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arXiv:cond-mat/0207111

Susy atomic model

John M. Hopkinson, Piers Coleman

Published Jul 3, 2002• 0 citations
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Abstract

We present the simplest model to which one can apply the supersymmetric Hubbard operators recently introduced. For the atomic model, $H=-E_d X_{00}$, where $X_{00} = |0><0|$ is a Hubbard operator and $E_d$ is the energy of the localized spin level, we show how one can develop exact solutions for the entropy and heat capacity as a function of temperature. With this gold standard we are able to develop a controlled approximation scheme to field theoretically treat the susy approximation at the level of mean field + gaussian corrections and test its accuracy against the widely used slave boson and slave fermion approximations. We find that in addition to slave boson and slave fermion solutions, a new class of solutions exists in the physical case Q = 1, N = 2 which can be properly treated by neither previously existing approach. The phase diagram generated by the mean field saddle-point bears a superficial resemblance to the V-shaped phase diagram common to systems close to a quantum critical point and may provide a natural starting point for investigations of strongly correlated models capturing this physics.

DOI
10.48550/arxiv.cond-mat/0207111

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We present the simplest model to which one can apply the supersymmetric Hubbard operators recently introduced. For the atomic model, H=−EdX00H=-E_d X_{00}H=−Ed​X00​, where X00=∣0><0∣X_{00} = |0><0|X00​=∣0><0∣ is a Hubbard operator and EdE_dEd​ is the energy of the localized spin level, we show how one can develop exact solutions for the entropy and heat capacity as a function of temperature. With this gold standard we are able to develop a controlled approximation scheme to field theoretically treat the susy approximation at the level of mean field + gaussian corrections and test its accuracy against the widely used slave boson and slave fermion approximations. We find that in addition to slave boson and slave fermion solutions, a new class of solutions exists in the physical case Q = 1, N = 2 which can be properly treated by neither previously existing approach. The phase diagram generated by the mean field saddle-point bears a superficial resemblance to the V-shaped phase diagram common to systems close to a quantum critical point and may provide a natural starting point for investigations of strongly correlated models capturing this physics.
We apply the recently proposed supersymmetric Hubbard operators [P. Coleman, C. P\'epin, and J. Hopkinson, Phys. Rev. B 63, 140411(R) (2001)] to an atomic model. In the limiting case of free spins, we derive exact results for the entropy which are compared with a mean-field + Gaussian corrections description. We show how these results can be extended to the case of charge fluctuations and calculate exact results for the partition function, free energy, and heat capacity of an atomic model for some simple examples. Wave-functions of possible states are listed. We compare the accuracy of large N expansions of the susy spin operators [P. Coleman, C. P\'epin, and A. M. Tsvelik, Phys. Rev. B 62, 3852 (2000); Nucl. Phys. B 586, 641 (2000)] with those obtained using ``Schwinger bosons'' and ``Abrikosov pseudofermions.'' For the atomic model, we compare results of slave boson, slave fermion, and susy Hubbard operator approximations in the physically interesting but uncontrolled limiting case of $\stackrel{\ensuremath{\rightarrow}}{N}2.$ For a mixed representation of spins, we estimate the accuracy of large N expansions of the atomic model. In the single box limit, we find that the lowest-energy susy saddle point reduces to simply either slave bosons or slave fermions, while for higher boxes this is not the case. The highest energy saddle point solution has the interesting feature that it admits a small region of a mixed representation, which bears a superficial resemblance to that observed experimentally close to an antiferromagnetic quantum critical point.
