Emptiness and Depletion Formation Probability in spin models with inverse square interaction

TL;DR

This paper calculates the probability of finding a completely empty region in the ground state of spin models with inverse square interactions, using hydrodynamic and instanton methods, revealing connections to spin-charge separation.

Contribution

It introduces a hydrodynamic instanton approach to compute the Emptiness Formation Probability in spin-Calogero and Haldane-Shastry models, highlighting novel analytical techniques.

Findings

EFP can be computed via classical instanton solutions with exponential accuracy.

Representation of spin hydrodynamics as two spin-less Calogero theories simplifies calculations.

Results suggest a form of spin-charge separation in the EFP expression.

Abstract

We calculate the Emptiness Formation Probability (EFP) in the spin-Calogero Model (sCM) and Haldane-Shastry Model (HSM) using their hydrodynamic description. The EFP is the probability that a region of space is completely void of particles in the ground state of a quantum many body system. We calculate this probability in an instanton approach, by considering the more general problem of an arbitrary depletion of particles (DFP). In the limit of large size of depletion region the probability is dominated by a classical configuration in imaginary time that satisfies a set of boundary conditions and the action calculated on such solution gives the EFP/DFP with exponential accuracy. We show that the calculation for sCM can be elegantly performed by representing the gradientless hydrodynamics of spin particles as a sum of two spin-less Calogero collective field theories in auxiliary variables. Interestingly, the result we find for the EFP can be casted in a form reminiscing of spin-charge separation, which should be violated for a non-linear effect such as this. We also highlight the connections between sCM, HSM and $\lambda=2$ spin-less Calogero model from a EFP/DFP perspective.