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Journal of Physics Condensed Matter

Ring exchange and the Heisenberg and Hubbard models

M W Long, C. W. M. Castleton, C A Hayward

Published Oct 31, 1994• 2 citations
Publisher

Abstract

We study ring-exchange or cyclic-permutation correlations in one-dimensional quantum spin-half systems. For the Heisenberg model we show numerically that these correlations decay as R n approximately 1/ square root n, although we can deduce nothing about any possible important logarithmic corrections. As such, ring-exchange correlations are much longer range than the more commonly considered spin-spin correlation functions. By considering the relationship between solitonic excitations and cyclic permutations, we suggest a way to predict the value of J 2 /J 1 at which the phase transition between a gapped and gapless phase occurs in the next-nearest-neighbour Heisenberg model, suggesting J 2 =4J 1 as the exact transition point. For the Hubbard model with a spin-charge-separated solution, we show that the occupation number, n k , is a 'convolution' of the cyclic-permutation correlations of the spin ground state with the anyonic occupation number of the charge ground state, with the integration being over statistical phase. We deduce the one-eighth singularity previously found for the U= infinity Hubbard model using this new route. We show that for the limit where nearest-neighbour hopping dominates longer-range hopping in the U= infinity Hubbard model, the single-particle correlation function, n k , for an infinitesimal concentration of holes in a half-filled system, is identical to the Fourier transform of the cyclic-exchange correlations of the corresponding spin wavefunction. For the elementary t 1 -t 2 model, we show a relationship between the singularity which occurs at the Fermi surface, the so-called Luttinger-liquid singularity, and the long-range Heisenberg-model cyclic-permutation correlations.

