Abstract
Using the Matrix Product State framework, we generalize the Affleck-Kennedy-Lieb-Tasaki (AKLT) construction to one-dimensional spin liquids with global color SU(N) symmetry, finite correlation lengths, and edge states that can belong to any self-conjugate irreducible representation (irrep) of SU(N). In particular, SU(2) spin-1 AKLT states with edge states of arbitrary spin s=1/2,1,3/2,⋯ are constructed, and a general formula for their correlation length is given. Furthermore, we show how to construct local parent Hamiltonians for which these AKLT states are unique ground states. This enables us to study the stability of the edge states by interpolating between exact AKLT Hamiltonians. As an example, in the case of spin-1 physical degrees of freedom, it is shown that a quantum phase transition of central charge c=1 separates the Symmetry Protected Topological (SPT) phase with spin-1/2 edge states from a topologically trivial phase with spin-1 edge states. We also address some specificities of the generalization to SU(N) with N>2, in particular regarding the construction of parent Hamiltonians. For the AKLT state of the SU(3) model with the 3-box symmetric representation, we prove that the edge states are in the 8-dimensional adjoint irrep, and for the SU(3) model with adjoint irrep at each site, we are able to construct two different reflection-symmetric AKLT Hamiltonians, each with a unique ground state which is either even or odd under reflection symmetry and with edge states in the adjoint irrep. Finally, examples of two-column and adjoint physical irreps for SU(N) with N even and with edge states living in the antisymmetric irrep with N/2 boxes are given, with a conjecture about the general formula for their correlation lengths.
Highlights
The Affleck-Kennedy-Lieb-Tasaki (AKLT) model of a spin-1 chain [1, 2], with a biquadratic interaction equal to a third of the bilinear one, has played an important role in proving Haldane’s conjecture that the Heisenberg spin1 chain is gapped [3, 4]
This can be conveniently reformulated in terms of a Matrix Product State (MPS), and in that respect the AKLT construction has played an important role in popularizing the MPS, which is nowadays the standard formulation of the Density Matrix Renormalization Group (DMRG) [10,11,12,13]
In conclusion we have found an optimal representation of the AKLT state introduced by Greiter and Rachel for which the nature of the edge states is manifest
Summary
The Affleck-Kennedy-Lieb-Tasaki (AKLT) model of a spin-1 chain [1, 2], with a biquadratic interaction equal to a third of the bilinear one, has played an important role in proving Haldane’s conjecture that the Heisenberg spin chain is gapped [3, 4]. In the AKLT construction the physical spins are written in terms of two virtual spin-1/2 degrees of freedom attached to each lattice site which, simultaneously, form maximally entangled bond singlets between neighboring sites This can be conveniently reformulated in terms of a Matrix Product State (MPS), and in that respect the AKLT construction has played an important role in popularizing the MPS, which is nowadays the standard formulation of the Density Matrix Renormalization Group (DMRG) [10,11,12,13]. This construction is most done in the MPS language, which we use throughout, and parent Hamiltonians are constructed in a systematic way This allows us for instance to discuss models of spin-1 chains with edge states of arbitrary spin s = 1/2, 1, 3/2, · · · , and to study the quantum phase transition between topological and trivial phases with half-integer respectively integer spin edge states.
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