Tensor network states and parton wave functions are two pivotal methods for studying quantum many-body systems. This work connects these two subjects as we demonstrate that a variety of parton wave functions, such as projected Fermi sea and projected fermionic or bosonic paired states, can be represented exactly as tensor networks. The results can be compressed into matrix product states with moderate bond dimensions so various physical quantities can be computed efficiently. For the projected Fermi sea, we develop an excellent compression scheme with high fidelity using maximally localized Wannier orbitals. Numerical calculations on two parton wave functions demonstrate that our method exceeds commonly adopted Monte Carlo methods in some aspects. It produces energy and correlation function with very high accuracy that is difficult to achieve using Monte Carlo method. The entanglement measures that were almost impossible to compute before can also be obtained easily using our method.
Tensor network states and parton wave functions are two pivotal methods for studying quantum many-body systems. This work connects these two subjects as we demonstrate that a variety of parton wave functions, such as projected Fermi sea and projected fermionic or bosonic paired states, can be represented exactly as tensor networks. The results can be compressed into matrix product states with moderate bond dimensions so various physical quantities can be computed efficiently. For the projected Fermi sea, we develop an excellent compression scheme with high fidelity using maximally localized Wannier orbitals. Numerical calculations on two parton wave functions demonstrate that our method exceeds commonly adopted Monte Carlo methods in some aspects. It produces energy and correlation function with very high accuracy that is difficult to achieve using Monte Carlo method. The entanglement measures that were almost impossible to compute before can also be obtained easily using our method.
Recent work by Wu et al. [arXiv:1910.11011] proposed a numerical method, so-called matrix product operator--matrix product state (MPO-MPS) method, by which several types of quantum many-body wave functions, in particular, the projected Fermi sea state, can be efficiently represented as a tensor network. In this paper, we generalize the MPO-MPS method to study Gutzwiller projected paired states of fermions, where the maximally localized Wannier orbitals for Bogoliubov quasiparticles/quasiholes have been adapted to improve the computational performance. The study of SO(3)-symmetric spin-1 chains reveals that this new method has better performance than variational Monte Carlo for gapped states and similar performance for gapless states. Moreover, we demonstrate that dynamic correlation functions can be easily evaluated by this method cooperating with other MPS-based accurate approaches, such as the Chebyshev MPS method.
Projected wave functions offer a means of incorporating local correlation effects in gapless electronic phases of matter, like metals. Although such wave functions can be readily specified formally, computing their associated physical observables is challenging. Tensor network approaches offer a modern numerical method for this task. In this paper, we develop and demonstrate a constructive tensor-network approach for obtaining physical quantities, like fermion two-point functions and density-density correlation functions, for one-dimensional projected Fermi sea states. We benchmark our method against exact analytical results for spin- 1 / 2 electrons subjected to the Gutzwiller projection, and then present results on spinless fermions with nearest-neighbor repulsion. For a state with two pairs of Fermi points, we reveal a correlation-dependent tuning of the characteristic wave vector of the charge density modulation in the system.
We propose an approach to study the ground state of quantum many-body systems in which tensor network states, specifically projected entangled pair states (PEPSs), and Green's function Monte Carlo (GFMC) are combined. PEPSs, by design, encode the area law which governs the scaling of entanglement entropy in quantum systems with short-range interactions but are hindered by the high computational complexity scaling with bond dimension D. GFMC is a highly efficient method, but it usually suffers from the infamous negative sign problem which can be avoided by the fixed-node approximation in which a guiding wave function is utilized to modify the sampling process. The trade-off for the absence of the negative sign problem is the introduction of systematic error by the guiding wave function. In this work, we combine these two methods, PEPS and GFMC, to take advantage of both of them. PEPSs are very accurate variational wave functions, while at the same time, only contractions of the single-layer tensor network are necessary in GFMC, which reduces the cost substantially. Moreover, energy obtained in GFMC is guaranteed to be variational and lower than the variational energy of the guiding PEPS wave function. Benchmark results of the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice are provided.
