# Theory of possible sliding regimes in twisted bilayer WTe2

Yi-Ming Wu  Stanford Institute for Theoretical Physics, Stanford University,
Stanford, California 94305, USA  Chaitanya Murthy  Department of Physics,
Stanford University, Stanford, California 94305, USA  Steven A. Kivelson
Department of Physics, Stanford University, Stanford, California 94305, USA

###### Abstract

Inspired by the observation of increasingly one-dimensional (1D) behavior with
decreasing temperature in small-angle twisted bilayers of WTe2 (tWTe2), we
theoretically explore the exotic sliding regimes that could be realized in
tWTe2. At zero displacement field, while hole-doped tWTe2 can be thought of as
an array of weakly coupled conventional two-flavor 1D electron gases (1DEGs),
the electron-doped regime is equivalent to coupled four-flavor 1DEGs, due to
the presence of an additional “valley” degree of freedom. In the decoupled
limit, the electron-doped system can thus realize phases with a range of
interesting ordering tendencies, including $4k_{F}$ charge-density-wave and
charge-$4e$ superconductivity. Dimensional crossovers and cross-wire transport
due to inter-wire couplings of various kinds are also discussed. We find that
a sliding Luther–Emery liquid with small inter-wire couplings is probably most
consistent with current experiments on hole-doped tWTe2.

Introduction.—One dimensional quantum electronic systems are the best
understood examples of non-Fermi liquids, i.e. metals without coherent
electronic quasiparticles. Indeed, the low energy elementary excitations in a
1D electron gas (1DEG) are all bosonic, and exhibit phenomena such as
interaction-dependent power-law scaling of correlation functions and spin-
charge separation [1, 2, 3, 4]. Besides the 1DEG, there also exist systems
with quasi-1D properties in higher dimension, such as organic
(super)conductors [5, 6, 7] and Cs3As3-chain based superconductors [8, 9, 10,
11]. Moreover, while the stripe phases of underdoped cuprates are not, in any
literal sense, quasi-1D, it has been suggested that their unusual non-Fermi
liquid and non-BCS properties can be obtained by considering adiabatic
continuity from an artificial strongly stripe-ordered quasi-1D limit [12, 13,
14, 15, 16].

At low enough temperatures, $T<T_{\text{2D}}$, almost all quasi-1D systems
ultimately crossover to states with some version of 2D coherence. However,
there can exist “sliding phases” (e.g. sliding Luttinger or Luther–Emery
liquids) [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 22, 27, 28, 29, 30, 31, 32]
in which there is a parametrically broad temperature window,
$T_{\text{2D}}<T<T_{\text{1D}}$, within which the intra-chain quantum dynamics
is highly coherent but all inter-chain effects are incoherent and
perturbative. Such a sliding metallic phase may have been observed in, e.g.
lithium purple bronze (LiPB) [33, 34, 35, 36].

One essential question is how far can the sliding metal phase persist down in
temperature, and might there exist circumstances in which it truly persists to
$T\to 0$? Recent experiments on small angle twisted bilayer WTe2 [tWTe2, see
Fig. 1(a)] have seemingly evinced an affirmative answer to this question [37,
38]. In common with other twisted bilayer transition-metal-dichalcogenides
(TMDs) [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55,
56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66], the properties of tWTe2 are
highly tunable (by varying twist angle and gate-controlled doping). This,
combined with its anisotropic electronic structure, make it an ideal system to
address these issues. Moreover, the nature of the sliding phases resulting
from the emergence of multiple moiré bands remains largely unexplored.

In this paper, we study tWTe2 from a theoretical perspective. As shown in Fig.
1(a), the stripe-like moiré pattern of tWTe2 naturally lends itself to a model
of weakly coupled 1D wires. We first explicitly construct the moiré band
structures and show that both hole- and electron-doped tWTe2 host quasi-1D
moiré bands, with the hole-regime having weaker inter-wire hopping $t_{\bot}$.
In the decoupled wire limit, while the hole-regime is a relatively
conventional two-flavor 1DEG, the electron-regime realizes a four-flavor 1DEG.
The existence of both spin and pseudospin flavors in the electron-regime
results in a rich variety of possible favored correlations, even in the
decoupled wire limit, including phases with dominant Wigner-crystal-like
$4k_{F}$ charge-density-wave (CDW) [67, 68, 69, 70, 71] or charge-$4e$
superconducting [72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82] correlations.
Moreover, from an analysis of the hole-doped regime, we conclude that the
experimentally established bound on $T_{\text{2D}}$ reported in Ref. [38], as
well as the most salient features of the anisotropic transport at the lowest
accessible temperatures, can most readily be understood as reflecting the
behavior of a sliding Luther–Emery (LE) liquid.

Model.—Monolayer 1T′-WTe2 has a rectangular unit cell, and its low energy
electrons are from two $d$-orbitals on W atoms and two $p$-orbitals on Te
atoms [83, 84], as shown in Fig. 1(a). The symmetries of the free-standing
monolayer include lattice translations, time-reversal $\mathcal{T}$, three-
dimensional inversion $\mathcal{P}$, and a glide mirror $\bar{M}_{y}$ that
sends $(x,y,z)\mapsto(x,-y+a_{y}/2,z)$, where $a_{y}$ is the lattice constant
in the $y$-direction. The combination of $\mathcal{P}$ and $\mathcal{T}$
ensures a two-fold (Kramers) degeneracy of each band. An $s_{z}$-conserving
spin-orbit-coupling (SOC) opens a band gap at charge neutrality, and turns
monolayer WTe2 into a $\mathbb{Z}_{2}$ topological insulator [85, 86, 87].
With electron doping it becomes superconducting [88, 89, 90, 91, 92]. In Fig.
1(b) we show the band structure and Fermi pockets at small doping of monolayer
WTe2 based on the tight-binding model of Ref. [84]. Note that for hole doping
there is a single Fermi pocket at the $\Gamma$ point, while for electron
doping there are two well-separated pockets related by
$\bar{M}_{y}\mathcal{P}\mathcal{T}$, which we label ‘u’ and ‘d’ respectively.

Figure 1: Emergent 1D physics in tWTe2. (a) Moiré pattern shows 1D stripes,
which we model as an array of weakly coupled wires. (b) Left panel - near zero
energy band structure of monolayer WTe2; red and blue are the valence and
conduction bands. Right panel - electron and hole pockets at the indicated
Fermi energies for doped WTe2 obtained from Ref.[84]. Typical Fermi surfaces
and reduced 1D model for the first moiré band(s) in hole- and electron-doped
tWTe2 ($\theta\approx 3^{\circ}$) are shown in (c) and (d) respectively.

In twisted bilayer WTe2, both $\mathcal{P}$ and $\bar{M}_{y}$ are broken, but
at small twist angle $\theta$ they remain approximate symmetries; in this
regime the distinction between $\bar{M}_{y}$ and regular mirror $M_{y}$ is
also unimportant at low energies. A displacement field breaks $\mathcal{P}$
but preserves $M_{y}$. In Figs. 1(c) and 1(d) we plot representative moiré
Fermi surfaces for hole- and electron-doped tWTe2 at $\theta\approx 3^{\circ}$
and zero displacement field, obtained using the continuum model approach [93,
94] [see the Supplementary Material (SM) [95] for details]. The hole regime
can be described as a weakly-coupled array of conventional spinful 1DEGs,
whose properties are characterized by charge and spin Luttinger parameters
$K_{\mathrm{c}}$ and $K_{\mathrm{s}}$. The sliding regimes and dimensional
crossovers in hole-doped tWTe2 are therefore readily understood based on
existing theoretical results [96, 97, 98, 99, 100, 101, 102, 103, 104]. The
electron regime hosts fermions with an additional “valley” pseudospin
$\tau_{z}=\mathrm{u},\mathrm{d}$ (which is conserved at small $\theta$ since
the u- and d-pockets are decoupled in this limit), and hence realizes an array
of four-flavor $\lambda=(s_{z},\tau_{z})$ 1DEGs. The symmetries ensure that
(at zero displacement field) $k_{F}$ is identical for all flavors in the
decoupled-wire limit [see Fig.1(d)].

The electron-doped regime of tWTe2 can in principle realize a rich variety of
phases due to its four flavors, and is the main focus of this paper. To
proceed with the analysis, we bosonize each wire via four density operators

$\displaystyle\hat{\rho}_{0}(x)=\tfrac{1}{2}\big{[}\hat{\rho}_{\uparrow,\mathrm{u}}(x)+\hat{\rho}_{\uparrow,\mathrm{d}}(x)+\hat{\rho}_{\downarrow,\mathrm{u}}(x)+\hat{\rho}_{\downarrow,\mathrm{d}}(x)\big{]},$
(1)
$\displaystyle\hat{\rho}_{1}(x)=\tfrac{1}{2}\big{[}\hat{\rho}_{\uparrow,\mathrm{u}}(x)+\hat{\rho}_{\uparrow,\mathrm{d}}(x)-\hat{\rho}_{\downarrow,\mathrm{u}}(x)-\hat{\rho}_{\downarrow,\mathrm{d}}(x)\big{]},$
$\displaystyle\hat{\rho}_{2}(x)=\tfrac{1}{2}\big{[}\hat{\rho}_{\uparrow,\mathrm{u}}(x)-\hat{\rho}_{\uparrow,\mathrm{d}}(x)+\hat{\rho}_{\downarrow,\mathrm{u}}(x)-\hat{\rho}_{\downarrow,\mathrm{d}}(x)\big{]},$
$\displaystyle\hat{\rho}_{3}(x)=\tfrac{1}{2}\big{[}\hat{\rho}_{\uparrow,\mathrm{u}}(x)-\hat{\rho}_{\uparrow,\mathrm{d}}(x)-\hat{\rho}_{\downarrow,\mathrm{u}}(x)+\hat{\rho}_{\downarrow,\mathrm{d}}(x)\big{]}.$

Here the subscripts $0$–$3$ label, respectively, the total charge, spin,
valley, and spin-valley densities. Their fluctuations are represented by
bosonic fields $\phi_{i}(x)$ via
$\delta\hat{\rho}_{i}(x)=-\partial_{x}\phi_{i}(x)/\pi$, with dual fields
$\theta_{i}(x)$ defined such that $\Pi_{i}(x)=\partial_{x}\theta_{i}(x)/\pi$
is the conjugate momentum.

We consider the generic case in which the band filling is incommensurate, so
umklapp scattering can be neglected, i.e. the Hamiltonian is invariant under
$\phi_{0}(x)\to\phi_{0}(x)+\mathrm{const}$. The most general form of the
effective Hamiltonian of a decoupled wire, respecting the symmetries of the
problem ($\mathcal{T}$, $\mathcal{P}$ and $M_{y}$, as well as conservation of
total $s_{z}$ and $\tau_{z}$) is then [95]

$\displaystyle H_{1D}$ $\displaystyle=\sum_{i=0}^{3}\frac{v_{i}}{2}\int
dx\left\\{\pi K_{i}[\Pi_{i}(x)]^{2}+\frac{1}{\pi
K_{i}}[\partial_{x}\phi_{i}(x)]^{2}\right\\}$
$\displaystyle-{\sum_{\\{ijk\\}}\frac{g_{i}}{(\pi\alpha)^{2}}}\int
dx\,\cos[2\phi_{j}(x)]\cos[2\phi_{k}(x)],$ (2)

where $v_{i}$ are the velocities of the four bosonic fields, $K_{i}$ are
Luttinger parameters incorporating all forward-scattering interactions,
$g_{i}$ are the strengths of the leading back-scattering interactions,
$\alpha$ is a short-distance cutoff, and $\\{i,j,k\\}$ runs over the three
cyclic permutations of $\\{1,2,3\\}$ 111Note that this form neglects a variety
of higher order (or at least less relevant) terms, including higher-derivative
terms and higher-order cosines of various sorts.. Note that the values of
$v_{i}$, $K_{i}$ and $g_{i}$ are not constrained by the symmetries; in
particular, the absence of $SU(2)$ symmetry for both spin and valley implies
there is no reason to expect the $K_{i}$’s to be near $1$.

Of possible inter-wire couplings, a subset are forward-scattering interactions
that couple the densities $\hat{\rho}_{i}$, or the conjugate currents, on
neighboring wires. In bosonized form, these interactions are marginal, and can
be treated exactly; they lead to a renormalization and proliferation of
effective Luttinger parameters, but preserve the “sliding” symmetry of the
system under arbitrary shifts of $\phi_{0}$ on each wire [18, 19, 20]. If
large, these terms can have a qualitative effect on the dimensional crossover.
However, these interactions are likely small in current experimental setups
due to the screening by top and bottom gates, and hence can be neglected.

Thus, the crossover to higher-dimensional behavior is induced by inter-wire
coupling of the form

$H_{c}=-\sum_{a}J_{a}\sum_{y}\int dx\
O_{a}^{\dagger}(x,y)\,O_{a}(x,y+1)+\text{h.c.},$ (3)

where $y\in\mathbb{Z}$ labels different wires and $a$ labels different
couplings. If the decoupled wire remains gapless, single-particle hopping
($J_{e}=t_{\bot}$) is usually the dominant process, where
$O_{e}^{\dagger}(x,y)\equiv\psi^{\dagger}(x,y)$ is just the fermion creation
operator. However, when there are interaction-induced gaps on each wire,
single-particle hopping is irrelevant at energies smaller than the gap, and
higher-order processes are the key actors. Inter-wire interactions between
particle-particle (e.g. charge-2e SC) or particle-hole (e.g. $2k_{F}$ CDW)
orders are generated to second order in $t_{\bot}$, giving rise to a “bare”
value of $J_{2e}\sim J_{2k_{F}}\sim t_{\bot}^{2}/w$ 222For the case of CDW
correlations, but not SDW or SC fluctuations, inter-chain Coulomb interactions
can also generate couplings, $J_{2k_{F}}\sim V$. However, there are reasons to
expect these to be small, not least if the distance to metallic gates is small
compared to the distance between wires.. We will also be interested in still
higher order terms, especially charge-$4e$ SC and $4k_{F}$ CDW, both of which
have bare values $J_{4e}\sim J_{4k_{F}}\sim t_{\bot}^{4}/w^{3}$.

Figure 2: (a) RG flows from Eq. (4) in the invariant symmetric subspace
$\tilde{g}_{j}=\tilde{g}$ and $K_{j}=K$. A fully gapless, stable Luttinger
liquid fixed line is shown in solid red. The dashed red is an unstable LL
fixed line. Beyond the separatrix (dotted blue) the system flows to strong
coupling, indicating a maximally gapped generalized LE phase. Small
perturbations that break the flavor symmetries scale away (flows converge on
the represented plane) within the green region. (b) Solution of $\delta_{i}$’s
in the unfrustrated fully gapped case obtained from Eq. (5). (c) Hierarchy in
$\delta_{i}$’s in the frustrated fully gapped case. Here for simplicity we
take $g_{1}=0.3v,g_{2}=0.25v$ and $v_{j}=v$.

Renormalization group (RG) analysis.—We first analyze the decoupled 1D wires.
From Eq. (2) we see that the charge field $\phi_{0}(x)$ is absent from the
cosine terms, and thus remains gapless, as required by the generalized
Luttinger theorem [72]. If any of the cosine terms is relevant, this typically
leads to the opening of a gap in the $\phi_{1}$, $\phi_{2}$, and/or $\phi_{3}$
sector without symmetry breaking, in which case the system can be viewed as a
multi-component generalization of the LE liquid. To quadratic order in the
dimensionless couplings $\tilde{g}_{i}\equiv g_{i}/\bar{v}$, where
$\bar{v}\equiv(v_{1}+v_{2}+v_{3})/3$, the RG equations at $T\to 0$ are [95]

$\displaystyle dK_{1}/dl$
$\displaystyle=-K_{1}^{2}\big{[}\tilde{g}^{2}_{2}(A_{3}K_{3}+A_{1}K_{1})+\tilde{g}^{2}_{3}(A_{2}K_{2}+A_{1}K_{1})\big{]},$
$\displaystyle d\tilde{g}_{1}/dl$
$\displaystyle=[2-(K_{2}+K_{3})]\tilde{g}_{1}+A_{1}^{\prime}\tilde{g}_{2}\tilde{g}_{3}K_{1},$
(4)

and four other equations obtained by cyclic permutations of the indices
$\\{1,2,3\\}$, where $A_{j}$ and $A_{j}^{\prime}$ are positive constants that
depend on the relative velocities $v_{j}/\bar{v}$ and on the cutoff scheme
333The precise values of both $A_{j}$ and $A_{j}^{\prime}$ depend on the
momentum cutoff procedure; see Ref. [116] for more discussions. Here in our
calculation we use $A_{j}\approx 0.02$ and $A_{j}^{\prime}\approx 0.56$; see
details in [95].. These non-linear equations define extremely complicated
flows in a 6-dimensional coupling space, which we have extensively explored in
various regimes. We find three qualitatively distinct sorts of “long RG-time”
($l\gg 1$) behavior (generalized basins of attraction) of these flows for
distinct ranges of initial values of the (bare) interactions, as discussed
below.

