#### 3.1. Formulation and Coupling Constants

We apply the renormalization group approach to our effective electron gas model described above. In the following, we sketch out only the main steps of the procedure that will be useful later on for applications [

17,

46,

47]. We follow Ref. [

46] and write the partition function in the functional integral form,

over a set anticommuting fermion variables

$\{{\psi}^{*},\psi \}$, where the measure is

and

$\overline{k}=\left(k,{\omega}_{n}=(2n+1)\pi T\right)$ (

${k}_{B}=1$ throughout). The action

$S={S}_{0}+{S}_{I}$ splits into a free-quadratic-

$\left({S}_{0}\right)$ and an interacting-quartic-(

${S}_{I}$) parts,

where

z and

${z}_{1,2,3}$ are in order the renormalization factors for the one-particle propagator

and the four-points electron-electron vertices

${\mathsf{\Gamma}}_{1,2,3}$. The

${z}_{i}$ are combined to

z to give the renormalization factors

${z}^{2}{z}_{i}$ for each coupling

${g}_{i=1,2,3}$. At the bare level, the couplings

${g}_{i}$ are defined at the band edge energy cutoff

${\mathsf{\Lambda}}_{0}(\equiv {E}_{F})$ above and below the Fermi level where both

z and the

${z}_{i}$’s equal unity.

The RG transformation is standard and consists in the succesive integrations of electronic degrees of freedom, denoted by

${\overline{\psi}}^{(*)}$, in outer energy shell of thickness

$\mathsf{\Lambda}\left(\ell \right)d\ell $ on both sides of the Femi level of the lower band, where

$\mathsf{\Lambda}\left(\ell \right)={\mathsf{\Lambda}}_{0}{e}^{-\ell}$ is the cutoff at the step

ℓ of the RG procedure. The integration of degrees of freedom from step

ℓ to

$\ell +d\ell $ is achieved perturbatively. This recursive transformation can be written in the form

where

${\langle {S}_{I}^{n}\rangle}_{0,d\ell}$ are outer shell free (loop) averages with external fermion legs in the inner energy shells at

$\mathsf{\Lambda}\le \mathsf{\Lambda}\left(\ell +d\ell \right)$. The effective-renormalized-action

$S{[{\psi}^{*},\psi ]}_{\ell +d\ell}$ at

$\ell +d\ell $ leads to the recursion transformation for the

${z}^{\prime}s$.

Thus for the one-particle propagator,

$z(\ell +d\ell )=z\left(\ell \right)z\left(d\ell \right)$, which leads to the familiar result at the two-loop level [

17,

46,

47],

which is independent of the phase of the Umklapp term. The recursion relations

${g}_{i}(\ell +d\ell )={z}_{i}\left(d\ell \right){z}^{2}\left(d\ell \right){g}_{i}\left(\ell \right)$ for the coupling constants lead to the two-loop flow equations

The first equation for

${\overline{g}}_{1}$ is connected to spin degrees of freedom and is decoupled from

$(2{\overline{g}}_{2}-{\overline{g}}_{1},{\overline{g}}_{3}^{p})$, which is connected to the charge. These extend the known flow equations of the electron gas model [

17,

46,

47] to the case of a complex

${g}_{3}^{p}$. Note that only the amplitude of Umklapp

$|{g}_{3}|$ renormalizes, whereas its phase

$\theta $ remains scale invariant [

$\mathrm{Im}({d}_{\ell}ln{g}_{3}^{p})=0$] and is then fixed at the bare level by the expression (

21). The renormalization of

${g}_{4}$ is here neglected. However the influence of this coupling has been incorporated through the normalization

${\overline{g}}_{1}={g}_{1}/\pi {v}_{\sigma}$ and

$(2{\overline{g}}_{2}-{\overline{g}}_{1},{\overline{g}}_{3}^{p})$ $=(2{g}_{2}-{g}_{1},{g}_{3}^{p})/\pi {v}_{\rho}$ for the decoupled spin (

$\sigma $) and charge (

$\rho $) interactions, respectively [

17], where

${v}_{\sigma ,\rho}={v}_{F}\mp {g}_{4}/2\pi $ are the spin and charge velocities.

