Abstract We study the tractability of classically simulating critical phenomena in the quench dynamics of one-dimensional transverse field Ising models (TFIMs) using highly truncated matrix product states (MPSs). We focus on two paradigmatic examples: a dynamical quantum phase transition (DQPT) that occurs in nonintegrable long-range TFIMs, and the infinite-time correlation length of the integrable nearest-neighbor TFIM when quenched to the critical point, where the quantities of interest involve equal time one- and two- point correlation functions, which we associate with macroproperties. For the DQPT, we show that the order parameters can be efficiently simulated with heavy truncation of the MPS bond dimension. This can be used to reliably extract critical properties of the phase transition, including critical exponents, even when the full many-body state is not simulated with high fidelity. The long-time correlation length near the critical point is more sensitive to the full many-body state fidelity, and generally requires a large bond dimension MPS. Nonetheless, this can still be efficiently simulated with strongly truncated MPS because it can be extracted from the short-time behavior of the dynamics where entanglement is low. Our results provide illustrations of scenarios where accurate calculation of the full many-body state (microstate) is intractable due to the volume-law growth of entanglement, yet a precise specification of an exact microstate may not be required when simulating macroproperties that play a role in phases of matter of many-body systems. We also study the tractability of simulation using truncated MPS based on quantum chaos and equilibration in the models. We find a counterintuitive inverse relationship, whereby local expectation values are most easily approximated for chaotic systems whose exact many-body state is most intractable.
Abstract We study the tractability of classically simulating critical phenomena in the quench dynamics of one-dimensional transverse field Ising models (TFIMs) using highly truncated matrix product states (MPSs). We focus on two paradigmatic examples: a dynamical quantum phase transition (DQPT) that occurs in nonintegrable long-range TFIMs, and the infinite-time correlation length of the integrable nearest-neighbor TFIM when quenched to the critical point, where the quantities of interest involve equal time one- and two- point correlation functions, which we associate with macroproperties. For the DQPT, we show that the order parameters can be efficiently simulated with heavy truncation of the MPS bond dimension. This can be used to reliably extract critical properties of the phase transition, including critical exponents, even when the full many-body state is not simulated with high fidelity. The long-time correlation length near the critical point is more sensitive to the full many-body state fidelity, and generally requires a large bond dimension MPS. Nonetheless, this can still be efficiently simulated with strongly truncated MPS because it can be extracted from the short-time behavior of the dynamics where entanglement is low. Our results provide illustrations of scenarios where accurate calculation of the full many-body state (microstate) is intractable due to the volume-law growth of entanglement, yet a precise specification of an exact microstate may not be required when simulating macroproperties that play a role in phases of matter of many-body systems. We also study the tractability of simulation using truncated MPS based on quantum chaos and equilibration in the models. We find a counterintuitive inverse relationship, whereby local expectation values are most easily approximated for chaotic systems whose exact many-body state is most intractable.
Ising model has been successful in describing ferromagnetism and its phase transition to paramagnet. At the critical point, the free energy density function and its derivatives diverge. Their behaviour near the critical point are described by power-laws with associated critical exponents. In many cases, the critical exponents can be determined from analytic solutions via conventional renormalization groups methods or from Monte Carlo simulations. However, for quantum many-body systems, very few are tractable to analytical solutions. The quantum many-body wavefunction belongs to large dimensional Hilbert space that increases exponentially with system size. If the Hamiltonian is gapped and only local interaction is considered, then the wavefunction can be efficiently truncated. Tensor Network formalism provides a scheme to truncate the less important degrees of freedom via Singular Value Decomposition (SVD) of the density matrix. In this study, we investigated the thermodynamic properties and phase transition of one-dimensional transverse-field quantum Ising model (1D-tQIM) under the finite-size effect and random coupling strength. Starting with Matrix Product States (MPS) as a wavefunction ansatz, the Density Matrix Renormalization Group algorithm is applied to the MPS. The variational algorithm, which iteratively performs SVDs and truncation at each bond, approximates the ground state MPS wavefunction. All quantum observables are calculated from the contraction of the resultant ground state MPS. Although theoretically, divergence at critical points only happen in an infinite system, one can obtain the critical exponents through simulation of finite-size 1D-tQIM. Using the analytic solution as a benchmark, we compared the finite-size effects of the system using finite-size scaling analysis and MPS methods. The critical exponents of 1D-tQIM are independently calculated and compared with the analytical results. Thermodynamic quantities such as magnetization, susceptibility and correlation function are calculated for system sizes of 20, 40, 60, 80, 100 and 120 spins. We determined the respective critical exponents:
