most fundamental correlation function.
We prove that for the short strings it can be expressed in terms of
the Riemann zeta function with odd arguments, logarithm ln 2 and
rational coefficients. This adds yet another link between statistical
mechanics and number theory. We have obtained an analytical formula
for *P*(5) for the first time. We have also calculated *P*(*n*) numerically
by the Density Matrix Renormalization Group. The results agree quite
well with the analytical ones. Furthermore we study asymptotic
behavior of *P*(*n*) at finite temperature by Quantum Monte-Carlo
simulation. It also agrees with our previous analytical results."
H.E. Boos, V.E. Korepin and F.A. Smirnov, "Emptiness
Formation Probability and Quantum Knizhnik-Zamolodchikov Equation", *Nucl. Phys.* B **658** (2003) 417-439
[abstract:] "We consider the one-dimensional XXX spin 1/2 Heisenberg antiferromagnet at zero temperature and zero magnetic field.
We are interested in a probability of formation of a ferromagnetic string in the antiferromagnetic ground-state. We call it
emptiness formation probability [EFP]. We suggest a new technique for computation of EFP in the inhomogeneous case. It is
based on quantum Knizhnik-Zamolodchikov equation. We evalauted EFP for strings of the length six in the inhomogeneous
case. The homogeneous limit confirms our hypothesis about the relation of quantum correlations to number theory. We also
make a conjecture about a general structure of EFP for arbitrary length of the string."
H. Boos, V. Korepin and F. Smirnov,
"New formulae for solutions
of quantum Knizhnik-Zamolodchikov equations on level -4", *J. Phys.* A **37** (2004) 323-336
[abstract:] "We present a new form of solution to the quantum Knizhnik-Zamolodchikov
equation [qKZ] on level -4 in a special case corresponding to the Heisenberg XXX spin chain.
Our form is equivalent to the integral representation obtained by Jimbo and Miwa in 1996 [JM].
An advantage of our form is that it is reduced to the product of single integrals. This fact is
deeply related to a cohomological nature of our formulae. Our approach is also based on the
deformation of hyper-elliptic integrals and their main property - deformed Riemann bilinear
relation. Jimbo and Miwa also suggested a nice conjecture which relates solution of the qKZ on
level -4 to any correlation function of the XXX model. This conjecture together with our form of
solution to the qKZ makes it possible to prove a conjecture that any correlation function of the
XXX model can be expressed in terms of the Riemann zeta-function at odd arguments and
rational coefficients suggested in [bk1], [bk2]. This issue will be discussed in our next
publication."
V.E. Korepin, S. Lukyanov, Y. Nishiyama and M. Shiroishi,
"Asymptotic behavior
of the emptiness formation probability in the critical phase of XXZ Spin Chain",
*Phys. Lett.* A **312** (2003) 21-26
[abstract:] "We study the Emptiness Formation Probability (EFP) for the spin 1/2 XXZ spin
chain. EFP P(n) detects a formation of ferromagnetic string of the length n in the ground
state. It is expected that EFP decays in a Gaussian way for large strings P(n) ~ n^{-gamma}
C^{-n^2}. Here, we propose the explicit expressions for the rate of Gaussian decay C as
well as for the exponent gamma. In order to confirm the validity of our formulas, we
employed an ab initio simulation technique of the density-matrix renormalization group to
simulate XXZ spin chain of sufficient length. Furthermore, we performed Monte-Carlo
integration of the Jimbo-Miwa multiple integral for P(n). Those numerical results for
P(n) support our formulas fairly definitely."
