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Journal of Chemical Theory and Computation

Spin-Projected Matrix Product States: Versatile Tool for Strongly Correlated Systems

Zhendong Li, Garnet Kin‐Lic Chan

Published May 3, 2017• 62 citations
Publisher

Abstract

We present a new wavefunction ansatz that combines the technique of spin projection with the language of matrix product states (MPS) and matrix product operators (MPO) as used in the density matrix renormalization group (DMRG). Specifically, spin-projected matrix product states (SP-MPS) are constructed as |Ψ^(N,S,M)_SP−MPS⟩=P_S|Ψ(N,M)_MPS⟩, where P_S is the spin projector for total spin S and |Ψ(N,M)MPS⟩ is an MPS wavefunction with a given particle number N and spin projection M. This new ansatz possesses several attractive features: (1) It provides a much simpler route to achieve spin-adaptation (i.e. to create eigenfunctions of Ŝ^2) compared to explicitly incorporating the non-Abelian SU(2) symmetry into the MPS. In particular, since the underlying state |Ψ(N,M)MPS⟩ in the SP-MPS uses only Abelian symmetries, one does not need the singlet embedding scheme for non-singlet states, as normally employed in spin-adapted DMRG, to achieve a single consistent variationally optimized state. (2) SP-MPS are closely connected to broken symmetry mean-field states. This allows to straightforwardly generate the large number of broken symmetry guesses needed to explore complex electronic landscapes in magnetic systems. Further, this connection can be exploited in the future development of quantum embedding theories for open-shell systems. (3) The sum of MPOs representation for the Hamiltonian and spin projector P_S naturally leads to an embarrassingly parallel algorithm for computing expectation values and optimizing SP-MPS. (4) Optimizing SP-MPS belongs to the variation-after-projection (VAP) class of spin projected theories. Unlike usual spin projected theories based on determinants, the SP-MPS ansatz can be made essentially exact simply by increasing the bond dimensions in |Ψ(N,M)MPS⟩. Computing excited states is also simple by imposing orthogonality constraints, which are simple to implement with MPS. To illustrate the versatility of SP-MPS, we formulate algorithms for the optimization of ground and excited states, develop perturbation theory based on SP-MPS, and describe how to evaluate spin-independent and spin-dependent properties such as the reduced density matrices. We demonstrate the numerical performance of SP-MPS with applications to several models typical of strong correlation, including the Hubbard model, and [2Fe-2S] and [4Fe-4S] model complexes.