1) If $K_{i}+K_{j}>2$ for all $\\{i,j\\}$, all the $\tilde{g}_{k}$ are
irrelevant, meaning that the ground state is a fully gapless Luttinger liquid
(LL) with four gapless modes, each of which will generically have a different
velocity. The power law correlations in this phase are governed by four
continuously tunable (marginal) couplings, $K_{j}$ (including $K_{0}$).

2) If two factors $K_{i}+K_{j}>2$ and one is $<2$, the RG flows go to strong
coupling along a trajectory in which one $\tilde{g}_{j}$ grows while the other
two tend to zero, leading to a partially gapped generalized LE phase with two
of the $\phi_{j}$’s gapped. For example, consider the case in which
$K_{1}+K_{2}$ and $K_{1}+K_{3}>2$ but $K_{2}+K_{3}<2$. In this case, both
$\tilde{g}_{2}$ and $\tilde{g}_{3}$ are irrelevant and only weakly renormalize
$K_{1}$, so we need only consider the flows in the subspace
$\tilde{g}_{2,3}=0$ and $K_{1}=\text{constant}$. In this subspace,
$\epsilon\equiv 1/K_{3}-1/K_{2}$ is also an invariant. For $\epsilon=0$, the
flows are exactly the same as those for the conventional two-flavor 1DEG,
while for $\epsilon\neq 0$ they are qualitatively similar.

3) We have not found any condition in which only two of the $\tilde{g}_{j}$’s
flow to strong coupling. Thus, the third case arises when all three flow to
larger values, implying a maximally gapped generalized LE phase, in which only
$\phi_{0}$ remains gapless, and $K_{0}$ determines the power-law decay of
various correlation functions. The nature of the flow to strong coupling
depends qualitatively on
$\eta\equiv\operatorname{sgn}(\tilde{g}_{1}\tilde{g}_{2}\tilde{g}_{3})$. If
$\eta=1$, there exist stable (attractive) trajectories along which all
$|\tilde{g}_{j}|$ are equal, whereas similar trajectories are unstable if
$\eta=-1$. Indeed for $\eta=-1$ the cosine terms in Eq. (2) resemble
antiferromagnetically coupled Ising spins on a triangular lattice, and are
thus frustrated, leading to the growth of an initial hierarchy among the
$|\tilde{g}_{j}|$’s under the RG. In Fig. 2(a) we show the flows within the
invariant symmetric subspace $\tilde{g}_{j}=\tilde{g}$ and $K_{j}=K$,
highlighting the stable region in green.

Self-consistent gap equations.—Complementary to the weak coupling RG analysis,
here we estimate the values of the gaps, $\Delta_{j}$, in the various strong-
coupling limits. We solve the problem variationally, using a trial Gaussian
action with mass terms as variational parameters [95]. In the case without
frustration ($\eta=1$), the result is a set of coupled self-consistency
conditions for the gaps:

$\delta_{1}^{2}=\frac{4K_{1}}{\pi
v_{1}}\,\delta_{1}^{K_{1}}\\!\left(|g_{2}|\delta_{3}^{K_{3}}+|g_{3}|\delta_{2}^{K_{2}}\right),$
(5)

and two similar equations obtained by cyclic permutations of the indices
$\\{1,2,3\\}$, where $\delta_{i}\equiv\Delta_{i}/(v_{i}/\alpha)\ll 1$ is the
dimensionless gap. The (always possible) solution to Eq. (5) with
$\delta_{1}=0$ is physically relevant if $K_{1}+K_{2}$ and $K_{1}+K_{3}>2$
(i.e. if $g_{2}$ and $g_{3}$ are both irrelevant) while the non-zero solution
is physical otherwise. Applying this same logic to all three gap equations we
find, consistent with the RG results, that two of the three modes are gapped
if only one $g_{j}$ is relevant; for instance, if only $g_{1}$ is relevant,
then $\delta_{1}=0$ and
$\delta_{2}\sim\delta_{3}\sim|g_{1}|^{1/(2-K_{2}-K_{3})}$ with
$\delta_{2}/\delta_{3}=\sqrt{K_{2}v_{3}/K_{3}v_{2}}$. All three modes are
gapped if either two or three of the $g_{j}$’s is relevant, and the induced
gaps $\delta_{1,2,3}$ are generally comparable in magnitude, as shown in Fig.
2(b). In the case with frustration ($\eta=-1$), the gap equations are the same
except that the smallest $|g_{j}|$ is replaced by $-|g_{j}|$ 444The self-
consistent Gaussian analysis is more complicated in the (fine-tuned) case in
which $\eta=-1$ and the smallest $|g_{j}|$ is not unique; we exclude this
case.. Frustration suppresses the gaps unequally, resulting in a hierarchy in
which two gaps are parametrically larger than the third [see Fig. 2(c) and the
SM for more details].

Figure 3: Phases of decoupled wires. (a) Schematic of the spatial order
identified as WC, where red and blue represent u and d. (b) Phase diagram of
the fully gapped system with phases identified with the most divergent
susceptibility as $T\to 0$, with subleading divergences noted in parentheses.
(c) Phase diagrams for partially gapped cases when the relevant $g_{i}>0$.
Here CDW, SDW, VDW and SVDW are abbreviations for charge, spin, valley and
spin-valley $2k_{F}$ density waves respectively.

Order parameters.—We now identify the order parameters with the slowest-
decaying correlations, which determine the character of the 2D long-range
order at $T\to 0$ when inter-wire couplings are included. A complete analysis
of all such order parameters, up to fourth-order in fermion operators, is
presented in the SM [95]. In different regimes, we find the leading orders to
be various kinds of momentum-$2k_{F}$ and -$4k_{F}$ density waves
($O_{2k_{F}}\sim\psi_{R}^{\dagger}\psi_{L}$,
$O_{4k_{F}}\sim\psi_{R}^{\dagger}\psi_{R}^{\dagger}\psi_{L}\psi_{L}$, where
$\psi_{R,L}$ are right or left moving fermions with flavor indices omitted),
charge-$2e$ and -$4e$ superconductivities ($O_{2e}\sim\psi_{R}\psi_{L}$,
$O_{4e}\sim O_{2e}O_{2e}^{\prime}$), and zero momentum (Pomeranchuk)
instabilities ($O_{\text{PI}}\sim O_{2e}^{\dagger}O_{2e}^{\prime}$). Some
representative order parameters of each type in bosonized form are

$\displaystyle
O_{2k_{F}\text{-CDW}}\propto\left(\textstyle\prod_{j=1}^{3}\cos\phi_{j}-i\prod_{j=1}^{3}\sin\phi_{j}\right)e^{i\phi_{0}-i2k_{F}x},$
$\displaystyle
O_{2e,j}\propto\cos(\phi_{k}\pm\phi_{i})e^{i(\theta_{0}\pm\theta_{j})},$
$\displaystyle O_{{4k_{F}}_{j}}\propto\cos(2\phi_{j})e^{i2\phi_{0}-i4k_{F}x},$
$\displaystyle
O_{4e,j}\propto\cos(\phi_{k}+\phi_{i})\cos(\phi_{k}-\phi_{i})e^{i2\theta_{0}}$
$\displaystyle
O_{\text{PI},j}\propto\cos(\phi_{k}+\phi_{i})\cos(\phi_{k}-\phi_{i})e^{i2\theta_{j}}.$
(6)

Another interesting $4k_{F}$-order, which we identify as a “Wigner crystal”
order $O_{\text{WC}}\propto\sin(2\phi_{1})e^{i2\phi_{0}-i4k_{F}x}$, is
invariant under the combination of $x\to x+\pi/(4k_{F})$ and inversion, as
shown in Fig. 3(a).

When $\phi_{1,2,3}$ are all gapped, the connected (fluctuation) correlation
functions involving $\phi_{j}$, and the full correlation functions involving
the dual fields $\theta_{j}$, decay exponentially with distance [109]. Hence,
all order parameters containing $e^{i\theta_{j}}$ become irrelevant, as do
those containing $\sin(\phi_{j})$ or $\cos(\phi_{j})$, depending on whether
$\langle\phi_{j}\rangle=0$ or $\pi/2$, which in turn depends on the signs of
the $g_{i}$’s. Therefore, the only leading order parameters in this case are
$2k_{F}$-DW or $4e$-SC; these have scaling dimensions $K_{0}/4$ and $1/K_{0}$
respectively, so that the corresponding susceptibilities are

$\displaystyle\chi_{2k_{F}}$
$\displaystyle\sim\Delta_{1}^{K_{1}/2}\Delta_{2}^{K_{2}/2}\Delta_{3}^{K_{3}/2}\,T^{K_{0}/2-2},$
(7) $\displaystyle\chi_{\text{4e},i}$
$\displaystyle\sim\Delta_{j}^{2K_{j}}\Delta_{k}^{2K_{k}}\,T^{2/K_{0}-2}.$

Note that $\chi_{\text{4e}}$ is divergent for $K_{0}>1$, and is more divergent
than $\chi_{2k_{F}}$ for $K_{0}>2$. Similar analysis shows that
$\chi_{4k_{F}}\sim T^{2K_{0}-2}$ diverges for $K_{0}<1$, but is always less
divergent than $\chi_{2k_{F}}$. The phase diagram for the fully gapped case
with all $g_{i}>0$ is depicted in Fig. 3(b).

Phase diagrams for the leading orders in partially gapped cases are shown in
Fig. 3(c). Depending on which $g_{i}$ flows to strong coupling, there are
three different cases, but the phase boundaries for $2k_{F}$, $4k_{F}$, $2e$,
$4e$ and PI orders are the same in each case. The specific orders, however,
indeed depend on which $g_{i}$ is relevant and also on the sign of $g_{i}$.
Figure 3(c) presents the leading orders when the revelant $g_{i}>0$, with a
complete discussion given in the SM [95]. Expressions for the various
susceptibilities, such as those in Eq. (7), in a partially gapped phase or in
a fully gapped phase at temperatures large compared to a particular gap, can
be obtained from those of the fully gapped phase by replacing factors of
$\Delta_{i}\to T$.

flavor number |  $\alpha_{2k_{F}}$ |  $\alpha_{4k_{F}}$ |  $\alpha_{e}$ |  $\alpha_{2e}$ |  $\alpha_{4e}$
---|---|---|---|---|---
2 (gapless) | $K_{c}+K_{s}$ | $4(K_{c}+K_{s})$ | $\frac{1}{4}\sum_{i=c,s}(K_{i}+K_{i}^{-1})$ | $K_{s}+K_{c}^{-1}$ | $4(K_{s}+K_{c}^{-1})$
2 (spin gapped) | $K_{c}$ | $4K_{c}$ | – | $K_{c}^{-1}$ | $4K_{c}^{-1}$
4 (gapless) | $\frac{1}{2}\sum_{j=0}^{3}K_{j}$ | $2(K_{0}+K_{j})$ | $\frac{1}{8}\sum_{i=0}^{3}(K_{i}+K_{i}^{-1})$ | $\frac{1}{2}(K_{0}^{-1}+K_{i}+K_{j}^{-1}+K_{k})$ | $2(K_{0}^{-1}+K_{i}+K_{k})$
4 ($\Delta_{i,j}\sim\Delta,\Delta_{k}=0$) | $\frac{1}{2}(K_{0}+K_{k})$ | $2(K_{0}+K_{k})$ | – | $\frac{1}{2}(K_{0}^{-1}+K_{k}^{-1})$ | $2K_{0}^{-1}$
4 ($\Delta_{1,2,3}\sim\Delta$) | $\frac{1}{2}K_{0}$ | $2K_{0}$ | – | – | $2K_{0}^{-1}$
Table 1: Scaling dimensions for various interchain couplings. Those which are
irrelevant in the gapped regime are not shown. In the 4-flavor cases, the
indices $\\{i,j,k\\}\in\\{1,2,3\\}$.

Interwire effects— Finally, we address the effect of weak inter-wire
couplings. We are primarily interested in an intermediate range of
temperatures, low compared to the bandwidth of each wire, $w$, but large
compared to a dimensional crossover scale, $T_{2D}$. In this “sliding regime,”
$T_{\text{2D}}<T\ll w$, the inter-wire couplings can be treated
perturbatively. The cross-wire conductivity $\sigma_{\bot}(T)$ has a power-law
dependence on $T$, and can be expressed as a sum of single-particle and $n=$
even particle (Josephson) tunneling processes with amplitudes $J_{ne}$:
$\sigma_{\bot}=\sigma_{\bot,e}+\sigma_{\bot,2e}+\cdots$, where
$\sigma_{\bot,ne}\sim|J_{ne}|^{2}$. The crossover scale $T_{\text{2D}}$ marks
the point where the product of the most relevant coupling $J_{a}$ and the
associated susceptibility $\chi_{a}(T)$ becomes order 1. Thus, $T_{\text{2D}}$
is parametrically smaller than $w$ when the inter-wire coupling is weak. For
the cases considered in this paper [110, 111, 112, 113],

$\displaystyle\frac{T_{2D}}{w}$
$\displaystyle\sim\left(\frac{J_{a}}{w}\right)^{1/(2-\alpha)}\left(\frac{\Delta}{w}\right)^{\alpha^{\prime}/(2-\alpha)},$
(8) $\displaystyle\sigma_{\bot,ne}(T)$
$\displaystyle\sim\left(\frac{J_{ne}}{w}\right)^{2}\left(\frac{\Delta}{w}\right)^{2\alpha^{\prime}}\left(\frac{T}{w}\right)^{2\alpha_{ne}-3},$

where
$\alpha=\min\\{\alpha_{2k_{F}},\alpha_{4k_{F}},\alpha_{e},\alpha_{2e},\alpha_{4e},...\\}$,
with the appropriate scaling dimensions in various situations listed are in
Table 1. If the 1D system remains gapless, $\alpha^{\prime}=0$, and
$\sigma_{\bot}(T)$ is usually dominated by single particle tunneling
$\sigma_{\bot,e}(T)$. The system can crossover to Fermi liquid behavior below
$T_{\text{2D}}$ if $J_{a}=J_{e}=t_{\perp}$ is the most relevant coupling. When
the system is gapped (which in the four-flavor case can mean either partially
or fully gapped), single particle tunneling is suppressed, and
$\sigma_{\bot}(T)$ is dominated by $\sigma_{\bot,2e}$ or in some circumstances
$\sigma_{\bot,4e}$. In this case,
$\alpha^{\prime}=\alpha_{2e}(\Delta=0)-\alpha_{2e}(\Delta\neq 0)$ for order
parameters like $2k_{F}$-CDW or $2e$-SC, while
$\alpha^{\prime}=\alpha_{4e}(\Delta=0)-\alpha_{4e}(\Delta\neq 0)$ for order
parameters like $4k_{F}$-CDW or $4e$-SC. This $\alpha^{\prime}$ ensures that
in the gapped case, $\sigma_{\bot,ne}(T)$ at $T\sim\Delta$ smoothly crosses
over to its behavior in the gapless case at $T>\Delta$.

