The contraction of tensor networks is a central task in the application of tensor network methods to the study of quantum and classical many-body systems. In this paper, we investigate the impact of gauge degrees of freedom in the virtual indices of the tensor network on the contraction process, specifically focusing on boundary matrix product state methods for contracting two-dimensional tensor networks. We show that the gauge transformation can affect the entanglement structures of the eigenstates of the transfer matrix and change how the physical information is encoded in the eigenstates, which can influence the accuracy of the numerical simulation. We demonstrate this effect by looking at two different examples. First, we focus on the local gauge transformation, and analyze its effect by viewing it as an imaginary-time evolution governed by a diagonal Hamiltonian. As a specific example, we perform a numerical analysis in the classical Ising model on the square lattice. Second, we go beyond the scope of local gauge transformations and study the antiferromagnetic Ising model on the triangular lattice. The partition function of this model has two tensor network representations connected by a nonlocal gauge transformation, resulting in distinct numerical performances in the boundary matrix product state calculation.
The contraction of tensor networks is a central task in the application of tensor network methods to the study of quantum and classical many-body systems. In this paper, we investigate the impact of gauge degrees of freedom in the virtual indices of the tensor network on the contraction process, specifically focusing on boundary matrix product state methods for contracting two-dimensional tensor networks. We show that the gauge transformation can affect the entanglement structures of the eigenstates of the transfer matrix and change how the physical information is encoded in the eigenstates, which can influence the accuracy of the numerical simulation. We demonstrate this effect by looking at two different examples. First, we focus on the local gauge transformation, and analyze its effect by viewing it as an imaginary-time evolution governed by a diagonal Hamiltonian. As a specific example, we perform a numerical analysis in the classical Ising model on the square lattice. Second, we go beyond the scope of local gauge transformations and study the antiferromagnetic Ising model on the triangular lattice. The partition function of this model has two tensor network representations connected by a nonlocal gauge transformation, resulting in distinct numerical performances in the boundary matrix product state calculation.
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.
We adapt the bialgebra and Hopf relations to expose internal structure in the ground state of a Hamiltonian with Z 2 topological order. Its tensor network description allows for exact contraction through simple diagrammatic rewrite rules. The contraction property does not depend on specifics such as geometry, but rather originates from the non-trivial algebraic properties of the constituent tensors. We then generalise the resulting tensor network from a spin-1/2 lattice to a class of exactly contractible states on spin- S degrees of freedom, yielding the most efficient tensor network description of finite Abelian lattice gauge theories. We gain a new perspective on these states as examples of two-dimensional quantum states with algebraically contractible tensor network representations. The introduction of local perturbations to the network is shown to reduce the von Neumann entropy of string-like regions, creating an unentangled sub-system within the bulk in a certain limit. We also show how local perturbations induce finite-range correlations in this system. This class of tensor networks is readily translated onto any lattice, and we differentiate between the physical consequences of bipartite and non-bipartite lattices on the properties of the corresponding quantum states. We explicitly show this on the hexagonal, square, kagome and triangular lattices.
This dissertation contributes to the ongoing effort of understanding the origins and applications of real-space renormalization group methods in tensor network representations of classical and quantum many-body systems. First, we construct a matrix product operator ansatz to coarse-grain real-space transfer matrices of matrix product state descriptions of one-dimensional quantum spin chains. By treating the physical spin as an impurity, we unravel the virtual entanglement degrees of freedom of matrix product states into a layered structure to reveal an inherent renormalization group scale. Secondly, we rephrase tensor network renormalization for two-dimensional classical lattice models in a manifestly nonnegative way. The resulting real-space renormalization group flow preserves positivity and hence yields an interpretation in terms of Hamiltonian flows, reconciling modern real-space tensor network renormalization methods with traditional block-spin approaches. Thirdly, we study non-local symmetries in tensor networks by expressing two-dimensional classical partition functions in terms of strange correlators of judiciously chosen product states and string-net wave functions. We exhibit and exploit the emerging non-local symmetries of the partition function at criticality and highlight parallels between topological sectors and conformal primary fields in the shared framework of matrix product operator symmetries. Additionally, we provide a complementary perspective on real-space renormalization by recognizing known tensor network renormalization methods as the approximate truncation of an exactly coarse-grained strange correlator.
Frustration is a ubiquitous phenomenon in many-body physics that influences the nature of the system in a profound way with exotic emergent behavior. Despite its long research history, the analytical or numerical investigations on frustrated spin models remain a formidable challenge due to their extensive ground-state degeneracy. In this paper, we propose a unified tensor network theory to numerically solve the frustrated classical spin models on various two-dimensional (2D) lattice geometry with high efficiency. We show that the appropriate encoding of emergent degrees of freedom in each local tensor is of crucial importance in the construction of the infinite tensor network representation of the partition function. The frustrations are thus relieved through the effective interactions between emergent local degrees of freedom. Then the partition function is written as a product of a one-dimensional (1D) transfer operator, whose eigenequation can be solved by the standard algorithm of matrix product states rigorously, and various phase transitions can be accurately determined from the singularities of the entanglement entropy of the 1D quantum correspondence. We demonstrated the power of our general theory by numerically solving 2D fully frustrated XY spin models on the kagome, square, and triangular lattices, giving rise to a variety of thermal phase transitions from infinite-order Brezinskii-Kosterlitz-Thouless transitions, second-order transitions, to first-order phase transitions. Our approach holds the potential application to other types of frustrated classical systems like Heisenberg spin antiferromagnets.
We introduce contraction tree boundary matrix product states (CT-bMPS), a scalable approximate contraction method for general closed tensor networks. CT-bMPS represents the boundary tensor and its complementary environment at each edge of a contraction tree as matrix product states. This representation allows local compression based on a reduced transition matrix to incorporate environment information along arbitrary contraction trees, while keeping the time and memory costs polynomial in the network size and bond dimension. We benchmark the method on a variety of models and applications, including Ising partition functions, random tensor networks, kicked Ising dynamics, and quantum error correction decoding. These benchmarks demonstrate improved accuracy and computational efficiency over existing general-purpose approximate contraction methods, together with the ability to handle larger and more complex networks. Optimized contraction trees and environment updates substantially improve accuracy without increasing the bond dimension. These results establish CT-bMPS as a general and scalable approximation framework with flexible contraction orders and environment-based compression for a wide range of applications.
We use the formalism of tensor network states to investigate the relation between static correlation functions in the ground state of local quantum many-body Hamiltonians and the dispersion relations of the corresponding low-energy excitations. In particular, we show that the matrix product state transfer matrix (MPS-TM)—a central object in the computation of static correlation functions—provides important information about the location and magnitude of the minima of the low-energy dispersion relation(s), and we present supporting numerical data for one-dimensional lattice and continuum models as well as two-dimensional lattice models on a cylinder. We elaborate on the peculiar structure of the MPS-TM’s eigenspectrum and give several arguments for the close relation between the structure of the low-energy spectrum of the system and the form of the static correlation functions. Finally, we discuss how the MPS-TM connects to the exact quantum transfer matrix of the model at zero temperature. We present a renormalization group argument for obtaining finite bond dimension approximations of the MPS, which allows one to reinterpret variational MPS techniques (such as the density matrix renormalization group) as an application of Wilson’s numerical renormalization group along the virtual (imaginary time) dimension of the system.