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arXiv:cond-mat/0306316 · Physical review. B, Condensed matter

Spin-charge separation and kinetic energy in the t−J model

Sanjoy Kumar Sarker

Published Aug 14, 2003• 3 citations
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Abstract

The effect of spin-charge separation on the kinetic energy (KE) of the two-dimensional $t\ensuremath{-}J$ model is examined. Using a sum rule, we derive an exact expression for the lowest possible KE ${(E}_{\mathrm{bound}})$ for any state without doubly occupied sites. The kinetic energies of the relevant slave-boson and Schwinger-boson mean-field (MF) states are found to be considerably larger than ${E}_{\mathrm{bound}}.$ The MF states exhibit complete spin-charge separation, and form the basis of a number of microscopic theories. An examination of the momentum distribution reveals that the large increase in KE of the MF states is due to excessive depletion of electrons from the bottom of the band (Schwinger boson states) and holes from the top of the band (slave boson states). To see whether the excess KE is simply due to the poor treatment of the local constraints, we solve the constraint problem analytically for the Schwinger boson MF states in the $J=0$ limit. This restores gauge invariance, incorrectly violated in the MF theories. The resulting state is a generalization of the Hartree-Fock state in the Hubbard model, but one that includes spin-wave excitations, removing a deficiency of the simple HF theory. Even after the constraints are imposed correctly, the MF kinetic energy is found to be much larger than ${E}_{\mathrm{bound}}.$ These results support the notion, advanced in earlier papers, that spin-charge separation in the MF state costs excessive kinetic energy, and makes the state unstable toward recombination processes which lead to superconductivity in $d=2$ and a Fermi liquid state in higher dimensions.

