We study an effective action of bosons interacting with a gauge field, which is derived from the t-J model in the slave-boson approach. We use a renormalization-group (RG) scheme to calculate the temperature
TBE below which the bosons start to condense in a superfluid state. One expects that a strongly fluctuating gauge field tends to destroy this superfluid state. Our numerical calculations indeed show that the presence of a gauge field suppresses the onset of superfluidity. This suppression is partly due to quantum fluctuations, and partly due to thermal excitations of the gauge field. An analytical calculation of the flow in the classical regime T(l)\ensuremath{\gg}1 shows that our perturbative RG scheme provides an expansion of the suppression of
TBE in powers of
q002/
u0, where
q002 is the coupling constant of the gauge field, and
u0 is the short-range repulsion between the bosons. The RG scheme breaks down if
q002/
u0\ensuremath{\gtrsim}1, in which case one gets into the strong-coupling limit. Thus unfortunately the RG scheme is not reliable for the strong-coupling problem, which is appropriate for the t-J model with t/J\ensuremath{\sim}1.
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 consider the quantum corrections to the conductivity of fermions interacting via a Chern--Simons gauge field and concentrate on the Hartree-type contributions. The first-order Hartree approximation is only valid in the limit of weak coupling $\ensuremath{\lambda}⪡{g}^{\ensuremath{-}1/2}$ to the gauge field (
g⪢1 is the dimensionless conductance) and results in an antilocalizing conductivity correction $\ensuremath{\sim}{\ensuremath{\lambda}}^{2}g\text{ }{\text{ln}}^{2}\text{ }T$. In the case of strong coupling, an infinite summation of higher-order terms is necessary, which includes both the virtual (renormalization of the frequency) and real (dephasing) processes. At intermediate temperatures,
T0⪡T⪡gT0, where ${T}_{0}\ensuremath{\sim}1/{g}^{2}\ensuremath{\tau}$ and $\ensuremath{\tau}$ is the elastic scattering time, the
T dependence of the conductivity is determined by the Hartree correction, $\ensuremath{\delta}{\ensuremath{\sigma}}^{H}(T)\ensuremath{-}\ensuremath{\delta}{\ensuremath{\sigma}}^{H}(g{T}_{0})\ensuremath{\propto}{g}^{1/2}\ensuremath{-}{(T/{T}_{0})}^{1/2}[1+\text{ln}{(g{T}_{0}/T)}^{1/2}]$, so that $\ensuremath{\sigma}(T)$ increases with lowering
T. At low temperatures,
T⪡T0, the temperature-dependent part of the Hartree correction assumes a logarithmic form with a coefficient of order unity, $\ensuremath{\delta}{\ensuremath{\sigma}}^{H}\ensuremath{\propto}\text{ln}(1/T)$. As a result, the negative exchange contribution $\ensuremath{\delta}{\ensuremath{\sigma}}^{\text{ex}}\ensuremath{\propto}\ensuremath{-}\text{ln}\text{ }g\text{ }\text{ln}(1/T)$ becomes dominant, which yields localization in the limit of $T\ensuremath{\rightarrow}0$. We further discuss dephasing at strong coupling and show that the dephasing rates are of the order of
T, owing to the interplay of inelastic scattering and renormalization. On the other hand, the dephasing length is anomalously short, ${L}_{\ensuremath{\varphi}}⪡{L}_{T}$, where
LT is the thermal length. For the case of composite fermions with long-range Coulomb interaction, the gauge-field propagator is less singular. The resulting Hartree correction has the usual sign and temperature dependence, $\ensuremath{\delta}{\ensuremath{\sigma}}^{H}\ensuremath{\propto}\text{ln}\text{ }g\text{ }\text{ln}(1/T)$, and for realistic
g is overcompensated by the negative exchange contribution due to the gauge-boson and scalar parts of the interaction. In this case, the dephasing length ${L}_{\ensuremath{\varphi}}$ is of the order of
LT for not too low temperatures and exceeds
LT for $T\ensuremath{\lesssim}g{T}_{0}$.
The Chern-Simons bosonization with $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ gauge field is applied to the two-dimensional $t\ensuremath{-}J$ model in the limit $t\ensuremath{\gg}J,$ to study the normal-state properties of underdoped cuprate superconductors. We prove the existence of an upper bound on the partition function for holons in a spinon background, and we find the optimal spinon configuration saturating the upper bound on average---a coexisting flux phase and
s+id-like resonating-valence-bond state. After neglecting the feedback of holon fluctuations on the U(1) field B and spinon fluctuations on the SU(2) field
V, the holon field is a fermion and the spinon field is a hard-core boson. Within this approximation we show that the B field produces a \ensuremath{\pi} flux phase for the holons, converting them into Dirac-like fermions, while the
V field, taking into account the feedback of holons produces a gap for the spinons vanishing in the zero-doping limit. The nonlinear-\ensuremath{\sigma} model with a mass term describes the crossover from the short-ranged antiferromagnetic (AF) state in doped samples to long-range AF order in reference compounds. Moreover, we derive a low-energy effective action in terms of spinons, holons and a self-generated U(1) gauge field. Neglecting the gauge fluctuations, the holons are described by the Fermi-liquid theory with a Fermi surface consisting of four ``half-pockets'' centered at (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2) and one reproduces the results for the electron spectral function obtained in the mean-field approximation, in agreement with the photoemission data on underdoped cuprates. The gauge fluctuations are not confining due to coupling to holons, but nevertheless yield an attractive interaction between spinons and holons leading to a bound state with electron quantum numbers. The renormalization effects due to gauge fluctuations give rise to non-Fermi-liquid behavior for the composite electron, in certain temperature range showing the linear in T resistivity. This formalism provides a new interpretation of the spin gap in the underdoped superconductors (mainly due to the short-ranged AF order) and predicts that the minimal gap for the physical electron is proportional to the square root of the doping concentration. Therefore the gap does not vanish in any direction. All these predictions can be checked explicitly in experiment.
It is well known that two qualitatively different pictures arise in the description of the motion of an electron in an ionic crystal. In the first case, when the electron interacts weakly with the ionic lattice of the crystal, its motion is the same as the motion of a free band electron whose energy is shifted downward with respect to the bottom of the conduction band, and the effective mass is replaced by the renormalized mass (weak-coupling polaron). In the other limiting case, when the electron interacts strongly with the ionic crystal, an entire series of different self-consistent states of the electron and lattice, each of which has its own effective mass and radius (strong-coupling polaron), arises. Although the motion of an electron in an ionic crystal is described in both limiting cases with the help of the Pekar-Frohlich Hamiltonian, each case requires its own special methods of analysis. Thus, for example, in the strong-coupling theory a special form of adiabatic perturbation theory has been developed in which the trans-lational degeneracy is removed before the perturbation expansion is made. It has been possible to relate both descriptions within the framework of Feynman's varia-tional approach, in which a smooth upper bound was obtained for the energy of the ground state of the polaron for all values of the electron-phonon coupling constant a (see Fig. 1). We call attention in this connection to Ref. 6, where the thermodynamic properties are calculated with the help of a special method for performing the calculations and for finding averages of T-products. The approach developed there is more convenient for performing specific calculations than is the use of path integrals. It seems to us that it has definite advantages in the construction of general proofs concerning the choice of the approximating Hamiltonians (this is illustrated in Ref. 6 for the example of the linear Bogolyubov model), in the investigation of the characteristics of the behavior of a polaron in external fields, etc.