Unconventional behavior of Dirac fermions in three-dimensional gauge theory
We study unconventional behavior of massless Dirac fermions due to interaction with a U(1) gauge field in three time-space dimensions. At zero chemical potential, the longitudinal and transverse components of gauge interaction are both long-ranged. There is no fermion velocity renormalization since the system respects Lorentz invariance. At finite chemical potential, the Lorentz invariance is explicitly broken by the finite Fermi surface. The longitudinal gauge interaction is statically screened and becomes unimportant, whereas the transverse gauge interaction remains long-ranged and leads to singular renormalization of fermion velocity. The anomalous dimension of fermion velocity is calculated by means of the renormalization group method. We then examine the influence of singular velocity renormalization on several physical quantities, and show that they exhibit different behavior at zero and finite chemical potential.
pacs:11.10.Hi, 11.10.Kk, 71.10.Hf
Quantum electrodynamics of massless Dirac fermions defined in three time-space dimensions (QED) has been extensively investigated for 30 years (1); (2); (3); (4); (5); (6); (7); (8); (9); (10); (11); (12); (13); (14); (15); (16); (17); (18); (19); (20); (21); (22); (23). Different from its four-dimensional counterpart, QED is a super-renormalizable field theory, so its ultraviolet behavior can be well controlled. This gauge field theory is known to exhibit asymptotic freedom (10), which means the gauge interaction becomes stronger at lower energies. Because of this feature, intriguing non-perturbative phenomena are expected to occur in the low-energy regime. When Dirac fermions are strictly massless, the Lagrangian respects a continuous chiral symmetry. However, the strong gauge interaction may trigger fermion-antifermion vacuum condensation, , which generates a finite fermion mass and induces dynamical chiral symmetry breaking (DCSB). Appelquist el al. first found that DCSB can occur when the fermion flavor is smaller than some critical value (3). Motivated by this very interesting prediction, intense theoretical effort has been devoted to studying this problem (4); (5); (6); (7); (9); (10); (11); (12); (13); (14); (15); (16); (17); (18); (19); (20). Despite some debate (1); (6); (7), most of these analytical and numerical calculations agree that a critical value exists at roughly . A remarkable consequence of DCSB is that it leads to weak confinement (8); (9).
Apart from exhibiting many properties interesting in the context of elementary particle physics, QED also has extensive applications in condensed matter physics. Specifically, it serves as an effective low-energy theory of -wave high-temperature superconductor (24); (25); (26); (27); (28); (29); (30); (31); (32); (33); (36); (37); (34); (14); (35); (38) and some quantum spin liquid state (39). The occurrence of DCSB in QED corresponds to the formation of two-dimensional long-range antiferromagnetic order (28); (32); (33); (14); (35). Recently, it has been proposed that QED may be simulated on optical lattice (40), which provides an opportunity of measuring DCSB experimentally.
When the fermion flavor is beyond the threshold, , no dynamical fermion mass can be generated. There is no fermion vacuum condensation, i.e., , and the continuous chiral symmetry is preserved. However, the absence of chiral condensation in the vacuum does not mean that the chiral symmetric phase is trivial. On the contrary, many highly nontrivial features can emerge in the symmetric phase of QED. In particular, the gauge interaction is able to cause breakdown of Fermi liquid (FL). In 1973, Holstein et al. showed (41) that the unscreened transverse component of electromagnetic field in (3+1)-dimensional non-relativistic electron gas leads to unusual logarithmic specific heat, , which is apparently out of the scope of FL theory. This discovery have stimulated extensive investigations of non-FL behavior in various gauge theories, in the contexts of both condensed matter physics (42); (43); (46); (44); (49); (47); (48); (45); (50); (51); (27); (52); (53) and particle physics (54); (55); (58); (56); (57). The non-FL behavior of massless Dirac fermions in QED has also been discussed (31); (21); (22). In addition, QED can be used to describe some intriguing states, such as algebraic spin liquid state (29); (30); (31); (32); (33); (36) and algebraic charge liquid state (37).
