Spin wave theory of the Heisenberg model for large spin and the classical limit

Niels Benedikter

J. Zittartz

Joseph G. Conlon, Jan Philip Solovej

The authors discuss various asymptotic limits of the classical and quantum Heisenberg model. They give a new proof that the thermodynamic free energy of the quantum model converges to the free energy of the classical model in the limit of large spins. They also obtain Gaussian and free Bose gas limits for the classical and quantum models respectively.

Fabián H. Gaioli, Edgardo T. Garcia Alvarez

Larry Engelhardt, Marshall Luban, Christian Schröder

We determine the temperature range over which classical Heisenberg spin models closely reproduce the zero-field susceptibility of the corresponding quantum Heisenberg models for a finite number $N$ of interacting quantum spins $s$. Using mostly quantum and classical Monte Carlo methods, as well as analytical methods where applicable, we have explored a variety of geometries, including polygons, open chains, and all Platonic and several Archimedean polytopes. These systems range in size from $N=2$ to 120, and we have considered values of $s$ from $1∕2$ to 50 for both antiferromagnetic and ferromagnetic exchange. Particular attention is devoted to quantifying the slow convergence of the large $s$ quantum data to the limiting classical data. This is motivated by the desire to define conditions where classical Monte Carlo methods can provide useful predictions for finite quantum Heisenberg spin systems.

Fakher F. Assaad

We investigate the phase diagram of the half-filled $\mathrm{SU}(N)$ Hubbard-Heisenberg model with hopping $t$, exchange $J$, and Hubbard $U$, on a two-dimensional square lattice. In the large-$N$ limit, and as a function of decreasing values of $t∕J$, the model shows a transition from a $d$-density wave state to a spin dimerized insulator. A similar behavior is observed at $N=6$ whereas at $N=2$ a spin density wave insulating ground state is stabilized. The $N=4$ model, has a $d$-density wave ground state at large values of $t∕J$ which as a function of decreasing values of $t∕J$ becomes unstable to an insulating state with no apparent lattice and spin broken symmetries. In this state, the staggered spin-spin correlations decay as a power law, resulting in gapless spin excitations at $\stackrel{P\vec}{q}=(\ensuremath{\pi},\ensuremath{\pi})$. Furthermore, low lying spin modes with small spectral weight are apparent around the wave vectors $\stackrel{P\vec}{q}=(0,\ensuremath{\pi})$ and $\stackrel{P\vec}{q}=(\ensuremath{\pi},0)$. This gapless spin liquid state is equally found in the SU(4) Heisenberg $(U∕t\ensuremath{\rightarrow}\ensuremath{\infty})$ model in the self-adjoint antisymmetric representation. An interpretation of this state in terms of a $\ensuremath{\pi}$-flux phase is offered. Our results stem from projective $(T=0)$ quantum Monte Carlo simulations on lattice sizes ranging up to $24\ifmmode\times\else\texttimes\fi{}24$.