Out-of-equilibrium dynamics of non-integrable Hamiltonian many-body quantum systems are characterized by highly entangled wave functions. Near-maximal entanglement arises in systems exhibiting thermalization or pre-thermalization, where the system converges to a steady state with a fixed energy density. Classical simulation of the time dependence of such wave functions requires exponential resources. However, typical computations aim to estimate expectation values of local operators and correlation functions to some expected precision. For thermalizing systems at sufficiently high energy densities, such computations can be done without storing the full wave function by instead simulating the evolution of the local operator, which requires significantly fewer resources. Nonetheless, constructing such resource-efficient classical algorithms remains a challenge for intermediate energy densities, where simulating both the wave function and operator evolution is costly. In this paper, we propose a heuristic approach to accelerate the convergence of Matrix Product State (MPS) simulations of expectation values, applicable across a broad range of energy densities. We estimate the desired observables by rescaling the MPS results at low bond dimensions with a factor that depends on the fidelity of the MPS wave function. Using this technique, we simulated the dynamics of the two-dimensional Transverse-Field Ising Model (TFIM) on a 7×8 grid with periodic boundary conditions, using a maximum bond dimension of χ=4096 on a single A100 GPU, as well as the dynamics of the two-dimensional XY model on grids of size up to 9×9. We compare our TFIM results to similar simulations on a digital quantum processor [R. Haghshenas et al., Nature 653, 56 (2026)], demonstrating excellent agreement and confirming the predictive power of our method.
Theoretical understanding of strongly correlated systems in one spatial dimension has been greatly advanced by the density-matrix renormalization group (DMRG) algorithm, which is a variational approach using a class of entanglement-restricted states called matrix product states (MPSs). However, DMRG suffers from inherent accuracy restrictions when multiple states are involved due to multi-state targeting and also the approximate representation of the Hamiltonian and other operators. By formulating the variational approach of DMRG explicitly for MPSs one can avoid errors inherent in the multi-state targeting approach. Furthermore, by using the matrix product operator (MPO) formalism, one can exactly represent the Hamiltonian and other operators relevant for the calculation. The MPO approach allows one-dimensional (1D) Hamiltonians to be templated using a small set of finite state automaton rules without reference to the particular microscopic degrees of freedom. We present two algorithms which take advantage of these properties: eMPS to find excited states of 1D Hamiltonians and tMPS for the time evolution of a generic time-dependent 1D Hamiltonian. We properly account for time-ordering of the propagator such that the error does not depend on the rate of change of the Hamiltonian. Our algorithms are applicable to microscopic degrees of freedom of any variety, and do not require Hamiltonian-specialized implementation. We benchmark our algorithms with a case study of the Ising model, where the critical point is located using entanglement measures. We then study the dynamics of this model under a time-dependent quench of the transverse field through the critical point. Finally, we present studies of a dipolar, or long-range Ising model, again using entanglement measures to find the critical point and study the dynamics of a time-dependent quench through the critical point.
By means of the discrete truncated Wigner approximation, we study dynamical phase transitions arising in the steady state of transverse-field Ising models after a quantum quench. Starting from a fully polarized ferromagnetic initial condition, these transitions separate a phase with nonvanishing magnetization along the ordering direction from a disordered symmetric phase upon increasing the transverse field. We consider two paradigmatic cases, a one-dimensional long-range model with power-law interactions $\ensuremath{\propto}1/{r}^{\ensuremath{\alpha}}$ decaying algebraically as a function of distance r and a two-dimensional system with short-range nearest-neighbor interactions. In the former case, we identify dynamical phase transitions for $\ensuremath{\alpha}\ensuremath{\lesssim}2$ and we extract the critical exponents from a data collapse of the steady-state magnetization for up to 1200 lattice sites. We find identical exponents for $\ensuremath{\alpha}\ensuremath{\lesssim}0.5$, suggesting that the dynamical transitions in this regime fall into the same universality class as the nonergodic mean-field limit. The two-dimensional Ising model is believed to be thermalizing, which we also confirm using exact diagonalization for small system sizes. Thus, the dynamical transition is expected to correspond to the thermal phase transition, which is consistent with our data upon comparing to equilibrium quantum Monte Carlo simulations. We further test the accuracy of the discrete truncated Wigner approximation by comparing against numerically exact methods such as exact diagonalization, tensor network, as well as artificial neural network states and we find good quantitative agreement on the accessible time scales. Finally, our work provides an additional contribution to the understanding of the range and the limitations of qualitative and quantitative applicability of the discrete truncated Wigner approximation.
