Magic phase transition and non-local complexity in generalized $W$ State

TL;DR

The paper uses Stabilizer Rényi Entropy (SRE) to characterize a quantum phase transition in a 1D topologically frustrated XYZ spin chain under Frustrated Boundary Conditions, where the ground state changes from unique to a degenerate manifold with finite opposite momenta. By mapping ground states to generalized $W$-states via a Clifford circuit, the authors analytically quantify a jump in SRE that accompanies the transition, while bipartite entanglement remains continuous. In the thermodynamic limit, the SRE discontinuity is $\log_2(7/6)$, and the ground-state SRE of the frustrated model decomposes into the sum of the non-frustrated ground state’s SRE and that of the corresponding $W_p$ state, with entanglement insensitive to momentum. The results establish a genuine 'magic' phase transition detectable by SRE but invisible to entanglement, highlighting non-local complexity as a diagnostic tool for unconventional quantum phase transitions and revealing a deep connection between generalized $W$-states and frustrated ground-state structure.

Abstract

We employ the Stabilizer Renyi Entropy (SRE) to characterize a quantum phase transition that has so far eluded any standard description and can thus now be explained in terms of the interplay between its non-stabilizer properties and entanglement. The transition under consideration separates a region with a unique ground state from one with a degenerate ground state manifold spanned by states with finite and opposite (intensive) momenta. We show that SRE has a jump at the crossing points, while the entanglement entropy remains continuous. Moreover, by leveraging on a Clifford circuit mapping, we connect the observed jump in SRE to that occurring between standard and generalized $W$-states with finite momenta. This mapping allows us to quantify the SRE discontinuity analytically.

Figures (5)

• Figure 1: Value of $h^*$ as function of $J_y$ and $J_z$ ($J_x$ is assumed to be equal to 1) for the Hamiltonian in eq. \ref{['XYZ']}. The data is obtained numerically looking at the momentum of the ground state for a system made of $L=15$ spins. For $h^{*} > 0$, choosing $\vert h \vert < h^*$ the ground state manifold has a dimension equal to 2 and is spanned by states with finite, opposite momenta $p\neq 0$.
• Figure 2: The ratio $R(p,L)$ defined in eq. \ref{['rfun']}, as a function of $L$ for different sets of parameters. In both cases, we observe a power-law convergence of $R(p,L)\to1$ for $L\to\infty$, as highlighted in the inset plot, where we plot $1-R(p,L)$ as a function of $L^{-1}$ in log-log scale.
• Figure 3: Finite-size scaling analysis of the discontinuity in the SRE (top) and in the entanglement (bottom) for different sets of anisotropies ($J_x=1$ in all analyted cases ). Both the two quantities show a power-law decay to the thermodynamic values that are, respectively $\log_2(7/6)$ and 0. The entanglement is evaluated with the 2-Rényi entropy and the data plotted are associated at the partition $(A|B)$ in which $A$ is a connected subsystem made of $(L-1)/2$ spins.
• Figure 4: Left Panel: Dependence on the position of the entanglement properties for the states $\ket{\phi(p,\theta)}$ in \ref{['state_breaking_invariance']}. The entanglement is evaluated considering a partition (A|B) where $A$ is made by $(L-1)/2$ contiguous spins of which the first is $k^*$, $p=2\pi/L$ and $\theta=(L-1)/4$. Left Panel: Dependence of the amplitude of the oscillations of the entanglement properties on the dimension of the systems. The difference between the maximum and the minimum of $S(A,k^*)$ is plotted as a function of $1/L$ to highlight the fact that in the thermodynamic limit it vanishes.
• Figure 5: Dependence on the position of the entanglement properties for the ground states that preserve the mirror symmetry for different system lengths. The ground states considered in this plot are obtained by the DMRG algorithm setting $J_x=1$ and $J_y=0.33$. With these parameters, the ground states analyzed are the symmetric linear combination of the two ground states with momentum equal to $p=2\pi/L$. The entanglement is obtained considering a partition (A|B) where $A$ is made by $(L-1)/2$ contiguous spins of which the first is $k^*$.