Projector formalism for kept and discarded spaces of matrix product states

Abstract

Any matrix product state $|\Psi ⟩$ has a set of associated kept and discarded spaces, needed for the description of $|\Psi ⟩$, and changes thereof, respectively. These induce a partition of the full Hilbert space of the system into mutually orthogonal spaces of irreducible $n$-site variations of $|\Psi ⟩$. Here, we introduce a convenient projector formalism and diagrammatic notation to characterize these $n$-site spaces explicitly. This greatly facilitates the formulation of MPS algorithms that explicitly or implicitly employ discarded spaces. As an illustration, we derive an explicit expression for the $n$-site energy variance and evaluate it numerically for a model with long-range hopping. We also describe an efficient algorithm for computing low-lying $n$-site excitations above a finite MPS ground state.

arXiv comment14 pages, 2 figures

References in corpus (33)

• The density-matrix renormalization group in the age of matrix product states
• Efficient classical simulation of slightly entangled quantum computations
• Efficient simulation of one-dimensional quantum many-body systems
• Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• Time-dependent variational principle for quantum lattices
• Unifying time evolution and optimization with matrix product states
• Time-evolution methods for matrix-product states
• Real-time dynamics in Quantum Impurity Systems: A Time-dependent Numerical Renormalization Group Approach
• Sum-rule Conserving Spectral Functions from the Numerical Renormalization Group
• Dynamical density-matrix renormalization-group method
• Dynamical Correlation Functions using the Density Matrix Renormalization Group
• Variational optimization algorithms for uniform matrix product states
• Non-abelian symmetries in tensor networks: a quantum symmetry space approach
• A Numerical Renormalization Group approach to Green's Functions for Quantum Impurity Models
• Density matrix renormalization group algorithms with a single center site
• Post-Matrix Product State Methods: To tangent space and beyond
• Tangent-space methods for uniform matrix product states
• Density matrix algorithm for the calculation of dynamical properties of low dimensional systems
• Time integration of tensor trains
• A Strictly Single-Site DMRG Algorithm with Subspace Expansion
• Variational matrix product ansatz for dispersion relations
• Chebyshev matrix product state approach for spectral functions
• Elementary excitations in gapped quantum spin systems
• Tensor networks and the numerical renormalization group
• Error estimates for extrapolations with matrix-product states
• X-Symbols for Non-Abelian Symmetries in Tensor Networks
• Multipoint correlation functions: spectral representation and numerical evaluation
• Tensor network representations of parton wave functions
• Efficient MPS methods for extracting spectral information on rings and cylinders
• Exact dynamical structure factor of the degenerate Haldane-Shastry model
• Generating Function for Tensor Network Diagrammatic Summation
• Exact Dynamics of the SU(K) Haldane-Shastry Model

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• Searching for superconductivity in doped triangular lattice Kitaev magnets
• Thermal Tensor Network Simulations of Lattice Fermions with Fixed Filling