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Physical review. B./Physical review. B

Wetting transition in the transverse-field spin- 12 XY model with boundary fields

Kun Hu, Xintian Wu

Published Apr 25, 2023• 3 citations
Publisher

Abstract

The wetting transition in the transverse-field spin-$\frac{1}{2}$ XY model with opposite boundary fields ${h}_{L}^{x}{h}_{R}^{x}<0$ 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 $|{h}_{L}^{x}|,|{h}_{R}^{x}|<{h}_{w}$, a first-order phase transition occurs at ${h}_{L}^{x}=\ensuremath{-}{h}_{R}^{x}$, where ${h}_{w}$ is the continuous wetting transition point. For $|{h}_{R}^{x}|$ larger than ${h}_{w}$, the continuous wetting transition occurs at ${h}_{L}^{x}={h}_{w}$, 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 $z=2$. 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.

DOI
10.1103/physrevb.107.134433

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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 hLx=\ensuremath−hRx{h}_{L}^{x}=\ensuremath{-}{h}_{R}^{x}hLx​=\ensuremath−hRx​, 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≠1\ensuremath−\ensuremathγ2g\ensuremath{\ne}1\ensuremath{-}{\ensuremath{\gamma}}^{2}g\ensuremath=1\ensuremath−\ensuremathγ2, the wetting transition is second order, and commensurate and incommensurate phases occur for g<1\ensuremath−\ensuremathγ2g<1\ensuremath{-}{\ensuremath{\gamma}}^{2}g<1\ensuremath−\ensuremathγ2 and g>1\ensuremath−\ensuremathγ2g>1\ensuremath{-}{\ensuremath{\gamma}}^{2}g>1\ensuremath−\ensuremathγ2, respectively. For g=1\ensuremath−\ensuremathγ2g=1\ensuremath{-}{\ensuremath{\gamma}}^{2}g=1\ensuremath−\ensuremathγ2, 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 \ensuremathν=2\ensuremath{\nu}=2\ensuremathν=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 hLx=\ensuremath−hRx{h}_{L}^{x}=\ensuremath{-}{h}_{R}^{x}hLx​=\ensuremath−hRx​ and its finite-size scaling is discussed.
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 hL{h}_{L}hL​ and hR{h}_{R}hR​, respectively. For $|{h}_{L}|,|{h}_{R}|<\sqrt{1\ensuremath{-}g}$, where ggg is the transverse field and hLhR<0{h}_{L}{h}_{R}<0hL​hR​<0, a first-order phase transition occurs with the changing hL{h}_{L}hL​ or hR{h}_{R}hR​. 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 hLhR<0{h}_{L}{h}_{R}<0hL​hR​<0, there exists a continuous phase transition with changing hL{h}_{L}hL​ 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.
The quantum wetting transition in the one-dimensional transverse-field Ising model with random bonds is studied. Opposite boundary fields are applied at the two ends of the Ising chain. The interface between two domains with oppositely oriented magnetization is localized near the boundary or in the middle of the chain. The wetting transition refers to the phase transition of localization and delocalization of the interface. The tuning parameter hL{h}_{L}hL​ is the boundary field at one end, e.g., the left end. At the transition point hL=hw{h}_{L}={h}_{w}hL​=hw​, the wetting transition occurs. First, we study the wetting transition with one defect bond. It is a first-order transition, and the interface jumps from the left end of the chain to the defect bond at the transition point. Second, we study the wetting transition with two defect bonds and find that the weaker defect bond dominates the phase transition. The wetting transition is still first order, and the interface is localized at the left end of the chain or at the weaker defect bond. Lastly, the random bond case is studied. The random bonds have a rectangular distribution. The wetting transition is still first order. The finite-size effects are studied. The statistics of the transition points and the energy gaps are obtained. We study lattices with sizes N=100N=100N=100, 150, 200, 250, 300, and 350. The deviation of the phase transition points hw{h}_{w}hw​ does not decrease as the lattice size increases. It is argued that the transition point is sample dependent, i.e., the variation in the transition point hw{h}_{w}hw​ does not approach zero, even at the thermodynamic limit.
