Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0007495 · Physical review. B, Condensed matter

Supersymmetric Hubbard operators

Piers Coleman, C. Pépin, John M. Hopkinson

Published Mar 21, 2001• 21 citations
arXivHTMLPDF

Abstract

We develop a supersymmetric representation of the Hubbard operator which unifies the slave boson and slave fermion representations into a single $U(1)\ifmmode\times\else\texttimes\fi{}SU(1|1)$ gauge theory, a group with larger symmetry than the product of two $U(1)$ gauge groups. These representations of the Hubbard operator can be used to incorporate strong Hund's interactions in multielectron atoms as a constraint. We show how this method can be combined with the $\mathrm{SP}(N)$ group to yield a locally supersymmetric large-$N$ formulation of the $t\ensuremath{-}J$ model.

DOI
10.1103/physrevb.63.140411

Related papers

Supersymmetric approach to the infinite U Hubbard Model

Piers Coleman, C. Pépin

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

•Jan 1, 2001•4 citations

Atomic model of supersymmetric Hubbard operators

John M. Hopkinson, Piers Coleman

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.

arXiv:cond-mat/0202060•Physical review. B, Condensed matter•Feb 27, 2003•2 citations

Susy atomic model

John M. Hopkinson, Piers Coleman

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.

arXiv:cond-mat/0207111•Jul 3, 2002•0 citations

Supersymmetric formulation of t-J model

Chi-Ho Cheng

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.

•Jan 1, 2000•1 citation

Supersymmetry in models with strong on-site Coulomb repulsion: Application to the Heisenberg model

Tai Kai Ng, Chi-Ho Cheng

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-$U$ Anderson models. In particular, we study the Heisenberg model carefully in two dimensions where results on spin excitation spectrum are presented.

arXiv:cond-mat/9802080•Physical review. B, Condensed matter•Mar 1, 1999•8 citations

Canonical representation for electrons and its application to the Hubbard model

Brijesh Kumar

A new representation for electrons is introduced, in which the electron operators are written in terms of a spinless fermion and the Pauli operators. This representation is canonical, invertible, and constraint-free. Importantly, it simplifies the Hubbard interaction. On a bipartite lattice, the Hubbard model is reduced to a form in which the exchange interaction simply emerges by decoupling the Pauli subsystem from the spinless fermion bath. This exchange correctly reproduces the large $U$ superexchange. Also derived, for $U=\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$, is the Hamiltonian to study Nagaoka ferromagnetism. In this representation, the infinite-$U$ Hubbard problem becomes elegant and easier to handle. Interestingly, the ferromagnetism in the Hubbard model is found to be related to the gauge invariance of the spinless fermions. A generalization of this representation for the multicomponent fermions, a new representation for bosons, the notion of a ``soft-core'' fermion, and some interesting unitary transformations are introduced and discussed in the appendixes.

arXiv:0807.0972•Physical Review B•May 19, 2008•29 citations
We develop a supersymmetric representation of the Hubbard operator which unifies the slave boson and slave fermion representations into a single U(1)\ifmmode×\else\texttimes\fiSU(1∣1)U(1)\ifmmode\times\else\texttimes\fi{}SU(1|1)U(1)\ifmmode×\else\texttimes\fiSU(1∣1) gauge theory, a group with larger symmetry than the product of two U(1)U(1)U(1) gauge groups. These representations of the Hubbard operator can be used to incorporate strong Hund's interactions in multielectron atoms as a constraint. We show how this method can be combined with the SP(N)\mathrm{SP}(N)SP(N) group to yield a locally supersymmetric large-NNN formulation of the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model.
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
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 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.
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.
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.
A new representation for electrons is introduced, in which the electron operators are written in terms of a spinless fermion and the Pauli operators. This representation is canonical, invertible, and constraint-free. Importantly, it simplifies the Hubbard interaction. On a bipartite lattice, the Hubbard model is reduced to a form in which the exchange interaction simply emerges by decoupling the Pauli subsystem from the spinless fermion bath. This exchange correctly reproduces the large UUU superexchange. Also derived, for $U=\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$, is the Hamiltonian to study Nagaoka ferromagnetism. In this representation, the infinite-UUU Hubbard problem becomes elegant and easier to handle. Interestingly, the ferromagnetism in the Hubbard model is found to be related to the gauge invariance of the spinless fermions. A generalization of this representation for the multicomponent fermions, a new representation for bosons, the notion of a ``soft-core'' fermion, and some interesting unitary transformations are introduced and discussed in the appendixes.