In the fully gapped regime of the four-flavor 1DEG, Eq. (8) applies when all
induced gaps are of the same order $\Delta$. However, if there is a hierarchy
in the induced gaps such that $\Delta_{\text{max}}\gg\Delta_{\text{min}}$ as
illustrated in Fig. 2(c), and if in addition $\Delta_{\text{max}}\gg
T_{\text{2D}}\gg\Delta_{\text{min}}$, the system cannot be distinguished from
a partially gapped one, since at $T>T_{\text{2D}}$ the smaller gap
$\Delta_{\text{min}}$ can never be seen. But if $\Delta_{\text{min}}\gg
T_{\text{2D}}$, the system crosses over from one particular sliding LE liquid
at $\Delta_{\text{max}}\gg T\gg\Delta_{\text{min}}$ to another fully gapped
sliding LE liquid at $T\ll\Delta_{\text{min}}$.

Connection to experiment in hole doped WTe2.—If we assume the gapless (LL)
scenario proposed in Ref. [38], the observed $T$ dependence of $\sigma_{\bot}$
is dominated by $\sigma_{\bot,e}$, which implies $\alpha_{e}\approx 2.26>2$
and hence $J_{e}=t_{\bot}$ is irrelevant. Although the estimated
$T_{\text{2D}}$ from this scenario might be consistent with experiment, one
would need to assume $K_{c}$ and/or $K_{s}$ are unusually far from 1 (for
instance, for $K_{s}\approx 1$ one would need $K_{c}\approx 6.9$ or $0.14$).
This is difficult to reconcile with short range interactions [114], and
moreover would imply either $2k_{F}$-CDW or $2e$-SC would be strongly
relevant. However, if we assume there is a spin gap, we obtain $K_{c}\approx
0.44$ by comparing $\sigma_{\bot,2e}$ with Ref. [38]. Taking
$J_{2k_{F}}\approx J_{e}^{2}/w\approx 0.04^{2}w$, $\Delta_{s}/w=0.05$ and
$w\approx 4meV$ for estimation 555Here $w\approx 4meV$ is consistent with the
moiré band structure calculation in the SM. From both band structure
calculation and comparing $\sigma_{\bot,e}$ to experiment (under the
assumption that the system is gapless), we find $J_{e}\sim 10^{-2}w$. The gap
$\Delta_{s}$ is taking to be the possible minimal value that is above the
highest temperature $\sim 2K$ in experiment., it is easy to see that
$\Delta_{s}\approx 2.3K$ and, as long as $K_{s}>1.4$, the crossover
temperature $T_{\text{2D}}<50mK$. In this scenario the consistency with the
experiment could be realized with less extreme values of $K_{c}$ and $K_{s}$.

Acknowledgements.—We thank Sid Parameswaran, Trithep Devakul, and Sangfeng Wu
for getting us interested in this problem and providing essential guidance.
Y.-M.W. and C.M. acknowledge support from the Gordon and Betty Moore
Foundation’s EPiQS Initiative through GBMF8686. SAK and C.M. were supported in
part by the Department of Energy, Office of Basic Energy Sciences, under
contract No. DEAC02-76SF00515.

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SUPPLEMENTARY MATERIAL FOR
Theory of possible sliding regimes in twisted bilayer WTe2

Yi-Ming Wu,1, Chaitanya Murthy2 and Steven A. Kivelson2

1 Stanford Institute for Theoretical Physics, Stanford University, Stanford,
California 94305, USA 2 Department of Physics, Stanford University, Stanford,
California 94305, USA

In this Supplemental Material we show i) the construction of moiré narrow
bands of twsited bilayer WTe2 from a continuum model description, ii) explicit
bosonization procedures for the four-flavor one-dimensional electron gas, iii)
discussion about vairous order parameters and correlation functions, iv)
explicit steps to obtain the renormalization group equations and self-
consistent gap equations and v) the calculations for the cross-wire
conductivity.

##  I I. Emergence of moiré Fermi lines

In this section we first compare two existing tight-binding models for
monolayer WTe2 from Refs. [83] and [84], and then construct the moiré band
structure for small-angle twisted bilayer WTe2. The DFT results in both
references are consistent, but the tight binding models are a bit different.
Both models have inversion symmetry and time-reversal symmetry, which ensures
that each band from both models has two-fold spin degeneracy.

The first tight binding model (model I) is proposed in Ref. [83]. The lattice
model is depicted in Fig.1(a) in the main text. Note that our choice of unit
cell differs from that in Ref. [83] by $90^{\circ}$ rotation. Each unit cell
contains four orbitals for each spin projection. The tight binding model is

$H(\bm{k})=\hat{s}_{0}\otimes\hat{h}(\bm{k})+V_{\text{soc}}\hat{s}_{z}\otimes\hat{\sigma}_{z}\otimes\hat{l}_{y}$
(9)

where $\hat{s}$, $\hat{\sigma}$ and $\hat{l}$ acts on spin, sublattice and
orbital subspace respectively, and

$\hat{h}(\bm{k})=\begin{pmatrix}\epsilon_{d}(\bm{k})&0&\tilde{t}_{d}g_{k_{y}}e^{ik_{x}a_{x}}&\tilde{t}_{0}f_{k_{y}}\\\
0&\epsilon_{p}(\bm{k})&-\tilde{t}_{0}f_{k_{y}}&\tilde{t}_{p}g_{k_{y}}\\\
\tilde{t}_{d}g_{k_{y}}^{*}e^{-ik_{x}a_{x}}&-\tilde{t}_{0}f_{k_{y}}^{*}&\epsilon_{d}(\bm{k})&0\\\
\tilde{t}_{0}f_{k_{y}}^{*}&\tilde{t}_{p}g_{k_{y}}^{*}&0&\epsilon_{p}(\bm{k})\end{pmatrix}$
(10)

with $f_{k_{y}}=1-e^{-ik_{y}a_{y}}$, $g_{k_{y}}=1+e^{-ik_{y}a_{y}}$ and

$\epsilon_{l}=\mu_{l}+2t_{l}\cos[k_{y}a_{y}]+2t_{l}^{\prime}\cos[2k_{y}a_{y}]~{}~{}~{}\text{for}~{}l=p,d.$
(11)

The parameters used for calculation are (in units of eV)

$\displaystyle\mu_{d}=0.4935,~{}t_{d}=-0.28,~{}t_{d}^{\prime}=0.075,$ (12)
$\displaystyle\mu_{p}=-1.3265,~{}t_{p}=0.93,~{}t_{p}^{\prime}=0.075,$
$\displaystyle\tilde{t}_{d}=0.52~{},\tilde{t}_{p}=0.40,~{}\tilde{t}_{0}=1.02,~{}V_{\text{soc}}=0.115.$

In Fig.4(a)-(c) we show the band structure and the constant energy contours
obtained from this model. The presence of the spin-orbit coupling opens a gap
along the $\Gamma-$Y line. For slightly hole doped monolayers, there are two
hole pockets: one is along $\Gamma$-Y line, the other is located in the
opposite position. However, we emphasize that the existence of these two hole
pockets does not faithfully represent the DFT results in Ref. [83], which
shows the valence band top is still at $\Gamma$ point. For the electron-doped
monolayer, there are two electron-pockets located along $\Gamma$-Y line. These
two pockets contribute to what we call ‘u’ and ‘d’ fermions, which are related
by $C_{2z}$ rotation.

Figure 4: (a)-(c), band structure of monolayer WTe2 from the tight binding
model in Ref.[83]. The energy dispersion along a particular trajectory
$Y$-$\Gamma$-$M$ is shown in (a), the energy contours for the valence band and
conduction band are shown in (b) and (c) respectively. (d)-(f), band structure
of monolayer WTe2 from the tight binding model in Ref.[84]. The energy
dispersion along a particular trajectory $Y$-$\Gamma$-$M$ is shown in (d), the
energy contours for the valence band and conduction band are shown in (e) and
(f) respectively. The conduction band for these two models are quite similar.
For the valence band, the model in Ref.[83] indicates that the band top is
shifted away from $\Gamma$ point, while the valence band top obtained from the
model in Ref.[84] is still at $\Gamma$ point. This difference will have huge
impact on the hole-doped moiré bands, which are constructed using the valence
band dispersion. For the conduction band, both models contain two separated
electron pockets which we dub as ‘u’ and ‘d’ fermions, and therefore the
resulting electron-doped moiré bands from these two models are similar.

The second tight-binding model (model II) is proposed in Ref. [84], which we
also copy for convenience here. The first term is $H_{0}s_{0}$ where

$\displaystyle H_{0}$
$\displaystyle=[\frac{\mu_{p}}{2}+t_{px}\cos(ak_{x})+t_{py}\cos(bk_{y})]\Gamma_{1}^{-}+[\frac{\mu_{d}}{2}+t_{dx}\cos(ak_{x})]\Gamma_{1}^{+}$
(13) $\displaystyle+$ $\displaystyle
t_{dAB}e^{-ibk_{y}}(1+e^{iak_{x}})e^{i\bm{k}\cdot\Delta_{1}}\Gamma_{2}^{+}+t_{pAB}(1+e^{iak_{x}})e^{i\bm{k}\cdot\Delta_{2}}\Gamma_{2}^{-}+t_{0AB}(1-e^{iak_{x}})e^{i\bm{k}\cdot\Delta_{3}}\Gamma_{3}$
$\displaystyle-$ $\displaystyle
2it_{0x}\sin(ak_{x})[e^{i\bm{k}\cdot\Delta_{4}}\Gamma_{4}^{+}+e^{-i\bm{k}\cdot\Delta_{4}}\Gamma_{4}^{-}]+t_{0ABx}(e^{-iak_{x}}-e^{2iak_{x}})e^{i\bm{k}\cdot\Delta_{3}}\Gamma_{3}+\text{h.c.}$

The SOC part is

$\displaystyle H_{soc}$
$\displaystyle=[(\lambda_{dx}^{z}s_{z}+\lambda_{dx}^{y}s_{y})\sin(ak_{x})]\Gamma_{5}^{+}+[(\lambda_{px}^{z}s_{z}+\lambda_{px}^{y}s_{y})\sin(ak_{x})]\Gamma_{5}^{-}$
(14) $\displaystyle-$ $\displaystyle
i\lambda_{0AB}^{y}s_{y}(1+e^{iak_{x}})e^{i\bm{k}\cdot\Delta_{3}}\Gamma_{6}-i(\lambda_{0}^{z}s_{z}+\lambda_{0}^{y}s_{y})(e^{i\bm{k}\cdot\Delta_{4}}{\Gamma_{4}^{+}}-e^{-i\bm{k}\cdot\Delta_{4}}\Gamma_{4}^{-})$
$\displaystyle-$ $\displaystyle
i(\lambda^{{}^{\prime}z}_{0}s_{z}+\lambda_{0}^{{}^{\prime}y}s_{y})(e^{-ibk_{y}}e^{i\bm{k}\cdot\Delta_{4}}\Gamma^{+}-e^{ibk_{y}}e^{-i\bm{k}\cdot\Delta_{4}}\Gamma^{-})+\text{h.c.}$

Here $a=3.477$Å and $b=6.249$Å. $\Delta_{1}=\bm{r}_{Ad}-\bm{r}_{Bd}$,
$\Delta_{2}=\bm{r}_{Ap}-\bm{r}_{Bp}$, $\Delta_{3}=\bm{r}_{Ad}-\bm{r}_{Bp}$ and
$\Delta_{4}=\bm{r}_{Ad}-\bm{r}_{Ap}$, with $\bm{r}_{Ad}=(-0.25a,0.32b)$,
$\bm{r}_{Bp}=(0.25a,0.07b)$, $\bm{r}_{Ap}=(-0.25a,-0.07b)$ and
$\bm{r}_{Bd}=(0.25a,-0.32b)$. The $\Gamma$ matrices are defined as

$\displaystyle\Gamma_{0}$
$\displaystyle=\sigma_{0}l_{0},~{}\Gamma_{1}^{\pm}=\frac{\sigma_{0}}{2}(l_{0}\pm
l_{3}),~{}\Gamma_{2}^{\pm}=\frac{1}{4}(\sigma_{1}+i\sigma_{2})(l_{0}\pm
l_{3}),~{}\Gamma_{3}=\frac{1}{2}(\sigma_{1}+i\sigma_{2})il_{2}$ (15)
$\displaystyle\Gamma_{4}^{\pm}$
$\displaystyle=\frac{1}{4}(\sigma_{0}\pm\sigma_{3})(l_{1}+il_{2}),~{}\Gamma_{5}^{\pm}=\frac{\sigma_{3}}{2}(l_{0}\pm
l_{3}),~{}\Gamma_{6}=\frac{1}{2}(\sigma_{1}+i\sigma_{2})l_{1}.$

And the parameters are

$\displaystyle\mu_{p}=-1.75,~{}~{}~{}$
$\displaystyle\lambda_{0AB}^{y}=0.011{}{}{}$
$\displaystyle\mu_{d}=0.74,~{}~{}~{}$ $\displaystyle\lambda_{0}^{y}=0.051$
(16) $\displaystyle t_{px}=1.13,~{}~{}~{}$
$\displaystyle\lambda_{0}^{z}=0.012{}{}{}$ $\displaystyle
t_{dx}=-0.41,~{}~{}~{}$ $\displaystyle\lambda_{0}^{{}^{\prime}y}=0.05$
$\displaystyle t_{pAB}=0.4,~{}~{}~{}$
$\displaystyle\lambda_{0}^{{}^{\prime}z}=0.012{}{}{}$ $\displaystyle
t_{dAB}=0.51,~{}~{}~{}$ $\displaystyle\lambda_{px}^{y}=-0.04$ $\displaystyle
t_{0AB}=0.39,~{}~{}~{}$ $\displaystyle\lambda_{px}^{z}=-0.01{}{}{}$
$\displaystyle t_{0ABx}=0.29,~{}~{}~{}$ $\displaystyle\lambda_{dx}^{y}=-0.031$
$\displaystyle t_{0x}=0.14,~{}~{}~{}$
$\displaystyle\lambda_{dx}^{z}=-0.008{}{}{}$ $\displaystyle t_{py}=0.13$

In Fig. 4(d)-(f) we show the band structure and the constant energy contours
obtained from model II. Note in this model, the valence band top is at
$\Gamma$ point, which is consistent with the DFT calculations in Ref. [83],
and is also consistent with the ARPES data in Ref. [85]. For the electron-
doped monolayer, there are two electron-pockets located along $\Gamma$-Y line,
which agrees with model I and justifies our division of low energy fermions
into ‘u’ and ‘d’ sectors related by $C_{2z}$ rotation.

Figure 5: Emergence of quasi-1D moiré band structure from twisted bilayer
WTe2, obtained from both model I and model II using $w=5$meV and
$\theta=2.12^{\circ}$. (a) Cutoff procedure for constructing the moiré band
structures from the tight banding models. For electron-doped tWTe2, since the
low energy fermions are located in two separate positions, we divide the
monolayer Brillouin zone into two part which we call u- and d-fermions. Moiré
bands in this case are obtained using u- and d-fermions separately, assuming
the coupling between them is negligible. For hole-doped tWTe2, we use the band
fermions inside the blue region as a whole to construct moiré bands. (b) Band
structure for hole-doped tWTe2 based on the model I [83]. The first two bands
are energetically close to each other, which is due to the fact that in model
I there are two maxima for the valence band. The right panel shows energy
contours for the topmost moiré band. (c) Band structure for electron-doped
tWTe2 based on the model I. The two bottom bands are energetically close to
each other, which are from u- and d-fermions. The right panel shows energy
contours for the bottommost moiré bands. (d) Band structure for hole-doped
tWTe2 based on the model II [84]. Note there is only one topmost moiré band.
The right panel shows energy contours for the topmost moiré band. (c) Band
structure for electron-doped tWTe2 based on the model II. The two bottom bands
are energetically close to each other, which are from u- and d-fermions. The
right panel shows energy contours for the bottommost moiré bands.

