Toward precision Fermi liquid theory in two dimensions

In two spatial dimensions ($d=2+1$), the Lagrangian with only the $C_{0}$ interaction is scale invariant⁵⁵5In fact, this theory is also invariant under non-relativistic conformal transformations, see Ref. Jackiw:1991je .. This is most easily seen by rescaling the field and spatial coordinates by $\psi\rightarrow M^{1/2}\psi$ and ${\bf x}\rightarrow M^{-1/2}{\bf x}$, which removes all dimensionful parameters from the Hamiltonian obtained from Eq. (1). The resulting theory has a marginal contact interaction whose strength is proportional to the dimensionless coupling $MC_{0}$. This symmetry does not survive quantization and is broken by the regularization of the singular $C_{0}$ interaction which necessarily introduces a scale into the theory. This constitutes a fundamental difference between the many-body physics of two and three dimensions.

The $\ell=0$ (s-wave) scattering amplitude is then

The effective range expansion takes the conventional form

where $a_{2}$ is the scattering length⁷⁷7Another common convention for the definition of the scattering length coincides with the diameter $a_{2D}$ in the case of the hard-disk potential, corresponding to $a_{2D}=2a_{2}\exp{(-\gamma)}$, where $\gamma$ is the Euler-Mascheroni constant. and $\sqrt{|\sigma_{2}|}$ is the effective range. This form of the expansion appears odd from the EFT perspective since the leading effect at low-$k$ is non-analytic in $k$. As will be seen below, this purely quantum mechanical effect occurs because of strong infrared effects in two dimensions.

The $\ell=1$ (p-wave) scattering amplitude is

and the p-wave effective range expansion can be written as

where $\sigma_{p}$ is a scattering volume (units of area), and $a_{p}$ is a length scale that characterizes higher order effects in the momentum expansion. For $\sigma_{p}k^{2}\ll 1$, the scattering amplitude can be expanded in perturbation theory to give

In the next subsection, these scattering amplitudes will be recovered in the EFT.

where

$\mu$ is the dimensional regularization (DR) scale, and $\epsilon\equiv 3-d\,$⁹⁹9In DR the couplings are multiplied by $\left(\frac{\mu}{2}\right)^{\epsilon}$ to keep the action dimensionless. In the EFT, this is equivalent to multiplying the divergent loop integrals by $\left(\frac{\mu}{2}\right)^{\epsilon}$.. As perturbative physics can always be treated nonperturbatively, it is convenient to neglect the $C_{2}$ contribution, and sum the bubble chain to all orders, giving

A useful integral is:

where $n$ is a non-negative integer and the logarithmic divergence has been captured by the $1/\epsilon$ pole.

Using Eq. (11) and Eq. (12) gives

Defining the renormalized EFT coefficient $C_{0}(\mu)$ with $\overline{MS}$ results in

This exact renormalization condition then gives the physical scattering amplitude,

with $C_{0}(\mu)$ treated to all orders. For the perturbative calculations presented below, it is useful to expand Eq. (14) formally to third order in the renormalized coupling,

Now consider matching Eq. (15) to the effective range expansion given in Eq. (5). In order to include effective range corrections via the $C_{2}$ operator, it is convenient to note that in DR all contact interactions can be formally summed to give

with

Matching Eq. (15), with the $C_{2}$ contribution included, to the effective range expansion then gives the EFT-inspired form:

where

As intuited above from general scaling arguments, it is clear that $\alpha(\mu)$ is a dimensionless scale-dependent coupling constant which is the natural expansion parameter in the EFT.

Evidently the condition for a bound state, $\cot\delta_{0}(i\gamma_{B})=i$ with binding momentum $\gamma_{B}>0$, is satisfied both for attractive coupling, $C_{0}<0$, and for repulsive coupling, $C_{0}>0$. This is due to the strong infrared quantum effects which give rise to the logarithm when $C_{0}$ is treated to all orders. Neglecting range corrections, the binding momentum is given by:

with binding energy $\varepsilon_{B}=-\gamma_{B}^{2}/M$.

The RG evolution of $\alpha(\mu)$ clarifies the distinction between attraction and repulsion. Consider the $C_{0}$ beta function,

Solving this equation gives the RG evolution,

In the attractive case, $\alpha(\nu)=-|\alpha(\nu)|$, and the coupling is asymptotically free, whereas in the repulsive case, $\alpha(\nu)=|\alpha(\nu)|$, and the coupling increases monotonically with scale until it hits the Landau pole at $\nu=\mu\exp\left({1}/{\alpha(\mu)}\right)$ which coincides with the binding momentum. This is illustrated in Fig. 3 with the choice $\mu=1$ and $|\alpha(1)|=1$¹⁰¹⁰10Note that here natural units are chosen such that $\mu=1$ corresponds to a typical infrared physical scale of the system.. One sees that in the repulsive case, the singularity of the $S$-matrix coincides with the position of the Landau pole, which marks the upper limit of the perturbative description in terms of $\alpha$, and is therefore unphysical. The physical cutoff of the EFT is therefore determined by the smaller of this scale and the scale $R^{-1}$ which characterizes the range of the interaction. In the attractive case, the universal interaction is UV complete, and the EFT is valid below the scale $R^{-1}$.