The discovery of dynamical quantum phase transitions (DQPTs) has fundamentally challenged the traditional view that phase transitions only occur in thermal equilibrium. Experimental platforms and 1D numerical methods, like matrix product states (MPS), have made great progress. However, exploring true 2D DQPTs remains difficult due to finite-size limitations and the geometric biases of quasi-1D cylinder mappings. Here, we bypass these limitations by deploying a tree tensor network (TTN) approach. This allows us to directly compute the quench dynamics of the transverse-field Ising model (TFIM) on an open 2D square lattice. Because the TTN architecture naturally mirrors 2D lattice connectivity, we can extract the global Loschmidt echo. Our simulations reveal that while deep quenches yield standard DQPTs, quenching within the ferromagnetic phase produces an anomalous dynamical response. In this regime, the rate function exhibits sharp non-analytic peaks even as the macroscopic order parameter maintains its initial sign. This decoupled behavior strongly indicates that local spin excitations drive 2D DQPTs, rather than the macroscopic domain-wall motions seen in 1D chains. These results provide a quantitative numerical baseline for understanding non-equilibrium quantum matter in higher dimensions.
The study of low-dimensional, strongly-interacting quantum systems has been of interest in the past few decades due to new and exotic physics that show great potential in the advancement of fundamental physics as well as technological applications. In order to investigate such systems, a vast number of techniques have been developed throughout the years, all of which have their respective strengths and weaknesses. In the world of one-dimensional (1D) quantum systems, the density-matrix renormalization group (DMRG) algorithm and matrix product states (MPS) have proven to be powerful and reliable tools that provide invaluable information while providing a wide range of data that support both theoretical and experimental studies. Coupled with the development and refinement of tensor networks - an ansatz for quantum many-body wavefunctions, in particular MPS, the DMRG algorithm has seen a spike of improvements that has pushed the boundaries of its application into more sophisticated and complex systems.This thesis broadly covers the study of several 1D many-body quantum systems represented by infinite MPS (iMPS) - a class of ansatz that represent translationally-invariant, 1D many-body quantum states. More specifically, it is divided into two main parts.The first part investigates the 1D topological Kondo insulator (TKI). This is an effective model designed to investigate how strong interactions between conduction electrons and localized moments result in a symmetry-protected topological (SPT) phase, specifically the Haldane phase. By studying the effect of the electron repulsion as well as the competition between the local and non-local couplings between the electrons and the local moments, it is found that a topological phase transition from the SPT phase to a topologically trivial phase occurs with the assistance of electron interaction. The properties of both topologically trivial and nontrivial groundstates were revealed via means of quantum information theoretic quantities such as the entanglement entropy, entanglement spectrum and non-local order parameters. These quantities unveil the nature, structure and the effect that the different interactions have on the groundstate properties. Besides the phases, the properties of the critical point was studied and classified. By obtaining the critical exponents and central charge, the universality class of this transition was found to be that of 1D free fermions/bosons. Furthermore, the presence of electron interactions was found not to alter this universality class.The second part of this thesis is a project that stems from the necessity to overcome the inherent limitation of an iMPS in representing a system at a cusp of a quantum phase transition (QPT), specifically, in determining the critical point and exponents. In the thermodynamic limit, the laws of statistical mechanics dictates that the system’s correlation length diverges at the critical point. However, when representing a critical state with an iMPS, the finite bond dimension implies that the correlation length of the iMPS remains finite, albeit large. This presents a fundamental challenge in extracting critical data since the critical exponents are defined for systems in the thermodynamic limit. To overcome this shortcoming, finite-entanglement scaling (FES) - the analog of finite-size scaling, has previously been used to locate the critical point and extract the critical exponents for systems represented by finite-bond dimensional iMPS. While FES serves its purpose sufficiently well, the computational cost is high for just about any moderately complex, real-world system. To overcome such difficulties, the second part of this thesis combines new ideas of using higher-order cumulants of the order parameter with previously-known iMPS techniques to determine the critical point and extract the exponents, thus relating statistical mechanics of QPTs and tensor networks of critical states. Some of the main methods used and developed here are FES, the application of the Binder cumulant in iMPS and the new cumulant exponent relation. Putting these tools together provides a scheme that can be used to efficiently locate the critical point and extract the critical exponents at a lower computational cost than previously-known methods. This is shown through comparisons of several exemplary models such as the 1D and quasi-1D transverse field Ising model, the 1D TKI, the S = 1 Heisenberg chain with single-ion anisotropy and the 1D Bose-Hubbard model.