We now discuss the band structures for twisted bilayer WTe2, using the
continuum model analogous to the twisted bilayer graphene [93] and other
twisted bilayer TMD [57, 58] systems. In practice, we need to distinguish
between electron doped and hole doped tWTe2, this is because for the hole-
doped system, the low energy fermions are from the hole pocket centered at
$\Gamma$ point, but for electron-doped system the low energy fermions are from
both u- and d-pockets. If we neglect the coupling between u and d fermions, we
can use each of the electron pocket to construct moiré band individually, then
the resulting moiré band fermions acquire additional pseudospin indices
$\tau=$u,d, and the particle number for u and d fermions are individually
conserved, and $\tau=$u,d effectively becomes a good quantum numbers in
addition to the physical spin. In Fig. 5(a) we show the momentum cutoff
procedures when constructing the moiré bands. For hole-doped systems, we
consider a large region centered at the $\Gamma$ point, while for electron
doped systems, we separately consider two regions centered at either u- or
d-pockets.

Let’s discuss the electron-doped system for instance. The continuum limit
Hamiltonian in this case can be written as (for each spin projection)

$H_{\tau}=\begin{pmatrix}\varepsilon_{\tau,t}(\bm{k})&T(\bm{r})\\\
T^{\dagger}(\bm{r})&\varepsilon_{\tau,b}(\bm{k}).\end{pmatrix}$ (17)

where $\epsilon_{\tau,t/b}(\bm{k})$ is the monolayer band dispersion rotated
by $\theta/2$ and $-\theta/2$. Technically, this can be obtained by first
diagonalizing the tight binding Hamiltonian and rotating by $\pm\theta/2$ on a
meshgrid, and then using numerical interpolation to obtain energies at
arbitrary $\bm{k}$. The interlayer coupling
$T(\bm{r})=2w[\cos(\bm{q}_{1}\cdot\bm{r})+\cos(\bm{q}_{2}\cdot\bm{r})]$ where
the two vectors $\bm{q}_{1}$ and $\bm{q}_{2}$ are shown in Fig. 5(a) and $w$
is the interlayer tunneling strength. As is also shown in Fig. 5(a), we divide
the momentum space into two parts by setting some momentum cutoff: the orange
region containing u fermions is used for construction of $H_{\text{u}}$, while
the purple region contains d fermions and is used for $H_{\text{d}}$. We also
choose some specific twist angles such that the width of the original
Brillouin zone is approximately integer multiples of the length of the
resulting moiré Brillouin zone.

Figure 6: Moiré band structure obtained from model II at $\theta=2.89$ and
$w=8$mEV for (a) hole-doping and (b) electron-doping.

In Fig. 5(b-e) we present the moiré band structure and constant energy
contours for the first moiré band calculated from both model I and model II.
Here we have chosen $w=5$meV and $\theta=2.12^{\circ}$. We note the precise
value of $w$ does not lead to any qualitative difference and other choices of
$w$ of the same order yields similar results. The hole-doped moiré bands
obtained from model I is shown in Fig. 5(b). The first two (top) bands are
energetically close to each other, a direct consequence of the fact that model
I has two valence band maximums located along $\Gamma$-Y line. But as we noted
above, this behavior is not consistent with DFT and ARPES results. The
electron-doped moiré bands obtained from model I is shown in Fig. 5(c). The
first two moiré bands are highlighted in blue, and the corresponding constant
energy contours shows that u- and d-fermions are indeed related by $C_{2z}$
rotation. The same moiré bands obtained from model II are shown in Fig. 5(d)
and (c). For the hole-doped bands, we see that the first two bands are well
separated in energy scale, which is consistent with the fact that in model II
the valence band maximum is unique and located at $\Gamma$. The electron doped
moiré bands are similar to those obtained from model I, which also reflects
the $C_{2z}$ symmetry relation between u- and d-fermions. This symmetry can
also be seen at the Hamiltonian level. Since we have
$\varepsilon_{\text{u},t}(\bm{k})=\varepsilon_{\text{d},t}(-\bm{k})$ ($\bm{k}$
here is measured from $\Gamma$ point) and $T(\bm{r})$ is invariant under
inversion,

$H_{\text{u}}(\bm{k})\leftrightarrow H_{\text{d}}(-\bm{k})$ (18)

under the inversion operation. This indicates the resulting moiré band
structure also has this relation $E_{\text{u}}(\bm{k})=E_{\text{d}}(-\bm{k})$.

From the above comparison, we believe model II is more faithful in
representing the low energy band structures of WTe2. In Fig. 6 we shown
additional results of both hole-doped and electron-doped moiré bands obtained
from model II with $w=8$meV and a larger twist angle $\theta=2.89^{\circ}$.
The moiré bands in this case are quite similar to those obtained at
$\theta=2.12^{\circ}$. Again we can clearly see the hole-doped side is more
quasi-1D, but the electron-doped side has additional pseudo-spin flavor.

##  II II. Bosonization of the Multi-component 1D system

As discussed in the main text and in the above section that we model the
decoupled 1D wire system with multi components labeled by
$\lambda=(s_{z},\tau_{z})$, where $s_{z}=\uparrow,\downarrow$ is the physical
spin and $\tau_{z}=\text{u,d}$ is Fermi line index. The spin degeneracy
ensures that the spin-up fermions and spin-down fermions have the same Fermi
momentum $k_{F}$. Furthermore, according to the inversion symmetry between u
and d fermions discussed above, the u and d fermions should also have the same
$k_{F}$ if we require each wire to respect this inversion symmetry. For
simplicity, we also assume that $k_{F}$ is incommensurate so we neglect
Umklapp scattering. The total Hamiltonian for each wire can be written as

$H_{1D}=H_{0}+H_{I}$ (19)

where

$H_{0}=\sum_{k,r,\lambda}v_{F}(rk-
k_{F})\psi_{r,\lambda}^{\dagger}(k)\psi_{r,\lambda}(k)=2\pi
v_{F}\sum_{q>0,r,\lambda}\rho_{r,\lambda}(q)\rho_{r,\lambda}(-q)$ (20)

Here $L$ is the wire length, $v_{F}$ is the Fermi velocity and $r=\pm$ denotes
right ($+$) and left ($-$) movers. The momentum summation is
$\sum_{q}\equiv\frac{1}{L}\sum_{q}$. The interaction part is written as
(assuming short range interaction)

$\displaystyle H_{I}$
$\displaystyle=\frac{1}{L}\sum_{k,p,q}\sum_{\lambda\neq\lambda^{\prime}}V_{1,\lambda\lambda^{\prime}}\psi^{\dagger}_{+,\lambda}(k)\psi^{\dagger}_{-,\lambda^{\prime}}(p)\psi_{+,\lambda^{\prime}}(p+q)\psi_{-,\lambda}(k-q)$
(21)
$\displaystyle+\frac{1}{L}\sum_{k,p,q}\sum_{r,\lambda=\lambda^{\prime}}V_{1,\lambda\lambda^{\prime}}\psi^{\dagger}_{r,\lambda}(k)\psi^{\dagger}_{-r,\lambda^{\prime}}(p)\psi_{r,\lambda^{\prime}}(p+q)\psi_{-r,\lambda}(k-q)$
$\displaystyle+\frac{1}{L}\sum_{q,r,\lambda,\lambda^{\prime}}V_{2,\lambda\lambda^{\prime}}\rho_{r,\lambda}(q)\rho_{-r,\lambda^{\prime}}(-q)+\frac{1}{L}\sum_{r,q,\lambda.\lambda^{\prime}}V_{4,\lambda\lambda^{\prime}}\rho_{r,\lambda}(q)\rho_{r,\lambda^{\prime}}(-q).$

Here we have adopted the notations from previous literatures such that $V_{1}$
is for backscatterting, and $V_{2}$ and $V_{4}$ are two forward scatterings.
We assume the system is away from commensurate filling such that $4k_{F}$ is
not a reciprocal lattice vector, so we neglect the umklapp scattering $V_{3}$.
In analogy to two-component 1D systems, it will be convenient to further
distinguish $V_{i,\lambda\lambda}$ according to different spin and orbit
configurations. To this end, we define,

$\displaystyle
V_{i,\lambda\lambda^{\prime}}=V_{i,1},~{}~{}~{}\text{if}~{}~{}s_{z}=s_{z}^{\prime},~{}\tau_{z}=\tau_{z}^{\prime};$
(22) $\displaystyle
V_{i,\lambda\lambda^{\prime}}=V_{i,2},~{}~{}~{}\text{if}~{}~{}s_{z}\neq
s_{z}^{\prime},~{}\tau_{z}=\tau_{z}^{\prime};$ $\displaystyle
V_{i,\lambda\lambda^{\prime}}=V_{i,3},~{}~{}~{}\text{if}~{}~{}s_{z}=s_{z}^{\prime},~{}\tau_{z}\neq\tau_{z}^{\prime};$
$\displaystyle
V_{i,\lambda\lambda^{\prime}}=V_{i,4},~{}~{}~{}\text{if}~{}~{}s_{z}\neq
s_{z}^{\prime},~{}\tau_{z}\neq\tau_{z}^{\prime}.$

Like the the spin and charge densities in conventional 1DEG, for the four-
component system it’s useful to define the following four chiral-densities,

$\displaystyle\rho_{0}$
$\displaystyle=\frac{1}{2}\left(\rho_{\uparrow\text{u}}+\rho_{\downarrow\text{u}}+\rho_{\uparrow\text{d}}+\rho_{\downarrow\text{d}}\right),$
(23) $\displaystyle\rho_{1}$
$\displaystyle=\frac{1}{2}\left(\rho_{\uparrow\text{u}}-\rho_{\downarrow\text{u}}+\rho_{\uparrow\text{d}}-\rho_{\downarrow\text{d}}\right),$
$\displaystyle\rho_{2}$
$\displaystyle=\frac{1}{2}\left(\rho_{\uparrow\text{u}}+\rho_{\downarrow\text{u}}-\rho_{\uparrow\text{d}}-\rho_{\downarrow\text{d}}\right),$
$\displaystyle\rho_{3}$
$\displaystyle=\frac{1}{2}\left(\rho_{\uparrow\text{u}}-\rho_{\downarrow\text{u}}-\rho_{\uparrow\text{d}}+\rho_{\downarrow\text{d}}\right),$

where the $r$-index is omitted for brevity. Using Eq.(22) and (23), we can
rewrite $H_{0}$, the two forward-scattering terms ($V_{2}$ and $V_{4}$) and
the backward-scattering with $\lambda=\lambda^{\prime}$ ($V_{1,1}$) in $H_{I}$
as

$\displaystyle\sum_{q}$
$\displaystyle[(V_{2,1}-V_{1,1})(\rho_{r,0}\rho_{-r,0}+\rho_{r,1}\rho_{-r,1}+\rho_{r,2}\rho_{-r,2}+\rho_{r,3}\rho_{-r,3})+V_{2,2}(\rho_{r,0}\rho_{-r,0}-\rho_{r,1}\rho_{-r,1}+\rho_{r,2}\rho_{-r,2}-\rho_{r,3}\rho_{-r,3})$
(24) $\displaystyle+$ $\displaystyle
V_{2,3}(\rho_{r,0}\rho_{-r,0}+\rho_{r,1}\rho_{-r,1}-\rho_{r,2}\rho_{-r,2}-\rho_{r,3}\rho_{-r,3})+V_{2,4}(\rho_{r,0}\rho_{-r,0}-\rho_{r,1}\rho_{-r,1}-\rho_{r,2}\rho_{-r,2}+\rho_{r,3}\rho_{-r,3})]$
$\displaystyle+\sum_{r,q}$ $\displaystyle[(\pi
v_{F}+V_{4,1})(\rho_{r,0}^{2}+\rho_{r,1}^{2}+\rho_{r,2}^{2}+\rho_{r,3}^{2})+V_{4,2}(\rho_{r,0}^{2}-\rho_{r,1}^{2}+\rho_{r,2}^{2}-\rho_{r,3}^{2})$
$\displaystyle+$ $\displaystyle
V_{4,3}(\rho_{r,0}^{2}+\rho_{r,1}^{2}-\rho_{r,2}^{2}-\rho_{r,3}^{2})+V_{4,4}(\rho_{r,0}^{2}-\rho_{r,1}^{2}-\rho_{r,2}^{2}+\rho_{r,3}^{2})].$

The other backward-scattering terms are cast into cosine terms by introducing
the following bosonization formula,

$\psi_{r,\lambda}(x)=\lim_{\alpha\to
0}\frac{U_{r,\lambda}}{\sqrt{2\pi\alpha}}\exp\left[i\theta_{\lambda}(x)+ir(k_{F}x-\phi_{\lambda}(x))\right]$
(25)

where $\alpha$ is a UV cutoff and $U_{r,\lambda}$ are the Klein factors. It’s
also convenient at this point to introduce $\phi_{i}$ in terms of
$\phi_{\lambda}$, completely parallel to Eq.(23),

$\displaystyle\phi_{0}$
$\displaystyle=\frac{1}{2}\left(\phi_{\uparrow\text{u}}+\phi_{\downarrow\text{u}}+\phi_{\uparrow\text{d}}+\phi_{\downarrow\text{d}}\right),$
(26) $\displaystyle\phi_{1}$
$\displaystyle=\frac{1}{2}\left(\phi_{\uparrow\text{u}}-\phi_{\downarrow\text{u}}+\phi_{\uparrow\text{d}}-\phi_{\downarrow\text{d}}\right),$
$\displaystyle\phi_{2}$
$\displaystyle=\frac{1}{2}\left(\phi_{\uparrow\text{u}}+\phi_{\downarrow\text{u}}-\phi_{\uparrow\text{d}}-\phi_{\downarrow\text{d}}\right),$
$\displaystyle\phi_{3}$
$\displaystyle=\frac{1}{2}\left(\phi_{\uparrow\text{u}}-\phi_{\downarrow\text{u}}-\phi_{\uparrow\text{d}}+\phi_{\downarrow\text{d}}\right).$

The inverse relations are

$\displaystyle\phi_{\uparrow\text{u}}$
$\displaystyle=\frac{1}{2}\left(\phi_{0}+\phi_{1}+\phi_{2}+\phi_{3}\right),$
(27) $\displaystyle\phi_{\downarrow\text{u}}$
$\displaystyle=\frac{1}{2}\left(\phi_{0}-\phi_{1}+\phi_{2}-\phi_{3}\right),$
$\displaystyle\phi_{\uparrow\text{d}}$
$\displaystyle=\frac{1}{2}\left(\phi_{0}+\phi_{1}-\phi_{2}-\phi_{3}\right),$
$\displaystyle\phi_{\downarrow\text{d}}$
$\displaystyle=\frac{1}{2}\left(\phi_{0}-\phi_{1}-\phi_{2}+\phi_{3}\right).$

Using this relation, it is straightforward to rewrite the back-scattering
terms as

$\displaystyle\frac{V_{1,2}}{(2\pi\alpha)^{2}}\int
dx\left\\{U^{\dagger}_{R\uparrow u}U^{\dagger}_{L\downarrow u}U_{R\downarrow
u}U_{L\uparrow u}e^{i2(\phi_{2}+\phi_{4})}+sU^{\dagger}_{R\uparrow
d}U^{\dagger}_{L\downarrow d}U_{R\downarrow d}U_{L\uparrow
d}e^{i2(\phi_{2}-\phi_{4})}+\text{h.c.}\right\\}$ (28) $\displaystyle+$
$\displaystyle\frac{V_{1,3}}{(2\pi\alpha)^{2}}\int
dx\left\\{U^{\dagger}_{R\uparrow u}U^{\dagger}_{L\uparrow d}U_{R\uparrow
d}U_{L\uparrow u}e^{i2(\phi_{3}+\phi_{4})}+sU^{\dagger}_{R\downarrow
u}U^{\dagger}_{L\downarrow d}U_{R\downarrow d}U_{L\downarrow
u}e^{i2(\phi_{3}-\phi_{4})}+\text{h.c.}\right\\}$ $\displaystyle+$
$\displaystyle\frac{V_{1,4}}{(2\pi\alpha)^{2}}\int
dx\left\\{U^{\dagger}_{R\uparrow u}U^{\dagger}_{L\downarrow d}U_{R\downarrow
d}U_{L\uparrow u}e^{i2(\phi_{2}+\phi_{3})}+sU^{\dagger}_{R\uparrow
d}U^{\dagger}_{L\downarrow u}U_{R\downarrow u}U_{L\uparrow
d}e^{i2(\phi_{2}-\phi_{3})}+\text{h.c.}\right\\}$

