String orders in the Luttinger liquid phase of one-dimensional spin-1/2 systems

TL;DR

This paper investigates string orders in the Luttinger liquid phase of one-dimensional spin-1/2 systems, revealing the presence of even and odd string orders, their destruction at critical fields, and their relation to entanglement and thermal effects.

Contribution

It provides a detailed analysis of string orders in various 1D spin-1/2 models within the LL phase, including effects of magnetic fields and thermal fluctuations, which is a novel exploration.

Findings

String orders with even number of spins exist in the LL phase of the XX model.

Application of transverse magnetic field induces odd-string orders in the LL phase.

All string orders are destroyed at the quantum critical transverse field, $h_c$.

Abstract

Luttinger liquid (LL) phase refers to a quantum phase which emerges in the ground state phase diagram of quite often low-dimensional quantum magnets as spin-1/2 XX, XYY and frustrated chains. It is believed that the quasi long-range order exists between particles forming the system in the LL phase. Here, at the first step we concentrate on the study of correlated spin particles in the one-dimensional (1D) spin-1/2 XX model which is exactly solvable. We show that the spin-1/2 particles form string orders with an even number of spins in the LL phase of the 1D spin-1/2 XX model. As soon as the transverse magnetic field is applied to the system, string orders with an odd number of spins induce in the LL phase. All ordered strings of spin-1/2 particles will be destroyed at the quantum critical transverse field, $h_c$. No strings exist in the saturated ferromagnetic phase. At the second step we focus on the LL phase in the ground state phase diagram of the 1D spin-1/2 XYY and frustrated ferromagnetic models. We show that the even-string orders exist in the LL phase of the 1D spin-1/2 XYY model but in the LL phase of the 1D spin-1/2 frustrated ferromagnetic model we found all kind of strings. In addition, the existence of a clear relation between the long-distance entanglement and string orders in the LL phase is shown. Also, the effect of the thermal fluctuations on the behavior of the string orders is studied.