Quantum Heisenberg antiferromagnet in a field on the Tasaki square lattice

TL;DR

The paper studies the low-temperature thermodynamics of the $S=1/2$ Heisenberg antiferromagnet on the Tasaki square lattice near the saturation field, where a flat magnon band dominates. It maps the degenerate ground-state subspaces to configurations of hard squares on an auxiliary lattice, reducing the quantum problem to a classical grand partition function $\\\\Xi(z,\\\cal N)$ with $z=e^{\\\mu/T}$ and $\\\mu=h_{ m sat}-h$, $h_{ m sat}=-\\\epsilon_0$. Small-cluster exact diagonalization confirms the mapping and reproduces the magnetization and specific-heat behavior, including a magnetization jump of $\\Delta m=1/3$ at $h=h_{ m sat}$. Large-system classical Monte Carlo simulations reveal a finite-temperature order-disorder transition in the 2D Ising universality class, with a critical activity $z_c\\approx3.7962$ and $T_c(h)= (h_{ m sat}-h)/\\\ln z_c$, consistent with the hard-square Baxter model. The approach provides a tractable framework to study flat-band quantum magnets and suggests extensions to other Tasaki lattices and potential cold-atom realizations.

Abstract

We consider the $S=1/2$ Heisenberg antiferromagnet on the Tasaki square lattice (flat-band spin system) and study its low-temperature thermodynamics around the saturation magnetic field. To this end, we construct a mapping of the ground states in the subspaces with total $S^z=N/2,\ldots,N/3$ ($N$ is the number of lattice sites) on the hard squares on an auxiliary square lattice and use classical Monte Carlo simulations to examine the latter classical system. The most prominent feature of the $S=1/2$ Heisenberg antiferromagnet on the Tasaki square lattice is an order-disorder phase transition which occurs at a low temperature just below the saturation magnetic field and belongs to the 2D Ising universality class.

Figures (6)

• Figure 1: (Top) Tasaki square lattice. Here, ${\bm m}=m_x{\bm i}+m_y{\bm j}$, $m_x$ and $m_y$ are integers, enumerates the unit cells, each of which contains three sites enumerated by $\alpha=1,2,3$. For antiferromagnetic couplings $J_2=4J_1>0$, the lowest-energy one-magnon band is dispersionless (flat). (Bottom) Auxiliary square lattice used for representation of localized magnons. It consists of two sublattices, denoted as $A$ and $B$, and all sites of the sublattice $A$ are surrounded by the sites of the sublattice $B$ and vice versa.
• Figure 2: One-magnon energies $\Lambda^{(1)}_{\bm q}$, $\Lambda^{(2)}_{\bm q}$, $\Lambda^{(3)}_{\bm q}$ given in Eq. (\ref{['04']}) for $J_1=1$ and $J_2=4$.
• Figure 3: $c(T)$ for the Tasaki square lattice, $J_1=1$, $J_2=4$, $h=11.88,12,12.12$, (top) $N=12$, periodic boundary conditions, and (bottom) $N=18$, twisted/periodic boundary conditions, see Sec. \ref{['s31']}. The results of the localized-magnon description ($\mu=-0.12,0,0.12$ and ${\cal N}=4$ and ${\cal N}=6$) are shown by lines for comparison.
• Figure 4: $m(h)$ for the Tasaki square lattice, $J_1=1$, $J_2=4$, $T=0.01,0.1,0.5$, (top) $N=12$, periodic boundary conditions, and (bottom) $N=18$, twisted/periodic boundary conditions, see Sec. \ref{['s31']}. The results of the localized-magnon description (${\cal N}=4$ and ${\cal N}=6$) are shown by solid lines for comparison. Inset: $N=18$ (${\cal N}=6$), periodic boundary conditions, see Sec. \ref{['s31']}.
• Figure 5: (Top) Expected $c(T)$ for $J_1=1$, $J_2=4$, $h=0.99h_{\rm sat},1.01h_{\rm sat}$, $h_{\rm sat}=12$; classical Monte Carlo simulations for ${\cal N}={\cal L}^2$, ${\cal L}$ up to $1\,000$. (Middle) Expected temperature dependence of the sublattice occupation densities $\rho_A=\langle {n}_A\rangle/{\cal N}$ and $\rho_B=\langle {n}_B\rangle/{\cal N}$ at $h=0.99h_{\rm sat}$. (Bottom) Expected temperature dependence of the order parameter $\eta=\vert \rho_{A}-\rho_{B}\vert$ and the total occupation density $\rho= \rho_{A}+\rho_{B}$ at $h=0.99h_{\rm sat}$.