UT-Komaba 95-9

Finite Temperature Properties of

the Gauge Theory of Nonrelativistic Fermions

Masaru Onoda,
Ikuo Ichinose^{1}^{1}1e-mail
address: , and
Tetsuo Matsui^{2}^{2}2e-mail
address:

Institute of Physics, University of Tokyo, Komaba, Tokyo, 153 Japan

Department of Physics, Kinki University, Higashi-Osaka, 577 Japan

Abstract

We study the finite temperature properties of the gauge theory of nonrelativistic fermions introduced by Halperin, Lee, and Read. This gauge theory is relevant to two interesting systems: high- superconductors in the anomalous metallic phase and a two-dimensional electron system in a strong magnetic field at the Landau filling factor . We calculate the self-energies of both gauge bosons and fermions by the random-phase approximation, showing that the dominant term at low energies is generated by the gauge-fermion interaction. The current-current correlation function is also calculated by the ladder approximation. We confirm that the electric conductivity satisfies the Drude formula and obtain its temperature dependence, which is of a non-Fermi-liquid.

## 1 Introduction

In the last several years, it has been well recognized that gauge field theories play important roles in some interesting topics in condensed matter physics, like the fractional quantum Hall effect (FQHE) and the high- superconductivity. One of the canonical low-energy models describing such electron systems is believed to be a U(1) gauge theory of nonrelativistic fermions, although its concrete form has not been identified yet.

In the strongly-correlated electron systems like the high- superconductors, it is expected that the phenomenon of charge-spin separation (CSS) takes place at low temperatures () [1], that is, the charge and spin degrees of freedom of electrons behave independently. Various experiments are explained consistently by assuming the CSS. In the previous papers [2], two of the present authors showed that the CSS can be explained very naturally by a confinement-deconfinement phase transition of strong-coupling gauge theory. As demonstrated there, the CSS occurs at low and the quasi-excitations there are holons, spinons, and gauge bosons.

For the FQHE, a Ginzburg-Landau (GL) theory has been proposed [3], which explains various experimental results. This GL theory is a Chern-Simons (CS) gauge theory coupled with a complex boson field; so-called bosonized electrons, and the FQH state is characterized as a condensation of the bosonized electrons. Motivated by the success of the GL theory as well as Jain’s idea of composite fermions for the FQHE [4], Halperin, Lee, and Read [5] studied the system of electrons at the Landau filling factor by introducing and analyzing a U(1) gauge theory of nonrelativistic fermions. This theory contains a parameter , which controls the strength of gauge-field fluctuations [see (2.1),(2.3) in Sect.2].

Because of gauge invariance, the transverse component of gauge boson may survive as a massless mode, i.e., not shielded by vacuum polarization due to fermions, and so fermions interacting massless gauge bosons may have non-Fermi-liquid behavior at low energies. With this expectation, the effect of gauge field on the low-energy fermionic excitations has been studied by the random-phase approximation (RPA) [5, 6] and by the renormalization-group (RG) equation [7]. The dynamics of transverse gauge field is controlled by the Landau dissipative (damping) term in its propagator. At , the fermions exhibit marginal-Fermi-liquid like behavior [8] due to the coupling to this gauge field. This phenomenon bears a close resemblance to the coherent soft-photon dressing of electrons in quantum electrodynamics (QED). The latter is known to be crucial for resolving the problem of infrared singularities in QED.

Therefore, ample systematic studies have been carried out so far for this gauge theory in case of . In this paper, we shall study its finite-temperature properties. There have appeared some studies on similar topics: Lee and Nagaosa [9] calculated the conductivity in the uniform RVB mean-field theory plus gauge field fluctuations of the t-J model of high- superconductivity. This case corresponds to the special value . Kim et al. [10] calculated the current-current correlation functions at at the two-loop level, and get the conductivity at finite by assuming the Drude formula and certain scaling arguments. We shall compare our methods and results with theirs in some details. They are summarized in Sect.4.3 and in Sect. 5.