These Klein factors in thermodynamic limit can be replaced by Majorana fermion
operators, given that their change on the particle number can be neglected. So
we will introduce $R_{\lambda}$ and $L_{\lambda}$ to denote the Majoranas
which satisfy $R_{\lambda}R_{\lambda}=L_{\lambda}L_{\lambda}=1$. They are
important in order to properly identify the order parameters, as we will see
in the next section. At this point, we need to use them to determine $s$. The
strategy is to come up with some auxiliary order parameter, for instance (the
boson parts are omitted),

$\displaystyle R_{\uparrow u}L_{\downarrow u}R_{\downarrow u}L_{\uparrow
u}+sR_{\uparrow d}L_{\downarrow d}R_{\downarrow d}L_{\uparrow d}$ (29)
$\displaystyle+$ $\displaystyle R_{\uparrow u}L_{\uparrow d}R_{\uparrow
d}L_{\uparrow u}+sR_{\downarrow d}L_{\downarrow u}R_{\downarrow
u}L_{\downarrow d}.$

Suppose we evaluate this operator perturbatively in the presence of $V_{1,4}$,
so we need to expand $e^{-\beta H_{I}}$ to the first order of $V_{1,4}$. In
order to maintain the form of the order parameter, we need to set $s=-1$. As a
result, the bosonized form of the interactions is given by

$-\frac{4V_{1,2}}{(2\pi\alpha)^{2}}\int
dx\sin(2\phi_{1})\sin(2\phi_{3})-\frac{4V_{1,3}}{(2\pi\alpha)^{2}}\int
dx\sin(2\phi_{2})\sin(2\phi_{3})-\frac{4V_{1,4}}{(2\pi\alpha)^{2}}\int
dx\sin(2\phi_{1})\sin(2\phi_{2}).$ (30)

Note in the main text we have introduced new definitions for these
backscattering terms as follows,

$g_{2}:=V_{1,2},~{}~{}g_{1}:=V_{1,3},~{}~{}g_{3}:=V_{1,4}.$ (31)

Alternatively, we can use $-V_{1,2}$, $-V_{1,3}$ and $-V_{1,4}$ in Eq.(28),
and similar reasoning leads to $s=1$ instead. Therefore in this convention the
interaction is given by

$-\frac{4g_{2}}{(2\pi\alpha)^{2}}\int
dx\cos(2\phi_{1})\cos(2\phi_{3})-\frac{4g_{1}}{(2\pi\alpha)^{2}}\int
dx\cos(2\phi_{2})\cos(2\phi_{3})-\frac{4g_{3}}{(2\pi\alpha)^{2}}\int
dx\cos(2\phi_{1})\cos(2\phi_{2}),$ (32)

where we have used the new convention defined in Eq.(31).

The field $\phi_{i}$ is related to $\rho_{i}$ as follows

$-\frac{1}{\pi}\partial_{x}\phi_{i}(x)=\rho_{+,i}(x)+\rho_{-,i}(x)-\rho_{0,i}.$
(33)

where $\rho_{0}$ is the average density. So we can identify

$\displaystyle\text{Charge density
fluctuations}:~{}\frac{1}{\pi}\partial_{x}\phi_{0}(x),$
$\displaystyle~{}~{}~{}~{}\text{Spin density
fluctuations}:~{}\frac{1}{\pi}\partial_{x}\phi_{1}(x),$ (34)
$\displaystyle\text{Valley density
fluctuations}:~{}\frac{1}{\pi}\partial_{x}\phi_{2}(x),$
$\displaystyle~{}~{}~{}~{}\text{Spin-valley density
fluctuations}:~{}\frac{1}{\pi}\partial_{x}\phi_{3}(x).$

The conjugate field $\Pi_{i}$ is

$\Pi_{i}(x)=\rho_{+,i}(x)-\rho_{-,i}(x).$ (35)

It is related to the $\theta(x)$ in Eq.(25) via

$\Pi_{i}(x)=\frac{1}{\pi}\partial_{x}\theta(x).$ (36)

Expressing the fermion density operators in terms of $\Pi_{i}(x)$ and
$\phi_{i}(x)$ we have,

$\displaystyle{}^{2}+[\rho_{-,i}(x)]^{2}$
$\displaystyle=\frac{[\frac{1}{\pi}\partial_{x}\phi_{i}(x)-\rho_{0,i}]^{2}+[\Pi_{i}(x)]^{2}}{2}$
(37) $\displaystyle\rho_{+,i}(x)\rho_{-,i}(x)+\rho_{-,i}(x)\rho_{+,i}(x)$
$\displaystyle=\frac{[\frac{1}{\pi}\partial_{x}\phi_{i}(x)-\rho_{0,i}]^{2}-[\Pi_{i}(x)]^{2}}{2}.$

Substituting these relations to Eq.(24) and combining it with Eq.(32), we
arrive at the final bosonized Hamiltonian (up to some constant)

$\displaystyle H=$ $\displaystyle\sum_{i=0}^{3}\int dx\left\\{\frac{\pi
v_{i}K_{i}}{2}\left[\Pi_{i}(x)\right]^{2}+\frac{v_{i}}{2\pi
K_{i}}\left[\partial_{x}\phi_{i}(x)\right]^{2}\right\\}$ (38)
$\displaystyle-\frac{4g_{2}}{(2\pi\alpha)^{2}}\int
dx\cos(2\phi_{1})\cos(2\phi_{3})-\frac{4g_{1}}{(2\pi\alpha)^{2}}\int
dx\cos(2\phi_{2})\cos(2\phi_{3})-\frac{4g_{3}}{(2\pi\alpha)^{2}}\int
dx\cos(2\phi_{1})\cos(2\phi_{2}),$

where $\phi_{i}\equiv\phi_{i}(x)$ and the renormalized velocities $u_{i}$ and
Luttinger parameters $K_{i}$ are given by

$v_{i}=\sqrt{(v_{F}+G_{i}/\pi)(v_{F}+U_{i}/\pi)},~{}~{}K_{i}=\sqrt{\frac{\pi
v_{F}+U_{i}}{\pi v_{F}+G_{i}}}.$ (39)

Here $U_{i}$ and $G_{i}$ are determined by $V_{i,j}$:

$\displaystyle
V_{0}=V_{4,1}+V_{4,2}+V_{4,3}+V_{4,4}+V_{2,1}-V_{1,1}+V_{2,2}+V_{2,3}+V_{2,4},$
(40) $\displaystyle
U_{0}=V_{4,1}+V_{4,2}+V_{4,3}+V_{4,4}-V_{2,1}+V_{1,1}-V_{2,2}-V_{2,3}-V_{2,4},$
$\displaystyle
V_{1}=V_{4,1}-V_{4,2}+V_{4,3}-V_{4,4}+V_{2,1}-V_{1,1}-V_{2,2}+V_{2,3}-V_{2,4},$
$\displaystyle
U_{1}=V_{4,1}-V_{4,2}+V_{4,3}-V_{4,4}-V_{2,1}+V_{1,1}+V_{2,2}-V_{2,3}+V_{2,4},$
$\displaystyle
V_{2}=V_{4,1}+V_{4,2}-V_{4,3}-V_{4,4}+V_{2,1}-V_{1,1}+V_{2,2}-V_{2,3}-V_{2,4},$
$\displaystyle
U_{2}=V_{4,1}+V_{4,2}-V_{4,3}-V_{4,4}-V_{2,1}+V_{1,1}-V_{2,2}+V_{2,3}+V_{2,4},$
$\displaystyle
V_{3}=V_{4,1}-V_{4,2}-V_{4,3}+V_{4,4}+V_{2,1}-V_{1,1}-V_{2,2}-V_{2,3}+V_{2,4},$
$\displaystyle
U_{3}=V_{4,1}-V_{4,2}-V_{4,3}+V_{4,4}-V_{2,1}+V_{1,1}+V_{2,2}+V_{2,3}-V_{2,4}.$

In a special case when all $V_{i,j}=V_{i,j^{\prime}}\equiv V_{i}$ for every
$i$, $j$ and $j^{\prime}$, and neglecting the backscattering term $V_{1,i}$,
we obtain

$\displaystyle G_{0}=4(V_{4}+V_{2}),~{}~{}U_{0}=4(V_{4}-V_{2}),$ (41)
$\displaystyle G_{1}=U_{1}=G_{2}=U_{2}=G_{3}=U_{3}=0.$

Then it is easy to see from Eq.(39) that $v_{1}=v_{2}=v_{3}=v_{F}$ and
$K_{1}=K_{2}=K_{3}=1$. When the backscattering is included, as we will see
below that for repulsive interaction, the system still flows to this fixed
point, while for attraction, the system tends to gap opening.

Without the backscattering (cosine) terms, Eq.(38) remains unchanged via

$\theta_{i}(x)\leftrightarrow\phi_{i}(x)~{}~{}\text{and}~{}~{}K_{i}\leftrightarrow\frac{1}{K_{i}}$
(42)

which is called T-duality. This observation will be used later to evaluate the
correlation functions.

The Hamiltonian in Eq.(38) leads to the following action:

$S_{tot}=\sum_{i=0}^{3}S_{i}+S_{I}$ (43)

where

$\displaystyle S_{i}$ $\displaystyle=\frac{1}{2\pi K_{i}}\int
dxd\tau\left\\{\frac{1}{v_{i}}\left(\partial_{\tau}\phi_{i}\right)^{2}+v_{i}\left(\partial_{x}\phi_{i}\right)^{2}\right\\}$
(44) $\displaystyle=\frac{1}{2\pi K_{i}}\int
dxdy\left\\{\tilde{v}_{i}\left(\partial_{x}\phi_{i}\right)^{2}+\frac{1}{\tilde{v}_{i}}\left(\partial_{y}\phi_{i}\right)^{2}\right\\}$

with $y:=\bar{v}\tau$ and $\tilde{v}_{i}=v_{i}/\bar{v}$ with
$\bar{v}=(v_{1}+v_{2}+v_{3})/3$. The integration limits will be the same in
zero temperature limit. Similarly,

$S_{I}=-\frac{4}{(2\pi\alpha)^{2}}\int
d^{2}\bm{r}\left[\frac{g_{2}}{\bar{v}}\cos(2\phi_{1})\cos(2\phi_{3})+\frac{g_{1}}{\bar{v}}\cos(2\phi_{2})\cos(2\phi_{3})+\frac{g_{3}}{\bar{v}}\cos(2\phi_{1})\cos(2\phi_{2})\right].$
(45)

Without the cosine terms, the regularized Green’s function is given by

$G(x,t):=\braket{\phi_{j}(x,t)\phi_{j}(0,0)}=-\frac{K_{j}}{4}\ln\left[\frac{x^{2}+y^{2}+a^{2}}{R^{2}}\right]_{y\to
iv_{j}t+a}=-\frac{K_{j}}{4}\ln\left[\frac{x^{2}+(iv_{j}t+a)^{2}}{R^{2}}\right]$
(46)

where $a\to 0$ and $R\to\infty$ are the UV and IR cutoff respectively.
According to the T-duality we have

$G_{\theta\theta}(x,t):=\braket{\theta_{j}(x,t)\theta_{j}(0,0)}=-\frac{1}{4K_{j}}\ln\left[\frac{x^{2}+(iv_{j}t+a)^{2}}{R^{2}}\right].$
(47)

The Green’s function between $\phi_{j}$ field and $\theta_{j}$ field is
calculated from the original action when both of these two fields are present,

$G_{\phi\theta}(x,t):=\braket{\phi_{j}(x,t)\theta_{j}(0,0)}=-\frac{1}{4}\ln\frac{i(v_{j}t+x)+a}{i(v_{j}t-x)+a}.$
(48)

On the other hand, if the boson becomes massive, namely

$S_{i}=\frac{1}{2\pi K_{i}}\int
dxdy\left\\{\left(\partial_{x}\phi_{i}\right)^{2}+\left(\partial_{y}\phi_{i}\right)^{2}+m^{2}\phi_{i}^{2}\right\\},$
(49)

then the correlation function is given by

$\braket{\phi_{i}(\bm{r})\phi_{i}(0)}=\frac{K_{i}}{2}K_{0}(m|\bm{r}|)$ (50)

where $K_{0}(...)$ is the modified Bessel function of the second kind, which
decays exponentially at large distance,

$K_{0}(m|\bm{r}|)\sim e^{-m|\bm{r}|}.$ (51)

and behaves as logarithmic at short distances

$K_{0}(m|\bm{r}|)\sim-\frac{1}{2}\ln(m^{2}\bm{r}^{2}).$ (52)

Therefore, we can identify the correlation length as $\xi=1/m$ which is
finite. The correlation function in space-time coordinates in the massive case
is thus

$G(x,t)=\frac{K_{j}}{2}K_{0}\left(\frac{\sqrt{x^{2}+(iv_{j}t+a)^{2}}}{\xi}\right).$
(53)

##  III III. Order parameters and correlation functions at Gaussian fixed
point

To obtain the correlation functions, one can make use of the general formula
for the vertex operator correlation functions for a massless boson model
(let’s at this point neglect the logarithmic corrections due to finite
backscattering in the limit when all $K_{i}\to 1$),

$\displaystyle\braket{e^{i\beta_{1}\phi(\bm{r}_{1})}...e^{i\beta_{N}\phi(\bm{r}_{N})}}$
$\displaystyle=\exp\left(-\sum_{i>j}\beta_{i}\beta_{j}G(\bm{r}_{i},\bm{r}_{j})\right)\exp\left(-\frac{1}{2}\sum_{i}\beta_{i}^{2}G(\bm{r}_{i},\bm{r}_{i})\right)$
(54)

This equation can be proved by using the cumulant expansion formula for
Gaussian action.

###  III.1 Massless boson

If the boson model is massless, we expect that the expectation value of the
vertex operator vanishes. This is because

$\braket{e^{i\beta\phi_{j}(x)}}=\exp\left[-\frac{1}{2}\beta^{2}G(0)\right]=\left(\frac{a}{R}\right)^{\frac{\beta^{2}K_{j}}{4}},$
(55)

where $G(0)$ is the Green’s function in Eq.(46) in the IR limit. Apparently,
this results vanishes in the limit when $R\to\infty$. This results is
consistent with the fact that in the massless case the expectation value of
order paramters remains zero.

For multi-point correlation functions, we use the explicit formula for the
Green’s function in Eq.(46) in the massless case and obtain

$\displaystyle\braket{e^{i\beta_{1}\phi(\bm{r}_{1})}...e^{i\beta_{N}\phi(\bm{r}_{N})}}$
$\displaystyle=\prod_{i>j}\left[\frac{(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}+a^{2}}{R^{2}}\right]^{\frac{\beta_{i}\beta_{j}K}{4}}\prod_{i}\left[\frac{a^{2}}{R^{2}}\right]^{\frac{\beta_{i}^{2}K}{8}}$
(56)
$\displaystyle=\prod_{i>j}\left[\frac{(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}+a^{2}}{a^{2}}\right]^{\frac{\beta_{i}\beta_{j}K}{4}}\left[\frac{a}{R}\right]^{\frac{K}{8}(\sum_{i}\beta_{i})^{2}}.$

The correlation function does not vanish in the limit $R\to\infty$ only if
$\sum_{i}\beta_{i}=0$. From this general expression for correlations
functions, it is easy to obtain

$\displaystyle\braket{e^{i\phi_{j}(x,t)-i\phi_{j}(0)}}$
$\displaystyle=\left(\frac{a^{2}}{x^{2}+(iv_{j}t+a)^{2}}\right)^{\frac{K_{j}}{4}},$
(57) $\displaystyle\braket{\cos\phi_{j}(x,t)\cos\phi_{j}(0,0)}$
$\displaystyle=\braket{\sin\phi_{j}(x,t)\sin\phi_{j}(0,0)}=\frac{1}{2}\left(\frac{a^{2}}{x^{2}+(iv_{j}t+a)^{2}}\right)^{\frac{K_{j}}{4}},$
$\displaystyle\braket{\cos\phi_{j}(x,t)\sin\phi_{j}(0,0)}$
$\displaystyle=\braket{\sin\phi_{j}(x,t)\cos\phi_{j}(0,0)}=0.$

The correlation functions for the $\theta$ field can be found by the duality
relation in Eq.(42).

