[Paper Review] Order and Disorder in AKLT Antiferromagnets in Three Dimensions

S. A. Parameswaran, S. L. Sondhi+1 more|arXiv (Cornell University)|Jul 21, 2008

Advanced Condensed Matter Physics4 citations

TL;DR

This paper investigates quantum antiferromagnetic ground states in three-dimensional AKLT models using a quantum-classical mapping, where AKLT ground states correspond to finite-temperature classical O(3) models with T = 1/M. It finds that the S=2 AKLT state on the diamond lattice and the S=3 state on the pyrochlore lattice are quantum disordered due to strong quantum fluctuations, expanding the known set of 3D quantum spin liquids without topological order.

ABSTRACT

The models constructed by Affleck, Kennedy, Lieb, and Tasaki describe a family of quantum antiferromagnets on arbitrary lattices, where the local spin S is an integer multiple M 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) model on the same lattice, where the temperature is given by T=1/M. In dimensions d=1 and d=2 this mapping implies that all AKLT states are quantum disordered. We consider AKLT states in d=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 S some form of Neel 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=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=3 state are disordered. The disordered AKLT models we report here are a significant addition to the catalog of magnetic Hamiltonians in d=3 with ground states known to lack order on account of strong quantum fluctuations.

Motivation & Objective

• To determine whether AKLT antiferromagnets in three dimensions exhibit long-range Néel order or remain quantum disordered due to quantum fluctuations.
• To extend the understanding of AKLT models beyond one and two dimensions, where quantum disorder is guaranteed by the Hohenberg-Mermin-Wagner theorem.
• To identify specific 3D lattices and spin values where quantum disordered ground states emerge despite the absence of the theorem’s protection.
• To provide a systematic analysis of the transition from disorder to Néel order in AKLT states on unfrustrated (cubic, diamond) and frustrated (pyrochlore) lattices.
• To contribute new examples of 3D spin Hamiltonians with analytically known, quantum disordered ground states for studies in quantum magnetism and topological quantum computing.

Proposed method

• Utilizing the AKLT quantum-classical correspondence, where the ground state of an AKLT model maps to the thermal equilibrium state of a classical O(3) spin model at temperature T = 1/M.
• Applying mean-field theory to the classical O(3) model on each lattice to estimate the critical temperature T_c^MF, which serves as an upper bound for the true transition temperature.
• Performing coarse Monte Carlo simulations on the classical model to estimate the true critical temperature T_c and infer the spin threshold S_c above which long-range order emerges.
• Analyzing the ground state manifold of the classical model on the pyrochlore lattice, which exhibits macroscopic degeneracy due to geometrical frustration.
• Using the self-consistency condition for the mean-field Hamiltonian to derive the effective field acting on each spin, accounting for the tetrahedral coordination and spin angles of 120°.
• Employing the order-by-disorder mechanism to explain the stabilization of a particular Néel-like state in the pyrochlore lattice despite macroscopic ground state degeneracy.

Experimental results

Research questions

• RQ1 Does the absence of the Hohenberg-Mermin-Wagner theorem in three dimensions allow for long-range Néel order in AKLT antiferromagnets?
• RQ2 What is the critical spin S_c (or M_c) above which AKLT states on the pyrochlore lattice transition from quantum disorder to Néel order?
• RQ3 How does lattice geometry—specifically coordination number and frustration—affect the stability of quantum disordered ground states in 3D AKLT models?
• RQ4 Can the AKLT construction yield quantum disordered ground states in three dimensions that are not topologically ordered?
• RQ5 To what extent do entropic effects (order-by-disorder) stabilize a particular Néel state in the highly degenerate ground state manifold of the pyrochlore lattice?

Key findings

• On the unfrustrated simple cubic lattice, all AKLT states are Néel ordered regardless of spin value, with the minimal S=3 state exhibiting long-range order.
• On the diamond lattice, the minimal S=2 AKLT state is quantum disordered due to enhanced quantum fluctuations, while all higher-spin states (S≥4) are Néel ordered.
• On the frustrated pyrochlore lattice, the S=3 AKLT state (M=1) is definitively quantum disordered, with the transition to order likely occurring above M=5 (S>15).
• The mean-field estimate for the critical temperature on the pyrochlore lattice gives T_c^MF = 2/3, suggesting that the M=1 state is disordered, though the true T_c is significantly lower.
• Coarse Monte Carlo simulations place an upper bound on the true critical temperature at T_c < 0.2, implying that the critical spin S_c > 15 for the pyrochlore lattice, corresponding to a highly complex Hamiltonian.
• The disordered AKLT states identified—S=2 on diamond and S=3 on pyrochlore—are not topologically ordered; they are described as fully symmetric valence bond solids or quantum paramagnets.