Magnetoelastic signatures of thermal and quantum phase transitions in a deformable Ising chain under a longitudinal and transverse magnetic field

Abstract

We investigate a deformable spin-1/2 Ising chain subjected to either a longitudinal or a transverse magnetic field, which incorporates a magnetoelastic coupling linearly dependent on a lattice distortion parameter. Within the harmonic and static adiabatic approximations, the variational Gibbs free energy is evaluated exactly using transfer-matrix and Jordan-Wigner fermionization techniques and then minimized self-consistently with respect to the lattice distortion parameter. This approach enables a unified description of magnetic and elastic properties including the magnetization, magnetic susceptibility, lattice distortion, inverse compressibility, and relative change in the sound velocity. In a longitudinal magnetic field, the deformable Ising chain displays a line of discontinuous thermal phase transitions terminating at a critical point. The discontinuous transitions are accompanied by metastable states, which give rise to a hysteresis loop at low temperatures. In contrast, the deformable Ising chain in a transverse field undergoes exclusively a continuous quantum phase transition at zero temperature with no indication of thermal phase transitions. The magnetic susceptibility and inverse compressibility exhibit cusp- and dip-like anomalies at discontinuous phase transitions, while a diverging susceptibility and vanishing inverse compressibility characterize the continuous phase transitions. An elastic softening of the deformable chain near thermal and quantum phase transitions manifest itself also through a significant sound attenuation.

Figures (10)

• Figure 1: Magnetic-field dependence of (a) the magnetization, (b) the magnetic susceptibility, (c) the distortion parameter, and (d) the inverse compressibility of the deformable spin-1/2 Ising chain in a longitudinal magnetic field at fixed pressure $p = 0$ and four distinct temperatures. For the lowest temperature $k_{\rm B}T/J_0 = 0.005$, the insets display additional results for two specific values of the nonzero pressure $p = \pm 0.1$ for comparison. To better visualize small dip singularities in the inset of Fig. \ref{['fig1']}(d), the vertical axis represents the deviation of the inverse compressibility from the reference value $\varkappa = 2$. A star symbol marks the critical point.
• Figure 2: (a) Isothermal field dependence of the magnetization of the deformable spin-1/2 Ising chain in a longitudinal magnetic field at fixed pressure $p=0$ and three distinct temperatures. Left-pointing (right-pointing) arrows indicate decreasing (increasing) field-sweep regime, whereas the curves without any directional arrow correspond to the thermodynamically stable solution obtained from the global minimum of the free energy; (b)-(d) Dependence of the variational Gibbs free energy on the distortion parameter at $p=0$ and three selected temperatures: (b) $k_{\rm B}T/J_0 = 0.005$; (c) $k_{\rm B}T/J_0 = 0.03$; (d) $k_{\rm B}T/J_0 = 0.0481$. Open circles correspond to local minima, filled circles to global minima, and half-filled circles indicate the degeneracy of the two minima. The minimum at the critical point is highlighted with a star symbol.
• Figure 3: Temperature dependence of (a) the magnetization, (b) the magnetic susceptibility, (c) the distortion parameter, and (d) the inverse compressibility of the deformable spin-1/2 Ising chain at fixed pressure $p=0$ and five selected strengths of the longitudinal magnetic field. A star symbol denotes the critical point.
• Figure 4: Density plots of (a) the magnetization, (b) the magnetic susceptibility, (c) the distortion parameter, and (d) the inverse compressibility of the deformable spin-1/2 Ising chain in a longitudinal magnetic field shown in the field-temperature plane at fixed pressure $p=0$. In panels (b) and (d), the dashed red curve marks the line of discontinuous thermal phase transitions terminating at the critical point indicated by the orange star symbol.
• Figure 5: Magnetic-field (a) and temperature (b) dependencies of the relative change of the sound velocity in the deformable spin-1/2 Ising chain under a longitudinal magnetic field at the fixed pressure $p = 0$. The inset shows in an enhanced scale a small dip, which appears near the transition field at the lowest considered temperature $k_{\rm B}T/J_0 = 0.005$.