Abstract We present a detailed study of the finite-size one-dimensional quantum XY chain in a transverse field in the presence of boundary fields coupled with the order-parameter spin operator. We consider fields located at the chain boundaries that have the same strength and that are oppositely aligned. We derive exact expressions for the gap Δ as a function of the model parameters for large values of the chain length L. These results allow us to characterize the nature of the ordered phases of the model. We find a magnetic (M) phase ( Δ ∼ e − a L ), a magnetic-incommensurate (MI) phase ( Δ ∼ e − a L f MI ( L ) ), a kink (K) phase ( Δ ∼ L − 2 ) and a kink-incommensurate (KI) phase ( Δ ∼ L − 2 f KI ( L ) ); f MI ( L ) and f KI ( L ) are bounded oscillating functions of L. We also analyze the behavior along the phase boundaries. In particular, we characterize the universal crossover behavior across the K-KI phase boundary. On this boundary, the dynamic critical exponent is z = 4, i.e. Δ ∼ L − 4 for large values of L.
We performed extensive simulations of the random-bond Ising model confined between walls where competitive surface fields act. By properly taking the thermodynamic limit we unambiguously determined wetting transition points of the system, as extrapolation of localization-delocalization transitions of the interface between domains of different orientation driven by the respective fields. The finite-size scaling theory for wetting with short-range fields [E. V. Albano and K. Binder, Phys. Rev. Lett. 109, 036101 (2012)] establishes that the average magnetization of the sample, with critical exponent $\ensuremath{\beta}$, is the proper order parameter for the study of wetting. While the hyperscaling relationship given by $\ensuremath{\gamma}+2\ensuremath{\beta}={\ensuremath{\nu}}_{\ensuremath{\parallel}}+{\ensuremath{\nu}}_{\ensuremath{\perp}}$ requires $\ensuremath{\beta}=1/2$ ($\ensuremath{\gamma}=4, {\ensuremath{\nu}}_{\ensuremath{\parallel}}=3$, and ${\ensuremath{\nu}}_{\ensuremath{\perp}}=2$), the thermodynamic scaling establishes that ${\mathrm{\ensuremath{\Delta}}}_{s}=\ensuremath{\gamma}+\ensuremath{\beta}$, which in contrast requires $\ensuremath{\beta}=0$ (${\mathrm{\ensuremath{\Delta}}}_{s}=4$), where $\ensuremath{\gamma}, {\ensuremath{\nu}}_{\ensuremath{\parallel}}, {\ensuremath{\nu}}_{\ensuremath{\perp}}$, and ${\mathrm{\ensuremath{\Delta}}}_{s}$ are the critical exponents of the susceptibility, the correlation lengths parallel and perpendicular to the interface, and the gap exponent, respectively. So, we formulate a finite-size scaling theory for wetting without hyperscaling and perform numerical simulations that provide strong evidence of hyperscaling violation (i.e., $\ensuremath{\beta}=0$) and a direct measurement of the susceptibility critical exponent $\ensuremath{\gamma}/{\ensuremath{\nu}}_{\ensuremath{\perp}}=2.0\ifmmode\pm\else\textpm\fi{}0.2$, in agreement with theoretical results for the strong fluctuation regime of wetting transitions with quenched noise.
The global phase diagram of wetting in the two-dimensional Ising model is obtained through the exact calculation of the surface excess free energy. In addition to a surface field for inducing wetting, a surface-coupling enhancement is also included. The wetting transition (of second order) is critical for any finite ratio of surface coupling J_{s} to bulk coupling J, and becomes of first order in the limit J_{s}/J→∞. However, for J_{s}/J≫1, the critical region is exponentially small and is practically invisible to numerical studies. A distinct preasymptotic regime exists in which the transition displays first-order character. In this regime, surprisingly, the surface susceptibility and surface specific heat develop a divergence and show anomalous scaling with an exponent equal to 3/2.
We present a detailed study of the finite one-dimensional quantum Ising chain in a transverse field in the presence of boundary magnetic fields coupled with the order-parameter spin operator. We consider two magnetic fields located at the boundaries of the chain that have the same strength and that are aligned in the same or in the opposite direction. We derive analytic expressions for the gap in all phases for large values of the chain length L , as a function of the boundary field strength. We also investigate the behaviour of the chain in the quantum ferromagnetic phase for oppositely aligned fields, focusing on the magnet-to-kink transition that occurs at a finite value of the magnetic field strength. At this transition we compute analytically the finite-size crossover functions for the gap, the magnetisation profile, the two-point correlation function, and the density of fermionic modes. As the magnet-to-kink transition is equivalent to the wetting transition in two-dimensional classical Ising models, our results provide new analytic predictions for the finite-size behaviour of Ising systems in a strip geometry at this transition.