$\displaystyle\braket{e^{i\theta_{j}(x,t)-i\theta_{j}(0)}}$
$\displaystyle=\left(\frac{a^{2}}{x^{2}+(iv_{j}t+a)^{2}}\right)^{\frac{1}{4K_{j}}},$
(58) $\displaystyle\braket{\cos\theta_{j}(x,t)\cos\theta_{j}(0,0)}$
$\displaystyle=\braket{\sin\theta_{j}(x,t)\sin\theta_{j}(0,0)}=\frac{1}{2}\left(\frac{a^{2}}{x^{2}+(iv_{j}t+a)^{2}}\right)^{\frac{1}{4K_{j}}},$
$\displaystyle\braket{\cos\theta_{j}(x,t)\sin\theta_{j}(0,0)}$
$\displaystyle=\braket{\sin\theta_{j}(x,t)\cos\theta_{j}(0,0)}=0.$

###  III.2 Massive boson

If we use the massive action with the Green’s function given by Eq.(53), and
if the expectation value of $\phi_{j}$ is $\braket{\phi_{j}}=0$, we obtain for
the expectation value

$\braket{e^{i\beta\phi_{j}(x)}}=\exp\left[-\frac{1}{2}\beta^{2}G(0)\right]=\left(\frac{a}{\xi}\right)^{\frac{\beta^{2}K_{j}}{4}},$
(59)

which remains finite as long as $\xi=1/m$ is finite.

The two point correlation function
$\braket{e^{i\phi_{j}(\bm{r})-i\phi_{j}(0)}}$ is of long range, and the
fluctuation

$\displaystyle\braket{e^{i\phi_{j}(\bm{r})-i\phi_{j}(0)}}-\braket{e^{i\phi_{j}(\bm{r})}}\braket{e^{-i\phi_{j}(0)}}$
(60) $\displaystyle=e^{G(\bm{r})-G(0)}-e^{-G(0)}\approx
e^{-G(0)}G(\bm{r})=\left(\frac{a}{\xi}\right)^{\frac{K_{j}}{2}}\frac{K_{j}}{2}e^{-\frac{r}{\xi}}$

has exponential decaying behavior at large distances, due to the presence of
the mass term.

When a mass term of $\phi_{j}(x)$ field develops, the correlation functions of
$e^{i(...)\theta_{j}(x)}$ has exponentially decaying behavior. A direct
calculation based on the effective $\theta$-only action is difficult, but some
arguments based on symmetry and continuity all supports this result[109].

###  III.3 Order parameters

With the additional pseudo-spin degrees of freedom, there exist many order
parameters which we list below (now setting $s_{z}=\pm$ and $\tau_{z}=\pm$):

  1. 1.

Charge-density-wave:

$\displaystyle O_{\text{CDW}}$
$\displaystyle=\sum_{\lambda}\psi^{\dagger}_{+,\lambda}\psi_{-,\lambda}\propto
e^{-i2k_{F}x}\left(R_{\uparrow u}L_{\uparrow
u}e^{i(\phi_{0}+\phi_{1}+\phi_{2}+\phi_{3})}+R_{\downarrow u}L_{\downarrow
u}e^{i(\phi_{0}-\phi_{1}+\phi_{2}-\phi_{3})}\right.$ (61)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\left.R_{\uparrow
d}L_{\uparrow d}e^{i(\phi_{0}+\phi_{1}-\phi_{2}-\phi_{3})}+R_{\downarrow
d}L_{\downarrow d}e^{i(\phi_{0}-\phi_{1}-\phi_{2}+\phi_{3})}\right)$

Using the convention of Eq.(28) with $s=1$, which leads to Eq.(32), we find
that the Klein factors are consistent with the following convention for the
CDW order parameter:

$\displaystyle O_{\text{CDW}}$
$\displaystyle=\sum_{\lambda}\psi^{\dagger}_{+,\lambda}\psi_{-,\lambda}\propto
e^{-i2k_{F}x}\left(e^{i(\phi_{0}+\phi_{1}+\phi_{2}+\phi_{3})}+e^{i(\phi_{0}-\phi_{1}+\phi_{2}-\phi_{3})}+e^{i(\phi_{0}+\phi_{1}-\phi_{2}-\phi_{3})}+e^{i(\phi_{0}-\phi_{1}-\phi_{2}+\phi_{3})}\right)$
(62)
$\displaystyle=4e^{-i2k_{F}x}\left(\cos\phi_{1}\cos\phi_{2}\cos\phi_{3}-i\sin\phi_{1}\sin\phi_{2}\sin\phi_{3}\right)e^{i\phi_{0}}$

Below we just list the final results after properly removing the Klein
factors.

  2. 2.

Spin-density-wave ($z$-component):

$\displaystyle
O_{\text{SDW}}=\sum_{\lambda}s_{z}\psi^{\dagger}_{+,\lambda}\psi_{-,\lambda}\propto
4ie^{-i2k_{F}x}\left(\sin\phi_{1}\cos\phi_{2}\cos\phi_{3}+i\cos\phi_{1}\sin\phi_{2}\sin\phi_{3}\right)e^{i\phi_{0}}$
(63)

  3. 3.

Pseudospin(valley)-density-wave ($z$-component):

$O_{\text{VDW}}=\sum_{\lambda}\tau_{z}\psi^{\dagger}_{+,\lambda}\psi_{-,\lambda}\propto
4ie^{-i2k_{F}x}\left(\cos\phi_{1}\sin\phi_{2}\cos\phi_{3}+i\sin\phi_{1}\cos\phi_{2}\sin\phi_{3}\right)e^{i\phi_{0}}$
(64)

  4. 4.

Spin-Pseudospin(valley)-density-wave ($zz$-component):

$O_{\text{SVDW}}=\sum_{s_{z},\tau_{z}}s_{z}\tau_{z}\psi^{\dagger}_{+,s_{z},\tau_{z}}\psi_{-,s_{z},\tau_{z}}\propto
4ie^{-i2k_{F}x}\left(\cos\phi_{1}\cos\phi_{2}\sin\phi_{3}+i\sin\phi_{1}\sin\phi_{2}\cos\phi_{3}\right)e^{i\phi_{0}}$
(65)

  5. 5.

Spin-singlet Pseudospin-singlet superconductivity:

$\displaystyle O_{SS}$
$\displaystyle=\sum_{s_{z},\tau_{z}}s_{z}\tau_{z}\psi_{+,s_{z},\tau_{z}}\psi_{-,-s_{z},-\tau_{z}}\propto\left(R_{\uparrow
u}L_{\downarrow d}e^{i(\theta_{0}+\theta_{3}-\phi_{1}-\phi_{2})}-R_{\uparrow
d}L_{\downarrow u}e^{i(\theta_{0}-\theta_{3}-\phi_{1}+\phi_{2})}\right.$ (66)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\left.R_{\downarrow
u}L_{\uparrow d}e^{i(\theta_{0}-\theta_{3}+\phi_{1}-\phi_{2})}+R_{\downarrow
d}L_{\uparrow u}e^{i(\theta_{0}+\theta_{3}+\phi_{1}+\phi_{2})}\right)$
$\displaystyle\propto\left(e^{i(\theta_{0}+\theta_{3}-\phi_{1}-\phi_{2})}-e^{i(\theta_{0}-\theta_{3}-\phi_{1}+\phi_{2})}+e^{i(\theta_{0}-\theta_{3}+\phi_{1}-\phi_{2})}-e^{i(\theta_{0}+\theta_{3}+\phi_{1}+\phi_{2})}\right)$
$\displaystyle=4\left(\sin\phi_{1}\cos\phi_{2}\sin\theta_{3}-i\cos\phi_{1}\sin\phi_{2}\cos\theta_{3}\right)e^{i\theta_{0}}$

The following superconducting order parameters are obtained in similar way.

  6. 6.

Spin-singlet Pseudospin-triplet superconductivity:

$\displaystyle O_{ST_{1}}$
$\displaystyle=\sum_{s_{z}}s_{z}\psi_{+,s_{z},+}\psi_{-,-s_{z},+}\propto
2\cos(\phi_{1}+\phi_{3})e^{i(\theta_{0}+\theta_{2})},$ (67) $\displaystyle
O_{ST_{-1}}$
$\displaystyle=\sum_{s_{z}}s_{z}\psi_{+,s_{z},-}\psi_{-,-s_{z},-}\propto
2\cos(\phi_{1}-\phi_{3})e^{i(\theta_{0}-\theta_{2})},$ $\displaystyle
O_{ST_{0}}$
$\displaystyle=\sum_{s_{z},\tau_{z}}s_{z}\psi_{+,s_{z},\tau_{z}}\psi_{-,-s_{z},-\tau_{z}}\propto
4\left(\cos\phi_{1}\cos\phi_{2}\cos\theta_{3}-i\sin\phi_{1}\sin\phi_{2}\sin\theta_{3}\right)e^{i\theta_{0}}.$

  7. 7.

Spin-triplet Pseudospin-singlet superconductivity:

$\displaystyle O_{T_{1}S}$
$\displaystyle=\sum_{\tau_{z}}\tau_{z}\psi_{+,+,\tau_{z}}\psi_{-,+,-\tau_{z}}\propto
2\cos(\phi_{2}+\phi_{3})e^{i(\theta_{0}+\theta_{1})},$ (68) $\displaystyle
O_{T_{-1}S}$
$\displaystyle=\sum_{\tau_{z}}\tau_{z}\psi_{+,-,\tau_{z}}\psi_{-,-,-\tau_{z}}\propto
2\cos(\phi_{2}-\phi_{3})e^{i(\theta_{0}-\theta_{1})},$ $\displaystyle
O_{T_{0}S}$
$\displaystyle=\sum_{s_{z},\tau_{z}}\tau_{z}\psi_{+,s_{z},\tau_{z}}\psi_{-,-s_{z},-\tau_{z}}\propto
4i\left(\cos\phi_{1}\cos\phi_{2}\sin\theta_{3}+i\sin\phi_{1}\sin\phi_{2}\cos\theta_{3}\right)e^{i\theta_{0}}.$

  8. 8.

Spin-triplet Pseudospin-triplet superconductivity:

$\displaystyle O_{T_{0}T_{0}}$
$\displaystyle=\sum_{s_{z},\tau_{z}}\psi_{+,s_{z},\tau_{z}}\psi_{-,-s_{z},-\tau_{z}}\propto-4i\left(\sin\phi_{1}\cos\phi_{2}\cos\theta_{3}+i\cos\phi_{1}\sin\phi_{2}\sin\theta_{3}\right)e^{i\theta_{0}},$
(69) $\displaystyle O_{T_{0}T_{1}}$
$\displaystyle=\sum_{s_{z}}\psi_{+,s_{z},+}\psi_{-,-s_{z},+}\propto
2\sin\left(\phi_{1}+\phi_{3}\right)e^{i\left(\theta_{0}+\theta_{2}\right)}$
$\displaystyle O_{T_{0}T_{-1}}$
$\displaystyle=\sum_{s_{z}}\psi_{+,s_{z},-}\psi_{-,-s_{z},-}\propto
2\sin\left(\phi_{1}-\phi_{3}\right)e^{i\left(\theta_{0}-\theta_{2}\right)}$
$\displaystyle O_{T_{1}T_{0}}$
$\displaystyle=\sum_{\tau_{z}}\psi_{+,+,\tau_{z}}\psi_{-,+,-\tau_{z}}\propto
2\sin\left(\phi_{2}+\phi_{3}\right)e^{i\left(\theta_{0}+\theta_{1}\right)}$
$\displaystyle O_{T_{1}T_{1}}$ $\displaystyle=\psi_{+,+,+}\psi_{-,+,+}\propto
e^{i\left(\theta_{0}+\theta_{1}+\theta_{2}+\theta_{3}\right)}$ $\displaystyle
O_{T_{1}T_{-1}}$ $\displaystyle=\psi_{+,+,-}\psi_{-,+,-}\propto
e^{i\left(\theta_{0}+\theta_{1}-\theta_{2}-\theta_{3}\right)}$ $\displaystyle
O_{T_{-1}T_{0}}$
$\displaystyle=\sum_{\tau_{z}}\psi_{+,-,\tau_{z}}\psi_{-,-,-\tau_{z}}\propto
2\sin\left(\phi_{2}-\phi_{3}\right)e^{i\left(\theta_{0}-\theta_{1}\right)}$
$\displaystyle O_{T_{-1}T_{1}}$ $\displaystyle=\psi_{+,-,+}\psi_{-,-,+}\propto
e^{i\left(\theta_{0}-\theta_{1}+\theta_{2}-\theta_{3}\right)}$ $\displaystyle
O_{T_{-1}T_{-1}}$ $\displaystyle=\psi_{+,-,-}\psi_{-,-,-}\propto
e^{i\left(\theta_{0}-\theta_{1}-\theta_{2}+\theta_{3}\right)}$

In addition to these two particle order parameters, we can also consider four-
fermion order parameters. One apparent example is the charge-4e
superconductivity, whose order parameters can be

$\displaystyle O_{4e,1}$
$\displaystyle=O_{T_{1}S}O_{T_{-1}S}=\sum_{\tau_{z},\tau_{z}^{\prime}}\tau_{z}\tau_{z}^{\prime}\psi_{+,+,\tau_{z}}\psi_{-,+,-\tau_{z}}\psi_{+,-,\tau_{z}^{\prime}}\psi_{-,-,-\tau_{z}^{\prime}}\propto
4\cos(\phi_{2}+\phi_{3})\cos(\phi_{2}-\phi_{3})e^{i2\theta_{0}}$ (70)
$\displaystyle O_{4e,2}$
$\displaystyle=O_{ST_{1}}O_{ST_{-1}}=\sum_{s_{z},s_{z}^{\prime}}s_{z}s_{z}^{\prime}\psi_{+,s_{z},+}\psi_{-,-s_{z},+}\psi_{+,s_{z}^{\prime},-}\psi_{-,-s_{z}^{\prime},-}\propto
4\cos(\phi_{1}+\phi_{3})\cos(\phi_{1}-\phi_{3})e^{i2\theta_{0}},$
$\displaystyle O_{4e,3}$
$\displaystyle=\frac{1}{4}(O_{ST_{0}}+O_{T_{0}S})(O_{ST_{0}}-O_{T_{0}S})\propto
4\cos(\phi_{1}+\phi_{2})\cos(\phi_{1}-\phi_{2})e^{i2\theta_{0}}$
$\displaystyle O_{4e,1^{\prime}}$
$\displaystyle=O_{T_{1}T_{0}}O_{T_{-1}T_{0}}\propto
4\sin(\phi_{2}+\phi_{3})\sin(\phi_{2}-\phi_{3})e^{i2\theta_{0}}$
$\displaystyle O_{4e,2^{\prime}}$
$\displaystyle=O_{T_{0}T_{1}}O_{T_{0}T_{-1}}\propto
4\sin(\phi_{1}+\phi_{3})\sin(\phi_{1}-\phi_{3})e^{i2\theta_{0}},$
$\displaystyle O_{4e,3^{\prime}}$
$\displaystyle=\frac{1}{4}(O_{SS}+O_{T_{0}T_{0}})(O_{SS}-O_{T_{0}T_{0}})\propto
4\sin(\phi_{1}+\phi_{2})\sin(\phi_{1}-\phi_{2})e^{i2\theta_{0}}.$

In the density-wave channel, there are also various kinds of $4k_{F}$ density
waves, such as

$\displaystyle
O_{4k_{F},1}=\sum_{s_{z},\tau_{z}}\psi^{\dagger}_{+,s_{z},\tau_{z}}\psi^{\dagger}_{+,s_{z},-\tau_{z}}\psi_{-,s_{z},-\tau_{z}}\psi_{-,s_{z},\tau_{z}}\propto
4e^{-i4k_{F}x}\cos(2\phi_{1})e^{i2\phi_{0}},$ (71) $\displaystyle
O_{4k_{F},2}=\sum_{s_{z},\tau_{z}}\psi^{\dagger}_{+,s_{z},\tau_{z}}\psi^{\dagger}_{+,-s_{z},\tau_{z}}\psi_{-,-s_{z},\tau_{z}}\psi_{-,s_{z},\tau_{z}}\propto
4e^{-i4k_{F}x}\cos(2\phi_{2})e^{i2\phi_{0}},$ $\displaystyle
O_{4k_{F},3}=\sum_{s_{z},\tau_{z}}\psi^{\dagger}_{+,s_{z},\tau_{z}}\psi^{\dagger}_{+,-s_{z},-\tau_{z}}\psi_{-,-s_{z},-\tau_{z}}\psi_{-,s_{z},\tau_{z}}\propto
4e^{-i4k_{F}x}\cos(2\phi_{3})e^{i2\phi_{0}}.$

There are of course many other charge-4e and $4k_{F}$-density wave orders, but
they all contain $e^{i2\theta_{0}}$ and $e^{i2\phi_{0}}$.

