Effective free-fermionic form factors and the XY spin chain

Abstract

We introduce effective form factors for one-dimensional lattice fermions with arbitrary phase shifts. We study tau functions defined as series of these form factors. On the one hand we perform the exact summation and present tau functions as Fredholm determinants in the thermodynamic limit. On the other hand simple expressions of form factors allow us to present the corresponding series as integrals of elementary functions. Using this approach we re-derive the asymptotics of static correlation functions of the XY quantum chain at finite temperature.

Correlation functions and transport coefficients in generalised hydrodynamics

“…An important result in that aspect was derivation of the large-time and long-distance asymptotics of the two-point functions in the Lieb-Liniger model [31] and XXZ spin chain in the massless regime [32]. Traces of the resummation can be also identified in the perturbative expansion of the Lieb-Liniger gas in the Tonks-Girardeau regime [33] and in the form-factor approach to correlation functions in the XY model [136].…”

The relevant excitations for the one-body function in the Lieb-Liniger model

“…On more technical level, an open question is how to analytically perform the soft-mode summation dressing the two-spinon excitations. Similar problems were recently successfully addressed in the Lieb-Liniger model [71,72] and in the spin chains [73,74]. We hope to bring those techniques to this context.…”

Mobile impurity in a one-dimensional gas at finite temperatures

“…The kernel K in ( 17) is nothing but a generalized sine-kernel and the asymptotic of det(1 + σ K) can be found by solving the corresponding Riemann-Hilbert problem (RHP) [65,66]. In principle, accounting for the Ŵ can be also done via RHP, however, this approach is technically involved and we prefer to employ instead recently developed heuristic methods of the effective form factors [57][58][59]. In this approach an exact form factor (spectral) series that describes the correlation function at finite temperature and that in the thermodynamic limit leads to ( 17) is replaced with the effective one.…”

“…We find that similar to [56] the answer in the thermodynamic limit can be expressed via the Fredholm determinants that additionally have to be integrated over an additional degree of freedom related to the impurity's momentum (the spin rapidity). We also explore large distance asymptotic of these Fredholm determinants by employing the effective form factors methods [57][58][59]. This allows us to find analytically the prefactor and the correlation length (before integration of the spin rapidity).…”