K. Sakai, M. Shiroishi, Y. Nishiyama and M. Takahashi,
"Third Neighbor
Correlators of Spin-1/2 Heisenberg Antiferromagnet", *Phys. Rev.* E **67**
(2003) 65-101
[abstract:] "We exactly evaluate the third neighbor correlator < *S*_{j}^{z} S_{j}+3^{z} >
and all the possible non-zero correlators <*S*^{alpha}_{j} S^{beta}_{j}+1*S*^{gamma}
_{j}+2*S*^{delta}_{j}+3 > of the spin-1/2 Heisenberg XXX antiferromagnet in the ground state without
magnetic field. All the correlators are expressed in terms of certain combinations of logarithm ln2, the Riemann zeta function zeta(3), zeta(5)
with rational coefficients. The results accurately coincide with the numerical ones obtained by the density-matrix renormalization group method
and the numerical diagonalization."
G. Mussardo, A. Trombettoni and Z. Zhang, "Prime suspects in a quantum ladder" (preprint 05/2020)
[abstract:] "In this paper we set up a suggestive number theory interpretation of a quantum ladder system made of $\mathcal{N}$ coupled chains of spin $1/2$. Using the hard-core boson representation, we associate to the spins $\sigma_a$ along the chains the prime numbers $p_a$ so that the chains become quantum registers for square-free integers. The Hamiltonian of the system consists of a hopping term and a magnetic field along the chains, together with a repulsion rung interaction and a permutation term between
next neighborhood chains . The system has various phases, among which there is one whose ground state is a coherent superposition of the first $\mathcal{N}$ prime numbers. We also discuss the realization of such a model in terms of an open quantum system with a dissipative Lindblad dynamics."
W.J. Caspers, M. Kuma, B. Lulek and T. Lulek, "Magnons
in the Heisenberg model with a dynamical scaling symmetry", *Physica A: Statistical and
Theoretical Physics* **252** 477-487
[abstract:] "A special Heisenberg model is considered for which the exchange integral takes
on the same value *J* not only for geometrically equivalent neighbours, but also for such
*j*-neighbours which constitute an orbit of a 'hidden' symmetry group of scaling
transformations. The dispersion law for magnons for this model constitutes a reproduction of
some rules of arithmetic number theory. The extra symmetry is illustrated by a chain of 12
spins, which may be shown to be equivalent to a toroidal $4\times 3$-periodic crystal."
A. Klümper, D. Nawrath and J. Suzuki, "Correlation functions of the integrable isotropic spin-1 chain: algebraic expressions for arbitrary temperature" (preprint 04/2013)
[abstract:] "We derive algebraic formulas for the density matrices of finite segments of the integrable $\mathrm{SU}(2)$ isotropic spin-1 chain in the thermodynamic limit. We give explicit results for the 2 and 3 site cases for arbitrary temperature $T$ and zero field. In the zero temperature limit the correlation functions are given in elementary form in terms of Riemann's zeta function at even integer arguments."
G. A. P. Ribeiro and A. Klümper, "Correlation functions of the integrable spin-$s$ chain" (preprint 02/2016)
[abstract:] "We study the correlation functions of $\mathrm{SU}(2)$ invariant spin-$s$ chains in the thermodynamic limit. We derive non-linear integral equations for an auxiliary correlation function $\omega$ for any spin $s$ and finite temperature $T$. For the spin-$3/2$ chain for arbitrary temperature and zero magnetic field we obtain algebraic expressions for the reduced density matrix of two-sites. In the zero temperature limit, the density matrix elements are evaluated analytically and appear to be given in terms of Riemann's zeta function values of even and odd arguments."
G.A.P. Ribeiro and A. Klümper, "Correlation functions of the integrable SU(n) spin chain" (preprint 04/2018)
[abstract:] "We study the correlation functions of $SU(n)$ $n>2$ invariant spin chains in the thermodynamic limit. We formulate a consistent framework for the computation of short-range correlation functions via functional equations which hold even at finite temperature. We give the explicit solution for two- and three-site correlations for the $SU(3)$ case at zero temperature. The correlators do not seem to be of factorizable form. From the two-sites result we see that the correlation functions are given in terms of Hurwitz' zeta function, which differs from the $SU(2)$ case where the correlations are expressed in terms of Riemann's zeta function of odd arguments."
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