DOI
10.1021/acs.jctc.7b00270

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We present a new wavefunction ansatz that combines the technique of spin projection with the language of matrix product states (MPS) and matrix product operators (MPO) as used in the density matrix renormalization group (DMRG). Specifically, spin-projected matrix product states (SP-MPS) are constructed as |Ψ^(N,S,M)_SP−MPS⟩=P_S|Ψ(N,M)_MPS⟩, where P_S is the spin projector for total spin S and |Ψ(N,M)MPS⟩ is an MPS wavefunction with a given particle number N and spin projection M. This new ansatz possesses several attractive features: (1) It provides a much simpler route to achieve spin-adaptation (i.e. to create eigenfunctions of Ŝ^2) compared to explicitly incorporating the non-Abelian SU(2) symmetry into the MPS. In particular, since the underlying state |Ψ(N,M)MPS⟩ in the SP-MPS uses only Abelian symmetries, one does not need the singlet embedding scheme for non-singlet states, as normally employed in spin-adapted DMRG, to achieve a single consistent variationally optimized state. (2) SP-MPS are closely connected to broken symmetry mean-field states. This allows to straightforwardly generate the large number of broken symmetry guesses needed to explore complex electronic landscapes in magnetic systems. Further, this connection can be exploited in the future development of quantum embedding theories for open-shell systems. (3) The sum of MPOs representation for the Hamiltonian and spin projector P_S naturally leads to an embarrassingly parallel algorithm for computing expectation values and optimizing SP-MPS. (4) Optimizing SP-MPS belongs to the variation-after-projection (VAP) class of spin projected theories. Unlike usual spin projected theories based on determinants, the SP-MPS ansatz can be made essentially exact simply by increasing the bond dimensions in |Ψ(N,M)MPS⟩. Computing excited states is also simple by imposing orthogonality constraints, which are simple to implement with MPS. To illustrate the versatility of SP-MPS, we formulate algorithms for the optimization of ground and excited states, develop perturbation theory based on SP-MPS, and describe how to evaluate spin-independent and spin-dependent properties such as the reduced density matrices. We demonstrate the numerical performance of SP-MPS with applications to several models typical of strong correlation, including the Hubbard model, and [2Fe-2S] and [4Fe-4S] model complexes.
We present a new wavefunction ansatz that combines the strengths of spin projection with the language of matrix product states (MPS) and matrix product operators (MPO) as used in the density matrix renormalization group (DMRG). Specifically, spin-projected matrix product states (SP-MPS) are constructed as ∣ΨSP−MPS(N,S,M)⟩=PS∣ΨMPS(N,M)⟩|Ψ^{(N,S,M)}_{SP-MPS}\rangle=\mathcal{P}_S|Ψ_{MPS}^{(N,M)}\rangle∣ΨSP−MPS(N,S,M)​⟩=PS​∣ΨMPS(N,M)​⟩, where PS\mathcal{P}_SPS​ is the spin projector for total spin SSS and ∣ΨMPS(N,M)⟩|Ψ_{MPS}^{(N,M)}\rangle∣ΨMPS(N,M)​⟩ is an MPS wavefunction with a given particle number NNN and spin projection MMM. This new ansatz possesses several attractive features: (1) It provides a much simpler route to achieve spin-adaptation (i.e. to create eigenfunctions of S^2\hat{S}^2S^2) compared to explicitly incorporating the non-Abelian SU(2) symmetry into the MPS. In particular, since the underlying state in the SP-MPS uses only Abelian symmetries, one does not need the singlet embedding scheme for non-singlet states, as normally employed in spin-adapted DMRG, to achieve a single consistent variationally optimized state. (2) Due to the use of ∣ΨMPS(N,M)⟩|Ψ_{MPS}^{(N,M)}\rangle∣ΨMPS(N,M)​⟩ as its underlying state, the SP-MPS can be closely connected to broken symmetry mean-field states. This allows to straightforwardly generate the large number of broken symmetry guesses needed to explore complex electronic landscapes in magnetic systems. Further, this connection can be exploited in the future development of quantum embedding theories for open-shell systems. (3) The sum of MPOs representation for the Hamiltonian and spin projector PS\mathcal{P}_SPS​ naturally leads to an embarrassingly parallel algorithm for computing expectation values and optimizing SP-MPS. (4) Optimizing SP-MPS belongs to the variation-after-projection (VAP) class of spin projected theories.
During the past 15 years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. The underlying matrix product state (MPS) ansatz is a low-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS controls the size of the corner of the many-body Hilbert space that can be reached. Whereas the MPS ansatz will only yield an efficient description for noncritical one-dimensional systems, it can still be used as a variational ansatz for other finite-size systems. Rather large virtual dimensions are then required. The two most important aspects to reduce the corresponding computational cost are a proper choice and ordering of the active space orbitals, and the exploitation of the symmetry group of the Hamiltonian. By taking care of both aspects, DMRG becomes an efficient replacement for exact diagonalization in quantum chemistry. DMRG and Hartree-Fock theory have an analogous structure. The former can be interpreted as a self-consistent mean-field theory in the DMRG lattice sites, and the latter in the particles. It is possible to build upon this analogy to introduce post-DMRG methods. Based on an approximate MPS, these methods provide improved ansätze for the ground state, as well as for excitations. Exponentiation of the single-particle (single-site) excitations for a Slater determinant (an MPS with open boundary conditions) leads to the Thouless theorem for Hartree-Fock theory (DMRG), an explicit nonredundant parameterization of the entire manifold of Slater determinants (MPS wavefunctions). This gives rise to the configuration interaction expansion for DMRG. The Hubbard-Stratonovich transformation lies at the basis of auxiliary field quantum Monte Carlo for Slater determinants. An analogous transformation for spin-lattice Hamiltonians allows to formulate a promising variant for MPSs.
We describe how to efficiently construct the quantum chemical Hamiltonian operator in matrix product form. We present its implementation as a density matrix renormalization group (DMRG) algorithm for quantum chemical applications. Existing implementations of DMRG for quantum chemistry are based on the traditional formulation of the method, which was developed from the point of view of Hilbert space decimation and attained higher performance compared to straightforward implementations of matrix product based DMRG. The latter variationally optimizes a class of ansatz states known as matrix product states, where operators are correspondingly represented as matrix product operators (MPOs). The MPO construction scheme presented here eliminates the previous performance disadvantages while retaining the additional flexibility provided by a matrix product approach, for example, the specification of expectation values becomes an input parameter. In this way, MPOs for different symmetries - abelian and non-abelian - and different relativistic and non-relativistic models may be solved by an otherwise unmodified program.
We present an implementation of the relativistic quantum-chemical density matrix renormalization group (DMRG) approach based on a matrix-product formalism. Our approach allows us to optimize matrix product state (MPS) wave functions including a variational description of scalar-relativistic effects and spin–orbit coupling from which we can calculate, for example, first-order electric and magnetic properties in a relativistic framework. While complementing our pilot implementation ( Knecht, S. J. Chem. Phys. 2014, 140, 041101 ), this work exploits all features provided by its underlying nonrelativistic DMRG implementation based on an matrix product state and operator formalism. We illustrate the capabilities of our relativistic DMRG approach by studying the ground-state magnetization, as well as current density of a paramagnetic f 9 dysprosium complex as a function of the active orbital space employed in the MPS wave function optimization.
Neural-network quantum states (NNQS) provide a flexible variational framework for many-electron wave functions, but their performance in quantum chemistry depends strongly on whether the ansatz encodes the physical structure of the electronic Hilbert space. In this work, we introduce symmetry-blocked matrix product states (MPS) as neural-network quantum-state ansatzes, denoted QiankunNet-MPS, for ab initio quantum chemistry, explicitly enforcing the U(1) ⊗ U(1) particle-number symmetry of electronic Hamiltonians to eliminate unphysical configurations and reduce the number of variational parameters. We further develop batched autoregressive sampling for canonical MPS representations and a two-site sweeping optimization scheme that combines stochastic energy gradients with singular value decomposition (SVD)-based bond adaptation. Benchmarks on small molecules show ground-state energies comparable with density matrix renormalization group (DMRG) at the same bond dimensions, while Fe2S2 calculations illustrate how DMRG-initialized bond expansion and variance extrapolation can be used to assess the large-bond-dimension trend in a strongly correlated transition-metal active space.