We will also be interested in some zero-charge, zero momentum orders, which we
call Pomeranchuk instabilities (PI). Some representative examples are given by
fusions of charge-$2e$ orders as well:

$\displaystyle O_{\text{PI,1}}$
$\displaystyle=O_{T_{1}S}^{\dagger}O_{T_{-1}S}\propto
4\cos(\phi_{2}+\phi_{3})\cos(\phi_{2}-\phi_{3})e^{-i2\theta_{1}}$ (72)
$\displaystyle O_{\text{PI,2}}$
$\displaystyle=O_{ST_{1}}^{\dagger}O_{ST_{-1}}\propto
4\cos(\phi_{1}+\phi_{3})\cos(\phi_{1}-\phi_{3})e^{-i2\theta_{2}},$
$\displaystyle O_{\text{PI,3}}$
$\displaystyle=\frac{1}{4}(O_{ST_{0}}^{\dagger}+O_{T_{0}S}^{\dagger})(O_{ST_{0}}-O_{T_{0}S})\propto
4\cos(\phi_{1}+\phi_{2})\cos(\phi_{1}-\phi_{2})e^{-i2\theta_{3}}$
$\displaystyle O_{\text{PI,1}^{\prime}}$
$\displaystyle=O_{T_{1}T_{0}}^{\dagger}O_{T_{-1}T_{0}}\propto
4\sin(\phi_{2}+\phi_{3})\sin(\phi_{2}-\phi_{3})e^{-i2\theta_{1}}$
$\displaystyle O_{\text{PI,2}^{\prime}}$
$\displaystyle=O_{T_{0}T_{1}}^{\dagger}O_{T_{0}T_{-1}}\propto
4\sin(\phi_{1}+\phi_{3})\sin(\phi_{1}-\phi_{3})e^{-i2\theta_{2}},$
$\displaystyle O_{\text{PI,3}^{\prime}}$
$\displaystyle=\frac{1}{4}(O_{SS}^{\dagger}+O_{T_{0}T_{0}}^{\dagger})(O_{SS}-O_{T_{0}}O_{T_{0}})\propto
4\sin(\phi_{1}+\phi_{2})\sin(\phi_{1}-\phi_{2})e^{-i2\theta_{3}}.$

In table 2 we discuss the leading order parameters in the partially gapped
casese. Here the leading orders are the ones which have the most divergent
susceptibility, and their correlation functions have the slowest decay rate.
In the partically gapped case, only one of the couplings is relevant. The
leading orders depend on the which $g_{i}$ flows to strong coupling, and also
on the sign of the relevant $g_{i}$. Table 2 summarizes different cases.

|  | $\Delta_{1}=0$ |  |  |  | $\Delta_{2}=0$ |  |  |  | $\Delta_{3}=0$ |
---|---|---|---|---|---|---|---|---|---|---|---
| $g_{1}>0$ |  | $g_{1}<0$ |  | $g_{2}>0$ |  | $g_{2}<0$ |  | $g_{3}>0$ |  | $g_{3}<0$
$2k_{F}$ | $O_{\text{CDW}},O_{\text{SDW}}$ |  | $O_{\text{VDW}},O_{\text{SVDW}}$ |  | $O_{\text{CDW}},O_{\text{VDW}}$ |  | $O_{\text{SDW}},O_{\text{SVDW}}$ |  | $O_{\text{CDW}},O_{\text{SVDW}}$ |  | $O_{\text{SDW}},O_{\text{VDW}}$
$4k_{F}$ | $O_{4k_{F},1}$ |  | $O_{4k_{F},1}$ |  | $O_{4k_{F},2}$ |  | $O_{4k_{F},2}$ |  | $O_{4k_{F},3}$ |  | $O_{4k_{F},3}$
$2e$ | $O_{T_{\pm 1}S}$ |  | $O_{T_{\pm 1}T_{0}}$ |  | $O_{ST_{\pm 1}}$ |  | $O_{T_{0}T_{\pm 1}}$ |  | $\frac{1}{2}(O_{ST_{0}}\pm O_{T_{0}S})$ |  | $\frac{1}{2}(O_{SS}\pm O_{T_{0}T_{0}})$
$4e$ | $O_{4e,1}$ |  | $O_{4e,1^{\prime}}$ |  | $O_{4e,2}$ |  | $O_{4e,2^{\prime}}$ |  | $O_{4e,3}$ |  | $O_{4e,3^{\prime}}$
PI | $O_{\text{PI},1}$ |  | $O_{\text{PI},1^{\prime}}$ |  | $O_{\text{PI},2}$ |  | $O_{\text{PI},2^{\prime}}$ |  | $O_{\text{PI},3}$ |  | $O_{\text{PI},3^{\prime}}$
Table 2: Leading order parameters for the partially gapped case when one of
the three $\phi_{1,2,3}$ remains gapless but the other two develop finite
gaps.

##  IV IV. Renormalization group analysis

To proceed with the RG analysis, we reformulate the action as follows

$\displaystyle S$ $\displaystyle=\sum_{i}\frac{1}{2}\int
d^{2}\bm{r}\left(\tilde{v}_{i}(\partial_{x}\phi_{i})^{2}+\frac{1}{\tilde{v}_{i}}(\partial_{y}\phi_{i})^{2}\right)$
(73) $\displaystyle+\int
d^{2}\bm{r}\left[u_{2}\cos(\beta_{1}\phi_{1})\cos(\beta_{3}\phi_{3})+u_{1}\cos(\beta_{2}\phi_{2})\cos(\beta_{3}\phi_{3})+u_{3}\cos(\beta_{1}\phi_{1})\cos(\beta_{2}\phi_{2})\right]$
$\displaystyle=S_{0}[\phi_{i}]+S_{I}[\phi_{i}]$

where we have introduced

$u_{i}=\frac{-g_{i}}{\pi^{2}\alpha^{2}\bar{v}},~{}~{}\beta_{i}=\sqrt{4\pi
K_{i}}$ (74)

Each of these fields $\phi_{i}(x)$ can be separated into slow mode
$\phi_{i}^{s}(x)$ and fast mode $h_{i}(x)$

$\phi_{i}(x)=\phi_{i}^{s}(x)+h_{i}(x)$ (75)

and the Gaussian part of the action is also separable

$S_{0}[\phi_{i}]=S_{0}[\phi_{i}^{s}]+S_{0}[h_{i}].$ (76)

The partition function is then

$\displaystyle Z$
$\displaystyle=\int\prod_{i}\mathcal{D}[\phi_{i}^{s}]e^{-S_{0}[\phi_{i}^{s}]}\prod_{i}\mathcal{D}[h_{i}]e^{-S_{0}[h_{i}]}e^{-S_{I}[\phi_{i}^{s}+h_{i}]}$
(77)
$\displaystyle=Z_{h}\int\prod_{i}\mathcal{D}[\phi_{i}^{s}]e^{-S_{0}[\phi_{i}^{s}]}\braket{e^{-S_{I}[\phi_{i}^{s}+h_{i}]}}_{h}$
$\displaystyle=Z_{h}\int\prod_{i}\mathcal{D}[\phi_{i}^{s}]e^{-S_{\text{eff}}[\phi_{i}^{s}]}$

where

$S_{\text{eff}}[\phi_{i}^{s}]=S_{0}[\phi_{i}^{s}]+\braket{S_{I}[\phi_{i}^{s}+h_{i}]}_{h}-\frac{1}{2}\left(\braket{S_{I}^{2}[\phi_{i}^{s}+h_{i}]}_{h}-\braket{S_{I}[\phi_{i}^{s}+h_{i}]}_{h}^{2}\right)$
(78)

and $Z_{h}$ is the partition function coming from fast mode which is
unimportant, and $\braket{...}_{h}$ means average over all the fast mode under
$e^{-S_{0}[h_{i}]}$.

We begin with the first order corrections, i.e. the term
$\braket{S_{I}[\phi_{i}^{s}+h_{i}]}_{h}$. As an example, we discuss only one
of the three interacting terms,

$\braket{u_{2}\int
d^{2}\bm{r}\cos[\beta_{1}(\phi_{1}^{s}+h_{1})]\cos[\beta_{3}(\phi_{3}^{s}+h_{3})]}_{h}=u_{2}\int
d^{2}\bm{r}\cos(\beta_{1}\phi_{1}^{s})\braket{e^{i\beta_{1}h_{1}}}_{h_{1}}\cos(\beta_{3}\phi_{3}^{s})\braket{e^{i\beta_{3}h_{3}}}_{h_{3}}.$
(79)

The average of the vertex $e^{i\beta_{1}h_{1}}$ and $e^{i\beta_{3}h_{3}}$ is
evaluated as follows,

$\braket{e^{i\beta_{1}h_{1}}}_{h_{1}}\approx
1-\frac{1}{2}\beta_{1}^{2}\braket{h_{1}^{2}}_{h_{1}}$ (80)

and in the $T\to 0$ limit,

$\displaystyle\braket{h_{1}^{2}}_{h_{1}}$
$\displaystyle=\int_{\Lambda^{\prime}<\sqrt{k^{2}+(\omega/\bar{v})^{2}}<\Lambda}\frac{dkd\omega}{(2\pi)^{2}}\frac{1}{v_{1}k^{2}+\omega^{2}/v_{1}}$
(81)
$\displaystyle=\int_{\Lambda^{\prime}<\sqrt{k^{2}+(\omega/\bar{v})^{2}}<\Lambda}\frac{dkd(\omega/\bar{v})}{(2\pi)^{2}}\frac{v_{1}}{\bar{v}}\frac{1}{(v_{1}/\bar{v})^{2}k^{2}+(\omega/\bar{v})^{2}}$
$\displaystyle=\int_{\Lambda^{\prime}<p<\Lambda}\frac{pdpd\theta}{(2\pi)^{2}}\frac{v_{1}}{\bar{v}}\frac{1}{(v_{1}/\bar{v})^{2}p^{2}\cos^{2}\theta+p^{2}\sin^{2}\theta}=\frac{dl}{2\pi}\frac{v_{1}}{\bar{v}}\int\frac{d\theta}{2\pi}\frac{1}{(v_{1}/\bar{v})^{2}\cos^{2}\theta+\sin^{2}\theta}=\frac{dl}{2\pi}.$

where we use $dl=d\Lambda/\Lambda>0$. Note this result is independent of
$v_{1}$ in the zero temperature limit, so for other sectors it remains the
same. Using this result, Eq.(79) becomes

$u_{2}\left(1-\frac{\beta_{1}^{2}+\beta_{3}^{2}}{4\pi}dl\right)\int
d^{2}\bm{r}\cos(\beta_{1}\phi_{1}^{s})\cos(\beta_{3}\phi_{3}^{s})$ (82)

and other terms in the first order correction are obtained in exactly the same
way. To compare with the original action, we need to rescale the real-space
$\phi_{i}^{s}$ back to $\phi$. From the Gaussian level, we see $\phi_{i}^{s}$
does not change, but since the small distance cutoff is now $1+dl$ larger than
the last step, we need to multiply $(1+dl)^{2}\approx(1+2dl)$ to Eq.(82), so
the resulting RG equation for the interactions are [making use of Eq.(74) we
see that $\beta_{j}^{2}/(4\pi)=K_{j}$]

$\displaystyle\frac{du_{2}}{dl}$
$\displaystyle=\left[2-(K_{1}+K_{3})\right]u_{2},$ (83)
$\displaystyle\frac{du_{1}}{dl}$
$\displaystyle=\left[2-(K_{2}+K_{3})\right]u_{1},$
$\displaystyle\frac{du_{3}}{dl}$
$\displaystyle=\left[2-(K_{1}+K_{2})\right]u_{3},$

In the next order corrections, we first evaluate

$\displaystyle\braket{S_{I}[\phi_{i}^{s}+h_{i}]}_{h}^{2}=\int
d^{2}\bm{r}_{1}d^{2}\bm{r}_{2}\left\\{u_{2}^{2}(1-2B_{13}dl)\cos[\beta_{1}\phi_{1}^{s}(\bm{r}_{1})]\cos[\beta_{1}\phi_{1}^{s}(\bm{r}_{2})]\cos[\beta_{3}\phi_{3}^{s}(\bm{r}_{1})]\cos[\beta_{3}\phi_{3}^{s}(\bm{r}_{2})]\right.$
(84)
$\displaystyle+u_{1}^{2}(1-2B_{23}dl)\cos[\beta_{2}\phi_{2}^{s}(\bm{r}_{1})]\cos[\beta_{2}\phi_{2}^{s}(\bm{r}_{2})]\cos[\beta_{3}\phi_{3}^{s}(\bm{r}_{1})]\cos[\beta_{3}\phi_{3}^{s}(\bm{r}_{2})]$
$\displaystyle+u_{3}^{2}(1-2B_{12}dl)\cos[\beta_{1}\phi_{1}^{s}(\bm{r}_{1})]\cos[\beta_{1}\phi_{1}^{s}(\bm{r}_{2})]\cos[\beta_{2}\phi_{2}^{s}(\bm{r}_{1})]\cos[\beta_{2}\phi_{2}^{s}(\bm{r}_{2})]$
$\displaystyle+2u_{2}u_{1}[1-(B_{13}+B_{23})dl]\cos[\beta_{1}\phi_{1}^{s}(\bm{r}_{1})]\cos[\beta_{3}\phi_{3}^{s}(\bm{r}_{1})]\cos[\beta_{2}\phi_{2}^{s}(\bm{r}_{2})]\cos[\beta_{3}\phi_{3}^{s}(\bm{r}_{2})]$
$\displaystyle+2u_{2}u_{3}[1-(B_{13}+B_{12})dl]\cos[\beta_{1}\phi_{1}^{s}(\bm{r}_{1})]\cos[\beta_{3}\phi_{3}^{s}(\bm{r}_{1})]\cos[\beta_{1}\phi_{1}^{s}(\bm{r}_{2})]\cos[\beta_{2}\phi_{2}^{s}(\bm{r}_{2})]$
$\displaystyle+\left.2u_{1}u_{3}[1-(B_{12}+B_{23})dl]\cos[\beta_{2}\phi_{2}^{s}(\bm{r}_{1})]\cos[\beta_{3}\phi_{3}^{s}(\bm{r}_{1})]\cos[\beta_{1}\phi_{1}^{s}(\bm{r}_{2})]\cos[\beta_{2}\phi_{2}^{s}(\bm{r}_{2})]\right\\}$

where we have used

$B_{ij}=\frac{\beta_{i}^{2}+\beta_{j}^{2}}{4\pi}=K_{i}+K_{j}.$ (85)

