Effects of disorder on interacting fermionic systems in one dimension

Abstract

The work begins with constructing an XXX matrix whose diagonal elements represent the momenta of a classical NNN-particle system, while off-diagonal elements are odd functions of coordinates. Summing over rows of this matrix leads to differential equations driven by Gaussian random forces, forming a Langevin equation. Using Parisi-Wu stochastic quantization, a Hamiltonian for a quantum NNN-particle system at T=0T = 0T=0 is derived, which is inherently supersymmetric. The symmetry generators are row/column sums of the XXX matrix combined with Fermi operators. Restricting interactions to two-body types reveals that elliptic interactions provide the most general solution, making the 1/r21/r^21/r2 model universal for disordered overdamped systems governed by Langevin dynamics. Additionally, the explicit Lax XXX and mmm matrices required for proving integrability of the bosonic quantum system naturally emerge in the fermionic part of the supersymmetric Hamiltonian, highlighting their origin and role in establishing integrability.