Universal time evolution of string order parameter in quantum critical systems with boundary invertible or non-invertible symmetry breaking

TL;DR

The paper develops a universal framework to detect and characterize symmetry-breaking boundaries or interfaces in (1+1)D quantum critical systems via non-local string order parameters built from symmetry operators. By combining boundary conformal field theory analyses with lattice simulations (notably free-fermion models and Ising/Dirac examples), it demonstrates that type-I string orders defined over the whole system exhibit exponential decay after global quenches and power-law or logarithmic behavior after local quenches, with universal coefficients tied to boundary defect energies. Type-II string orders defined on subsystems reveal universal time evolution that persists irrespective of global symmetry breaking, linking to full counting statistics; their behavior is confirmed across semi-infinite and finite geometries and is corroborated by lattice computations. Overall, the work connects BCFT predictions with concrete lattice realizations to reveal robust, geometry-dependent universal signatures of boundary symmetry breaking in non-equilibrium quantum critical dynamics, and outlines promising directions for higher dimensions and non-unitary or driven settings.

Abstract

The global symmetry, either invertible or non-invertible, has been extensively studied in two dimensional conformal field theories in recent years. When the theory is defined on a manifold with open boundaries, however, many interesting conformal boundary conditions will fully or partially break such global symmetry. In this work, we study the effect of symmetry-breaking boundaries or interfaces when the system is out of equilibrium. We show that the boundary or interface symmetry-breaking can be detected by the time evolution of string order parameters, which are constructed from the symmetry operators that implement the symmetry transformations. While the string order parameters are independent of time if the symmetry is preserved over the whole system, they evolve in time in a universal way if the boundary or interface breaks the symmetry. More explicitly, in the presence of boundary or interface symmetry-breaking, the string order parameters decay exponentially in time after a global quantum quench, and decay as a power-law in time after a local quantum quench. We also generalize our study to the case when the string order parameters are defined in a subsystem, which are related to the full counting statistics. It is found there are also universal features in the time evolution of string order parameters in this case. We verify our field theory results by studying the time evolution of these two different types of string order parameters in lattice models.

Figures (28)

• Figure 1: Two types of string operators, which are illustrated here in the ground state of a finite system with open boundary conditions in the two dimensional Euclidean spacetime $z=x+i\tau$. The physical boundaries are along vertical solid lines. The global symmetry is (partially) broken along the boundary (blue solid lines) or along the conformal interface (blue dashed lines).
• Figure 2: Type-I string order parameter for a semi-infinite system after a global quench. The gray lines denote the symmetry preserved conformal boundary states that are used to define the initial state. The blue line denote the symmetry-breaking boundary at the left end of the system, and the red line denotes the string operator $\mathcal{L}$, which is defined along the path $C=\{i\tau+x,x\ge 0\}$. A small half-disk of radius $\epsilon$ is removed to introduce a UV cutoff, with a conformal boundary condition $|b\rangle$ imposed along its boundary.
• Figure 3: Type-I string order parameter after the conformal mapping in \ref{['ConformalMap_Global1']}. One can deform the string operator (red line) freely from the curve (left plot) to a straight line (right plot), since only the boundaries at the top and bottom of this $w$-strip break the symmetry.
• Figure 4: Lattice calculation of type-I string order parameter evolution after a global quantum quench. We choose the total system size $L=600$, and the strength of symmetry breaking $h = 0.5$ in \ref{['H1_Global_semi']}. (a) String order parameter evolution for different mass terms $m$ in \ref{['H0_globalQuench']} with a fixed $\theta$ in $U(\theta)$. (b) String order parameter evolution for different $\theta$ with a fixed mass term $m$.
• Figure 5: Type-I string order parameter for an infinite system after a global quench. The dashed blue line corresponds to the conformal interface that breaks the global symmetry, and the red line corresponds to the string operator. A small disk of radius $\epsilon$ is removed to introduce a UV cutoff, with a conformal boundary condition $|b\rangle$ imposed along its boundary.