Universal Short-Imaginary-Time Quantum Critical Dynamics Near Boundaries

Technical Summary: Universal Short-Imaginary-Time Quantum Critical Dynamics Near Boundaries

Problem Statement
While imaginary-time evolution is a standard paradigm for preparing ground states in numerical simulations and quantum devices, its intrinsic dynamical properties have been largely overlooked. Although short-time critical dynamics are well-established for bulk quantum phase transitions and boundary criticality is a core problem in classical statistical physics, the specific behavior of short-imaginary-time critical dynamics near the boundaries of quantum systems remains unexplored. Specifically, it is unclear how boundary universality classes (ordinary, special, and extraordinary) influence the early-time relaxation and critical initial slip behaviors in quantum systems, and whether standard quantum-classical mappings apply to these dynamic boundary exponents.

Methodology
The authors develop a general scaling theory for universal critical dynamics near boundaries under different initial conditions, extending the Short-Distance Expansion (SDE) from classical boundary critical dynamics to the quantum short-imaginary-time regime. To verify this theory, they perform numerical simulations using the Projector Quantum Monte Carlo (QMC) method on the two-dimensional (2D) quantum Ising model.

The study focuses on:

1. Model: A 2D quantum Ising model on a square lattice with open boundary conditions along the $y$-direction and periodic conditions along $x$. The Hamiltonian includes a transverse field $h$ and distinct coupling strengths for bulk bonds ($J$) and boundary bonds ($J_s$).
2. Universality Classes: The simulations investigate the ordinary transition (weak boundary coupling) and the special transition (critical boundary coupling $J_s^*$), which separates the ordinary and extraordinary regimes.
3. Initial States: Two distinct initial conditions are employed:
   • Ordered: A fully ordered state to probe the decay of the boundary order parameter.
   • Disordered: Uncorrelated spin configurations with zero initial magnetization to probe autocorrelation, spatial correlations, and critical initial slip.
4. Observables: The study analyzes the time evolution of the boundary order parameter ($M_s$), transverse and longitudinal spatial correlations ($C_\perp, C_\parallel$), and boundary autocorrelation functions ($A_s$).

Key Contributions and Results

• Scaling Theory for Boundary Dynamics: The authors establish that for an ordered initial state, the boundary order parameter decays as $M_s \propto \tau^{-\beta_1/\nu z}$, where $\beta_1$ is the boundary order parameter exponent, $\nu$ is the correlation length exponent, and $z$ is the dynamic exponent. This confirms that boundary dynamics are governed by boundary-specific static exponents even in the dynamic regime.

• Novel Boundary Initial-Slip Exponents: For disordered initial states, the study identifies a new critical exponent, $\theta_1$, governing the boundary autocorrelation function ($A_s \propto \tau^{\theta_1 - d/z}$). A related exponent, $\theta'_1$, characterizes the initial slip of the boundary order parameter in the presence of a small initial magnetization.

  • Ordinary Transition: The boundary initial-slip exponent $\theta'_1$ is found to be negative ($\theta'_1 \approx -0.532$). This indicates that critical fluctuations dominate over domain formation at the boundary, causing the order parameter to decay in the short-time regime.
  • Special Transition: In contrast, $\theta'_1$ is positive ($\theta'_1 \approx 0.376$). Here, strong boundary coupling reinforces domain formation, leading to growth behavior similar to, but enhanced compared to, the bulk.

• Breakdown of Quantum-Classical Mapping: A significant finding is that the boundary initial-slip exponent $\theta_1$ does not follow the conventional quantum-classical correspondence. While static exponents like $\beta_1$ map between $d$-dimensional quantum and $(d+1)$-dimensional classical systems, the dynamic exponent $\theta_1$ takes distinct values in 2D quantum systems compared to 3D classical systems. For instance, the calculated $\theta_1$ for the ordinary transition in the 2D quantum model ($\approx -1.295$) differs significantly from the 3D classical Ising value ($\approx -0.635$).

• Spatial Correlations: The study verifies the scaling forms for transverse and longitudinal correlations near the boundary, confirming the predicted exponential dependence on $L/\tau^{1/z}$ and the specific anomalous dimensions $\eta_\perp$ and $\eta_\parallel$ derived from the SDE.

Significance
The paper claims to provide a new physical framework for understanding boundary critical dynamics in quantum many-body systems. By uncovering rich dynamic critical behaviors dictated by boundary universality classes, the work demonstrates that boundary effects introduce unique dynamical signatures not present in bulk systems. The discovery of the negative initial-slip exponent in the ordinary transition highlights a regime where boundary fluctuations suppress ordering, a phenomenon distinct from bulk behavior. Furthermore, the identification of $\theta_1$ as a non-mapping dynamic exponent suggests that boundary criticality in quantum systems possesses intrinsic dynamical properties that cannot be fully captured by static classical analogs. The authors suggest this framework offers a novel route for probing exotic boundary critical behaviors, such as those found in topological boundaries or the extraordinary-log universality class, though these specific cases are noted as areas for future exploration rather than results of the current study.