Universal shape-dependence of quantum entanglement in disordered magnets

Abstract

Disordered quantum magnets are not only experimentally relevant, but offer efficient computational methodologies to calculate the low energy states as well as various measures of quantum correlations. Here, we present a systematic analysis of quantum entanglement in the paradigmatic random transverse-field Ising model in two dimensions, using an efficient implementation of the asymptotically exact strong disorder renormalization group method. The phase diagram is known to be governed by three distinct infinitely disordered fixed points (IDFPs) that we study here. For square subsystems, it has been recently established that quantum entanglement has a universal logarithmic correction due to the corners of the subsystem at all three IDFPs. This corner contribution has been proposed as an "entanglement susceptibility", a useful tool to locate the phase transition and to measure the correlation length critical exponent. Towards a deeper understanding, we quantify how the corner contribution depends on the shape of the subsystem. While the corner contribution remains universal, the shape-dependence is qualitatively different in each universality class, also confirmed by line segment subsystems, a special case of skeletal entanglement. Therefore, unlike in conformally invariant systems, in general different subsystem shapes are versatile probes to unveil new universal information on the phase transitions in disordered quantum systems.

Figures (8)

• Figure 1: Phase diagram of the RTFIM in $d\geq 2$. The three studied quantum phase transitions indicated in different colors. QMCP stands for the quantum multi-critical point at the junction of the percolation and generic quantum critical transitions.
• Figure 2: Locating the phase transition via the corner contribution. The corner contribution to the entanglement entropy for a regular square shows a sharp peak at a quantum phase transition, as a function of the control parameter ($p$ or $\theta$). Clockwise from top left: percolation QCPKovacs2012-percolation, generic QCP Kovacs2012-rtfim, QMCP ($h$-direction), QMCP ($p$-direction) Kovacs2024-qmcp.
• Figure 3: Angle dependence at the percolation QCP. Conformally invariant systems follow the Cardy-Peschel formula for the angle dependence in equation (\ref{['eq:shape-factor']}) cardy1988, as shown here for percolation, equation (\ref{['eq:cp']}). The inset shows an example of the studied sheared square subsystem geometry. The volume of the subsystem is $\ell^2=L^2/4$, equal to the volume of a square subsystem with the same linear extent, and its interior angles are $\gamma$ and $\pi-\gamma$.
• Figure 4: Critical and multi-critical ground state configurations. The ground state of the RTFIM consists of independent GHZ spin clusters in all three universality classes, as shown here for $64\times64$ systems. (Left: generic QCP, fixed-$h$ disorder; middle: QMCP, fixed-$h$ disorder; right: percolation QCP, bond percolation.) The largest cluster in each sample is shown in black. The generic and multi-critical clusters are generally geometrically disconnected, unlike the percolation clusters.
• Figure 5: Shape-dependence at the generic QCP and QMCP. Estimates for the generic QCP (top) and QMCP (bottom) prefactor $-b$ for sheared squares as a function of $\ln\gamma$. At each angle, the numerical results converge to the same asymptotic values as indicated by triangles and the corresponding standard error. The insets show the infinite size extrapolation at $\gamma=\tan^{-1}(1)$ (top) and $\gamma=\tan^{-1}(1/2)$ (bottom). For reference, the dashed line (C.P.) shows the conformal prediction based on the Cardy-Peschel formula, with an effective $c'(1)$ calculated using the regular square data.