Scaling of polarization amplitude in quantum many-body systems in one dimension

TL;DR

This paper investigates how the polarization amplitude in one-dimensional quantum many-body systems scales with system size, revealing power-law decay and the influence of Umklapp processes in the gapless phase.

Contribution

It provides analytical, exact, and numerical results on the scaling behavior of polarization amplitude in 1D systems, highlighting the role of Umklapp terms.

Findings

Polarization amplitude scales as a power law with system size.

The decay exponent depends on the model and interaction details.

Elimination of Umklapp terms changes the scaling exponent.

Abstract

Resta proposed a definition of the electric polarization in one-dimensional systems in terms of the ground-state expectation value of the large gauge transformation operator. Vanishing of the expectation value in the thermodynamic limit implies that the system is a conductor. We study Resta's polarization amplitude (expectation value) in the $S=1/2$ XXZ chain and its several generalizations, in the gapless conducting Tomonaga-Luttinger Liquid phase. We obtain an analytical expression in the lowest-order perturbation theory about the free fermion point (XY chain), and an exact result for the Haldane-Shastry model with long-range interactions. We also obtain numerical results, mostly using the exact diagonalization method. We find that the amplitude exhibits a power-law scaling in the system size (chain length) and vanishes in the thermodynamic limit. On the other hand, the exponent depends on the model even when the low-energy limit is described by the Tomonaga-Luttinger Liquid with the same Luttinger parameter. We find that a change in the exponent occurs when the Umklapp term(s) are eliminated, suggesting the importance of the Umklapp terms.