A noteworthy feature of the p-wave interaction, which is also the case in three dimensions, is that if the leading operators $C_{2}^{\prime}$ and $C_{4}^{\prime}$ are treated to all orders, then subleading counterterms are required Bertulani:2002sz . The highly-singular nature of the p-wave interaction renders the all-orders renormalization subtle, and analogous to all-orders renormalization of range corrections in the s-wave Beane:2021dab . In this paper, only the leading order in the perturbative expansion of the p-wave scattering amplitude will be considered. An important distinction from the s-wave is that the leading p-wave effect, due to $C_{2}^{\prime}$, does not run with the RG in $\overline{MS}$ when one-loop effects are included.

Feynman diagrams can be used to compute the effect of interactions on the energy density, and the relevant Feynman rules can be found in Refs. Hammer:2000xg ; Furnstahl:2006pa . In particular, the interaction vertices can be read off of Eq. (1) and, for corrections to the energy at weak coupling, internal lines are assigned propagators $iG_{0}(\widetilde{k})_{\alpha\gamma}$, where $\widetilde{k}\equiv(k_{0},{\bf k})$ is the three-momentum assigned to the line, $\alpha$ and $\gamma$ are spin indices, and

The second line breaks the propagator into “free” and “in-medium” components.

In medium, the free-space power counting of Eq. (2) gets simply modified by setting $E=0$ which gives

This contributes to the energy density at order $k_{\rm F}^{\chi}R^{\chi-d-1}$ where the powers of $R$ follow from dimensional analysis. For universal interactions ($C_{0}$ only), $\chi=d+1+(d-3)V_{0}^{2}$. Therefore, in three dimensions, $\chi=5+V_{0}^{2}$ and there is a perturbative expansion with a new power of $k_{\rm F}R$ accompanying each insertion of $C_{0}$. By contrast, in two dimensions, $\chi=4$, and the energy density is given by

where $f$ is a function of $\alpha$ such that ${\cal E}_{0}=k_{\rm F}^{4}\,f(0)$. In the Fermi liquid regime considered here, $f$ admits a power series expansion in $\alpha$, and the goal is to compute

up to $n_{max}$. This work will consider $n_{max}=3$ corresponding to three orders in $\alpha$.

The superfluid gap in two dimensions was originally computed in Refs. PhysRevLett.62.981 ; PhysRevB.41.327 . Here the s-wave gap in two dimensions is computed in the $\overline{MS}$ scheme in two ways: by a direct construction and minimization of the renormalized thermodynamic potential, following Ref. LOKTEV20011 , and via a direct solution of the self-consistent gap equation Papenbrock:1998wb ; Marini_1998 ; Schafer:2006yf ; Heiselberg:2000ya .

As long as $C_{0}<0$, it is necessary to treat the interactions which are kinematically enhanced by the BCS mechanism to all orders, in direct violation of the power-counting rules introduced above. For consideration of pairing phenomena, it is convenient to view the EFT somewhat more expansively. Beginning with the effective Lagrangian defined in Eq. (1), with universal interactions only and $g=2$, one goes to Euclidean space via $t\rightarrow-i\tau$, and ${\cal L}\rightarrow-{\cal L}_{E}$ to give

and a chemical potential, $\mu_{F}$, has been introduced for $\psi$ (not to be confused with the DR scale $\mu$). The partition function is then

Now if ${\cal Z}$ can be computed at finite temperature $T$, then the thermodynamic potential is known and given by

where $A$ is the area and $\beta\equiv 1/T$ with $k_{B}=1$. The solution can be found by introducing a complex auxiliary field $\Phi=C_{0}\psi_{\uparrow}\psi_{\downarrow}$ which decouples the four-Fermi interaction and whose expectation value gives the superfluid gap. The Euclidean action is now bilinear in the fermion fields and is formally solved in terms of a fermionic determinant.

Neglecting fluctuations in the fields, that is assuming $\Phi=\text{constant}\neq 0$, gives the bare thermodynamic potential at zero temperature LOKTEV20011

These divergent integrals may be evaluated with DR using the formula¹²¹²12Note that one may take a derivative of the integral with respect to $|\Phi|$, evaluate using DR and then integrate with respect to $|\Phi|$.

