We investigate the imaginary-time relaxation critical dynamics of the two-dimensional transverse-field Ising model using infinite projected entangled pair states (iPEPS) with the full-update strategy. Simulating directly in the thermodynamic limit, we explore the relaxation process near the critical point with two types of initial states: a fully polarized state and a product state with a small magnetization. For the fully polarized state, the magnetization shows a power law scaling $M\propto τ^{-β/(νz)}$ in the imaginary-time evolution, from which both the critical point and critical exponent can be determined with high accuracy. For the nearly paramagnetic state, the relaxation process exhibits a behavior of $M\propto τ^θ$ with $θ=0.1958$ being the critical initial-slip exponent, which is in good agreement with that obtained from the dynamic scaling of the self-correlation in quantum Monte Carlo method. These universal features emerge well before the system converges to the ground state, demonstrating the efficiency of imaginary-time evolution for probing quantum criticality. Our results demonstrate that iPEPS can serve as a robust and scalable method for studying dynamical critical phenomena in two-dimensional quantum many-body systems.
We investigate the imaginary-time relaxation critical dynamics of the two-dimensional transverse-field Ising model using infinite projected entangled pair states (iPEPS) with the full-update strategy. Simulating directly in the thermodynamic limit, we explore the relaxation process near the critical point with two types of initial states: a fully polarized state and a product state with a small magnetization. For the fully polarized state, the magnetization shows a power law scaling M∝τ−β/(νz) in the imaginary-time evolution, from which both the critical point and critical exponent can be determined with high accuracy. For the nearly paramagnetic state, the relaxation process exhibits a behavior of M∝τθ with θ=0.1958 being the critical initial-slip exponent, which is in good agreement with that obtained from the dynamic scaling of the self-correlation in quantum Monte Carlo method. These universal features emerge well before the system converges to the ground state, demonstrating the efficiency of imaginary-time evolution for probing quantum criticality. Our results demonstrate that iPEPS can serve as a robust and scalable method for studying dynamical critical phenomena in two-dimensional quantum many-body systems.
We investigate the short-time quantum critical dynamics in the imaginary-time relaxation processes of finite-size systems. Universal scaling behaviors exist in the imaginary-time evolution. In particular, the system undergoes a critical initial slip stage characterized by an exponent $\ensuremath{\theta}$, in which an initial power-law increase emerges in the imaginary-time correlation function when the initial state has a zero order parameter and a vanishing correlation length. Under different initial conditions, the quantum critical point and critical exponents can be determined from the universal scaling behaviors. We apply the method to the one- and two-dimensional transverse field Ising models using quantum Monte Carlo (QMC) simulations. In the one-dimensional case, we locate the quantum critical point at (h/J)c=1.00003(8) in the thermodynamic limit, and we estimate the critical initial slip exponent $\ensuremath{\theta}=0.3734(2)$ and the static exponent $\ensuremath{\beta}/\ensuremath{\nu}=0.1251(2)$ by analyzing data on chains of length L=32−−256 and 48--256, respectively. For the two-dimensional square-lattice system, the critical coupling ratio is given by 3.04451(7) in the thermodynamic limit, while the critical exponents are $\ensuremath{\theta}=0.209(4)$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.518(1)$ estimated by data on systems of size L=24--64 and 32--64, respectively. Remarkably, the critical initial slip exponents obtained in both models are notably distinct from their classical counterparts due to the essential differences between classical and quantum dynamics. The short-time critical dynamics and the imaginary-time relaxation QMC approach can be readily adapted to various models.
