Berezinskii-Kosterlitz-Thouless Quantum Supercriticality in XXZ Heisenberg Spin Chain

Abstract

Quantum fluctuations can give rise to a singular quantum critical point (QCP) in the ground state, whose influence extends to finite temperatures, forming a quantum critical regime (QCR). Recently, it has been shown that in the quantum Ising model, the symmetry-breaking, longitudinal field can induce a quantum supercritical regime (QSR) emanating from the QCP, which hosts a universally enhanced quantum supercritical magnetocaloric effect (MCE). In this paper, we show that the QSR also emerges in the spin-1/2 XXZ model, in both the form of Ising and Berezinskii-Kosterlitz-Thouless (BKT) supercriticality. Using ground-state and finite-temperature tensor-network methods, we investigate quantum supercritical phenomena near a BKT QCP. We reveal a quantum supercritical crossover scaling $T \propto h^{2/3}$ and a Grüneisen ratio scaling $Γ_h \propto T^{-3/2}$ for the BKT QCP, which differ from the corresponding Ising supercritical scalings. Nevertheless, we find that the scaling function $φ_Γ(x)$ of the singular Grüneisen ratio for both BKT and Ising cases can be approximately described by the same expression $φ_Γ(x) \approx x/(1+x^2)$. Our work extends the study of quantum supercritical phenomena from the Ising to the XXZ Heisenberg model, thereby revealing the presence of BKT quantum supercriticality and broadening the scope of quantum supercritical physics.

Figures (6)

• Figure 1: (a) Phase diagram of XXZ spin chain under the longitudinal ($h$) and transverse ($g$) fields. Vertical orange line represents the first-order transition line between the ferromagnetic (FM) and spin-flop (SF) phase. Black curve represents the second-order transition to the paramagnetic (PM) phase. The semitransparent orange plane inside FM phase identifies a first-order transition plane induced by $h$. Red star represents the Ising quantum critical point (QCP), while blue star represents the Berezinskii-Kosterlitz-Thouless (BKT) QCP. (b,c) Quantum supercritical regimes of the Ising QCP (red star) and the BKT QCP (blue star) in $h$-$T$ plane, respectively. Dashed lines represent the quantum supercritical crossover lines with universal scaling $T\propto h^{1/\sigma}$ ($\sigma = 15/8$ for the Ising QCP and $\sigma = 3/2$ for the BKT QCP).
• Figure 2: (a) Contour plot of the transverse susceptibility $\chi_g \equiv \partial M_x/\partial g$. The black solid line indicates the location of the maxima in $\chi_g$, representing the second-order Ising transition line. (b) $\chi_g$ as a function of $\tilde{g} \equiv (g-g_c)/g_c$. The red curve corresponds to the case with $\Delta = -1.5$, which passes through the Ising QCP at $g_c \approx 0.99$. The blue curve represents the case with $\Delta = -1$, crossing the BKT QCP at $g_c \approx 0.58$. (c) Contour plot of the nearest-neighbor correlation $\braket{S_{i}^z S_{i+1}^z}$. The orange line is the first-order transition line. (d) Order parameters $O_{\rm FM} \equiv \frac{1}{L}\sum_{i=1}^L\braket{S_i^z}$ and $O_{\rm SF} \equiv \frac{1}{L}\sum_{i=1}^L (-1)^i\braket{S_i^y}$ as a function of $\Delta$ at $g=0.3$.
• Figure 3: (a,b) Temperature dependence of the longitudinal and transverse magnetic susceptibilities, $\chi_{h}$ and $\chi_{g}$, at the Ising and BKT QCP, respectively. Dashed lines illustrate the low-temperature scaling laws. (c) Specific heat as a function of temperature at these QCPs. Dashed lines indicate the linear behavior at low temperatures.
• Figure 4: (a,b) Thermal entropy in the $h$-$T$ plane near the Ising QCP (red star, $\Delta = -1.5$, $g_c \approx 0.99$) and the BKT QCP (blue star, $\Delta = -1$, $g_c \approx 0.58$), respectively. Dashed lines represent the crossover lines of the QSR. (c,d) The Grüneisen ratio $\Gamma_h$ driven by $h$ near these QCPs. Solid dots indicate the peak positions, while hollow dots indicate the dip positions. Insets illustrate the quantum supercritical law $T\propto h^{1/\sigma}$ of the crossover lines ($\sigma = 15/8$ for the Ising QCP and $\sigma = 3/2$ for the BKT QCP). (e) Peak values of the Grüneisen ratio $\Gamma_h$. Inset shows the quantum supercritical scaling $\Gamma_h \propto T^{-\sigma}$ of these peak values. (f) Data collapse of the $\Gamma_h$ data for both the Ising case and the BKT case. The scaling functions are rescaled by $1.4 \phi_\Gamma(1.8x) \rightarrow \phi_\Gamma(x)$ for the Ising case, and $1.5 \phi_\Gamma(2.1x)\rightarrow \phi_\Gamma (x)$ for the BKT case. Black solid line represents the approximation $x/(1+x^2)$ of the scaling function of Grüneisen ratio.
• Figure 5: Order parameters of the $L=128$ spin chain calculated by DMRG method with the maximum bond dimension $D = 500$, under periodical boundary condition. $O_{\rm FM}$ is the ferromagnetic (FM) order parameter, while $O_{\rm SF}$ is the spin-flop (SF) order parameter. The finite-size critical field is about $g_c\approx 0.6$. (a,b) panels illustrate the $g < g_c$ case, and (c-f) panels exhibit the $g \geq g_c$ case.