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arXiv:cond-mat/0302565 · Physical review. B, Condensed matter

Possible Z2 phase and spin-charge separation in electron-doped cuprate superconductors

Tiago C. Ribeiro, Xiao-Gang Wen

Published Jul 1, 2003• 9 citations
arXivHTMLPDF

Abstract

The SU(2) slave-boson mean-field theory for the ${\mathrm{tt}}^{\ensuremath{'}}J$ model is analyzed. The role of next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$ on the phase diagram is studied. We find a pseudogap phase in hole-doped (HD) materials (where ${t}^{\ensuremath{'}}<0).$ The pseudogap phase is a U(1) spin liquid (the staggered-flux phase) with a U(1) gauge interaction and no fractionalization. This agrees with experiments on HD samples. The same calculation also indicates that a positive ${t}^{\ensuremath{'}}$ favors a ${Z}_{2}$ state with true spin-charge separation. The ${Z}_{2}$ state that exists when ${t}^{\ensuremath{'}}\ensuremath{\gtrsim}0.5J$ can be a candidate for the pseudogap phase of electron-doped (ED) cuprates (if such a phase exists). The experimental situation in ED materials is also addressed.

DOI
10.1103/physrevb.68.024501

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The SU(2) slave-boson mean-field theory for the tt\ensuremath′J{\mathrm{tt}}^{\ensuremath{'}}Jtt\ensuremath′J model is analyzed. The role of next-nearest-neighbor hopping t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ on the phase diagram is studied. We find a pseudogap phase in hole-doped (HD) materials (where t\ensuremath′<0).{t}^{\ensuremath{'}}<0).t\ensuremath′<0). The pseudogap phase is a U(1) spin liquid (the staggered-flux phase) with a U(1) gauge interaction and no fractionalization. This agrees with experiments on HD samples. The same calculation also indicates that a positive t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ favors a Z2{Z}_{2}Z2​ state with true spin-charge separation. The Z2{Z}_{2}Z2​ state that exists when t\ensuremath′\ensuremath≳0.5J{t}^{\ensuremath{'}}\ensuremath{\gtrsim}0.5Jt\ensuremath′\ensuremath≳0.5J can be a candidate for the pseudogap phase of electron-doped (ED) cuprates (if such a phase exists). The experimental situation in ED materials is also addressed.
The Gaussian fluctuations in slave-boson mean-field theory of the $t\text{\ensuremath{-}}J$ model are analyzed in this paper in the low-doping regime where the superconducting to normal (pseudogap phase) transition is driven by vanishing of bose-condensation amplitude. By eliminating the boson and constraint fields exactly in the linear-response regime, we show that the Gaussian theory describes an unconventional Fermi liquid superconductor where the superconducting to normal transition is a new type of phase transition characterized by a change of Landau parameter from ${F}_{1}>\ensuremath{-}1$ in the superconducting phase to ${F}_{1}=\ensuremath{-}1$ in the pseudo-gap phase, corresponding to a state with spin-charge separation. The consequences of this proposal are discussed.
We develop a slave-boson theory for the $t\ensuremath{-}J$ model at finite doping that respects an SU(2) symmetry: a symmetry previously known to be important at half filling. The mean-field phase diagram is found to be consistent with the phases observed in the cuprate superconductors, which contain ddd-wave superconductor, spin-gap, strange metal, and Fermi-liquid phases. The spin-gap phase is best understood as the staggered flux phase, which is nevertheless translationally invariant for physical quantities. The physical electron spectral function shows small Fermi segments at low doping that continuously evolve into the large Fermi surface at high-doping concentrations. The close relation between the SU(2) and the U(1) slave-boson theory is discussed. The low-energy effective theory for the low-lying fluctuations is derived and additional lying modes [which were overlooked in the U(1) theory] are identified.
We develop a slave-boson theory for the $t\ensuremath{-}J$ model at finite doping which respects an SU(2)\mathrm{SU}(2)SU(2) symmetry---a symmetry previously known to be important at half filling. The mean field phase diagram is found to be consistent with the phases observed in the cuprate superconductors, which contain ddd-wave superconductor, spin gap, strange metal, and Fermi liquid phases. The spin gap phase is best understood as the staggered flux phase, which is nevertheless translationally invariant for physical quantities. The electron spectral function shows small Fermi pockets at low doping which continuously evolve into the large Fermi surface at high doping concentrations.