The properties of the above flow equations are standard [

17,

47] and can be summarized as follows. In the spin sector for instance, the negative (positive) sign of

${g}_{1}$ determines the conditions for the flow to strong (weak) attractive coupling,

${\overline{g}}_{1}^{*}\to -2$ $({\overline{g}}_{1}^{*}\to 0)$, as

$\ell \to \infty $. In the attractive case, this indicates the emergence of a spin gap

${\mathsf{\Delta}}_{\sigma}$, whose scale is of the order of the cutoff energy

$2\mathsf{\Lambda}\left({\ell}_{\sigma}\right)$ at which the flow of

${\overline{g}}_{1}$ in (

28) becomes singular at the one-loop

$\mathcal{O}\left({\overline{g}}_{1}^{2}\right)$ level, namely

${\mathsf{\Delta}}_{\sigma}\sim 2{E}_{F}{e}^{-1/|{\overline{g}}_{1}|}$.

If we now consider the charge sector, the magnitude of

$2{\overline{g}}_{2}-{\overline{g}}_{1}$ with respect to

$|{\overline{g}}_{3}|$ at the bare level determines the conditions for strong coupling or a charge gap

${\mathsf{\Delta}}_{\rho}$. Thus for

${\overline{g}}_{1}-2{\overline{g}}_{2}\ge |{\overline{g}}_{3}|$, charge degrees of freedom remain gapless since

$|{\overline{g}}_{3}^{*}|\to 0$ and

${\overline{g}}_{1}^{*}-2{\overline{g}}_{2}^{*}$ is non universal as

$\ell \to \infty $. In the whole region where

${\overline{g}}_{1}-2{\overline{g}}_{2}<|{\overline{g}}_{3}|$, both

$2{\overline{g}}_{2}^{*}-{\overline{g}}_{1}^{*}\to 2$ and

$|{\overline{g}}_{3}^{*}|\to 2$ are marginally relevant and scale to strong coupling when

$\ell \to \infty $. An order of magnitude for the charge gap

${\mathsf{\Delta}}_{\rho}$ can be readily given by the singularities encountered at a finite

${\ell}_{\rho}$ in (

29) and (

30) at the one-loop,

$\mathcal{O}\left({\overline{g}}^{2}\right)$, level [

16]. For

$-|{\overline{g}}_{3}|<{\overline{g}}_{1}-2{\overline{g}}_{2}<|{\overline{g}}_{3}|$, one has

${\mathsf{\Delta}}_{\rho}\sim 2\mathsf{\Lambda}\left({\ell}_{\rho}\right)=2{E}_{F}exp[-c/\sqrt{|{\overline{g}}_{3}{|}^{2}-{(2{\overline{g}}_{2}-{\overline{g}}_{1})}^{2}}]$, where

$c=arccos[(2{\overline{g}}_{2}-{\overline{g}}_{1})/|{\overline{g}}_{3}|]$; for

${\overline{g}}_{1}-2{\overline{g}}_{2}=-|{\overline{g}}_{3}|$,

${\mathsf{\Delta}}_{\rho}=2{E}_{F}exp[-(1/|\overline{{g}_{3}}|)]$; and finally for

${\overline{g}}_{1}-2{\overline{g}}_{2}<-|{\overline{g}}_{3}|$, one has

${\mathsf{\Delta}}_{\rho}=2{E}_{F}exp[-{c}^{\prime}/\sqrt{{(2{\overline{g}}_{2}-{\overline{g}}_{1})}^{2}-{|{\overline{g}}_{3}|}^{2}}]$, where

${c}^{\prime}={cosh}^{-1}[(2{\overline{g}}_{2}-{\overline{g}}_{1})/|{\overline{g}}_{3}|]$.

We display in

Figure 3 the contour plot of the scale for the charge gap

${e}^{-{\ell}_{\rho}}={\mathsf{\Delta}}_{\rho}/2{E}_{F}$ at the one-loop level in the

$({\u03f5}_{0},\delta t)$ plane of alternating potentials and for repulsive

$(U,V)$ interactions and smaller modulations

$(\delta U,\delta V)$. In the first quadrant where both

$\delta t$ and

${\u03f5}_{0}$ are positive, the variation of

${\mathsf{\Delta}}_{\rho}$ is not monotonous; it first increases with

$\delta t$ and

${\u03f5}_{0}$ and then undergoes a smooth decreases. According to (

21), both the real and imaginary parts of Umklapp increase at relatively small

$\delta t$ and

${\u03f5}_{0}$; this is responsible for the increase of

${\mathsf{\Delta}}_{\rho}$. At sufficiently large

$\delta t$, however, a reduction of the imaginary part of

${g}_{3}^{p}$ becomes apparent and leads to the decrease of

${\mathsf{\Delta}}_{\rho}$. A similar variation of charge gap has been obtained in the bosonization approach to the alternating Hubbard model with positive

U [

28].