DOI
10.1088/0953-8984/6/44/016

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We study ring-exchange or cyclic-permutation correlations in one-dimensional quantum spin-half systems. For the Heisenberg model we show numerically that these correlations decay as R n approximately 1/ square root n, although we can deduce nothing about any possible important logarithmic corrections. As such, ring-exchange correlations are much longer range than the more commonly considered spin-spin correlation functions. By considering the relationship between solitonic excitations and cyclic permutations, we suggest a way to predict the value of J 2 /J 1 at which the phase transition between a gapped and gapless phase occurs in the next-nearest-neighbour Heisenberg model, suggesting J 2 =4J 1 as the exact transition point. For the Hubbard model with a spin-charge-separated solution, we show that the occupation number, n k , is a 'convolution' of the cyclic-permutation correlations of the spin ground state with the anyonic occupation number of the charge ground state, with the integration being over statistical phase. We deduce the one-eighth singularity previously found for the U= infinity Hubbard model using this new route. We show that for the limit where nearest-neighbour hopping dominates longer-range hopping in the U= infinity Hubbard model, the single-particle correlation function, n k , for an infinitesimal concentration of holes in a half-filled system, is identical to the Fourier transform of the cyclic-exchange correlations of the corresponding spin wavefunction. For the elementary t 1 -t 2 model, we show a relationship between the singularity which occurs at the Fermi surface, the so-called Luttinger-liquid singularity, and the long-range Heisenberg-model cyclic-permutation correlations.
We present variational calculations for quantum spin-half models based upon modifications to nearest-neighbour dimer states. Using a sequence of projections we can attempt a sort of finite-size scaling, in the number of variational parameters, towards the exact solution. For the simple nearest-neighbour Heisenberg model we achieve an error of only 0.005% in energy for our most accurate calculation, much better than an equivalent Lanczos calculation. With which we compare. Due to the fact that we control the total spin in our wavefunctions, we can evaluate the coherence lengths associated with cyclic-exchange permutations (the spin-half sector) and for spin-spin correlations (the spin-one sector), yielding, for example, a convincing numerical confirmation that the spin-one biquadratic-exchange Hamiltonian has a finite correlation length. We perform a study of the one-dimensional J 1 -J 2 model, trying to predict the onset of the phase transition by searching directly for the divergence of the correlation length. Our calculations predict the phase transition near J 2 /J 1 approximately 1/4, as expected. We also perform a study of the ladder geometry, giving further evidence that infinitesimal coupling between ladders yields an immediate gap in the spectrum. The only complication is that the minimization over our space of variational parameters is subtle: surprisingly complicated and "difficult to find" structure is observed in our low-energy solutions which might make our approach of limited use.
Using an extension of the Jordan-Wigner transformation, we solve the one-dimensional Hubbard model at U = infinity in the limit of dominant nearest-neighbour hopping combined with infinitesimal hopping over slightly longer ranges. We find several possible phases at zero temperature, including ferromagnetism, paramagnetism (with and without a spin gap), and even long-range antiferromagnetism with a particular limiting procedure. Our solutions are always spin-charge separated, and we give evidence that the charge degrees of freedom are best described by bosonic statistics.
The infinite-U one-dimensional Hubbard model is 'solvable' in the limit that the nearest-neighbour hopping is dominant and infinitesimal longer-range hopping lifts the one-dimensional spin degeneracy. In this limit the Hubbard model maps onto an effective spin interaction: cyclic or ring exchange with a variety of lengths and strengths. For bipartite geometries we find a region of stable ferromagnetism, which originates from the mechanism intrinsic to Nagaoka's ferromagnetism. We look at connectivities that limit exactly to the square-lattice connectivity as the range of the infinitesimal hopping diverges. When we extend this hopping range, we find that the stable region of ferromagnetism shrinks, eventually vanishing as the infinitesimal hopping range diverges and we limit to the square-lattice geometry. Since the calculation involves a non-trivial density of particles in the thermodynamic limit, it suggests how the proposed region of ferromagnetism may become lost for the case of the two-dimensional square lattice, as numerical simulations predict.
In the limit of dominant nearest-neighbour hopping and infinitesimal secondand third-nearest-neighbour hopping, the Hubbard model is solvable in the sense that it separates into two models which have previously been analysed in the literature. The solutions are spin-charge separated and involve highly correlated motion of the electrons. Although in general the solutions are too complicated to analyse, there are two quite specific states which are amenable to direct analytical attack: the phase boundary between the quantum paramagnet and the Nagaoka ferromagnet, and the 'dimer' phase where all spins pair up into nearest-neighbour singlets. We have been able to evaluate both (n(k)) and (S -k .S k ) accurately enough to deduce the qualitative behaviour for the two types of state in question. The second of these correlation functions is directly relevant to neutron-scattering studies. Although the solutions are highly correlated and spin-charge separated, the correlations are reminiscent of the non-interacting 'spinfull' free-electron gas solution, the spin response exhibiting a divergence where the noninteracting susceptibility does, for example.
Two aspects of strongly correlated electron systems are studied in this thesis. Hubbard-type models are thought to be at the basis of high-critical-temperature superconductivity. One interaction which has not had much study is the nearest-neighbour hopping of on-site singlet pairs. We refine earlier renormalization group arguments and, using the density-matrix renormalization group method, numerically confirm their prediction that a spin gap opens at infinitesimal pair-hopping amplitude V > 0 in the one-dimensional tight-binding model. We also find a phase separation transition at a finite V > 0 as well as a spin-gap transition at a finite V < 0. The exotic magnetic behaviour of heavy-fermion materials involves an interplay between the screening of local moments from each other and the formation of a magnetic state of long range order. While the single-impurity Kondo Hamiltonian is thought to model some aspects of this behaviour, the properties of the Kondo lattice model away from half-filling are largely unknown. We determine the presence of a spin-gap region in the phase diagram of the one-dimensional Heisenberg-Kondo lattice model and make predictions about certain concealed dimerization order parameters.
We investigate the phase diagram of the half-filled SU(N)\mathrm{SU}(N)SU(N) Hubbard-Heisenberg model with hopping ttt, exchange JJJ, and Hubbard UUU, on a two-dimensional square lattice. In the large-NNN limit, and as a function of decreasing values of t∕Jt∕Jt∕J, the model shows a transition from a ddd-density wave state to a spin dimerized insulator. A similar behavior is observed at N=6N=6N=6 whereas at N=2N=2N=2 a spin density wave insulating ground state is stabilized. The N=4N=4N=4 model, has a ddd-density wave ground state at large values of t∕Jt∕Jt∕J which as a function of decreasing values of t∕Jt∕Jt∕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)(T=0)(T=0) quantum Monte Carlo simulations on lattice sizes ranging up to $24\ifmmode\times\else\texttimes\fi{}24$.