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Physical review. B./Physical review. B

First- and second-order quantum phase transitions in the one-dimensional transverse-field Ising model with boundary fields

Kun Hu, Xintian Wu

Published Jan 8, 2021• 10 citations
Publisher

Abstract

The one-dimensional transverse field Ising model with boundary fields is studied analytically and numerically. The phase diagram in the ordered state is obtained. We find that there exist two types of phase transitions. The longitudinal boundary fields applied to the left and right ends of the Ising chain are ${h}_{L}$ and ${h}_{R}$, respectively. For $|{h}_{L}|,|{h}_{R}|<\sqrt{1\ensuremath{-}g}$, where $g$ is the transverse field and ${h}_{L}{h}_{R}<0$, a first-order phase transition occurs with the changing ${h}_{L}$ or ${h}_{R}$. The energy gap and boundary magnetization are solved exactly. The analytical expressions of the finite-size scaling for the first-order phase transition are obtained. For $|{h}_{R}|>\sqrt{1\ensuremath{-}g}$ and ${h}_{L}{h}_{R}<0$, there exists a continuous phase transition with changing ${h}_{L}$ and vice versa. This transition is identified as a quantum wetting transition. The singularity of the boundary magnetization in this phase transition is explicitly shown. A simple computational procedure with high accuracy and efficiency is proposed to calculate the magnetization. The magnetization profile, correlation functions, and wetting layer thickness are studied numerically.

DOI
10.1103/physrevb.103.024409

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The wetting transition in the transverse-field spin-12\frac{1}{2}21​ XY model with opposite boundary fields hLxhRx<0{h}_{L}^{x}{h}_{R}^{x}<0hLx​hRx​<0 is studied analytically and numerically. We find that the phase diagram is complex and that the wetting transition is of three types: first, second, and fourth order. The energy gap is obtained analytically, and the magnetization profile, correlation functions, and wetting layer thickness are obtained numerically. For ∣hLx∣,∣hRx∣<hw|{h}_{L}^{x}|,|{h}_{R}^{x}|<{h}_{w}∣hLx​∣,∣hRx​∣<hw​, a first-order phase transition occurs at ${h}_{L}^{x}=\ensuremath{-}{h}_{R}^{x}$, where hw{h}_{w}hw​ is the continuous wetting transition point. For ∣hRx∣|{h}_{R}^{x}|∣hRx​∣ larger than hw{h}_{w}hw​, the continuous wetting transition occurs at hLx=hw{h}_{L}^{x}={h}_{w}hLx​=hw​, and vice versa. For $g\ensuremath{\ne}1\ensuremath{-}{\ensuremath{\gamma}}^{2}$, the wetting transition is second order, and commensurate and incommensurate phases occur for $g<1\ensuremath{-}{\ensuremath{\gamma}}^{2}$ and $g>1\ensuremath{-}{\ensuremath{\gamma}}^{2}$, respectively. For $g=1\ensuremath{-}{\ensuremath{\gamma}}^{2}$, the wetting transition is fourth order. For this fourth-order phase transition, the third derivative of the surface magnetization oscillates and diverges near the transition point. The correlation length exponent is $\ensuremath{\nu}=2$, and the dynamic exponent is z=2z=2z=2. Thus, this fourth-order transition belongs to a new universality class. The wetting behavior is induced by asymmetric boundary fields ${h}_{L}^{x}=\ensuremath{-}{h}_{R}^{x}$ and its finite-size scaling is discussed.
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The characterization of quantum phase transitions is a fundamental task for the understanding of quantum phases of matter, with a number of potential applications in quantum technologies. In this work, we use digital quantum simulation as a resource to theoretically and experimentally study quantum phase transitions. More specifically, we implement the variational quantum eigensolver (VQE) algorithm for the one-dimensional spin-1/21/21/2 transverse-field Ising chain in the presence of boundary magnetic fields. Such fields can induce a rich phase diagram, including a first-order line and a continuous wetting transition, which is a quantum version of the classical wetting surface phenomenon. We present results for noiseless simulations of the associated quantum circuits as well as hardware results taken from a superconducting quantum processor. For different regions of the phase diagram, the quantum algorithm allows us to predict the critical value of the magnetic fields responsible for either the first- or second-order transitions occurring in the system.
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