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arXiv:cond-mat/9806116

Solution of quantum double exchange on the complete graph from spl(2,1) dynamical supersymmetry

Karlo Penc, R. Lacaze

Published Jun 9, 1998• 0 citations
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Abstract

The exact and analytical low energy spectrum of the ferromagnetic Kondo lattice model on a complete graph extended with on-site repulsion $U$ is obtained in terms of a dynamical spl(2,1) supersymmetry in the limit of infinitely strong Hund's coupling (double exchange). Furthermore, we show that for the particular value of $U=J_H/2$ the supersymmetry is not constrained to infinite $J_H/t$ only and we calculate the energy including the $t^2/J_H$ corrections analytically and we give numerical evidence which suggest that the Kondo Hamiltonian is itself supersymmetric for any $J_H/t$. On a $N$ site graph, the ferromagnetic ground state is realized for 1 and $N$+1 electrons only. In the leading order in the value of the core spin, the quantum and semiclassical spectra are identical.

DOI
10.48550/arxiv.cond-mat/9806116

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We study the ground-state properties of the SU(NNN) generalization of the Kondo lattice model in one dimension when the Kondo coupling JK{J}_{\text{K}}JK​ (both ferromagnetic and antiferromagnetic) is sufficiently strong. Both cases can be realized using alkaline-earth-metal-like cold gases in optical lattices. Specifically, we first carry out the strong-coupling expansion and identify two insulating phases [one of which is the SU(NNN) analog of the well-known gapped Kondo singlet phase]. We then rigorously establish that the ground state in the low-density (for JK<0{J}_{\text{K}}<0JK​<0) or the high-density (for JK>0{J}_{\text{K}}>0JK​>0) region is ferromagnetic. The results are accounted for by generalizing the double-exchange mechanism to SU(NNN) ``spins.'' Possible realizations of Bose-Fermi supersymmetry SU(N∣1N|1N∣1) in the (generalized) SU(NNN) Kondo lattice model are discussed as well.
We consider the transport properties in an applied magnetic field of the spin S=1/2S=1/2S=1/2 Kondo model with ferromagnetic exchange coupling to electronic reservoirs, a description relevant for the strong-coupling limit of underscreened spin S=1S=1S=1 Kondo impurities. Because the ferromagnetic Kondo interaction is marginally irrelevant, perturbative methods should prove accurate down to low energies. For the purpose of this study, we use a combination of Majorana diagrammatic theory with density matrix numerical renormalization group simulations. In the standard case of antiferromagnetic Kondo exchange, we first show that our simple analytical technique recovers results for the T\mathcal{T}T-matrix and spin relaxation at weak coupling (above the Kondo temperature), which were previously obtained using functional renormalization tools. Considering then the ferromagnetic case, we demonstrate how the low-energy Kondo anomaly splits for arbitrary small values of the Zeeman energy, in contrast to fully screened Kondo impurities near the strong-coupling Fermi liquid fixed point, and in agreement with recent experimental findings for spin S=1S=1S=1 molecular quantum dots.
A Bethe ansatz solution of a (modified) Hubbard chain with a Kondo impurity of arbitrary spin S at a highly symmetric line of parameter space is proposed and explored. Our results confirm the existence of a strong-coupling (line of) fixed-point(s) with ferromagnetic Kondo coupling as first hypothetized by Furusaki and Nagaosa on the basis of perturbative renormalization group calculations. For on-site Hubbard repulsion and ferromagnetic Kondo exchange, the ground state has spin S − 1/2, i.e. , is a singlet when S = 1/2. The contributions of the impurity to the magnetic susceptibility and low-temperature specific heat are discussed. While the Wilson ratio is unity in the half-filled band, it is found to be a function of density and interaction away from half-filling.
Excitation spectra in the SU(ν+1ν+1ν+1,1) supersymmetric t-J model with long-range exchange and transfer has quadratic dependence on spin and charge currents for all energies. After brief review on the supersymmetry, this paper gives a simple explanation for the spin-charge separation on the basis of a first-quantized representation. A useful identity is derived on permutation properties of SU(ννν) Jastrow-type wave functions, which constitute the ground states for a class of 1/r2r^2r2 interaction models.
As a prototype model of antiferromagnetism, we propose a repulsive Hubbard Hamiltonian defined on a graph $\ensuremath{\Lambda}=\mathcal{A}\ensuremath{\cup}\mathcal{B}$ with $\mathcal{A}\ensuremath{\cap}\mathcal{B}=\ensuremath{\emptyset},$ and bonds connecting any element of A\mathcal{A}A with all the elements of B.\mathcal{B}.B. Since all the hopping matrix elements associated with each bond are equal, the model is invariant under an arbitrary permutation of the A\mathcal{A}A sites and/or of the B\mathcal{B}B sites. This is the Hubbard model defined on the so-called (NA,NB){(N}_{A}{,N}_{B})(NA​,NB​) complete bipartite graph, NA(NB){N}_{A}{(N}_{B})NA​(NB​) being the number of elements in A(B).\mathcal{A}(\mathcal{B}).A(B). In this paper, we analytically find the exact ground state for NA=NB=N{N}_{A}{=N}_{B}=NNA​=NB​=N at half filling for any N; the repulsion has a maximum at a critical N-dependent value of the on-site Hubbard U. The wave function and the energy of the unique, singlet ground state assume a particularly elegant form for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.$ We also calculate the spin-spin correlation function and show that the ground state exhibits an antiferromagnetic order for any nonzero U even in the thermodynamic limit. We are aware of no previous explicit analytic example of an antiferromagnetic ground state in a Hubbard-like model of itinerant electrons. The kinetic term induces nontrivial correlations among the particles, and an antiparallel spin configuration in the two sublattices becomes energetically favored at zero temperature. On the other hand, if the thermodynamic limit is taken and then zero temperature is approached, a paramagnetic behavior results. The thermodynamic limit does not commute with the zero-temperature limit, and this fact can be made explicit by the analytic solutions.
A supersymmetric way of imposing the constraint of no double occupancy in models with strong on-site Coulomb repulsion is presented in this paper. In this formulation the physical operators in the constrainted Hilbert space are invariant under local unitary transformations mixing boson and fermion representations. To illustrate, the formulation is applied to the Heisenberg, $t\ensuremath{-}J,$ and infinite-UUU Anderson models. In particular, we study the Heisenberg model carefully in two dimensions where results on spin excitation spectrum are presented.