Full counting statistics of the subsystem energy in the free fermions and the quantum spin chains

TL;DR

This paper derives exact formulas for the full counting statistics of subsystem energy in free fermionic systems and quantum spin chains, revealing critical behavior and decay patterns consistent with conformal field theory.

Contribution

It introduces a determinant formula for the generating function of subsystem energy and applies it to free fermions and quantum XY and XX chains, highlighting critical and non-critical regimes.

Findings

Energy fluctuations decay as power-law at critical points

Energy fluctuations decay exponentially in non-critical regimes

Exact formulas for subsystem energy moments in free fermionic systems

Abstract

We calculate the full counting statistics (FCS) of a subsystem energy in free fermionic systems by means of the Grassmann variables. We demonstrate that the generating function of these systems can be written as a determinant formula with respect to the Hamiltonian couplings and by employing the Bell's polynomials, we derive exact formulas for the subsystem energy moments. In addition, we discuss the same quantities in the quantum XY spin chain, and we demonstrate that at the critical regimes the fluctuations of the energy moments decay like a power-law as we expect from the conformal field theory arguments, while in non-critical regimes, the decay is exponential. Furthermore, we discuss the full counting statistics of subsystem energy in the quantum XX chain.