Quantum Correlations and Number Theory

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.

8 pages, 2 figures

References in corpus (4)

• Multiple polylogarithms and mixed Tate motives
• Quantum Spin Chains and Riemann Zeta Function with Odd Arguments
• Spin chains and combinatorics: twisted boundary conditions
• Emptiness Formation Probability for the One-Dimensional Isotropic XY Model

Cited by in corpus (29)

• The density-matrix renormalization group
• New Trends in Density Matrix Renormalization
• Computation of dynamical correlation functions of Heisenberg chains: the gapless anisotropic regime
• Emptiness Formation Probability and Quantum Knizhnik-Zamolodchikov Equation
• Large distance asymptotic behavior of the emptiness formation probability of the XXZ spin-1/2 Heisenberg chain
• Asymptotics of Toeplitz Determinants and the Emptiness Formation Probability for the XY Spin Chain
• Factorization of the finite temperature correlation functions of the XXZ chain in a magnetic field
• Factorization of multiple integrals representing the density matrix of a finite segment of the Heisenberg spin chain
• Third-neighbor and other four-point correlation functions of spin-1/2 XXZ chain
• Asymptotic Behavior of the Emptiness Formation Probability in the Critical Phase of XXZ Spin Chain
• Next Nearest-Neighbor Correlation Functions of the Spin-1/2 XXZ Chain at Critical Region
• Third Neighbor Correlators of Spin-1/2 Heisenberg Antiferromagnet
• Correlation functions of the spin-1/2 anti-ferromagnetic Heisenberg chain: exact calculation via the generating function
• Next Nearest-Neighbor Correlation Functions of the Spin-1/2 XXZ Chain at Massive Region
• First principle approach to correlation functions of spin-1/2 Heisenberg chain : fourth-neighbor correlators
• Exact evaluation of density matrix elements for the Heisenberg chain
• Density matrix elements and entanglement entropy for the spin-1/2 XXZ chain at $\Delta$=1/2
• High temperature expansion of emptiness formation probability for isotropic Heisenberg chain
• Exact full counting statistics for the staggered magnetization and the domain walls in the XY spin chain
• Finite temperature density matrix and two-point correlations in the antiferromagnetic XXZ chain
• Emptiness formation probability at finite temperature for the isotropic Heisenberg chain
• Fifth-neighbor spin-spin correlator for the anti-ferromagnetic Heisenberg chain
• Six-vertex model on a finite lattice: integral representations for nonlocal correlation functions
• Dickman polylogarithms and their constants
• Integral formulas and antisymmetrization relations for the six-vertex model
• Evaluation of integrals for the emptiness formation probability in the square-ice model
• Correlation Function and Simplified TBA Equations for XXZ Chain
• Factorization of the Ising model form factors
• On Hydrodynamic Correlations in Low-Dimensional Interacting Systems