Universal Short-Imaginary-Time Quantum Critical Dynamics Near Boundaries

Abstract: While imaginary-time evolution has long served as a standard paradigm for ground-state preparation in numerical simulations and quantum devices, its intrinsic dynamical properties has been largely overlooked. Here, we investigate the short-imaginary-time critical dynamics in quantum systems with boundaries. A universal scaling theory is developed and verified in the two-dimensional quantum Ising model, uncovering rich dynamic critical behaviors dictated by boundary universality classes. For ordered initial states, the boundary order parameter $M_s$ decays with imaginary time $τ$ as $M_s \propto τ<sup>{-β_1/νz}$, where $β_1$ denotes the boundary order parameter exponent, and $ν$ and $z$ correspond to the correlation length exponent and the dynamic exponent, respectively. For disordered initial states, the autocorrelation of the boundary order parameter is governed by a novel critical exponent $θ_1$, which is closely related to the critical initial slip behavior of $M_s$ characterized by the corresponding exponent $θ_1&#39;$. In contrast to its positive bulk counterpart, the boundary initial-slip exponent $θ_1&#39;$ is negative for the ordinary transition while remaining positive for the special transition. Although the static universality classes of $d$-dimensional quantum phase transitions generally coincide with those of $(d+1)$-dimensional classical phase transitions, we show that $θ_1$ does not follow this conventional quantum-classical mapping. We further discuss the implications of our results for more exotic forms of boundary criticality. Our findings provide new physical insights into boundary critical dynamics and offer a novel route for probing exotic boundary critical behaviors in quantum many-body systems.

• The paper introduces a universal scaling theory that captures short-imaginary-time quantum critical dynamics near boundaries by identifying distinct decay exponents for ordered versus disordered initial states.
• It employs extensive projector QMC simulations on the 2D quantum Ising model to confirm power-law decay and scaling collapse of boundary order parameters across different system sizes.
• The findings reveal counterintuitive negative boundary initial slip exponents at ordinary transitions, offering novel insights for interpreting relaxation dynamics in quantum systems with open interfaces.

Universal Short-Imaginary-Time Quantum Critical Dynamics Near Boundaries

Introduction and Motivation

Quantum criticality in systems with boundaries has long been recognized as a source of rich and nontrivial universality, with boundary degrees of freedom introducing scaling phenomena not present in homogeneous bulks. While classical boundary criticality is well characterized, the dynamical aspect—particularly in the setting of quantum systems evolved in imaginary time—remains underexplored. This work systematically investigates universal short-imaginary-time quantum critical dynamics near boundaries, developing a scaling theory for such processes and verifying its predictions via extensive projector QMC simulations on the 2D quantum Ising model.

Scaling Theory for Boundary Quantum Critical Dynamics

The authors generalize the boundary short-distance expansion (SDE), originally established in classical contexts, to the early-stage imaginary-time evolution of quantum critical systems. For ordered initial states, the boundary order parameter $M_s$ exhibits a power-law decay in imaginary time: $M_s \sim \tau^{-\beta_1/\nu z}$. Here, $\beta_1$ is the boundary order parameter exponent, while $\nu$ and $z$ retain their meaning as the bulk correlation length and dynamical exponents respectively. For disordered initial states, new exponents $\theta_1$ and $\theta'_1$ emerge and govern the decay/growth rates of boundary autocorrelation and small-magnetization order parameter, distinct from their bulk counterparts.

The work demonstrates that, unlike the well-known static boundary exponents, the dynamic exponents $\theta_1$ and $\theta'_1$ deviate from the predictions of quantum-to-classical correspondence. Specifically, $\theta_1'$ is negative for the ordinary boundary transition but positive for the special boundary transition—signaling qualitative differences in the emergence of order and critical fluctuations at boundaries.

Numerical Verification in the 2D Quantum Ising Model

Ordered Initial State: Boundary Order Parameter Dynamics

QMC simulations starting from fully ordered product states show that $M_s \sim \tau^{-\beta_1/\nu z}$0 decays as $M_s \sim \tau^{-\beta_1/\nu z}$1 at short times, with $M_s \sim \tau^{-\beta_1/\nu z}$2 (ordinary) and $M_s \sim \tau^{-\beta_1/\nu z}$3 (special). Scaling collapse across different system sizes unambiguously verifies the predicted scaling form. The behavior persists until finite-size effects dominate at large $M_s \sim \tau^{-\beta_1/\nu z}$4.

