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Philosophical Magazine Letters

Total spin in the density matrix renormalization group algorithm

Ian P. McCulloch, M. Gulácsi

Published Jun 1, 2001• 29 citations
Publisher

Abstract

A new density matrix renormalization group (DMRG) algorithm is presented which conserves the total spin. This is the first time that standard DMRG is extended to exploit fully the symmetries of the Hamiltonian. The obtained general procedure works for any operator, even operators that do not commute with the Hamiltonian. The new method gives substantial improvements over the standard DMRG.

DOI
10.1080/09500830110040009

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Revista Mexicana de Física•Jan 1, 2001•0 citations
A new density matrix renormalization group (DMRG) algorithm is presented which conserves the total spin. This is the first time that standard DMRG is extended to exploit fully the symmetries of the Hamiltonian. The obtained general procedure works for any operator, even operators that do not commute with the Hamiltonian. The new method gives substantial improvements over the standard DMRG.
Substantial improvements in the computational effort in a density matrix renormalisation program can be made by utilising symmetries of the Hamiltonian. Extra quantum numbers are always desirable to include in the calculation, since it allows the Hilbert space of the superblock to be refined. Since the speed of the calculation is approximately O ( n 3 ) in superblock states, the speed increase in targeting a specific total spin state can be considerable. In this paper a new density matrix renormalisation algorithm is presented which conserves the total spin. The general procedure obtained works for any operator, even operators that do not commute with the Hamiltonian.
Abstract. – We describe here the extension of the density matrix renormalization group algorithm to the case where the Hamiltonian has a non-Abelian global symmetry group. The block states transform as irreducible representations of the non-Abelian group. Since the representations are multi-dimensional, a single block state in the new representation corresponds to multiple states of the original density matrix renormalization group basis. We demonstrate the usefulness of the construction via the the one-dimensional Hubbard model as the symmetry group is enlarged from U(1) × U(1), up to SU(2) × SU(2). In past years, the density matrix renormalization group (DMRG) method [1] has been extensively used to study one and two dimensional strongly correlated electron systems [2]. This method became very popular when it was realized that it enabled a level of numerical accuracy for one dimensional systems that was not possible using other methods [3]. One major drawback of DMRG is that calculations are performed in a subspace of purely Abelian symmetries, such as the U(1) symmetries of total particle number and the z component of the total spin. Thus one can only obtain a few states in different total particle
We present a spin-adapted density matrix renormalization group (DMRG) algorithm designed to target spin and spatial symmetry states that can be difficult to obtain while using a non-spin-adapted algorithm. The algorithmic modifications that have to be introduced into the usual density matrix renormalization group scheme in order to spin adapt it are discussed, and it is demonstrated that the introduced modifications do not change the overall scaling of the method. The new approach is tested on HNCO, a model system, that has a singlet-triplet curve crossing between states of the same symmetry. The advantages of the spin-adapted DMRG scheme are discussed, and it is concluded that the spin-adapted DMRG method converges better in almost all cases and gives more parallel curves to the full configuration interaction result than the non-spin-adapted method. It is shown that the spin-adapted DMRG energies can be lower than the ones obtained from the non-spin-adapted scheme. Such a counterintuitive result is explained by noting that the spin-adapted method is not a special case of the non-spin-adapted one; consequently, the spin-adapted result is not an upper bound for the non-spin-adapted energy.
We develop a density matrix renormalization group (DMRG) algorithm for constrained quantum lattice models that successfully {\it{implements the local constraints as symmetries in the contraction of the matrix product states and matrix product operators}}. Such an implementation allows us to investigate a quantum dimer model in DMRG for any lattice geometry wrapped around a cylinder with substantial circumference. We have thence computed the ground state phase diagram of the quantum dimer model on triangular lattice, with the symmetry-breaking characteristics of the columnar solid phase and 12×12\sqrt{12}\times\sqrt{12}12​×12​ valence bond solid phase fully captured, as well as the topological entanglement entropy of the Z2\mathbb{Z}_2Z2​ quantum spin liquid phase that extends to the RK point on non-bipartite lattice accurately revealed. Our DMRG algorithm on constrained quantum lattice models opens new opportunities for matrix and tensor-based algorithms for these systems that have immediate relevance towards the frustrated quantum magnets and synthetic quantum simulators.
In the nonrelativistic Schrödinger equation, the total spin S and spin projection M are good quantum numbers. In contrast, spin symmetry is lost in the presence of spin-dependent interactions, such as spin-orbit couplings in relativistic Hamiltonians. Therefore, the relativistic density matrix renormalization group algorithm (R-DMRG) only employing particle number symmetry is much more expensive than nonrelativistic DMRG. In addition, artificial breaking of Kramers degeneracy can happen in the treatment of systems with an odd number of electrons. To overcome these issues, we propose time-reversal symmetry adaptation for R-DMRG. Since the time-reversal operator is antiunitary, this cannot be simply achieved in the usual way. We introduce a time-reversal symmetry-adapted renormalized basis and present strategies to maintain the structure of basis functions during the sweep optimization. With time-reversal symmetry adaptation, only half of the renormalized operators are needed, and the computational costs of Hamiltonian-wavefunction multiplication and renormalization are reduced by half. The present construction of the time-reversal symmetry-adapted basis also directly applies to other tensor network states without loops.
A new approach to large-scale nuclear structure calculations, based on the density matrix renormalization group (DMRG), is described. The DMRG method, which was developed originally in the framework of low-dimensional quantum lattices, has recently been extended to finite Fermi systems. Following a general description of this extended DMRG methodology, we briefly review its first application to a pairing hamiltonian. We then discuss its application to a schematic nuclear model involving many identical nucleons constrained to move in a single large-j shell. In those cases in which exact diagonalization of the hamiltonian is possible, the method is able to reproduce the exact results with extreme precision. Results are also presented for a model problem in which exact solution is not feasible.