Renormalizing with $\overline{MS}$ using Eq. (14), and exchanging the renormalized coupling for the two-body binding energy using Eq. (21) then gives the renormalized thermodynamic potential LOKTEV20011

This remarkable formula immediately reveals that the superfluid state is energetically favorable and is intrinsically related to two-body binding PhysRevLett.62.981 ; PhysRevB.41.327 . The minimum of the potential occurs at $\Phi=\Phi^{*}=\Delta_{LO}$ and defines the leading-order gap

The density is given by

and, after using Eq. (28), results in

Combining Eq. (41) and Eq. (43) gives finally, for $\mu_{F}>0$,

in agreement with Ref. PhysRevLett.62.981 ; PhysRevB.41.327 .

It is instructive to obtain this result using Feynman diagrams as they render in-medium corrections more transparent. The gap equation is shown diagrammatically in Fig. 5 where the $C_{0}$ vertex is represented by a dotted line to distinguish which fermion lines are being contracted alma991009084709703276 . This is evaluated to LO by taking the two-particle irreducible potential equal to the tree-level contact vertex, $V_{pp}=C_{0}$, and using a propagator which accounts for the gap Schafer:2006yf ; LOKTEV20011 . The result is

where $q_{3}$ is the third component of Euclidean momentum. The gap is the self-consistent solution to this equation, which treats $C_{0}$ to all orders even if it is arbitrarily weak because of the kinematical enhancement (BCS instability) of the loop function for $|{\bf q}|\sim k_{\rm F}$. Performing the $q_{3}$ integration leaves

Using Eq. (39) and renormalizing with $\overline{MS}$ using Eq. (14) then gives

which immediately recovers Eq. (41).

The energy density of the paired state is given by

In the literature this approximation of the energy is denoted mean-field BCS theory. The contribution of the gap to the energy-per-particle is thus given by Gorkov_ContributionTheorySuperfluidity_1961 ; Heiselberg:2000ya ; PhysRevA.67.031601 ; Resende:2012zs ; Schafer:2006yf

Therefore, the existence of a bound state is a necessary and sufficient condition for the existence of s-wave pairing, in contrast to the three dimensional case PhysRevLett.62.981 ; PhysRevB.41.327 . At weak, attractive coupling, the contribution of the gap to the energy is exponentially suppressed, which, as will be seen, allows a meaningful perturbative expansion in $\alpha$.

The NLO contribution to the gap takes into account the particle-hole (“ring”) correction to the two-particle irreducible potential $V_{pp}$ Gorkov_ContributionTheorySuperfluidity_1961 ; Heiselberg:2000ya ; PhysRevA.67.031601 ; Resende:2012zs ; Schafer:2006yf as shown in the bottom of Fig. 5. This accounts for the polarizibility of the finite-density medium which effectively screens the contact interaction. For the kinematics which lead to the BCS instability, ${\bf k}_{1}=-{\bf k}_{2}\equiv{\bf k}$, ${\bf k}_{1}^{\prime}=-{\bf k}_{2}^{\prime}\equiv{\bf k}^{\prime}$ and $k=k^{\prime}=k_{\rm F}$, the potential may be computed in the gapless EFT with $k_{\rm F}R\ll 1$ to give

where $\tilde{P}_{\pm}=(0,{\bf k}\pm{\bf k}^{\prime})$. Spin indices have temporarily been restored and $\alpha,\beta$ ($\gamma,\delta$) are the spin indices of the incoming (outgoing) particles. To project the potential onto the s-wave one must integrate over all $\cos\theta=\hat{\bf k}\cdot\hat{\bf k}^{\prime}$. However, it is straightforward to show that $V_{pp}$ computed to one loop contains no partial waves higher than $\ell=0$ PhysRevB.48.1097 . Therefore the two terms with $\tilde{P}_{\pm}$ contribute equally to $V_{pp}$ which becomes

where spin indices have again been suppressed. Evaluating the integral¹³¹³13This integral appears in the NNLO correction to the energy density and is evaluated below, see Eq. (79). results in

The gap equation then becomes

Neglecting the ${\it O}(C_{0})$ corrections on the left hand side then leads to the gap energy up to NLO

in agreement with Ref. PhysRevA.67.031601 .

As the omitted corrections to $V_{pp}$ are ${\it O}(C_{0}^{3})$, one expects that Eq. (54) is valid up to corrections of ${\it O}(\alpha\Delta_{{\rm NLO}})$. It is important to stress that while the computation of the gap is valid for all interparticle separations $k_{\rm F}^{-1}$, the EFT giving rise to this screening correction is strictly valid at large interparticle separations. Indeed, at strong coupling, the paired fermions are expected to become tightly bound, leading to the BCS-BEC crossover to a gas of repulsive bosons. Clearly, in this limit, screening effects will become negligible as the diameter of the pair will be much smaller than $k_{\rm F}^{-1}$, and therefore it is expected that the LO gap contribution to the energy per particle, ${\varepsilon}_{B}/2$, will be exact. Notice, however, that mean-field BCS theory in the strong coupling molecular limit misses the correct interaction energy between composite bosons PhysRevA.67.031601 ; BertainaS . In particular, the first term in Eq. (48) is not correct in the BEC limit because it should include the interaction energy of the composite bosons.