This paper is organized as follows. In Sect.2, we introduce the gauge model, which is, as announced, relevant to the metallic phase of high- superconductors and the electron system at . The RG studies of the model at [7] shows that there is a nontrivial infrared (IR) fixed point, whose location depends on the parameter . This fixed point describes a non-Fermi-liquid. In Sect.3, we calculate the self-energies of both fermion and gauge-field propagators by RPA, to show that the relevant term at low energies appears through the loop corrections. In Sect.4, the current-current correlation function is calculated by the ladder approximation (LA). It is shown that the Drude formula of the conductivity is derived. By using the Kubo formula, we obtain the -dependence of dc conductivity in the leading order of low . It exhibits non-Fermi-liquid behavior for . In short, the resistivity for behaves as . For , we employ the -expansion w.r.t. . For , it behaves as . For , the Fermi-liquid behavior is obtained, i.e., . Special attentions are paid to the gauge invariance of the results. Section 5 is devoted for conclusions. In Appendix, detailed calculations of the current-current correlation function and the conductivity are presented.

## 2 Model

We shall consider a two-dimensional system of nonrelativistic spinless fermion interacting with a dynamical gauge field . In the imaginary-time formalism, the action of the model at finite is given by

(2.1) | |||||

where . The covariant derivative and the magnetic field are given by

(2.2) |

The fluctuations of are controlled by the ”potential” function,

(2.3) | |||||

where the parameter is assumed to be in the region . For the high- superconductivity, is chosen as [9], and for the electron system of the half-filled Landau level, is related to the Coulombic-type repulsion between electrons [5, 7], but not yet fixed uniquely. We have also introduced the parameter for dimensional reason.

We shall take the Coulomb gauge . The vector potential is then expressed in terms of as

(2.4) |

(2.5) |

We treat as a fundamental dynamical field, instead of itself. By substituting (2.4) into (2.1),

(2.6) | |||||

We perform Fourier transformation for and ,

(2.7) | |||||

(2.8) |

where

(2.9) |

Then the action (2.6) becomes

(2.10) | |||||

where

(2.11) |

In Sect.3, we shall study how the fermion and the gauge field propagators are renormalized by the gauge-fermion interactions, and .

## 3 Self-energies of fermions and gauge bosons

In this section we calculate the self-energies of the fermion and gauge-field propagators at finite by employing the RPA. At , it has been shown that the loop corrections generate the relevant terms at low energies [5, 7].

From (2.10), the gauge field propagator at the tree level is given by

(3.1) |

In the RPA, the diagrams in Fig.1 are summed up as a geometric series in order to obtain the corrected propagator of the gauge field:

(3.2) | |||||

By the straightforward calculation, we obtain

(3.3) | |||||

(3.4) | |||||

(3.5) | |||||

where

(3.6) | |||||

(3.7) | |||||

(3.8) |

and

(3.9) | |||||

In the later discussion, we shall assume the conditions and . These imply that we consider low-energy excitations near the Fermi surface. In this case we have

where

(3.11) | |||||

For , is evaluated as follows:

(3.12) |

From (3.3), (LABEL:Pitilde) and (3.12), we obtain

(3.13) | |||||

where . Therefore behaves as

(1)

(3.14) |

(2)

(3.15) |

Eq.(3.14) is nothing but the Landau damping factor, which plays an important role at . The above result shows that, at low , the term (3.14) is dominant, because and the summation over goes up to . On the other hand, at high , the effect of the dissipative term is less efficient. For the high- superconductivity, the above remark is important for the discussion on the confinement-deconfinement phase transition (CDPT). Actually, by using the hopping expansion [2], it is shown that the CDPT occurs in the t-J model at a finite critical temperature, . This result is strongly related with the above remark on the dissipative term. The CDPT in the present model is under study, and the results will be reported in future publications.