An infinite projected entangled pair state (iPEPS) is a tensor network ansatz to represent a quantum state on an infinite 2D lattice whose accuracy is controlled by the bond dimension D. Its real, Lindbladian, or imaginary time evolution can be split into small time steps. Every time step generates a new iPEPS with an enlarged bond dimension ${D}^{\ensuremath{'}}>D$, which is approximated by an iPEPS with the original D. In P. Czarnik and J. Dziarmaga, Phys. Rev. B 98, 045110 (2018), an algorithm was introduced to optimize the approximate iPEPS by maximizing directly its fidelity to the one with the enlarged bond dimension ${D}^{\ensuremath{'}}$. In this paper, we implement a more efficient optimization employing a local estimator of the fidelity. For imaginary time evolution of a thermal state's purification, we also consider using unitary disentangling gates acting on ancillas to reduce the required D. We test the algorithm simulating Lindbladian evolution and unitary evolution after a sudden quench of transverse field hx in the 2D quantum Ising model. Furthermore, we simulate thermal states of this model and estimate the critical temperature with good accuracy: 0.1 for hx=2.5 and 0.5 for the more challenging case of hx=2.9 close to the quantum critical point at hx=3.04438(2).
We study second-order finite-temperature phase transitions of the two-dimensional quantum Ising and interacting honeycomb fermion models using infinite projected entangled pair states (iPEPS). We obtain an iPEPS thermal state representation by variational tensor network renormalization. We find that at the critical temperature Tc the iPEPS correlation length is finite for the computationally accessible values of the iPEPS bond dimension D. Motivated by this observation, we investigate the application of finite correlation length scaling (FCLS), which has been previously used for iPEPS simulations of quantum critical points at T=0, to obtain precise values of Tc and the universal critical exponents. We find that in the vicinity of Tc the behavior of observables follows well the one predicted by FCLS. Using FCLS we obtain Tc and the critical exponents in agreement with quantum Monte Carlo results except for couplings close to the quantum critical points where larger bond dimensions are required.
The infinite projected entangled pair states (iPEPS) technique [J. Jordan et al., Phys. Rev. Lett. 101, 250602 (2008)] has been widely used in the recent years to assess the properties of two-dimensional quantum systems, working directly in the thermodynamic limit. This formalism, which is based upon a tensor-network representation of the ground-state wave function, has several appealing features, e.g., encoding the so-called area law of entanglement entropy by construction; still, the method presents critical issues when dealing with the optimization of tensors in order to find the best possible approximation to the exact ground state of a given Hamiltonian. Here, we discuss the obstacles that arise in the optimization by imaginary-time evolution within the so-called simple and full updates and connect them to the emergence of a sharp multiplet structure in the ``virtual'' indices of tensors. In this case, a generic choice of the bond dimension D is not compatible with the multiplets and leads to a symmetry breaking (e.g., generating a finite magnetic order). In addition, varying the initial guess, different final states may be reached, with very large deviations in the magnetization value. In order to exemplify this behavior, we show the results of the S=1/2 Heisenberg model on an array of coupled ladders, for which a vanishing magnetization below the critical interladder coupling is recovered only for selected values of D, while a blind optimization with a generic D gives rise to a finite magnetization down to the limit of decoupled ladders.
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 Ms decays with imaginary time τ as Ms∝τ−β1/νz, where β1 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 θ1, which is closely related to the critical initial slip behavior of Ms characterized by the corresponding exponent θ1′. In contrast to its positive bulk counterpart, the boundary initial-slip exponent θ1′ 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 θ1 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.
Infinite projected entangled-pair states (iPEPS) provide a powerful tool to study two-dimensional strongly correlated systems directly in the thermodynamic limit. In this work, we extend the iPEPS toolbox by a method to efficiently evaluate nonequal time two-point correlators, enabling the computation of spectral functions. It is based on an iPEPS ansatz of the ground state in a large unit cell, with an operator applied in the center of the cell, which is evolved in real-time using the fast-full update method. At every time step, the two-point correlators within a cell are computed based on the corner transfer matrix renormalization group method. Benchmark results for the two-dimensional transverse field Ising model show that the main features of the dynamical structure factor can already be reproduced at relatively small bond dimensions and unit cell sizes. The results for the magnon dispersion are found to be in good agreement with previous data obtained with the iPEPS excitation ansatz.