Figure 1: Dynamics of the boundary order parameter $M_s \sim \tau^{-\beta_1/\nu z}$5 from the ordered initial state for both ordinary and special transitions, showing power-law decay consistent with exponent $M_s \sim \tau^{-\beta_1/\nu z}$6.

Disordered Initial State: Two-Point Correlation Dynamics

From an uncorrelated state, both the transverse $M_s \sim \tau^{-\beta_1/\nu z}$7 and longitudinal $M_s \sim \tau^{-\beta_1/\nu z}$8 two-point correlation functions are characterized by nontrivial scaling dictated by the SDE:

• $M_s \sim \tau^{-\beta_1/\nu z}$9
• $\beta_1$0

For both boundary universality classes, the data demonstrate expected exponential decay and correct power-law prefactors, with system-size-independent coefficients in the exponential, matched by excellent scaling collapse.

Figure 2: Dynamics of the transverse correlation $\beta_1$1 from the disordered initial state reveal exponential decay and scaling collapse for both ordinary and special boundaries.

Figure 3: Dynamics of the longitudinal correlation $\beta_1$2 showing similar scaling with respect to $\beta_1$3 and collapse for both universality classes.

Critical Initial Slip and Autocorrelation

Critical initial slip at the boundary is characterized via the autocorrelation $\beta_1$4, which scales as $\beta_1$5. Notably, $\beta_1$6 is negative for the ordinary transition and positive for the special transition ($\beta_1$7 and $\beta_1$8, respectively), a stark contrast to the generic positive bulk value. Finite-size scaling further supports this nontrivial boundary dynamical behavior.

Figure 4: Dynamics of the boundary autocorrelation $\beta_1$9 from the disordered initial state display power-law decay with the sign of the scaling exponent dependent on the boundary universality class.

Additional Scaling Crossovers and Diagnoses

The authors extend their analyses to spatial crossovers from bulk to boundary regimes using the 1D quantum Ising model, further establishing the generality of their scaling forms. The full scaling functions interpolate correctly between bulk-like and boundary-dominated dynamics as a function of the ratio $\nu$0. Supplemental results demonstrate how two-time and mixed correlation functions ($\nu$1, $\nu$2, $\nu$3, $\nu$4) provide alternative measurements of the same exponents, confirming the robustness of the theoretical framework.

Figure 5: Scaling crossover of $\nu$5 in the 1D quantum Ising model, confirming bulk-to-boundary scaling interpolation for ordered initial states.

Figure 6: Scaling crossover of $\nu$6 in the 1D quantum Ising model with a disordered initial state, illustrating boundary-dominated decay at short distances.

Figure 7: Finite-size analysis of crossing points used to precisely locate the special transition.

Figure 8: Bulk local two-time correlation $\nu$7 showing scaling behavior.

Figure 9: Boundary local two-time correlation $\nu$8 verifying boundary scaling predictions.

Figure 10: Bulk mixed two-time correlation $\nu$9 demonstrating bulk initial slip scaling.

Figure 11: Boundary mixed two-time correlation $z$0 exhibiting the sign-changing boundary exponent $z$1 for ordinary and special transitions.

Implications and Future Directions

This study brings nonequilibrium boundary criticality in quantum many-body systems into sharp focus, providing a unified scaling description for both conventional and exotic universality classes. Crucially, the identification of negative boundary initial slip exponents at ordinary transitions has direct consequences for interpreting relaxation and equilibration processes in quantum devices with open boundaries or interfaces. The work also raises new questions regarding short-imaginary-time dynamical signatures in systems with topological boundaries and in models manifesting extraordinary-log universality, where logarithmic, rather than power-law, correlations may prevail.

Experimentally, the scaling relations established here suggest protocols for extracting boundary critical exponents from early-time relaxation of boundary observables, dramatically reducing computational and experimental requirements for exploring quantum critical boundary behavior in both numerics and quantum simulation platforms.

Conclusion

Universal short-imaginary-time quantum critical dynamics near boundaries is governed by a comprehensive scaling theory involving conventional static exponents as well as boundary-specific dynamical exponents $z$2 and $z$3. Numerical results in the 2D quantum Ising model fully support these predictions, including the counterintuitive emergence of negative exponents at ordinary transitions. These findings extend the paradigm of boundary criticality to nonequilibrium dynamical regimes and set the stage for future work on exotic boundary phenomena in quantum many-body dynamics.