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arXiv:1904.01822 · Nuclear Physics B

Novel families of SU(N) AKLT states with arbitrary self-conjugate edge states

Samuel Gozel, Didier Poilblanc, Ian K. Affleck, Frédéric Mila

Published Jun 4, 2019• 17 citations
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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.

DOI
10.1016/j.nuclphysb.2019.114663

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We study Z3{\mathbb{Z}}_{3}Z3​ symmetry-protected topological (SPT) phases in one-dimensional spin systems with ${Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3}$ symmetry. We construct ground-state wave functions of the matrix product form for nontrivial Z3{\mathbb{Z}}_{3}Z3​ phases and their parent Hamiltonian from a cocycle of the group cohomology ${H}^{2}({Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3},U(1))$. The Hamiltonian is an SU(3) version of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, consisting of bilinear and biquadratic terms of su(3) generators in the adjoint representation. A generalization to the SU(N)\mathrm{SU}(N)SU(N) case, the SU(N)\mathrm{SU}(N)SU(N) AKLT Hamiltonian, is also presented, which realizes nontrivial ZN{\mathbb{Z}}_{N}ZN​ SPT phases. We use the infinite-size variant of the density matrix renormalization group (iDMRG) method to determine the ground-state phase diagram of the SU(3) bilinear-biquadratic model as a function of the parameter $\ensuremath{\theta}$ controlling the ratio of the bilinear and biquadratic coupling constants. The nontrivial Z3{\mathbb{Z}}_{3}Z3​ SPT phase is found for a range of the parameter $\ensuremath{\theta}$ including the point of vanishing biquadratic term $(\ensuremath{\theta}=0)$ as well as the SU(3) AKLT point [$\ensuremath{\theta}=arctan(2/9)$]. A continuous phase transition to the SU(3) dimer phase takes place at $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.027\ensuremath{\pi}$, with a central charge $c\ensuremath{\approx}3.2$. For SU(3) symmetric cases, we define string order parameters for the Z3{\mathbb{Z}}_{3}Z3​ SPT phases in a similar way to the conventional Haldane phase. We propose simple spin models that effectively realize the SU(3) and SU(4) AKLT models.
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This thesis studies a pair of symmetry protected topological (SPT) phases which arise when considering one-dimensional quantum spin systems possessing a natural orthogonal group symmetry. Particular attention is given to a family of exactly solvable models whose ground states admit a matrix product state description and generalize the AKLT chain. We call these models ``SO(n)SO(n)SO(n) AKLT chains'' and the phase they occupy the ``SO(n)SO(n)SO(n) Haldane phase''. We present new results describing their ground state structure and, when nnn is even, their peculiar O(n)O(n)O(n)-to-SO(n)SO(n)SO(n) symmetry breaking. We also prove that these states have arbitrarily large correlation and injectivity length by increasing nnn, but all have a 2-local parent Hamiltonian, in contrast to the natural expectation that the interaction range of a parent Hamiltonian should diverge as these quantities diverge. We extend Ogata's definition of an SPT index for a split state for a finite symmetry group GGG to an SPT index for a compact Lie group GGG. We then compute this index, which takes values in the second Borel group cohomology H2(SO(n),U(1))H^2(SO(n),U(1))H2(SO(n),U(1)), at a single point in each of the SPT phases. The two points have different indices, confirming the two SPT phases are indeed distinct. Chapter 1 contains an introduction with a detailed overview of the contents of this thesis, which includes several chapters of background information before presenting new results in Chapter 7 and Chapter 8.