Based on an improved SU(2) slave-boson approach showing coupling between the charge and spin degrees of freedom, we derive a phase diagram of high-Tc{T}_{c}Tc​ cuprates which displays both the superconducting and pseudogap phases in the plane of temperature vs hole doping rate. It is shown that phase fluctuations in the order parameters result in a closer agreement with the observed phase diagram of an arch shape, by manifesting the presence of an optimal doping rate closer to observation, compared to the U(1) slave-boson theory.
The U(1) slave-boson mean-field theory of the t - J model has been used by many authors as a starting point for the theoretical analysis of high - Tc cuprate superconductors. With suitable refinements, the theory can explain a lot of qualitative features of cuprates. However, the theory does not produce a correct description of the low-energy quasi-particle properties in the under-doped regime. There exist several possible schemes to resolve this difficulty, among them, one generalizes the U(1) theory to SU(2) formulation and the other introduces confinement between spinons and holons. In this thesis, we generalize the Landau transport equations of the U(1) slave-boson mean-field theory of the t - J model to the SU(2) formulation. We first generalize the SU(2) slave-boson mean-field theory to the time-dependent regime. After eliminating the boson and constraint fields exactly in the linear-response regime at zero-temperature, we obtain a set of transport equations for quasi-particles, which has a similar structure as the Landau transport equations for Fermi-liquid superconductors, with all the Landau parameters explicitly given except that besides the usual effective interaction in the particle-hole channel in Fermi-liquid theory, we find an effective interaction in the particle-particle channel, which reflects the SU(2) symmetry between particles and holes. The transport equation is SU(2) invariant at half-filling. Around q⃗ = 0, the electron density couples only to the transverse part of the external field and the quasi-particle charge is still of order x. Besides the SU(2) formulation approach, we also investigate spinon-holon confinement in the U(1) slave-boson theory. The optical and thermal conduc-tivities of U (1) slave-boson theory with confinement are calculated. For optical conductivity, we consider only the contributions from static impurities. The qualitative behavior of our results agrees with experimental results. However, to explain the experiments quantitatively, we need to study further the con-tributions from phonons. We found a phenomenological effective Hamiltonian for the quasi-particle Green's functions in the U(1) slave-boson theory with spinon-hoion confinement. Starting from this effective Hamiltonian, we exam-ine the thermal conductivity within the theory. In particular, we compare the thermal conductivity for four different situations: (i) the ordinary d-wave su-perconducting state with four Dirac Fermi points; (ii) the normal metal state with a half-pocket Fermi surface; (iii) the spinon-holon confinement supercon-ducting state with a half-pocket Fermi surface; (iv) BCS superconducting state with a half-pocket Fermi surface.
Two of the iconic phases of the hole-doped cuprate materials are the intermediate temperature pseudogap metal and the lower temperature ddd-wave superconductor. Following the suggestion of P. W. Anderson, there were early theories of these phases as doped quantum spin liquids. However, these theories have had difficulties with two prominent observations: (i) angle-dependent magnetoresistance measurements (ADMR) in the pseudogap metal, including observation of the Yamaji effect, present convincing evidence of small hole pockets which can tunnel coherently between square lattice layers (Fang et al., Nature Physics 18, 558 (2022); Chan et al., Nature Physics 21, 1753 (2025)) and (ii) the velocities of the nodal Bogoliubov quasiparticles in the ddd-wave superconductor are highly anisotropic, with vF≫vΔv_F \gg v_ΔvF​≫vΔ​ (Chiao et al., Phys. Rev. B 62, 3554 (2000)). These notes review how the fractionalized Fermi Liquid (FL*) state, which dopes quantum spin liquids with gauge-neutral electron-like quasiparticles, resolves both difficulties. Theories of insulating quantum spin liquids employing fractionalization of the electron spin into bosonic or fermionic partons are discussed. Doping the bosonic parton theory leads to a holon metal theory: while not appropriate for the cuprate pseudogap, this theory is argued to apply to ultracold atom experiments on the Lieb lattice. Doping the fermionic parton theory leads to a ddd-wave superconductor with nearly isotropic quasiparticle velocities. The construction of the FL* state in a single band Hubbard-type model is described using a quantum dimer model, followed by a more realistic description using the Ancilla Layer Model (ALM), which is then used to obtain the theory of the pseudogap and the ddd-wave superconductor. The ALM also leads to a wavefunction for the FL* state of the Hubbard model, which agrees with ultracold atom observations.