A long-range model for the electron-nuclear coupling and two-stage order in TmVO$_4$

Abstract

We study an infinite-range coupled electronic-quadrupole and nuclear-spin model for ferro-quadrupolar and nuclear-spin ordering in TmVO$_4$ in external magnetic and strain fields. This material is an experimental realization of a Transverse-Field Ising Model, where the Ising degree of freedom is quadrupolar and non-magnetic, but a transverse component is magnetic and couples both to external magnetic fields and to the nuclear spins via a hyperfine coupling. In zero external magnetic-field, there is a well-separated two-step order of the electronic and nuclear degrees of freedom and the release of their respective entropies. A transverse magnetic-field polarizes the electronic orbital moments and also the nuclear spins via the hyperfine coupling. The quadrupolar ordering temperature is gradually reduced to zero. But, there is no longer a nuclear transition in non-zero fields. Quantum fluctuations are magnified near the phase transitions and lead to peaks in the magnetic susceptibility. The spectral functions reveal a softening of a low-energy mode near the quantum critical point, consistent with the closing of the excitation gap and its reopening in the disordered phase, providing direct dynamical signatures of the field-driven quantum critical phenomena.

Figures (10)

• Figure 1: In absence of the nuclear coupling $A=0$, the phase diagram is determined using Binder cumulant crossings for $N = 32, 64, 128, 256, 512, 1024$. The phase boundary agrees will with mean-field theory for transverse field Ising model and experiments - Ref. massat2022fieldcurro2024quantum.
• Figure 2: Entropy per site as a function of temperature for three fields $h_x/h_c^0 = 0$, $0.55$, and $0.82$. The solid line represents the simulated (written as 'S' in the figure) entropy from our proposed model with $N=1024$, while the circles indicate the experimental data (entropy estimated by integrating the experimental specific heat data from Ref. massat2022field).
• Figure 3: Temperature dependence of the transverse magnetic susceptibility $\chi_{xx}$ per site for representative values of the transverse field $h_x/h_c^0 = 0.1$, $0.5$, $0.9$, and $1.1$ ($N=1024$). The enhancement of $\chi_{xx}$ near intermediate temperatures reflects the increasing role of quantum fluctuations, which become most pronounced as the system approaches the field-driven quantum critical regime. The dotted line represents the mean field values of $\chi_{xx}$ at those fields. The arrows indicate the corresponding phase transition temperatures determined independently from the Binder cumulant analysis as shown in Fig.\ref{['fig:phase_diagram']}.
• Figure 4: Mean-field phase diagram. For $A/T_Q=0.01$, the boundary shifts inward by $2\%$ vs. $A/T_Q=0$, reducing the ordered phase. (The inset shows a zoomed-in view near $h_x/h_c^0 \to 1$, highlighting the bend-back phenomenon arising from nuclear coupling.)
• Figure 5: Entropy $S/N$ vs. $T$ for $A/T_Q=0.01$ with $N=160$ and various $h_x$. Black ($h_x=0$): two transitions are seen as entropy drops. The colored ($h_x>0$) curves show that the nuclear transition (black line) become crossovers for non-zero $h_x/h_c^0$.