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arXiv:2103.11819

Order, Disorder, and Transitions in Decorated AKLT States on Bethe Lattices

Nicholas Pomata

Published Mar 22, 2021• 0 citations
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Abstract

Returning to one of the original generalizations of the AKLT state, we extend prior analysis on the Bethe lattice (or Cayley tree) to a variant with a series of $n$ spin-1 decorations placed on each edge. The recurrence relations derived for this system demonstrate that such systems are critical for coordination numbers $z=3^{n+1}$, demonstrating order for greater and disorder for lesser coordination number. We then generalize further, effectively interpolating between systems with different values of $n$, using two realizations, one isotropic under local $SU(2)$ transformations and one anisotropic. Exact analysis of these recurrence relations allows us to deduce the location and behavior of order-disorder phase transitions for $z>4$.

DOI
10.48550/arxiv.2103.11819

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Returning to one of the original generalizations of the AKLT state, we extend prior analysis on the Bethe lattice (or Cayley tree) to a variant with a series of nnn spin-1 decorations placed on each edge. The recurrence relations derived for this system demonstrate that such systems are critical for coordination numbers z=3n+1z=3^{n+1}z=3n+1, demonstrating order for greater and disorder for lesser coordination number. We then generalize further, effectively interpolating between systems with different values of nnn, using two realizations, one isotropic under local SU(2)SU(2)SU(2) transformations and one anisotropic. Exact analysis of these recurrence relations allows us to deduce the location and behavior of order-disorder phase transitions for z>4z>4z>4.
By the decoration-iteration transformation, we find a general relation between interaction parameters of a general spin-1 model on any lattice with unique coordination number and on its decorated lattice. We obtain three exact phase transition points of the general spin-1 model on two-dimensional decorated lattices in terms of the exact results of the model on corresponding two-dimensional regular lattices, including the triangle, square, kagome and honeycomb lattices. We find the critical temperatures have no exact relation with the average coordination number and the type of coordination number for both the Ising model and the general spin-1 model.
We study AKLT models on locally treelike lattices of fixed connectivity and find that they exhibit a variety of ground states depending upon the spin, coordination, and global (graph) topology. We find (a) quantum paramagnetic or valence-bond solid ground states, (b) critical and ordered N\'eel states on bipartite infinite Cayley trees, and (c) critical and ordered quantum vector spin glass states on random graphs of fixed connectivity. We argue, in consonance with a previous analysis [C. R. Laumann, S. A. Parameswaran, and S. L. Sondhi, Phys. Rev. B 80, 144415 (2009)], that all phases are characterized by gaps to local excitations. The spin glass states we report arise from random long-ranged loops which frustrate N\'eel ordering despite the lack of randomness in the coupling strengths.
The models constructed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) [Phys. Rev. Lett. 59, 799 (1987)] describe a family of quantum antiferromagnets on arbitrary lattices, where the local spin SSS is an integer multiple MMM of half the lattice coordination number. The equal-time quantum correlations in their ground states may be computed as finite temperature correlations of a classical O(3)O(3)O(3) model on the same lattice, where the temperature is given by T=1/MT=1/MT=1/M. In dimensions d=1d=1d=1 and 2 this mapping implies that all AKLT states are quantum disordered. We consider AKLT states in d=3d=3d=3 where the nature of the AKLT states is now a question of detail depending upon the choice of lattice and spin; for sufficiently large SSS some form of N\'eel order is almost inevitable. On the unfrustrated cubic lattice, we find that all AKLT states are ordered, while for the unfrustrated diamond lattice the minimal S=2S=2S=2 state is disordered while all other states are ordered. On the frustrated pyrochlore lattice, we find (conservatively) that several states starting with the minimal S=3S=3S=3 state are disordered. The disordered AKLT models we report here are a significant addition to the catalog of magnetic Hamiltonians in d=3d=3d=3 with ground states known to lack order on account of strong quantum fluctuations.
The nonzero spectral gap of the original two-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) models has remained unproven for more than three decades. Recently, Abdul-Rahman et al. (arXiv:1901.09297) provided an elegant approach and proved analytically the existence of a nonzero spectral gap for the AKLT models on the decorated honeycomb lattice (for the number nnn of spin-1 decorated sites on each original edge no less than 3). We perform calculations for the decorated square lattice and show that the corresponding AKLT models are gapped if $n\ensuremath{\ge}4$. Combining both results, we also show that a family of decorated hybrid AKLT models, whose underlying lattice is of mixed vertex degrees 3 and 4, are also gapped for $n\ensuremath{\ge}4$. We develop a numerical approach that extends beyond what was accessible previously. Our numerical results further improve the nonzero gap to $n\ensuremath{\ge}2$, including the establishment of the gap for n=2n=2n=2 in the decorated triangular and cubic lattices. The latter case is interesting, as this shows that the AKLT states on the decorated cubic lattices are not N\'eel ordered, in contrast to the state on the undecorated cubic lattice.
Abstract. We study the structure of quantum ground states of simplex solid models, which are generalizations of the valence bond construction for quantum antiferromagnets originally proposed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) (1987 Phys. Rev. Lett . 59 799). Whereas the AKLT states are created by application of bond singlet operators for spins, the simplex solid construction is based on N -simplex singlet operators for spins. In both cases, a discrete one-parameter family of translationally-invariant models with exactly solvable ground states is defined on any regular lattice, and the equal time ground state correlations are given by the finite temperature correlations of an associated classical model on the same lattice, owing to the product form of the wave functions when expressed in a coherent state representation. We study these classical companion models via a mix of Monte Carlo simulations, mean-field arguments, and low-temperature effective field theories. Our analysis reveals that the ground states of edge- and face-sharing cubic lattice simplex solid models are long range ordered for sufficiently large values of the discrete parameter, whereas the ground states of the models on the kagome (2D) and hyperkagome (3D) lattices are always quantum disordered. The kagome simplex solid exhibits strong local order absent in its three-dimensional hyperkagome counterpart, a contrast that we rationalize with arguments similar to those leading to ‘order by disorder’.