By using the gauge field propagator (3.13) obtained by the RPA, we shall calculate the corrected fermion propagator . The corresponding diagram is given in Fig.2, which gives rise to

(3.16) | |||||

(3.17) | |||||

(3.18) |

We evaluate the -integral in (3.18) as follows. Let us assume that the dominant region satisfies . This gives rise to the peak of the integrand to be , which contradicts the assumption for low . On the other hand, if the dominant region is assumed to satisfy the opposite inequality, , the peak of the integrand, (for ) brings no incompatibility. With this assumption, the integral is simplified as follows:

(3.19) | |||||

where we have introduced the following dimensionless variables for later convenience:

(3.20) |

in the final line of (3.19) is given by

(3.21) | |||||

We get the second line of (3.19) by replacing the -sum with an integral and using the formula:

(3.22) | |||||

Especially for , we obtain

(3.23) |

By using the above results, we shall calculate the current-current correlation functions by the LA in the following section.

## 4 Current-current correlation function and the conductivity

In the previous section, we obtained the corrected gauge and the fermion propagators at finite . The loop corrections generate nontrivial relevant terms at low energies. The low-energy behavior of the fermion propagator has a branch cut rather than a pole in the frequency, and this behavior has a close resemblance to the 1D Luttinger liquid and the over-screened Kondo effect. Therefore, one can expect that the gauge-fermion interaction generates non-Fermi-liquid behavior also in gauge-invariant correlation functions. In these non-Fermi liquid systems, the -dependence of the resistivity behaves as , which is different from that of the usual Fermi liquid theory . We expect similar properties for the present gauge system.

In this section, we shall calculate the current-current correlation function (CCCF) at finite by the LA. At , this correlation was calculated by Kim et al. [10] at the two-loop order. They observed important cancellation of the leading singularities between the fermion self-energy and the vertex correction, due to the gauge invariance. In the LA below, we shall also evaluate the diagrams corresponding to their calculations, i.e., the fermion self-energy and the vertex correction.

### 4.1 Schwinger-Dyson equation

The gauge-invariant electromagnetic current is given by

(4.1) |

The effect of the second (contact) term in on the conductivity is less dominant at low . This can be seen in a straightforward manner from the calculations by Kim et al. [10]. Therefore we consider the CCCF for that is given by

(4.2) | |||||

It satisfies the ”Schwinger-Dyson (SD)” equation which is graphically depicted in Fig.3. To calculate the conductivity, we shall use the Kubo formula. In that calculation, only the limit of the above CCCF is needed. Therefore, we focus on the CCCF at zero-momenta below.

In order to solve the SD equation in the LA, it is useful to start with the following expression for the polarization tensor :

(4.3) |

In term of the above function , the SD equation is rewritten as

(4.4) | |||||

where

(4.5) |

and is the fermion propagator at the tree level defined in (3.6). To reduce the SD equation to more tractable form, we consider the following integral of the function :

(4.6) |

In terms of this , the SD equation (4.4) becomes

(4.7) | |||||

where we write

(4.8) |

In the above we used the relation:

Now, let us make the following ansatz for to solve the SD equation (4.7),

(4.9) |

where is the unknown function to be determined. This form is natural owing to the rotational symmetry. Then, must satisfy

(4.10) | |||||

where we have used the fact that the following integral is proportional to ,

[In the last line, the definition of is used.] At this stage, the CCCF is expressed as

(4.11) |

Hence, by solving (4.10) for , we get a solution for CCCF.

To solve (4.10) we first note that the integral in (4.10) is dominated by the region due to the appearance of as long as is a smooth function of . Furthermore, we make an assumption that the -dependence of is weak. One shall see that this crucial assumption is satisfied in the final solution. So this is a self-consistent solution. With these simplifications, the solution of (4.10) is easily obtained as

(4.12) |

where

(4.13) | |||||

### 4.2 The case of

Below we consider the region of low to get the concrete results. we shall discuss the case first, because this case allows us to extract the leading nontrivial term of at low . Let us evaluate as follows: we present basic steps of calculations and explain the approximations involved. The reader who are interested in more details can find them in Appendix. First, by using the polar coordinate, and setting in , we get

(4.14) | |||||