Diagrammatics forSU(2) invariant matrix product states

Abstract

Abstract. We report on a systematic implementation of su(2) invariance for matrix product states (MPS) with concrete computations cast in a diagrammatic language. As an application we present a variational MPS study of a spin-1/2 quantum chain. For efficient computations we make systematic use of the su(2) symmetry at all steps of the calculations: (i) the matrix space is set up as a direct sum of irreducible representations, (ii) the local matrices with state-valued entries are set up as superposition of su(2) singlet operators, (iii) products of operators are evaluated algebraically by making use of identities for 3j and 6j symbols. The remaining numerical computations like the diagonalization of the associated transfer matrix and the minimization of the energy expectation value are done in spaces free of symmetry degeneracies. The energy expectation value is a strict upper bound of the true ground-state energy and yields definite conclusions about the accuracy of DMRG results reported in the literature. Furthermore, we present explicit results with accuracy better than 10 −4 for nearestand next-nearest neighbour spin correlators and for general dimer-dimer correlators in the thermodynamical limit of the spin-1 2 Heisenberg chain with frustration.

Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Supergeometry

“…The hallmark of the topological phase protected by the above discrete symmetries is that all entanglement levels are even-fold degenerate. In this formulation, the difference between the odd-S VBS states and the even-S ones is naturally understood in terms of the entanglement structure; the degenerate structure exists only for odd-S cases 35 . It should also be mentioned that the topological phases of one-dimensional gapped spin systems have been classified by group cohomology, [59,63] and the detailed analyses based on the Lie group symmetries are reported in Ref.…”

“…Before going into the detailed discussion, it would also be worthwhile here to give the derivation of the entanglement spectrum using MPS. Suppose we divide a system into the two parts A and 35 This does not mean that all the odd-S spin chains have the degenerate entanglement spectrum. We can construct an odd-S spin state without the even-fold degeneracy.…”

“…The key idea there is that for generic short-range interacting systems, only a small part of the entire Hilbert space is important and, depending on the problems in question, there are variety of ways to parametrize this physically relevant subspace. Although the Wigner's 3j-symbol may give a convenient description of SU (2)-invariant MPS states [33,34,35], we adopt in the present paper the Schwinger-particle formalism (see, for instance, chapters 7 and 19 of [36]) to emphasize the analogies to the lowest Landau level physics. The list of possible applications includes a convenient description of gapped quantum ground states [37,38,28] as well as its application to efficient simulations of dynamics [39,40,41,42,43,44] and variational calculations [45,46,47,48].…”

Matrix product states forsu(2) invariant quantum spin chains

“…The use of Lie algebra symmetries in DMRG calculations has some history [35] - [40]. In the paper [39] two of the current authors have started to formulate the local tensors of MPS and various associated objects like transfer matrices in a su(2) invariant manner by use of Wigner calculus. The work [39] was restricted to spin-1/2 in the quantum space.…”

“…In the paper [39] two of the current authors have started to formulate the local tensors of MPS and various associated objects like transfer matrices in a su(2) invariant manner by use of Wigner calculus. The work [39] was restricted to spin-1/2 in the quantum space. In this paper we present the generalization to arbitrary spin-s and report on concrete applications to the spin-1 bilinear-biquadratic quantum chain which we investigate in a large part of the Haldane phase.…”