Density matrix renormalization group (DMRG) or matrix product states (MPS) is the most effective and accurate method for studying one-dimensional quantum many-body systems. However, the application of DMRG to two-dimensional systems is not as successful because of the limited entanglement encoded in the wave-function ansatz. The fully augmented matrix product states (FAMPS), introduced recently in [Chin. Phys. Lett. 40, 057102 (2023)], extends MPS formalism to two dimensions and increases the entanglement in the wave-function ansatz, representing a significant advance in the simulation of two-dimensional quantum many-body physics. In the study of one-dimensional systems, the concept of a parent Hamiltonian for MPS has proven pivotal in the understanding of quantum entanglement. In this work, we extend this framework to two-dimensional systems. We illustrate the procedure to construct a two-dimensional Hamiltonian with given FAMPS as its exact ground state (the parent Hamiltonian for FAMPS). Additionally, through numerical simulations, we demonstrate the effectiveness of the algorithm outlined in [Chin. Phys. Lett. 40, 057102 (2023)] in precisely identifying the FAMPS for the constructed parent Hamiltonian. The introduction of FAMPS and its associated parent Hamiltonian provides a useful framework for the future investigations of two-dimensional quantum many-body systems.
Density matrix renormalization group (DMRG) or matrix product states (MPS) is the most effective and accurate method for studying one-dimensional quantum many-body systems. However, the application of DMRG to two-dimensional systems is not as successful because of the limited entanglement encoded in the wave-function ansatz. The fully augmented matrix product states (FAMPS), introduced recently in [Chin. Phys. Lett. 40, 057102 (2023)], extends MPS formalism to two dimensions and increases the entanglement in the wave-function ansatz, representing a significant advance in the simulation of two-dimensional quantum many-body physics. In the study of one-dimensional systems, the concept of a parent Hamiltonian for MPS has proven pivotal in the understanding of quantum entanglement. In this work, we extend this framework to two-dimensional systems. We illustrate the procedure to construct a two-dimensional Hamiltonian with given FAMPS as its exact ground state (the parent Hamiltonian for FAMPS). Additionally, through numerical simulations, we demonstrate the effectiveness of the algorithm outlined in [Chin. Phys. Lett. 40, 057102 (2023)] in precisely identifying the FAMPS for the constructed parent Hamiltonian. The introduction of FAMPS and its associated parent Hamiltonian provides a useful framework for the future investigations of two-dimensional quantum many-body systems.
Density matrix renormalization group (DMRG) and its extensions in the form of matrix product states are arguably the choice for the study of one-dimensional quantum systems in the last three decades. However, due to the limited entanglement encoded in the wave-function ansatz, to maintain the accuracy of DMRG with the increase of the system size in the study of two-dimensional systems, exponentially increased resources are required, which limits the applicability of DMRG to only narrow systems. We introduce a new ansatz in which DMRG is augmented with disentanglers to encode area-law-like entanglement entropy (entanglement entropy supported in the new ansatz scales as l for an l × l system). In the new method, the O(D 3) low computational cost of DMRG is kept (with an overhead of O(d 4) and d the dimension of the physical degrees of freedom). We perform benchmark calculations with this approach on the two-dimensional transverse Ising and Heisenberg models. This new ansatz extends the power of DMRG in the study of two-dimensional quantum systems.
The density matrix renormalization group (DMRG) is widely acknowledged as a highly effective and accurate method for solving one-dimensional quantum many-body systems. However, the direct application of DMRG to the study of two-dimensional systems encounters challenges due to the limited entanglement encoded in the underlying wave-function Ansatz, known as the matrix product state. Conversely, Clifford circuits offer a promising avenue for simulating states with substantial entanglement, albeit confined to stabilizer states. In this work, we present the seamless integration of Clifford circuits within the DMRG algorithm, leveraging the advantages of both Clifford circuits and DMRG. This integration leads to a significant enhancement in simulation accuracy with small additional computational cost. Moreover, this framework is useful not only for its current application but also for its potential to be easily adapted to various other numerical approaches.
