Stability of the symmetry-protected topological phase and Ising transitions in a disordered U(1) quantum link model on a ladder

TL;DR

The paper investigates the stability of Ising-type criticality and a symmetry-protected topological (SPT) phase in a disordered two-leg ladder quantum link model (QLM) that realizes a U(1) lattice gauge theory. Using density-matrix renormalization group techniques and finite-size scaling of entanglement entropy and magnetizations, it shows that the disorder-free system hosts Ising transitions with central charge $c=1/2$ and that rung-bond disorder preserves these Ising transitions up to moderate disorder, while zero-mass ($m=0$) supports a robust SPT phase. A field-theoretic argument based on Majorana modes explains the apparent violation of the Harris criterion for Ising criticality under rung disorder: the disorder couples to a gapped symmetric sector, leaving the gapless antisymmetric sector to govern the critical behavior. When disorder is applied to the leg bonds, Ising criticality is destroyed at $m>0$, whereas the $m=0$ SPT phase remains robust, indicating a delicate interplay between disorder type and symmetry in quasi-1D lattice gauge theories.

Abstract

We revisit the U(1) quantum link model in a ladder geometry, finding, by finite-size scaling, that the critical exponent $ν=1$ and the central charge $c=1/2$ are consistent with the Ising universality class for all phase transitions observed. A blind application of the Harris criterion would suggest that this criticality is lost in the presence of the disorder. It turns out not to be the case. For the disorder affecting ladder's rung hoppings only, we have found that the transitions survive disappearing only for quite strong disorder. The disorder in the ladder's legs destroys the nonzero mass phase criticality, while the symmetry-protected topological phase for zero mass survives a small disorder. The observed robustness against disorder is explained qualitatively using field-theoretic arguments.

Figures (10)

• Figure 1: (Color online.) (a) Schematic depiction of the ladder system described by Eq. \ref{['eq:hamil']}. Red and blue circles denote even and odd staggered fermion sites with masses $+m$ and $-m$, respectively. The bonds between sites host truncated U(1) gauge fields, represented by spin-1/2 degrees of freedom. Nearest-neighbor fermion tunneling is mediated by parallel transporters associated with the gauge fields on the intervening links, ensuring that Gauss's law is preserved. (b)–(c) Two distinct phases of the system in the disorder-free case, i.e., $J_{yi} = J_y$ for all $i$, in the large-mass limit $|m| \gg J_y$. For $m > 0$, fermions occupy the odd sites (blue circles), while the even sites (red circles) remain empty, corresponding to a filled Dirac sea. The horizontal red dashed lines indicate the partitions where we divide the system into two parts to calculate the entanglement entropy. (b) For smaller $J_y$, the spin configuration forms vortices and antivortices on alternating plaquettes, defining the VA phase. (c) For larger $J_y$, vortices (or antivortices) appear on alternating plaquettes, while the intermediate plaquettes carry zero vorticity, defining the V0 phase. In (b) and (c), grey arrows indicate the fixed virtual electric fields located just outside the ladder used to impose the boundary conditions. In our convention, $\hat{S}^z\ket{\uparrow} = +\frac{1}{2}\ket{\uparrow}$ and $\hat{S}^z\ket{\downarrow} = -\frac{1}{2}\ket{\downarrow}$ for vertical links, while $\hat{S}^z\ket{\rightarrow} = +\frac{1}{2}\ket{\rightarrow}$ and $\hat{S}^z\ket{\leftarrow} = -\frac{1}{2}\ket{\leftarrow}$ for horizontal links.
• Figure 2: (Color online.) Phase diagram of the QLM on a two-leg ladder (Eq. \ref{['eq:hamil']}) in the disorder-free limit ($\delta J_y = 0$). The diagram is symmetric under a sign flip of the fermion mass, $m \leftrightarrow -m$, so only the region $m \ge 0$ is shown. For $m > 0$, the system exhibits two quantum phases: the vortex–antivortex (VA) phase at smaller $J_y$ and the vortex–zero-flux (V0) phase at larger $J_y$. In the thermodynamic limit, the VA phase is characterized by vanishing leg magnetization $S_L$. The two phases are separated by a symmetry-protected topological (SPT) phase at $m = 0$, where both $S_L$ and rung magnetization $S_R$ as well as long-range parity order parameter $O_P$ vanish.
• Figure 3: (Color online.) Entanglement entropy (a) at the middle of the ladder, leg magnetization $S_L$ (b), and rung magnetization $S_R$ (c) for the disorder-free system of lengths $L=40,60,80,100$ at zero fermion mass ($m=0$). The entropy profile clearly reveals two phase transitions separating the three gapped phases present at zero mass. The inset of (a) shows the critical entanglement scaling according to Eq. \ref{['eq:cardy_calabrese']} at the two critical points. Panels (b) and (c) demonstrate that $S_L$ and $S_R$ serve as order parameters for the SPT-to-V0 and VA-to-SPT transitions, respectively. The insets in (b) and (c) display the data collapse obtained from the universal finite-size scaling (Eq. \ref{['eq:fss']}).
• Figure 4: (Color online.) Entanglement entropy (a) at the center of the ladder and leg magnetization $S_L$ (b) in the disorder-free ladder of lengths $L=40,60,80,100$ at fermion mass $m=0.25$. The entropy indicates the presence of only two quantum phases, VA and V0, separated by a single transition. The inset of (a) shows the scaling of the block entanglement entropy with the chord length $\log\!\left[\tfrac{2L}{\pi} \sin(\pi l/L)\right]$. In contrast to the case in Fig. \ref{['fig:m=0_dJy=0']}, the block-size dependence at criticality exhibits Friedel oscillations due to OBC. The central charge is extracted by fitting the scaling form separately to the two sublattices. The inset of (b) presents the finite-size scaling collapse of $S_L$ near the direct VA-to-V0 transition.
• Figure 5: (Color online.) Disorder-averaged entanglement entropy $\bar{\mathcal{S}}$ at the center of the ladder and disorder-averaged leg magnetization $\bar{S}_L$ for system sizes $L=40, 80, 120, 160$ at fermion mass $m=0.25$ and disorder strength $\delta J_y=0.1$. Shaded regions indicate error bars obtained from sample averaging. The entropy clearly signals a direct VA-to-V0 transition in the presence of disorder. The inset of (a) shows the logarithmic scaling of $\bar{\mathcal{S}}$ with chord length at the critical point. The Friedel oscillations due to OBC are clearly observed. The critical coupling $\bar{J}_y^c$ and the associated universal exponents are determined from the finite-size scaling collapse of $\bar{S}_L$, as displayed in the inset of (b).