Interestingly, if we broaden the analysis situation where positive

$\delta U>0$ and

$\delta V>0$ are considered, the competition between a positive

${\u03f5}_{0}$ and negative

$\delta t$ can bring both the real and imaginary parts of

${g}_{3}^{p}$ and in turn

${\mathsf{\Delta}}_{\rho}$ to zero, as shown in the second quadrant of

Figure 3. Around this point, the behavior of Equations (

29) and (

30), as

$\ell \to {\ell}_{\rho}$ at the one-loop level shows that the gap vanishes following the power law

${\mathsf{\Delta}}_{\rho}/2{E}_{F}\sim [\frac{1}{2}|{\overline{g}}_{3}{|/(2{\overline{g}}_{2}-{\overline{g}}_{1})]}^{1/(2{\overline{g}}_{2}-{\overline{g}}_{1})}$.

#### 3.2. Response Functions and Phase Diagram

In order to analyze the nature of correlations and the possible phases of the above model, we proceed to the calculation of susceptibilities. To do so, we follow Ref. [

46] and add to the action a linear coupling to a set of infinitesimal source fields

$\left\{h\right\}$ to fermion pair fields. These are associated to susceptibilities that can become singular in the

$2{k}_{F}$ density-wave and superconducting channels. As infinitesimal terms, they can be combined at the bare level to the interaction term

${S}_{I}$ and treated as a perturbation. The action becomes

where

In the superconducting channel, the pair fields are

for

$\mu =0$ singlet (SS) and

$\mu =1,2,3$ triplet (TS) superconductivity. Here

${\sigma}_{0}=\mathbf{1}$,

${\sigma}_{1,2,3}$ are the Pauli matrices, and

$\overline{q}=(q,{\omega}_{m}=2\pi mT)$. The initial pair renormalization factors at

$\ell =0$ are

${z}_{\mu}^{\mathrm{s}}=1$.

In the

$2{k}_{F}$ density-wave channel, the presence of a complex Umklapp interaction term

${g}_{3}^{p}$ in (

21), which can be written as

${g}_{3}^{\pm}=\pm |{g}_{3}|{e}^{i{\theta}_{\pm}}$, where

${\theta}_{+}=\theta $ and

${\theta}_{-}=\theta -\pi $, introduces spin and charge density-wave correlations with a particular phase relation with respect to the lattice. For

$2{k}_{F}$ charge-density-wave (CDW), the pair field can be written in terms of two independent stationary waves,

where

for

$\overline{q}=(q,{\omega}_{m})$. The phase relation of CDW

${}_{{\theta}_{\pm}}$ maxima and minima with respect to the lattice is shown in

Figure 4 at

$q=2{k}_{F}$. In the absence of

${\u03f5}_{0}$ and

$\delta U$, for instance, the imaginary part of

${g}_{3}^{p}$ vanishes and

${\theta}_{-}=\pi $ and CDW

${}_{{\theta}_{-}}$ correlations are centered on bonds between dimers, whereas

${\theta}_{+}=0$ refers to CDW

${}_{{\theta}_{+}}$ whose maxima are centered on dimers. In the presence of a finite site potential

${\u03f5}_{0}$ and/or

$\delta U$, the inversion symmetry within the dimers is broken and the position of maxima for CDW

${}_{{\theta}_{\pm}}$ move accordingly (see

Figure 4).

