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Cambridge University Press eBooks

Continuous Matrix Product States

Tao Xiang

Published Aug 31, 2023• 0 citations
Publisher

Abstract

This chapter constructs the MPS representation of a quantum state in the continuous limit. It starts with an MPS representation for the corresponding state in a discretized lattice system. Then the limit of the discretized lattice constant going to zero is taken to obtain its continuous presentation. The formulas for determining the expectation values are also derived. Finally, we discuss the scheme of canonicalization and the method for optimizing the local tensors of the continuous MPS.

DOI
10.1017/9781009398671.014

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This chapter constructs the MPS representation of a quantum state in the continuous limit. It starts with an MPS representation for the corresponding state in a discretized lattice system. Then the limit of the discretized lattice constant going to zero is taken to obtain its continuous presentation. The formulas for determining the expectation values are also derived. Finally, we discuss the scheme of canonicalization and the method for optimizing the local tensors of the continuous MPS.
By combining the continuous matrix product state (cMPS) representation for quantum fields in the continuum with standard optimization techniques for matrix product states (MPS) on the lattice, we obtain an approximation $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$, directly in the continuum, of the ground state of nonrelativistic quantum field theories. This construction works both for translation-invariant systems and in the more challenging context of inhomogeneous systems, as we demonstrate for an interacting bosonic field in a periodic potential. Given the continuum Hamiltonian HHH, we consider a sequence of discretized Hamiltonians ${{H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})}}_{\ensuremath{\alpha}=1,2,\ensuremath{\cdots},p}$ on increasingly finer lattices with lattice spacing ${\ensuremath{\epsilon}}_{1}>{\ensuremath{\epsilon}}_{2}>\ensuremath{\cdots}>{\ensuremath{\epsilon}}_{p}$. We first use energy minimization to optimize an MPS approximation $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{1})\ensuremath{\rangle}$ for the ground state of $H({\ensuremath{\epsilon}}_{1})$. Given the MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})\ensuremath{\rangle}$ optimized for the ground state of $H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})$, we use it to initialize the energy minimization for Hamiltonian $H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}+1})$, resulting in the optimized MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{\ensuremath{\alpha}+1})\ensuremath{\rangle}$. By iteration we produce an optimized MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{p})\ensuremath{\rangle}$ for the ground state of $H({\ensuremath{\epsilon}}_{p})$, from which we finally extract the cMPS approximation $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$ for the ground state of HHH. Two key ingredients of our proposal are as follows: (i) a procedure to discretize HHH into a lattice model where each site contains a two-dimensional vector space (spanned by vacuum $|0\ensuremath{\rangle}$ and one boson $|1\ensuremath{\rangle}$ states), and (ii) a procedure to map MPS representations from a coarser lattice to a finer lattice.
This chapter discusses the properties of matrix product state (MPS). It starts with a simple proof that the wave function generated by DMRG is an MPS. Then three different but equivalent canonical forms or representations of MPS are introduced. An MPS generally has redundant gauge degrees of freedom on each bond linking two neighboring local tensors. One can convert it into a canonical form by taking a canonical transformation to remove the gauge redundancy in the local tensors. Finally, the implementation of symmetries, including both the U(1) and SU(2) symmetries, is discussed.
This chapter discusses the properties of infinite MPS and their associated transfer matrices. The formulas for determining the expectation values of physical observables are derived and expressed using the leading eigenvectors of the transfer matrix. The concept of the string order parameter is introduced and exemplified with the AKLT state, followed by a statement on the condition for the existence of string order. Furthermore, the procedure of canonicalizing an infinite MPS with one or more than one site in a unit cell is discussed.
We discuss various properties of the variational class of continuous matrix product states, a class of Ansatz states for one-dimensional quantum fields that was recently introduced as the direct continuum limit of the highly successful class of matrix product states. We discuss both attributes of the physical states, e.g., by showing in detail how to compute expectation values, as well as properties intrinsic to the representation itself, such as the gauge freedom. We consider general translation noninvariant systems made of several particle species and derive certain regularity properties that need to be satisfied by the variational parameters. We also devote a section to the translation invariant setting in the thermodynamic limit and show how continuous matrix product states possess an intrinsic ultraviolet cutoff. Finally, we introduce a new set of states, which are tangent to the original set of continuous matrix product states. For the case of matrix product states, this construction has recently proven relevant in the development of new algorithms for studying time evolution and elementary excitations of quantum spin chains. We thus lay the foundation for similar developments for one-dimensional quantum fields.
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.