We present a preliminary discussion on the use of supersymmetric representation of the Hubbard operator which unifies the slave boson and slave fermion representations into a single U(1) × SU(1|1) gauge theory to treat the physics of the infinite U Hubbard model. By looking for solutions to the Hamiltonian in which the spins can exist as both condensed ordered moments and as mobile charged carriers, we examine the possibility of a Nagaoka ferromagnetic phase at finite doping with a quantum critical point into the paramagnetic phase. Key words: Hubbard model, Supersymmetry, Ferromagnetism The infinite U Hubbard model[1] is a prototype model for strongly correlated electron systems. Interest in this venerable model has grown in recent times because of its link with one of the basic models of high temperature superconductivity, the t − J model.[2] The infinite U Hubbard model corresponds to the t − J model without antiferromagnetic interactions (J → 0). Despite the absence of antiferromagnetic interactions, evidence from finite temperature Lanzcos studies[3] suggests that many of high temperature properties associated with the cuprate metals are already present in the infinite U Hubbard model. The U = ∞ Hubbard hamiltonian is written HK = − t
The supersymmetric formulation of the t - J model is studied in this paper at the mean-field level where the δ- T phase diagram is determined. We found that the slave-fermion–like spiral phase is stable at low doping concentration, and the slave-boson–like d -wave fermionic spin pairing state becomes energetically favourable when δ ⩾ 0.23. An improvement in free energy using Gutzwiller's method lowers the transition doping concentration to δ ≃ 0.06. We also point out the existence of new branches of excitations in the supersymmetric theory.
A new kind of symmetry, called Supersymmetry, is introduced in t-J model and its corresponding Supersymmetric Formulation is constructed. The new formulation unifies the so-called slave-fermion and slave-boson formulations to the model in a single framework and is able to incorporate the advantage of the two approaches. The formulation is analysed at mean-field level where we found that antiferromagnetism described by slave-fermion-like state is stable at low doping regime, and the slave-boson-like d-wave RVB superconducting state becomes favorable at optimal doping. The global phase diagram is determined. To remedy the relaxation of non-double-occupancy constraint in a simple mean-field theory, the Gutzwiller's approximation is applied. Surprisingly the resulting phase diagram is in rather close agreement with experiment. Several physical quantities such as spin-spin correlation lengths, electron spectral functions, magnetic dynamical structure factors and spin susceptibilities are also calculated at the mean-field level where the new effect coming from finite spin and charge excitations of opposite statistics allowed by supersymmetry is pointed out. Finally, future research direction of the formulation is discussed.
The attractive Hubbard model for normal and superconducting ground states is examined for arbitrary electron concentration by the use of the slave-boson mean-field approximation (SBMFA) technique. This approach, at a saddle-point level, is equivalent to the Gutzwiller approximation. Several superfluid characteristics of the model are shown for hypercubic lattices of all dimensions, including , and a comparison with the Bardeen-Cooper-Schrieffer Hartree-Fock approximation (BCS-HFA) calculations and exact results is made. Our results show quantitative and qualitative corrections of the SBMFA to the HFA. The improvement of the SBMFA over the HFA diminishes with increasing lattice dimensionality. We have also evaluated the energy difference between the superconducting and the normal states; this is compared with available results for the superconducting critical temperatures in various dimensions and Uemura-type plots are obtained. The results confirm and substantiate the assertion of a continuous evolution of the superfluid properties of the model from the weak-coupling (BCS-like) to the strong-coupling (composite-boson superconductivity) limit.
We perform a comparative study of the finite temperature behavior of ultracold Bose atoms in optical lattices by the slave fermion and the slave boson approaches to the Bose-Hubbard model. The phase diagram of the system is presented. Although both approaches are equivalent without approximations, the mean field theory based on the slave fermion technique is quantitatively more appropriate. Conceptually, the slave fermion approach automatically excludes the double occupancy of two identical fermions on the same lattice site. By comparing to known results in limiting cases, we find the slave fermion approach better than the slave boson approach. For example, in the non-interacting limit, the critical temperature of the superfluid-normal liquid transition calculated by the slave fermion approach is closer to the well-known ideal Bose gas result. At zero-temperature limit of the critical interaction, strength from the slave fermion approach is also closer to that from the direct calculation using a zero-temperature mean field theory.