The density matrix renormalization group (DMRG) is a celebrated tensor network algorithm, which computes the ground states of one-dimensional quantum many-body systems very efficiently. Here we propose an improved formulation of continuous tensor network algorithms, which we name a matrix product renormalization group (MPRG). MPRG is a universal quantum many-body solver, which potentially works at both zero and finite temperatures, in two and higher dimensions, and is even applicable to open quantum systems. Furthermore, MPRG does not rely on any variational principles and thus supports any kind of non-Hermitian systems in any dimension. As a demonstration, we present critical properties of the Yang-Lee edge singularity in one dimension as a representative non-Hermitian system.
The density matrix renormalization group (DMRG) is a celebrated tensor network algorithm, which computes the ground states of one-dimensional quantum many-body systems very efficiently. Here we propose an improved formulation of continuous tensor network algorithms, which we name a matrix product renormalization group (MPRG). MPRG is a universal quantum many-body solver, which potentially works at both zero and finite temperatures, in two and higher dimensions, and is even applicable to open quantum systems. Furthermore, MPRG does not rely on any variational principles and thus supports any kind of non-Hermitian systems in any dimension. As a demonstration, we present critical properties of the Yang-Lee edge singularity in one dimension as a representative non-Hermitian system.
The density-matrix renormalization group (DMRG) method has established itself over the last decade as the leading method for the simulation of the statics and dynamics of one-dimensional strongly correlated quantum lattice systems. The DMRG is a method that shares features of a renormalization group procedure (which here generates a flow in the space of reduced density operators) and of a variational method that operates on a highly interesting class of quantum states, so-called matrix product states (MPSs). The DMRG method is presented here entirely in the MPS language. While the DMRG generally fails in larger two-dimensional systems, the MPS picture suggests a straightforward generalization to higher dimensions in the framework of tensor network states. The resulting algorithms, however, suffer from difficulties absent in one dimension, apart from a much more unfavourable efficiency, such that their ultimate success remains far from clear at the moment.
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) method has established itself over the last decade as the leading method for the simulation of the statics and dynamics of one-dimensional strongly correlated quantum lattice systems. The DMRG is a method that shares features of a renormalization group procedure (which here generates a flow in the space of reduced density operators) and of a variational method that operates on a highly interesting class of quantum states, so-called matrix product states (MPSs). The DMRG method is presented here entirely in the MPS language. While the DMRG generally fails in larger two-dimensional systems, the MPS picture suggests a straightforward generalization to higher dimensions in the framework of tensor network states. The resulting algorithms, however, suffer from difficulties absent in one dimension, apart from a much more unfavourable efficiency, such that their ultimate success remains far from clear at the moment.
Tensor network algorithms have been remarkably successful solving a variety of problems in quantum many-body physics. However, algorithms to optimize two-dimensional tensor networks known as PEPS lack many of the aspects that make the seminal density matrix renormalization group (DMRG) algorithm so powerful for optimizing one-dimensional tensor networks known as matrix product states. We implement a framework for optimizing two-dimensional PEPS tensor networks which includes all of steps that make DMRG so successful for optimizing one-dimension tensor networks. We present results for several 2D spin models and discuss possible extensions and applications.
Abstract The physical properties of a quantum many-body system can, in principle, be determined by diagonalizing the respective Hamiltonian, but the dimensions of its matrix representation scale exponentially with the number of degrees of freedom. Hence, only small systems that are described through simple models can be tackled via exact diagonalization. To overcome this limitation, numerical methods based on the renormalization group paradigm that restrict the quantum many-body problem to a manageable subspace of the exponentially large full Hilbert space have been put forth. A striking example is the density-matrix renormalization group (DMRG), which has become the reference numerical method to obtain the low-energy properties of one-dimensional quantum systems with short-range interactions. Here, we provide a pedagogical introduction to DMRG, presenting both its original formulation and its modern tensor-network-based version. This colloquium sets itself apart from previous contributions in two ways. First, didactic code implementations are provided to bridge the gap between conceptual and practical understanding. Second, a concise and self-contained introduction to the tensor-network methods employed in the modern version of DMRG is given, thus allowing the reader to effortlessly cross the deep chasm between the two formulations of DMRG without having to explore the broad literature on tensor networks. We expect this pedagogical review to find wide readership among students and researchers who are taking their first steps in numerical simulations via DMRG. Graphic abstract
The field of tensor networks, kicked off in 1992 by Steve White's invention of the spectacularly successful density matrix renormalization group (DMRG) algorithm, has exploded in popularity in recent years. Tensor networks are poised to play a role in helping us solve some of the greatest open physics problems of our time, such as understanding the nature of high-temperature superconductivity and illuminating a theory of quantum gravity. DMRG and extensions based on a class of variational states known as tensor network states have been indispensable tools in helping us understand both numerically and theoretically the properties of complicated classical and quantum many-body systems. However, practical challenges to these techniques still remain, and algorithmic developments are needed before tensor network algorithms can be applied to more physics problems. In this thesis we present a variety of recent advancements to tensor network algorithms. First we describe a DMRG-like algorithm for noninteracting fermions. Noninteracting fermions, naturally being gapless and therefore having high levels of entanglement, are actually a challenging setting for standard DMRG algorithms, and we believe this new algorithm can help with tensor network calculations in that setting. Next we explain a new algorithm called the variational uniform matrix product state (VUMPS) algorithm that is a DMRG-like algorithm that works directly in the thermodynamic limit, improving upon currently available MPS-based methods for studying infinite 1D and quasi-1D quantum many-body systems. Finally, we describe a variety of improvements to algorithms for contracting 2D tensor networks, a common problem in tensor network algorithms, for example for studying 2D classical statistical mechanics problems and 2D quantum many-body problems with projected entangled pair states (PEPS). One is a new variant of the corner transfer matrix renormalization group (CTMRG) algorithm of Nishino and Okunishi that improves the numerical stability for contracting asymmetric two-dimensional tensor networks compared to the most commonly used method. Another is the application of the VUMPS algorithm to contracting 2D tensor networks. The last is a new alternative to CTMRG, where the tensors are solved for with eigenvalue equations instead of a power method, which we call the fixed point corner method (FPCM). We present results showing the transfer matrix VUMPS algorithm and FPCM significantly improve upon the convergence time of CTMRG. We expect these algorithms will play an important role in expanding the set of 2D classical and 2D quantum many-body problems that can be addressed with tensor networks.