In some many-particle systems described by QED, there is a finite density of massless Dirac fermions. The finite fermion density is usually represented by chemical potential . An interesting question is how this chemical potential affects the physical properties of QED. The impacts of finite chemical potential on DCSB have been addressed in (17); (18); (20), where it is found that the critical flavor is lowered as chemical potential grows and DCSB is completely suppressed when chemical potential is sufficiently large. In the chiral symmetric phase with strictly massless Dirac fermions, the gauge interaction also leads to different properties at zero and finite chemical potential. For instance, the Dirac fermion damping rate behaves as at zero chemical potential (21) and at finite chemical potential (22).
In this paper, we consider chiral symmetric phase of QED and study unconventional behavior of massless Dirac fermions at finite chemical potential. We assume a relatively large fermion flavor so that the fermions are strictly massless. One important effect of a finite chemical potential is that it explicitly breaks the Lorentz invariance. As a consequence, the longitudinal component of gauge field develops an effective mass that is proportional to the chemical potential , which is analogous to the static screening of Coulomb interaction. However, the transverse component of gauge field remains massless due to the gauge invariance. Because of such difference in the longitudinal and transverse components of gauge field, the temporal and spatial components of fermion self-energy are no longer identical, which in turn gives rise to nontrivial renormalization of fermion velocity.
We shall analyze fermion velocity renormalization by performing a renormalization group (RG) calculation (59); (60); (61); (62). The fermion velocity remains a constant at zero chemical potential after including the gauge interaction corrections since the Lorentz invariance is preserved. However, it acquires strong momentum dependence after developing an anomalous dimension at finite chemical potential. The appearance of nonzero anomalous dimension is a consequence of Lorentz symmetry breaking and gauge symmetry. Therefore, QED with a finite chemical potential is fundamentally different from that defined at zero chemical potential. We then evaluate specific heat, density of states (DOS), and compressibility of massless fermions using obtained in the RG analysis. After analytical and numerical calculations, we demonstrate that massless Dirac fermions exhibits unconventional, non-FL like, behavior at finite chemical potential.
The rest of this paper is organized as follows. We define the Lagrangian and then perform RG analysis of fermion velocity renormalization in Sec. II. A finite anomalous dimension of fermion velocity is obtained. We calculate specific heat in Sec. III, and DOS and compressibility in Sec. IV. In section V, we briefly summarize the results.
Ii Renormalization group analysis of Fermi velocity renormalization
The Lagrangian density for with -flavor Dirac fermions is given by
where represents the chemical potential and is the constant fermion velocity. The Dirac fermions can be described by a four-component spinor field and . The matrices can be chosen as: , which satisfy the standard Clifford algebra with metric . In this paper, we consider a large and perform expansion. For convenience, we work in units with and restore them whenever necessary.
It is now helpful to further remark on the physical meaning of fermion velocity . There are two ways to define the velocity . If we regard QED as a standard relativistic quantum field theory, the velocity of massless particles is simply the velocity of light, i.e., . If, on the other hand, we consider the effective QED theory that is derived from a microscopic model of some condensed matter system (for instance, Hubbard model of high-temperature superconductor) (24); (25); (26), the fermion velocity should be calculated from the band structure of the corresponding microscopic model. In the latter case, the fermion velocity is no longer equal to the velocity of light, . Its magnitude is strongly material dependent, but is always much smaller than (24); (25); (26). In the present paper, we assume a constant velocity and study its renormalization due to gauge interaction at finite chemical potential. The main conclusion depends only on the momentum dependence of the renormalized velocity, but not on the concrete magnitude of the constant .
In Euclidian space, the free propagator of massless Dirac fermions at zero is
At finite , it proves convenient to work in the Matsubara formalism and write the fermion propagator as
where the fermion frequency is with being integers. For notational convenience, we use to denote the imaginary frequency , so that the fermion propagator can also be written as
We first consider the case of zero . The key physical quantity is the fermion self-energy function, which can be used to calculate the RG flow of fermion velocity (60); (61); (62). Following the RG strategy presented in Ref. (60), we introduce two cutoffs and , with being smaller than . To the leading order of expansion, the one-loop fermion self-energy at zero is
The full gauge field propagator is given by the Dyson equation
with free gauge field propagator
in the Landau gauge. To the leading order of expansion, the one-loop contribution to vacuum polarization tensor is
Here, it is convenient to define , which is fixed as taken to be large (3). Now the gauge field propagator becomes
where at zero temperature.
According to the Dyson equation, , the full fermion propagator is
Apparently, correspond to the wave function renormalization, whereas represents the product of the renormalization factors of wave function and fermion velocity.
