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arXiv:2403.09765

Perturbative description of high-$T_c$ superconductivity in the Hubbard model via unitary transformation and classical spins

Kirill Alpin

Published Mar 14, 2024• 0 citations
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Abstract

A unitary transformation is applied to the Hubbard model, which maps the Hubbard interaction to a single particle term. The resulting Hamiltonian consists of unconstrained fermions, which is then mapped to a Hamiltonian of spinless fermions coupled to pseudospins. The fermions are integrated out using second order perturbation theory in $1/U$, resulting in an effective spin Hamiltonian. An order parameter is identified, stabilizing d-wave superconductivity. The groundstate energy of classical spin configurations is minimized at a finite value of this order parameter after a critical chemical potential, resulting in d-wave superconductivity at non-zero doping. This suggests, that the onset of high-$T_c$ superconductivity is governed by the groundstate of a classical spin system.

DOI
10.48550/arxiv.2403.09765

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Perturbative description of high-TcT_cTc​ superconductivity in the Hubbard model via unitary transformation and classical spins
A unitary transformation is applied to the Hubbard model, which maps the Hubbard interaction to a single particle term. The resulting Hamiltonian consists of unconstrained fermions, which is then mapped to a Hamiltonian of spinless fermions coupled to pseudospins. The fermions are integrated out using second order perturbation theory in 1/U1/U1/U, resulting in an effective spin Hamiltonian. An order parameter is identified, stabilizing d-wave superconductivity. The groundstate energy of classical spin configurations is minimized at a finite value of this order parameter after a critical chemical potential, resulting in d-wave superconductivity at non-zero doping. This suggests, that the onset of high-TcT_cTc​ superconductivity is governed by the groundstate of a classical spin system.
We present a controlled perturbative approach to the low-temperature phase diagram of highly inhomogeneous Hubbard models in the limit of small coupling, ${t}^{\ensuremath{'}}$, between clusters. We apply this to the dimerized and checkerboard models. The dimerized model is found to behave like a doped semiconductor, with a Fermi-liquid groundstate with parameters (e.g., the effective mass) which are smooth functions of the Hubbard interaction, UUU. By contrast, the checkerboard model has a Fermi liquid phase at large U>Uc=4.58U>{U}_{c}=4.58U>Uc​=4.58, a ddd-wave superconducting state with a full gap for Uc>U>0{U}_{c}>U>0Uc​>U>0, and a narrow strip of an intermediate ddd-wave superconducting phase with gapless ``nodal'' quasiparticles for $\ensuremath{\mid}U\ensuremath{-}{U}_{c}\ensuremath{\mid}<\mathcal{O}({t}^{\ensuremath{'}})$.
We describe a unitary transformation which eliminates terms coupling states with differing numbers of doubly occupied sites from the Hamiltonian of the Hubbard model. The SSS matrix for the transformation, and the transformed Hamiltonian, ${H}^{\ensuremath{'}}$, are generated by an iterative procedure which results in an expansion in powers of the hopping integral ttt divided by the on-site energy UUU. For a half-filled band and in the space with no doubly occupied sites, ${H}^{\ensuremath{'}}$ is equivalent to a spin Hamiltonian. We discuss the implications of our results for ${H}^{\ensuremath{'}}$ on theories of high-temperature superconductivity.
A Hamiltonian for the motion of oxygen holes strongly coupled to the Cu spins in a one-dimensional (1D) Cu-O lattice is solved exactly for all dopings. This Hamiltonian retains the dominant O-hole hopping term, that is, spin-exchange hopping of the hole spin with the intervening Cu spin and neglects the weaker non-spin-exchange hopping term. The excitations naturally separate into two independent quasiparticles: spinons and holons. It is this Hamiltonian that should be taken as the correct zeroth-order Hamiltonian from which perturbation theory should be applied. We show that the spinon spectrum is the spectrum of the 1D antiferromagnet incommensurate with the Cu lattice and the holons are spinless noninteracting fermions in a cosine band. The Cu-Cu spin correlation in the ground state increases linearly with doping from antiferromagnetic (-0.443) at x=0 to ferromagnetic (\ensuremath{\approxeq}0.19) at x=1.
The electronic $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}U\text{\ensuremath{-}}J$ model for the CuO2\mathrm{Cu}{\mathrm{O}}_{2}CuO2​ plane in high-Tc{T}_{c}Tc​ superconducting cuprates is elaborated. It incorporates both charge transfer UUU and exchange correlation JJJ with direct ttt and diagonal ${t}^{\ensuremath{'}}$ hopping. Making use of the Yang concept of the off diagonal long range order (ODLRO) in the reduced density matrix of the electronic system we study the loss of quantum wave coherence (and thereby superconductivity) due to the Coulomb interaction. To this aim the second-quantized Hamiltonian of the model is translated to the phase representation with the help of the topologically constrained path integral formalism. As a result the electrons emerge as composite particles consisting of of spin-carrying neutral fermions and charged topological bosons in a form of ``flux tubes'' with the quantum phase variable dual to the local electron density. The charge carrying bose field forms a collective variable representing the correlated ``sliding'' motion of the entire electronic system. We found that the spin gap $\ensuremath{\Delta}$ which appears in one-particle spectrum is the result of valence bond pairing of neutral fermions without ODLRO and deduce the solution for $\ensuremath{\Delta}$ with ``ddd-wave'' symmetry. In the bosonic sector we derive the effective quantum rotor model along with microscopic phase stiffnesses setting the energy scale for superconductivity and point out the special role played by the diagonal hopping ${t}^{\ensuremath{'}}$. We explore the relationship of the quantum rotor phase model and the Bose-Hubbard system emphasizing the ring topology of quantum phase variable. Furthermore, we obtain the ground state phase diagrams which facilitate the Mott insulator-superconductor transition scenario. We elucidate the role of the chemical potential (doping) and the huge degeneracy of the effective bosonic system in achieving ODLRO with long-range phase coherence in the presence of large Coulomb repulsion. Finally, we discuss a number of relevant issues pertaining to the the present work which include: theoretical aspects of the hierarchy of the energy scales in cuprates, ``spin-charge separation'' concepts, ``local-pair'' and ``phase-fluctuation'' scenarios as well as experimental findings related to the role of the ``ddd-wave'' symmetry of the spin gap and diagonal hopping ${t}^{\ensuremath{'}}$ in cuprates.
In the model considered, the nonlocal interaction of the fermions in different sublattices of a bipartite lattice is introduced. It can also be regarded as local interaction of fermions with opposite ``hypercharge''. The corresponding term in the Hamiltonian is SU(4)-invariant and appears to be the most tractable version of the SO(5)-invariant model that unifies antiferromagnetic and d-wave superconducting order parameters. The model has been studied primarily in the weak interaction limit and in the mean field approximation. Near the half-filling the antiferromagnetic critical temperature has a peak. However, the superconducting transition takes place when the Fermi surface crosses the area where the density of states is of order of inverse coupling coefficient. Thus, in mean-field approximation, there exist an interval of values of the chemical potential, for which the system is a superconductor for arbitrary high temperatures. The temperature dependence of specific heat, Hall coefficient, and DC conductivity in the normal phase agrees with that experimentally observed in high-TcT_cTc​ cuprates.
Superfluid ground-state phase diagram of the 2d2d2d Hubbard Model in the emergent BCS regime
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