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arXiv:cond-mat/9402009

Exact Results for Spin and Charge Dynamics of Electrons with Supersymmetry

Y. KURAMOTO

Published Feb 3, 1994• 0 citations
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Abstract

Excitation spectra in the SU($ν+1$,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($ν$) Jastrow-type wave functions, which constitute the ground states for a class of 1/$r^2$ interaction models.

DOI
10.48550/arxiv.cond-mat/9402009

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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.
A one-dimensional electron has both spin and charge degrees of freedom. In the limit of strong on-site repulsion, double occupation of a site can be neglected since it has a large energy cost. If the number N e of electrons is less than the number N of lattice sites, vacant sites appear and the system acquires both spin and charge degrees of freedom. The simplest model to incorporate these degrees of freedom is called the t – J model, which is discussed in this chapter. We begin by reviewing the SU(2,1) supersymmetry (SUSY) as an extension of the SU(2) symmetry in the spin chain. With the supersymmetry, the hopping and exchange in the t – J model can be treated in a unified way as the graded permutation. Then the ground state of the SUSY t – J model is derived in Section 6.2 by the coordinate representation of wave functions by using the fully polarized state as the reference. The static structure factor at zero temperature is completely determined by the wave function at the ground state. The spin and charge components are derived using the determinant and its generalizations in Section 6.3. In Section 6.4, we proceed to derive the spectrum of the SUSY t – J model. The elementary excitations consist of spinons, holons, and their antiparticles. It is proved in Section 6.5 that these excitations span the complete set. The argument relies on a generalization of Young diagrams, which are called ribbon diagrams. At the same time, the degeneracy of energy levels beyond the global SU(2) symmetry is ascribed to the presence of a Yangian supersymmetry Y ( sl 2|1 ).
Dynamical charge structure factor N(Q,ω)N(Q,ω)N(Q,ω) with QQQ smaller than the Fermi wave number is derived analytically for the one-dimensional supersymmetric t-J model with 1/r21/r^21/r2 interaction. Strong spin-charge separation in dynamics is found. N(Q,ω)N(Q,ω)N(Q,ω) with given electron density does not depend on the spin polarization for arbitrary size of the system. In the thermodynamic limit, only two holons and one antiholon contribute to N(Q,ω)N(Q,ω)N(Q,ω). These results together with generalized ones for the SU(KKK,1) supersymmetric t-J model are derived via solution of the Sutherland model in the strong coupling limit.
Thermodynamics is derived rigorously for the 1D supersymmetric {\it t-J} model and its SU(K,1K,1K,1) generalization with inverse-square exchange. The system at low temperature is described in terms of spinons, antispinons, holons and antiholons obeying fractional statistics. They are all free and make the spin susceptibility independent of electron density, and the charge susceptibility independent of magnetization. Thermal spin excitations responsible for the entropy of the SU(K,1K,1K,1) model are ascribed to free para-fermions of order K−1K-1K−1.
A set of generalized Gutzwiller-type wave functions with spin and charge currents and/or spin polarization are shown to be exact eigenstates of the supersymmetric t - J -type model with long-range interactions. The increment of energy in the presence of currents is proportional to the square of the currents without any higher-order corrections for arbitrary size of the system. The exact excited states are identified by the absence of statistical fluctuations in the variational Monte Carlo calculation. The spin susceptibility is found to be independent of the size, as in the case of the charge susceptibility.
Supersymmetry is an algebraic property of a quantum Hamiltonian that, by giving every boson a fermionic superpartner and vice versa, may underpin physics beyond the Standard Model. Fractional bosonic and fermionic quasiparticles are familiar in condensed matter, as in the spin and charge excitations of the ttt-JJJ model describing electron dynamics in one-dimensional materials, but this type of symmetry is almost unknown. However, the triplet excitations of a quantum spin ladder in an applied magnetic field provide a supersymmetric analogue of the ttt-JJJ chain. Here we perform neutron spectroscopy on the spin-ladder compounds (C5_55​D12_{12}12​N)2_22​CuBr4_44​ and (C5_55​D12_{12}12​N)2_22​CuCl4_44​ over a range of applied fields and temperatures, and apply matrix-product-state methods to the ladder and equivalent chain models. From the momentum-resolved dynamics of a single charge-like excitation in a bath of fractional spins, we find essential differences in thermal broadening between the supersymmetric and non-supersymmetric sectors. The persistence of a strict zone-centre pole at all temperatures constitutes an observable consequence of supersymmetry that marks the beginning of supersymmetric studies in experimental condensed matter.
Abstract Supersymmetry is an algebraic property of a quantum Hamiltonian that, by giving every boson a fermionic superpartner and vice versa, may underpin physics beyond the Standard Model. Fractional bosonic and fermionic quasiparticles are familiar in condensed matter, as in the spin and charge excitations of the t - J model describing electron dynamics in one-dimensional materials, but this type of symmetry is almost unknown. However, the triplet excitations of a quantum spin ladder in an applied magnetic field provide a supersymmetric analogue of the t - J chain. Here we perform neutron spectroscopy on the spin-ladder compounds (C 5 D 12 N) 2 CuBr 4 and (C 5 D 12 N) 2 CuCl 4 over a range of applied fields and temperatures, and apply matrix-product-state methods to the ladder and equivalent chain models. From the momentum-resolved dynamics of a single charge-like excitation in a bath of fractional spins, we find essential differences in thermal broadening between the supersymmetric and non-supersymmetric sectors. The persistence of a strict zone-centre pole at all temperatures constitutes an observable consequence of supersymmetry that marks the beginning of supersymmetric studies in experimental condensed matter.