Exact solution of Riemann–Hilbert problem for a correlation function of the XY spin chain

A correlation function of the XY spin chain is studied at zero temperature. This is called the Emptiness Formation Probability (EFP) and is expressed by the Fredholm determinant in the thermodynamic limit. We formulate the associated Riemann–Hilbert problem and solve it exactly. The EFP is shown to decay in Gaussian. In this letter we study the Riemann–Hilbert problem associated with a correlation function of the XY spin chain. The Hamiltonian of the model is given by HXY = N ∑ n=1 (σ nσ x n+1 + σ y nσ y n+1 − hσ z n). (1) Here σ n, σ y n, σ z n are the Pauli matrices acting on the n-th site and h indicates an external magnetic field. Let pk = (1 + σ z k)/2. Consider a correlation function, Pn = 〈p1 · · · pn〉, (2) which describes the probability of finding a string of n-adjacent parallel spins up on the ground state of the model for a given value of the external magnetic field h. This is called the Emptiness Formation Probability (EFP). In the thermodynamic limit (N → ∞), at zero temperature, the EFP is known to be expressed by the Fredholm determinant as follows [1]: Pn = Det(1−Kn), (3) where the kernel Kn is of the form, Kn(z, w) = f (z)g(w) z − w , (4)