DOI
10.1103/physrevb.68.085105

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The effect of spin-charge separation on the kinetic energy (KE) of the two-dimensional t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model is examined. Using a sum rule, we derive an exact expression for the lowest possible KE (Ebound){(E}_{\mathrm{bound}})(Ebound​) for any state without doubly occupied sites. The kinetic energies of the relevant slave-boson and Schwinger-boson mean-field (MF) states are found to be considerably larger than Ebound.{E}_{\mathrm{bound}}.Ebound​. The MF states exhibit complete spin-charge separation, and form the basis of a number of microscopic theories. An examination of the momentum distribution reveals that the large increase in KE of the MF states is due to excessive depletion of electrons from the bottom of the band (Schwinger boson states) and holes from the top of the band (slave boson states). To see whether the excess KE is simply due to the poor treatment of the local constraints, we solve the constraint problem analytically for the Schwinger boson MF states in the J=0J=0J=0 limit. This restores gauge invariance, incorrectly violated in the MF theories. The resulting state is a generalization of the Hartree-Fock state in the Hubbard model, but one that includes spin-wave excitations, removing a deficiency of the simple HF theory. Even after the constraints are imposed correctly, the MF kinetic energy is found to be much larger than Ebound.{E}_{\mathrm{bound}}.Ebound​. These results support the notion, advanced in earlier papers, that spin-charge separation in the MF state costs excessive kinetic energy, and makes the state unstable toward recombination processes which lead to superconductivity in d=2d=2d=2 and a Fermi liquid state in higher dimensions.
A fermion-spin transformation is used to implement the charge-spin separation, and is developed to study the low-dimensional t-J model. In this approach, the charge and spin degrees of freedom of the physical electron are separated, and the charge degree of freedom is represented by a spinless fermion while the spin degree of freedom is representd by a hard-core boson. The on-site local constraint for single occupancy is satisfied even in the mean-field approximation and the sum rule for the physical electron is obeyed. This approach can be applied to both one and two-dimensional systems. In the one-dimensional case, the spinon as well as the physical electron behave like Luttinger liquids. We have obtained a gapless charge and spin excitation spectrum, a good ground state energy, and a reasonable electron-momentum distribution within the mean-field approximation. The correct exponents of the correlation functions and momentum distribution are also obtained if the squeezing effect and rearrangement of the spin configurations are taken into account. In the two-dimensional case, within the mean-field approximation the magnetized flux state with a gap in the spinon spectrum has the lowest energy at half filling. The antiferromagnetic long-range order is destroyed by hole doping of the order \ensuremath{\sim}10-15 % for t/J=3-5 and a disordered flux state with gapless spinon spectrum becomes stable. The calculated specific heat is roughly consistent with observed results on copper oxide superconductors. The possible phase separation is also discussed at the mean-field level.
A self-consistent theory of fluctuations is developed for the t-J model within the Schwinger-boson framework. It is shown that binding of spin and charge into physical electrons helps destroy long-range magnetic order and restore the Luttinger-Fermi surface. The bound states, which appear as poles in the electron Green's function near the top and bottom of the band, begin to break up into spin and charge on approaching the Fermi surface. For low to moderate hole densities, the quasiparticles do not have a pole character near the Fermi surface and the system is not a Fermi liquid. Implications for the cuprate superconductors are discussed.
We reconsider the problem of separation of spin and charge in one-dimensional quantum antiferromagnets. We show that spin and charge separation in one-dimensional strongly correlated systems cannot be described by the slave-boson or fermion representation within any perturbative treatment of the interactions between the slave holons and slave spinons. The constraint of single occupancy must be implemented exactly. As a result the slave fermions and bosons are not part of the physical spectrum. Instead, the excitations that carry the separate spin and charge quantum numbers are solitons. To prove this result, it is sufficient to study the pure spinon sector in the slave-boson representation. We start with a short-range resonating-valence-bond state spin liquid mean-field theory for the frustrated antiferromagnetic spin-1/2 chain. We derive an effective theory for the fluctuations of the Affleck-Marston and Anderson order parameters. We show how to recover the phase diagram as a function of the frustration by treating the fluctuations nonperturbatively.
We reconsider the strong-coupling expansion for the Hubbard model recently introduced by Sarker and Pairault et al. By introducing slave particles that act as projection operators onto the empty, singly occupied and doubly occupied atomic states, the perturbation theory around the atomic limit distinguishes between processes that do conserve or do not conserve the total number of doubly occupied sites. This allows for a systematic t/U expansion that does not break down at low temperature (t being the intersite hopping amplitude and U the local Coulomb repulsion). The fermionic field becomes a two-component field, which reflects the presence of the two Hubbard bands. The single-particle propagator is naturally expressed as a function of a 2×2 matrix self-energy. Furthermore, by introducing a time- and space-fluctuating spin-quantization axis in the functional integral, we can expand around a "non-degenerate" ground-state where each singly occupied site has a well defined spin direction (which may fluctuate in time). This formalism is used to derive the effective action of charge carriers in the lower Hubbard band to first order in t/U. We recover the action of the t–J model in the spin-hole coherent-state path integral. We also compare our results with those previously obtained by studying fluctuations around the large-U Hartree–Fock saddle point.
We study the ``t-J'' model with strongly correlated electrons (infinite-U Hubbard-model kinetic energy plus superexchange J) by exact diagonalization on small two-dimensional clusters of six, eight, and twelve sites on a square lattice with periodic boundary conditions. We find an attractive pairing potential for two holes when J/t\ensuremath{\ge}(J/t)c{)}_{\mathit{c}})c​\ensuremath{\simeq}0.2, with t the hopping matrix element. Ground-state calculations show that extended s-wave and d-wave pairing susceptibilities and equal-time pairing correlations grow weakly with increasing J and system size at fixed filling n. Pairing correlations may be enhanced for (J/t)>(J/t)c{)}_{\mathit{c}})c​\ensuremath{\approxeq}0.2. We compare equal-time pairing correlations to a mean-field theory for J=0 and note potential difficulties with interpreting numerical results for such small systems. The widths of one- and two-hole energy bands in low-spin states are of order J; the one-hole effective mass is larger than the two-hole effective mass in such states. Spin-charge correlation functions for a single hole in the eight- and twelve-site lattices are consistent with the spin-charge decoupling scenario for resonating-valence-bond states.
Quasiparticle properties are explored in an effective theory of the $t\ensuremath{-}J$ model which includes two important components: spin-charge separation and unrenormalizable phase shift. We show that the phase shift effect indeed causes the system to be a non-Fermi liquid as conjectured by Anderson on general grounds. But this phase shift also drastically changes a conventional perception of quasiparticles in a spin-charge separation state: an injected hole will remain stable due to the confinement of spinon and holon by the phase shift field despite the background is a spinon-holon sea. True deconfinement only happens in the zero-doping limit where a bare hole will lose its integrity and decay into holon and spinon elementary excitations. The Fermi-surface structure is completely different in these two cases, from a large band-structure-like one to four Fermi points in one-hole case, and we argue that the so-called underdoped regime actually corresponds to a situation in between, where the ``gaplike'' effect is amplified further by a microscopic phase separation at low temperature. Unique properties of the single-electron propagator in both normal and superconducting states are studied by using the equation of motion method. We also comment on some influential ideas proposed in the literature related to the Mott-Hubbard insulator and offer a unified view based on the present consistent theory.