These notes provide an introduction to tensor-network methods in quantum many-body physics, with an emphasis on matrix-product states (MPS). They develop the basic tensor-network language, including graphical notation, virtual indices, bond dimensions, gauge freedom, canonical forms, QR and singular-value decompositions, and the role of entanglement in controlling the efficiency of the representation. The main MPS algorithms are then introduced, including contractions, correlation functions, matrix-product operators, DMRG, and time-evolution methods. The notes also briefly discuss projected entangled-pair states (PEPS) as a higher-dimensional generalization of MPS, together with the basic ideas behind approximate PEPS contraction. Finally, tensor-network representations of mixed states, quantum channels, and Lindblad dynamics are presented, with applications to thermal states and open quantum systems. The presentation is accompanied by short Julia code examples based on ITensor, ITensorMPS, and TensorMixedStates. These notes were written for the 9th Les Houches Summer School on Computational Physics: Open Quantum Systems, held in June 2026.
The study of interacting quantum many-body systems is one of the central challenges in modern physics. Interestingly, by understanding the physical properties of such systems from a microscopic description, one can explain many complex quantum phenomena such as the high-temperature superconductivity, fractional quantum Hall effect, etc. However, due to high complexity of these systems, theoretical investigations such as analytical and perturbation theories are limited to studying only a small subset of problems. In the past few decades, numerical simulation has emerged as an indispensable method for studying many phenomena, and has led to immense success and progress in recent years. The main obstacle in addressing quantum many-body systems is that the number of parameters required to represent a state of the system scales exponentially with the system size. However, one hardly needs the full Hilbert space but only a small subspace to describe physical properties of the system. From the point of view in quantum information theory, quantum entanglement is one of the main factors that is responsible for the complexity of studying quantum many-body systems. In particular, the amount of quantum entanglement in such system quantifies the correlation between constituents and therefore it determines how big the subspace is required to represent the system. In this thesis, we are interested in using tensor network states, a new emerging numerical technique to study dynamical and ground state properties of the system. The key ingredient is that a tensor network state which consists of a fairly small number of parameters can represent very well the complete wave function of a system. In one spatial dimension, Matrix Product State (MPS) has become a standard tool, combined with related algorithms such as Time Evolving Block Decimation algorithm (TEBD) to study dynamical properties of a locally-perturbed quantum lattice system. In this thesis, we introduce a technique called ``infinite boundary conditions'' that allows us to use a finite-size MPS to represent the state of an infinite-size system. As a result, one can simulate dynamics of a system for a longer time with a significant computational cost reduction compared to traditional methods. In two spatial dimensions, the tensor network known as Projected Entangled Pair States (PEPS) is emerging as a useful approach. However algorithms for optimizing the matrix elements of these tensors are complex and not very efficient. In particular, the efficiency of the optimization depends critically on the quality of the so-called ``environment'', which is expensive to compute. We have introduced a new method to accelerate the convergence of calculating the PEPS ground state by recycling the environment tensors for many optimization steps, vastly reducing the number of times that the full environment needs to be calculated.
We propose an efficient numerical method, which combines the advantages of recently developed tensor-network based methods and standard trial wave functions, to study the ground-state properties of quantum many-body systems. In this approach, we apply a projector in the form of a tensor-product operator to an input wave function, such as a Jastrow-type or Hartree-Fock wave function, and optimize the tensor elements via variational Monte Carlo. The entanglement already contained in the input wave function can considerably reduce the bond dimensions compared to the regular tensor-product state representation. In particular, this allows us to also represent states that do not obey the area law of entanglement entropy. In addition, for fermionic systems, the fermion sign structure can be encoded in the input wave function. We show that the optimized states provide good approximations of the ground-state energy and correlation functions in the cases of two-dimensional bosonic and fermonic systems.