A similar decomposition can be made for

$2{k}_{F}$ spin-density-wave (SDW

${}_{{\theta}_{\pm}}$) by introducing

for SDW

${}_{{\theta}_{\pm}}$, where

is the spin field at

$\overline{q}=(q,{\omega}_{m})$. When

${\u03f5}_{0}$ and

$\delta U$ are absent,

${g}_{3}^{p}$ is real and

${\theta}_{\pm}=0\left(\pi \right)$, so that

${\overrightarrow{S}}_{{\theta}_{\pm}}$ describe

$2{k}_{F}$ SDW with spin maxima centered on (between) the dimers, as shown in

Figure 5. In the same Figure, for finite and positive

${\u03f5}_{0}$ and/or

$\delta U$,

${\theta}_{\pm}$ moves away from

$0\left(\pi \right)$ alongside the maxima of spin density that move in (between) the unit cell.

Making the substitution

${S}_{I}\to {S}_{I}+{S}_{h}$ in the RG transformation (

26), the renormalized action at

$\mathsf{\Lambda}\left(\ell \right)$ reads

Here, the flows of renormalization factors in

${S}_{0}$ and

${S}_{I}$ coincide with those obtained previously in (

27)–(

30), whereas the

${z}_{\mu}^{r}$’s associated to the pair vertices in (

33) are governed at the two-loop level by an equation of the form [

17,

46,

47]

where for superconducting correlations

$\left(r=\mathrm{s}\right)$ the combinations of couplings

${g}_{\mu}^{r}$ are

${g}_{\mathrm{SS}}^{\mathrm{s}}=-{g}_{1}-{g}_{2}$ and

${g}_{\mathrm{TS}}^{\mathrm{s}}={g}_{1}-{g}_{2}$ for singlet and triplet superconductivity; for density-wave correlations, one has

${g}_{{\theta}_{\pm}}^{\mathrm{c}}={g}_{2}-2{g}_{1}\mp |{g}_{3}|$ for CDW

${}_{{\theta}_{\pm}}$ in the charge sector (

$r=\mathrm{c}$); and

${g}_{{\theta}_{\pm}}^{\sigma}={g}_{2}\pm |{g}_{3}|$ for SDW

${}_{{\theta}_{\pm}}$ in the spin sector

$(r=\sigma )$. The second term of (

39), which is common to all pair vertices, refers to the self-energy corrections of Equation (

27). According to (

39), the behavior of

${z}_{\mu}^{r}$ is well known and follows the power law

at large

ℓ. It signals a singularity when the exponent

${\gamma}_{\mu}^{r*}>0$. The expression for

$\frac{1}{2}{\gamma}_{\mu}^{r*}$ coincides with the right side expression of (

39) evaluated at the fixed points values of scaling Equations (

28)–(

30). A singular behavior will also be found in the corresponding expressions for susceptibilities, which are given by the quadratic field terms of (

38). These are generated by the RG transformation (

26) and take the form,

which are defined positive and evaluated in the static limit

${\overline{q}}_{\mu}=({q}_{\mu},0)$, for

${q}_{\mathrm{SS},\mathrm{TS}}=0$ and

${q}_{\mathrm{CDW},\mathrm{SDW}}=2{k}_{F}$. At large

ℓ, the susceptibilities will be governed by a power law

where

${c}_{\mu}^{r}$ is a positive constant.

The phase diagram determined by the dominant and subdominant singularities in the susceptibilities

${\chi}_{\mu}^{r}$ is shown in

Figure 6, as a function of initial

${g}_{i}$. Its structure necessarily presents many similarities with the known two-loop RG results of the electron gas model [

17,

47], but also some differences due to the presence of a complex

${g}_{3}^{p}$. In

Figure 6 the massive (

${\mathsf{\Delta}}_{\rho}\ne 0$) charge sector, delimited by the separatrix

${g}_{1}-2{g}_{2}=|{g}_{3}|$, is enlarged with

$\delta t$,

${\u03f5}_{0}$,

$\delta V$ and negative

$\delta U$, which is detrimental to the region of singular superconducting correlations on the left of this line. At

${g}_{1}>0$, this is also concomitant with the strengthening of dominant dimer or site like SDW

${}_{{\theta}_{+}}$, and subdominant interdimer or bond like CDW

${}_{{\theta}_{-}}$ singular correlations. In the attractive region where

${g}_{1}<0$, only the reinforcement of CDW

${}_{{\theta}_{-}}$ singular correlations is found on the right-hand side of the separatrix, where a gap in both spin and charge degrees of freedom occurs.