At finite chemical potential, QED theory can exhibit qualitatively new features. First of all, there appears a finite Fermi surface , which explicitly breaks the Lorentz invariance. In this case, the temporal and spatial components of fermion self-energy are no longer equivalent, so the fermion velocity may be singularly renormalized.
To proceed, we first need to discuss the effects of a finite chemical potential on the effective gauge interaction function. These effects are reflected in the vacuum polarization function . It is technically hard to obtain an entirely analytical expression for , so we will derive an approximate that captures the essential features of QED at finite .
Under the Coulomb gauge condition, , the vacuum polarization tensor can be decomposed into two independent parts (63):
They are orthogonal and related by
Now, the full gauge field propagator can be written as
where the functions and are related to the temporal and spatial components of vacuum polarization tensor by
with . Based on these quantities, it is convenient to write the fermion self-energy as
In an earlier publication (22), the full analytical expressions of polarization function was derived. These analytical expressions are too complicated and can not be directly used. It is necessary to make proper approximations. We first consider the temporal component . In order to simplify computations, we utilize the so-called instantaneous approximation (63), i.e., , and write approximately as
Apparently, the finite serves as an energy scale: behaves quite differently above and below . In the low-energy (long wavelength) limit, , we have
As a consequence, the temporal part of gauge interaction becomes short-ranged due to static screening. In other words, the temporal component of gauge field acquires a finite effective mass that is proportional to . Such short-ranged interaction does not lead to any singular contribution to fermion self-energy, and thus can be simply neglected.
We then consider the spatial component of polarization function . Similar to its temporal counterpart, also exhibit different behavior above and below the energy scale . When , we still use the instantaneous approximation and have
When , vanishes at zero energy , so it is not appropriate to use the instantaneous approximation. The energy dependence of should be explicitly maintained (22). As pointed out in Ref. (22), the fermion self-energy is dominated by the low-energy region of . We notice this approximation is widely used in the calculations of fermion self-energy due to gauge interaction (41); (43); (44); (47); (48) and critical ordering fluctuation (64). In this region, can be approximated as
At the lowest energy , we have
This fact implies that, the transverse component of gauge interaction remains long-ranged even when the dynamical screening effect is taken into account. It also explains why the instantaneous approximation can not be used in this region. Physically, the long-range property of the transverse gauge interaction is protected by the local gauge invariance.
The fermion self-energy is given by
According to the general RG scheme (59), one needs to integrate out the high-energy degrees of freedom (fast modes) step by step, until eventually reaching the lowest energy. Since behaves differently at high and low energies, the self-energy should be calculated separately.
From RG theory, we know that the fermion velocity can receive singular renormalization only when the self-energy contains a logarithmic term. One can check that there is no such term in the regime where . Nevertheless, a logarithmic term emerges as one goes to the low-energy regime where . After neglecting , it is straightforward to obtain
Paralleling the above analysis, the second term of Eq. (27) can be computed in an analogous manner:
Now the total self-energy is
This result is valid only when . As the energy scale decreases below , should be replaced by its low-energy expression. In this case, the first term of fermion self-energy becomes
In order to simplify calculations, we can divide momenta, frequency, and chemical potential by to make all these variables dimensionless. However, for notational convenience, we still denote as . Now, the above equation is rewritten as
In the low-energy region , this integral becomes
It seems easier to first integrate over and then integrate over ,
Analogously, we have
We finally obtain the total contribution
Here we keep only the logarithmic term, which survives at the lowest energy and corresponds to the stable fixed point produced by the long-ranged transverse gauge interaction.
Summarizing the above results, we obtain the following RG equation
This equation has the following solution,
where the constant defines an anomalous dimension of velocity and the dimensionless momenta actually corresponds to with being the UV cutoff. Obviously, the constant fermion velocity becomes strongly momentum dependent due to the long-range transverse gauge interaction. The -dependence of fermion velocity is shown in Fig. (1) at both zero and finite . If we define as the bare velocity, which is taken to be unity in the figure, then the effective velocity decreases monotonically when is lowering. It eventually vanishes as .
