## 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 < Sjz Sj+3z > and all the possible non-zero correlators <Salphaj Sbetaj+1Sgamma j+2Sdeltaj+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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