A key property of many-body localized Hamiltonians is the area law entanglement of even highly excited eigenstates. Matrix product states (MPS) can be used to efficiently represent low entanglement (area law) wave functions in one dimension. An important application of MPS is the widely used density matrix renormalization group (DMRG) algorithm for finding ground states of one-dimensional Hamiltonians. Here, we develop two algorithms, the shift-and-invert MPS (SIMPS) and excited state DMRG which find highly excited eigenstates of many-body localized Hamiltonians. Excited state DMRG uses a modified sweeping procedure to identify eigenstates, whereas SIMPS applies the inverse of the shifted Hamiltonian to a MPS multiple times to project out the targeted eigenstate. To demonstrate the power of these methods, we verify the breakdown of the eigenstate thermalization hypothesis in the many-body localized phase of the random field Heisenberg model, show the saturation of entanglement in the many-body localized phase, and generate local excitations.
We present an implementation of the relativistic quantum-chemical density matrix renormalization group (DMRG) approach based on a matrix-product formalism. Our approach allows us to optimize matrix product state (MPS) wave functions including a variational description of scalar-relativistic effects and spin–orbit coupling from which we can calculate, for example, first-order electric and magnetic properties in a relativistic framework. While complementing our pilot implementation ( Knecht, S. J. Chem. Phys. 2014, 140, 041101 ), this work exploits all features provided by its underlying nonrelativistic DMRG implementation based on an matrix product state and operator formalism. We illustrate the capabilities of our relativistic DMRG approach by studying the ground-state magnetization, as well as current density of a paramagnetic f 9 dysprosium complex as a function of the active orbital space employed in the MPS wave function optimization.
The simulation of strongly correlated quantum systems on classical computers is an essential tool for investigating and understanding quantum materials.Due to the inherent "curse of dimensionality" of quantum systems, such simulations pose a challenging task, and have spurred the development of several computational approaches and approximations.For quasi one-dimensional lattice systems, the density matrix renormalization group (DMRG) framework (Schollwöck, 2005;White, 1992White, , 1993) has emerged as one of the most successful methods.Modern formulations are based on matrix product states (MPS) and operators (MPO) (Schollwöck, 2011): briefly speaking, a matrix product state is an Ansatz for the quantum many-body wavefunction amendable to efficient numerical manipulation.The favorable approximation properties of matrix product states have been theoretically proven for ground states of gapped Hamiltonians with local interactions (Hastings, 2007;Verstraete & Cirac, 2006), and recent work has extended this to quantum systems at finite temperature (Barthel, 2017).Over the years many variations and generalizations of the DMRG framework have been devised (Barthel, 2013;Stoudenmire & White, 2010;Verstraete, Murg, & Cirac, 2008;Vidal, 2003Vidal, , 2008)), and further developments and improvements of associated algorithms remain an active field of research.
The density matrix renormalization group (DMRG) is one of the most powerful numerical methods available for many-body systems. It has been applied to solve many physical problems, including the calculation of ground states and dynamical properties. In this work, we develop a perturbation theory of the DMRG (PT-DMRG) to greatly increase its accuracy in an extremely simple and efficient way. Using the canonical matrix product state (MPS) representation for the ground state of the considered system, a set of orthogonal basis functions $\left\{|{\ensuremath{\psi}}_{i}\ensuremath{\rangle}\right\}$ is introduced to describe the perturbations to the ground state obtained by the conventional DMRG. The Schmidt numbers of the MPS that are beyond the bond dimension cutoff are used to define these perturbation terms. The perturbed Hamiltonian is then defined as ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}}_{ij}=\ensuremath{\langle}{\ensuremath{\psi}}_{i}|\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}|{\ensuremath{\psi}}_{j}\ensuremath{\rangle}$; its ground state permits us to calculate physical observables with a considerably improved accuracy compared to the original DMRG results. We benchmark the second-order perturbation theory with the help of a one-dimensional Ising chain in a transverse field and the Heisenberg chain, where the precision of the DMRG is shown to be improved O(10) times. Furthermore, for moderate L the errors of the DMRG and PT-DMRG both scale linearly with ${L}^{\ensuremath{-}1}$ (with L being the length of the chain). The linear relation between the dimension cutoff of the DMRG and that of the PT-DMRG at the same precision shows a considerable improvement in efficiency, especially for large dimension cutoffs. In the thermodynamic limit we show that the errors of the PT-DMRG scale with $\sqrt{{L}^{\ensuremath{-}1}}$. Our work suggests an effective way to define the tangent space of the ground-state MPS, which may shed light on the properties beyond the ground state. This second-order PT-DMRG can be readily generalized to higher orders, as well as applied to models in higher dimensions.