After this renormalization, the energy spectrum of Dirac fermions becomes
This type of unusual velocity renormalization is a characteristic feature of Dirac fermion systems, and has been studied extensively in the contexts of d-wave high-temperature superconductor (27); (61); (62) and graphene (60); (66); (65). The finite anomalous dimension of fermion velocity distinguishes the system at finite from that at zero . An interesting issue is how this anomalous dimension affects the observable quantities of Dirac fermions, which will be addressed in the following two sections.
It is interesting to note that the anomalous dimension is a universal constant for any given flavor . Although is finite only at finite , it does not explicitly depend on . It is the finiteness, rather than the precise value, of that is really important. A finite is generated as long as becomes finite, no matter how small it is. Therefore, the ground state changes fundamentally once one moves away from the neutral Dirac point.
Iii Specific heat of Dirac fermions
According to Landau FL theory, a weakly interacting fermion system is in one-to-one correspondence with a non-interacting fermion gas. There is a sharp Fermi surface in a FL with well-defined quasiparticles existing in the low energy regime. The properties of a FL can be manifested in a variety of physical quantities, such as spectral function, specific heat, DOS, and susceptibility. The Coulomb interaction does not destroy the stability of FL state in normal metals, because it is always statically screened by the collective particle-hole excitations. However, non-FL behavior would emerge when there is some kind of long-range gauge interaction. We now would like to examine the corrections of gauge interaction to several physical quantities of massless Dirac fermions. We will consider specific heat in this section, and consider DOS and compressibility in the next section.
In order to calculate specific heat, we first need to calculate the free energy. In the following, to facilitate calculations we make the rescaling transformations: , where is ultraviolet cutoff. When , the specific heat due to gauge interaction has already been studied in previous works (37); (38), which show that .
The free energy density will be computed using the methods given in Ref. (67). The partition function is
where the action is expressed as
Employing the functional integral formula for Grassmann variables,
it is easy to obtain
Applying the identities,
we are left with
after some straightforward algebra. Since the free energy density , we have
where the zero-point energy (the first term in ) has been discarded. The fermion specific heat can be obtained from free energy, namely
In the absence of gauge interaction, the fermion velocity takes its bare value and the anomalous dimension . Therefore, the free energy density is simply
It is easy to integrate over , and get
where is polylogarithmic function. The specific heat has the following form,
It is obvious that the specific heat at . At finite , the transverse gauge interaction induces an anomalous dimension for fermion velocity, which modifies the Dirac fermion energy spectrum. Now the free energy density becomes
where . It is convenient to define , and write the free energy as
Making derivatives of with respect to , we obtain
which is complicated and will be evaluated numerically.
Note the UV cutoff does not qualitatively affect our basic conclusion, which allows us to set . The specific heat explicitly depends on both chemical potential and temperature . Its -dependence shown in Fig. (2) for . When chemical potential , the specific heat behaves as . At finite , the specific heat deviates from the curve. The deviation becomes more significant for larger . At low temperature, the specific heat can be approximately written as power-law, , where the exponent is a function of . Such unconventional non-FL behavior arises from the anomalous dimension of fermion velocity, , which is generated by the long-ranged transverse gauge interaction at finite . In particular, the deviation occurs once becomes finite. This implies that the ground states of QED are very different at zero and finite .
In many works on non-FL behavior caused by singular interactions, the specific heat is expressed in a logarithmic function (41); (42); (46); (27); (65); (66), i.e., . Note that this logarithmic expression does not contradict our results. Actually, the power-law specific heat presented here amounts to a summation of all powers of (56); (57); (60); (68).
In the above calculations, the anomalous dimension plays an essential role. However, it is obtained by adopting certain approximations. To examine the reliability of our results, we suppose becomes or after including higher order corrections, and show the corresponding results in Fig. (3) and Fig. (4). Apparently, the basic conclusions are independent of the precise value of anomalous dimension.
Iv Density of states and Compressibility
We now turn to the interaction corrections to DOS and compressibility. At , the propagator of massless Dirac fermion is
After analytical continuation, , we have the following retarded propagator
The corresponding spectral function is given by
which then gives rise to the DOS,
Apparently, the DOS vanishes at the Fermi level, .
At finite chemical potential , the fermion propagator becomes
The DOS can be calculated similarly, with the expression
which approaches a constant proportional to as . We are interested in the gauge interaction corrections to the above expressions. At finite , the fermion velocity becomes -dependent, . Using this expression, we find that,