Stacked quantum Ising systems and quantum Ashkin-Teller model

TL;DR

This work analyzes two stacked quantum Ising subsystems coupled by a local, symmetry-preserving term, focusing on the ground-state correlations within one subsystem as a function of interchain coupling and the environment's state. Using renormalization-group arguments and extensive DMRG simulations, it shows that weak coupling to a noncritical environment shifts Ising transitions while preserving 2D Ising universality, and that when both subsystems are critical the equal-subsystem limit maps to the quantum Ashkin–Teller model with a line of critical points where the correlation-length exponent $\nu$ varies continuously with the intercoupling $w$. In 1D, a rich QAT critical line with a central charge $c=1$ and $\nu(w)$ governs the scaling of longitudinal and transverse correlations, with explicit FSS forms derived for $G$ and $F$, including boundary-sensitive corrections. In 2D, the stacked-Ising system exhibits multicritical XY behavior at bicritical points, signaling an effective enlargement of symmetry from ${\mathbb Z}_2\oplus{\mathbb Z}_2$ to O(2). Overall, the results illuminate how the quantum state of an environment shapes the critical and coherence properties of a coupled quantum system, with implications for understanding decoherence and environment-induced modifications of quantum criticality.

Abstract

We analyze the quantum states of an isolated composite system consisting of two stacked quantum Ising (SQI) subsystems, coupled by a local Hamiltonian term that preserves the $Z_2$ symmetry of each subsystem. The coupling strength is controlled by an intercoupling parameter $w$, with $w=0$ corresponding to decoupled quantum Ising systems. We focus on the quantum correlations of one of the two SQI subsystems, $S$, in the ground state of the global system, and study their dependence on both the state of the weakly-coupled complementary part $E$ and the intercoupling strength. We concentrate on regimes in which $S$ develops critical long-range correlations. The most interesting physical scenario arises when both SQI subsystems are critical. In particular, for identical SQI subsystems, the global system is equivalent to the quantum Ashkin-Teller model, characterized by an additional $Z_2$ interchange symmetry between the two subsystem operators. In this limit, one-dimensional SQI systems exhibit a peculiar critical line along which the length-scale critical exponent $ν$ varies continuously with $w$, while two-dimensional systems develop quantum multicritical behaviors characterized by an effective enlargement of the symmetry of the critical modes, from the actual $Z_2\oplus Z_2$ symmetry to a continuous O(2) symmetry.

Figures (7)

• Figure 1: Sketch of a system made of two SQI chains, weakly coupled by local and homogeneous interactions controlled by a single, generic, parameter $k$. One of the chains represents the subsystem ${\cal S}$, while the other one plays the role of the environment ${\cal E}$.
• Figure 2: The critical value $g_c$ of the Ising transition as a function of $w$ for a model of weakly coupled Ising chains, described by the Hamiltonian \ref{['twoisiham']}-\ref{['twoisiham1']}. The two data sets are for $g_e=0.5$, black dots, and $g_e=2$, red squares (the uncertainty on the estimates of $g_c$ are smaller than the symbols). The dashed lines show linear fits of the critical points for $w\lesssim 0.2$ to $g_c(w,g_e) = g_{\cal I} + c(g_e) w$, which is the expected behavior for sufficiently small values of $w$ (see text). The inset shows $R_\xi$ vs $L^{-3/4}$ at the crossing point between data for sizes $L$ and $L+4$, at fixed $g_e$ and $w$ (symbols correspond to sizes ranging from $L=9$ to $L=45$, in increments of four). Dashed curves are fits to the corresponding numerical data, for $L \geq 13$, according to the predicted behavior \ref{['rxisca2']}.
• Figure 3: The scaling behavior of the ratio $R_G(x_1=2\ell X,x_2=\ell/4)$, defined in Eq. (\ref{['rgdef']}). We report data for $w=-0.5$ (empty symbols) and $0.5$ (full symbols), up to $L=65$. They confirm the approach to asymptotic $w$-dependent FSS curves, as predicted by Eq. (\ref{['rgcrit']}). The full line shows the exact result for the critical quantum Ising chain ($w=0$), obtained by the CFT approach (see App. \ref{['isichain']}). The inset shows results for $G_0(\ell/4)$ vs $1/L$, for various values of $w$, confirming its asymptotic $L^{-2y_\phi}$ power-law scaling with $1/L$ corrections (dashed lines are linear fits to the numerical data for $L \geq 25$).
• Figure 4: The scaling behavior of the ratio $R_F(X,1/8)$, defined in Eq. \ref{['rfdef']}, for $w=-0.5$ and $0.5$. They confirm the approach to asymptotic $w$-dependent FSS curves, as predicted by Eq. (\ref{['fxyscaqat']}). The full line shows the exact result for the critical quantum Ising chain ($w=0$)---see App. \ref{['isichain']}. The inset shows results for $F_0(\ell/4)$ vs $1/L$, confirming its asymptotic $L^{-\kappa}$ power-law scaling with $1/L$ corrections, with $\kappa$ given by Eq. \ref{['kappahyp']} (dashed lines are linear fits to the numerical data for $L \geq 25$).
• Figure 5: The scaling behavior of the ratio $R_P(X,1/8)$, defined in Eq. (\ref{['rwdef']}), for $w=-0.5$ and $0.5$. They confirm the approach to asymptotic $w$-dependent FSS curves, as predicted by Eq. (\ref{['wsca']}). The full line shows the exact result for the critical quantum Ising chain ($w=0$)---see App. \ref{['isichain']}. The inset shows results for $P_0(\ell/4)$ vs $1/L$, confirming its asymptotic $L^{-2y_p}$ power-law scaling with $1/L$ corrections (dashed lines are linear fits to the numerical data for $L \geq 25$).