Similarly we can obtain

$\displaystyle\braket{S_{I}^{2}[\phi_{i}^{s}+h_{i}]}_{h}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$
(86) $\displaystyle=\int
d^{2}\bm{r}_{1}d^{2}\bm{r}_{2}\\{\frac{u_{2}^{2}}{4}(1-2B_{13}dl)$
$\displaystyle[\cos\beta_{1}(\phi_{1}^{s}(\bm{r}_{1})+\phi_{1}^{s}(\bm{r}_{2}))e^{-\beta_{1}^{2}\braket{h_{1}(\bm{r}_{1})h_{1}(\bm{r}_{2})}}+\cos\beta_{1}(\phi_{1}^{s}(\bm{r}_{1})-\phi_{1}^{s}(\bm{r}_{2}))e^{\beta_{1}^{2}\braket{h_{1}(\bm{r}_{1})h_{1}(\bm{r}_{2})}}]$
$\displaystyle[\cos\beta_{3}(\phi_{3}^{s}(\bm{r}_{1})+\phi_{3}^{s}(\bm{r}_{2}))e^{-\beta_{3}^{2}\braket{h_{3}(\bm{r}_{1})h_{3}(\bm{r}_{2})}}+\cos\beta_{3}(\phi_{3}^{s}(\bm{r}_{1})-\phi_{3}^{s}(\bm{r}_{2}))e^{\beta_{3}^{2}\braket{h_{3}(\bm{r}_{1})h_{3}(\bm{r}_{2})}}]$
$\displaystyle+\frac{u_{1}^{2}}{4}(1-2B_{23}dl)$
$\displaystyle[\cos\beta_{2}(\phi_{2}^{s}(\bm{r}_{1})+\phi_{2}^{s}(\bm{r}_{2}))e^{-\beta_{2}^{2}\braket{h_{2}(\bm{r}_{1})h_{2}(\bm{r}_{2})}}+\cos\beta_{2}(\phi_{2}^{s}(\bm{r}_{1})-\phi_{2}^{s}(\bm{r}_{2}))e^{\beta_{2}^{2}\braket{h_{2}(\bm{r}_{1})h_{2}(\bm{r}_{2})}}]$
$\displaystyle[\cos\beta_{3}(\phi_{3}^{s}(\bm{r}_{1})+\phi_{3}^{s}(\bm{r}_{2}))e^{-\beta_{3}^{2}\braket{h_{3}(\bm{r}_{1})h_{3}(\bm{r}_{2})}}+\cos\beta_{3}(\phi_{3}^{s}(\bm{r}_{1})-\phi_{3}^{s}(\bm{r}_{2}))e^{\beta_{3}^{2}\braket{h_{3}(\bm{r}_{1})h_{3}(\bm{r}_{2})}}]$
$\displaystyle+\frac{u_{3}^{2}}{4}(1-2B_{12}dl)$
$\displaystyle[\cos\beta_{1}(\phi_{1}^{s}(\bm{r}_{1})+\phi_{1}^{s}(\bm{r}_{2}))e^{-\beta_{1}^{2}\braket{h_{1}(\bm{r}_{1})h_{1}(\bm{r}_{2})}}+\cos\beta_{1}(\phi_{1}^{s}(\bm{r}_{1})-\phi_{1}^{s}(\bm{r}_{2}))e^{\beta_{1}^{2}\braket{h_{1}(\bm{r}_{1})h_{1}(\bm{r}_{2})}}]$
$\displaystyle[\cos\beta_{2}(\phi_{2}^{s}(\bm{r}_{1})+\phi_{2}^{s}(\bm{r}_{2}))e^{-\beta_{2}^{2}\braket{h_{2}(\bm{r}_{1})h_{2}(\bm{r}_{2})}}+\cos\beta_{2}(\phi_{2}^{s}(\bm{r}_{1})-\phi_{2}^{s}(\bm{r}_{2}))e^{\beta_{2}^{2}\braket{h_{2}(\bm{r}_{1})h_{2}(\bm{r}_{2})}}]$
$\displaystyle+\frac{u_{2}u_{1}}{2}(1-(B_{13}+B_{23})dl)$
$\displaystyle[\cos(\beta_{1}\phi_{1}^{s}(\bm{r}_{1})+\beta_{2}\phi_{2}^{s}(\bm{r}_{2}))+\cos(\beta_{1}\phi_{1}^{s}(\bm{r}_{1})-\beta_{2}\phi_{2}^{s}(\bm{r}_{2}))]$
$\displaystyle[\cos\beta_{3}(\phi_{3}^{s}(\bm{r}_{1})+\phi_{3}^{s}(\bm{r}_{2}))e^{-\beta_{3}^{2}\braket{h_{3}(\bm{r}_{1})h_{3}(\bm{r}_{2})}}+\cos\beta_{3}(\phi_{3}^{s}(\bm{r}_{1})-\phi_{3}^{s}(\bm{r}_{2}))e^{\beta_{3}^{2}\braket{h_{3}(\bm{r}_{1})h_{3}(\bm{r}_{2})}}]$
$\displaystyle+\frac{u_{2}u_{3}}{2}(1-(B_{13}+B_{12})dl)$
$\displaystyle[\cos(\beta_{3}\phi_{3}^{s}(\bm{r}_{1})+\beta_{2}\phi_{2}^{s}(\bm{r}_{2}))+\cos(\beta_{3}\phi_{3}^{s}(\bm{r}_{1})-\beta_{2}\phi_{2}^{s}(\bm{r}_{2}))]$
$\displaystyle[\cos\beta_{1}(\phi_{1}^{s}(\bm{r}_{1})+\phi_{1}^{s}(\bm{r}_{2}))e^{-\beta_{1}^{2}\braket{h_{1}(\bm{r}_{1})h_{1}(\bm{r}_{2})}}+\cos\beta_{1}(\phi_{1}^{s}(\bm{r}_{1})-\phi_{1}^{s}(\bm{r}_{2}))e^{\beta_{1}^{2}\braket{h_{1}(\bm{r}_{1})h_{1}(\bm{r}_{2})}}]$
$\displaystyle+\frac{u_{1}u_{3}}{2}(1-(B_{12}+B_{23})dl)$
$\displaystyle[\cos(\beta_{3}\phi_{3}^{s}(\bm{r}_{1})+\beta_{1}\phi_{1}^{s}(\bm{r}_{2}))+\cos(\beta_{3}\phi_{3}^{s}(\bm{r}_{1})-\beta_{1}\phi_{1}^{s}(\bm{r}_{2}))]$
$\displaystyle[\cos\beta_{2}(\phi_{2}^{s}(\bm{r}_{1})+\phi_{2}^{s}(\bm{r}_{2}))e^{-\beta_{2}^{2}\braket{h_{2}(\bm{r}_{1})h_{2}(\bm{r}_{2})}}+\cos\beta_{2}(\phi_{2}^{s}(\bm{r}_{1})-\phi_{2}^{s}(\bm{r}_{2}))e^{\beta_{2}^{2}\braket{h_{2}(\bm{r}_{1})h_{2}(\bm{r}_{2})}}]\\}.$

It is now clear that we can easily combine Eq.(84) and (86), and this requires
the evaluation of
$e^{-\beta_{j}^{2}\braket{h_{j}(\bm{r}_{1})h_{j}(\bm{r}_{2})}}$. In the case
of an abrupt cut-off, the correlation
$\braket{h_{j}(\bm{r})h_{j}(0)}=F_{j}(r)dl$. However, when a smooth cutoff is
used, $F_{j}(r)$ can be short range, i.e. $F_{j}(r)$ is negligible when
$r>1/\Lambda$, which validate the approximation that
$\bm{r}_{1}\approx\bm{r}_{2}$ in Eq.(84) and (86) so we can expand the results
in terms of the relative coordinate $\bm{r}=\bm{r}_{1}-\bm{r}_{2}$. Like in
Eq.(81), the correlation at finite distance is calculated as follows

$\displaystyle\braket{h_{j}(\bm{r})h_{j}(0)}_{h_{j}}$
$\displaystyle=\int_{\Lambda^{\prime}<p<\Lambda}\frac{dp/p}{2\pi}\int\frac{d\theta}{2\pi}\frac{e^{ipr\cos\theta}}{(v_{j}/\bar{v})\cos^{2}\theta+\sin^{2}\theta(\bar{v}/v_{j})}$
(87)
$\displaystyle=\int_{\Lambda^{\prime}<p<\Lambda}\frac{dp/p}{2\pi}\int\frac{d\theta}{2\pi}\frac{e^{ipr\cos\theta}}{1+\delta_{j}\cos
2\theta}$

where we have introduced $\delta_{j}=(v_{j}-\bar{v})/\bar{v}$. In weak
coupling limit, $\delta$ is a small number, so we can approximately have

$\displaystyle\braket{h_{j}(\bm{r})h_{j}(0)}_{h_{j}}\approx\int_{\Lambda^{\prime}<p<\Lambda}\frac{dp/p}{2\pi}\int\frac{d\theta}{2\pi}(1-\delta_{j}\cos
2\theta)e^{ipr\cos\theta}=\frac{dl}{2\pi}(J_{0}(\Lambda
r)+\delta_{j}J_{2}(\Lambda r))\equiv F_{j}(r)dl$ (88)

for all $j$ and $r<1/\Lambda$. Note $F_{j}$ can be different for different
sectors since they have different velocities. It will be shown later that what
are useful in deriving the RG equations are not these correlations, but some
skewed form instead. The skewed correlations are defined as

$\displaystyle\tilde{F}_{j}(r)dl=\braket{h_{j}(\sqrt{\tilde{v}_{j}}x,{y}/{\sqrt{\tilde{v}_{j}}})h_{j}(0)}_{h_{j}}$
$\displaystyle=\int_{\Lambda^{\prime}<k<\Lambda}\frac{dk_{x}dk_{y}}{(2\pi)^{2}}\frac{e^{i(\sqrt{\tilde{v}_{j}}k_{x}x+k_{y}y/\sqrt{\tilde{v}_{j}})}}{\tilde{v}_{j}k_{x}^{2}+k_{y}^{2}/{\tilde{v}_{j}}}$
(89) $\displaystyle=\int_{\text{ellipse
shell}}\frac{ds}{(2\pi)^{2}}\frac{e^{ikr\cos\theta}}{k^{2}}$

where the ellipse shell integration is shown in Fig.7. We can use the polar
coordinate formula for the ellipse shell. Next we assume
$\delta_{j}=\tilde{v}_{j}-1$ is small, so we have

$\tilde{F}_{j}(r)dl=\int_{0}^{2\pi}\frac{d\theta}{2\pi}\int_{\Lambda^{\prime}(\theta)}^{\Lambda(\theta)}\frac{dkk}{2\pi}\frac{\cos[kr\cos\theta]}{k^{2}}=\frac{dl}{2\pi}\int_{0}^{2\pi}\frac{d\theta}{2\pi}\cos[\Lambda(\theta)r\cos\theta]$
(90)

where

$\Lambda(\theta)=\frac{\Lambda}{\sqrt{(1+\delta_{j})\cos^{2}\theta+\sin^{2}\theta/(1+\delta_{j})}}.$
(91)

Expanding to the leading order in $\delta_{j}$ we obtain

$\tilde{F}_{j}(r)dl=\frac{dl}{2\pi}\left[J_{0}(\Lambda
r)+\delta_{j}\left(\frac{\Lambda r}{2}J_{1}(\Lambda r)-J_{2}(\Lambda
r)\right)\right]$ (92)

Figure 7: Ellipse momentum shell integration arsing from $v_{j}\neq\bar{v}$.

Based on this, it is straightforward to see (defining
$\bm{R}=(\bm{r}_{1}+\bm{r}_{2})/2$)

$\displaystyle\braket{S_{I}^{2}[\phi_{i}^{s}+h_{i}]}_{h}-\braket{S_{I}[\phi_{i}^{s}+h_{i}]}_{h}^{2}=\frac{u_{2}^{2}}{4}\int
d^{2}\bm{R}d^{2}\bm{r}$
$\displaystyle(1-2B_{13}dl)\\{\cos(2\beta_{1}\phi_{1}^{s}(\bm{R}))\cos(2\beta_{3}\phi_{3}^{s}(\bm{R}))(-\beta_{1}^{2}F_{1}(\bm{r})-\beta_{3}^{2}F_{3}(\bm{r}))dl$
(93) $\displaystyle+$
$\displaystyle\cos(2\beta_{1}\phi_{2}^{s}(\bm{R}))\cos(\beta_{3}\bm{r}\cdot\nabla_{R}\phi_{3}^{s}(\bm{R}))(\beta_{3}^{2}F_{3}(\bm{r})-\beta_{1}^{2}F_{1}(\bm{r}))dl$
$\displaystyle+$
$\displaystyle\cos(2\beta_{3}\phi_{3}^{s}(\bm{R}))\cos(\beta_{1}\bm{r}\cdot\nabla_{R}\phi_{1}^{s}(\bm{R}))(\beta_{1}^{2}F_{1}(\bm{r})-\beta_{3}^{2}F_{3}(\bm{r}))dl$
$\displaystyle+$
$\displaystyle\cos(\beta_{1}\bm{r}\cdot\nabla_{R}\phi_{1}^{s}(\bm{R}))\cos(\beta_{3}\bm{r}\cdot\nabla_{R}\phi_{3}^{s}(\bm{R}))(\beta_{1}^{2}F_{1}(\bm{r})+\beta_{3}^{2}F_{3}(\bm{r}))dl\\}$
$\displaystyle+$ $\displaystyle u_{1}^{2}[...]+u_{3}^{2}[...]$
$\displaystyle+\frac{u_{2}u_{1}}{2}(1-(B_{13}+B_{23})dl)\int
d^{2}\bm{R}d^{2}\bm{r}\\{\cos[\beta_{1}\phi_{1}^{s}(\bm{R})+$
$\displaystyle\beta_{2}\phi_{3}^{s}(\bm{R})+\beta_{1}\frac{\bm{r}}{2}\cdot\nabla_{R}\phi_{1}^{s}(\bm{R})-\beta_{2}\frac{\bm{r}}{2}\cdot\nabla_{R}\phi_{2}^{s}(\bm{R})]\cos[2\beta_{3}\phi_{3}^{s}(\bm{R})](-\beta_{3}^{2})F_{3}(\bm{r})dl$
$\displaystyle+\cos[\beta_{1}\phi_{1}^{s}(\bm{R})+\beta_{2}\phi_{2}^{s}(\bm{R})+$
$\displaystyle\beta_{1}\frac{\bm{r}}{2}\cdot\nabla_{R}\phi_{1}^{s}(\bm{R})-\beta_{2}\frac{\bm{r}}{2}\cdot\nabla_{R}\phi_{2}^{s}(\bm{R})]\cos[\beta_{3}\bm{r}\cdot\nabla_{R}\phi_{3}^{s}(\bm{R})]\beta_{3}^{2}F_{3}(\bm{r})dl$
$\displaystyle+\cos[\beta_{1}\phi_{1}^{s}(\bm{R})-\beta_{2}\phi_{2}^{s}(\bm{R})+$
$\displaystyle\beta_{1}\frac{\bm{r}}{2}\cdot\nabla_{R}\phi_{1}^{s}(\bm{R})+\beta_{2}\frac{\bm{r}}{2}\cdot\nabla_{R}\phi_{2}^{s}(\bm{R})]\cos[2\beta_{3}\phi_{3}^{s}(\bm{R})](-\beta_{3}^{2})F_{3}(\bm{r})dl$
$\displaystyle+\cos[\beta_{1}\phi_{1}^{s}(\bm{R})-\beta_{2}\phi_{2}^{s}(\bm{R})+$
$\displaystyle\beta_{1}\frac{\bm{r}}{2}\cdot\nabla_{R}\phi_{1}^{s}(\bm{R})+\beta_{2}\frac{\bm{r}}{2}\cdot\nabla_{R}\phi_{2}^{s}(\bm{R})]\cos[\beta_{3}\bm{r}\cdot\nabla_{R}\phi_{3}^{s}(\bm{R})]\beta_{3}^{2}F_{3}(\bm{r})dl\\}$
$\displaystyle+u_{2}u_{3}[...]+u_{1}u_{3}[...].$

From the above expressions, we need to identify the ones which renormalize the
gradient terms, i.e. $S_{0}[\phi_{i}]$. To this end, we assume we are close
the the critical point, and all terms containing
$\cos[2\beta_{j}\phi_{j}^{s}]$ are irrelevant. It’s easy to see only the
squared terms ($u_{1}^{2}$, $u_{2}^{2}$ and $u_{3}^{2}$) contribute to the
renormalization of $S_{0}$. In comparing to $S_{0}$ we note that
$\nabla_{R}\phi_{j}=(\partial_{R_{x}}\phi_{j},\partial_{R_{y}}\phi_{j})$ is
symmetric with respect to interchanging $R_{x}$ and $R_{y}$. However, we see
from above that with $v_{j}\neq\bar{v}$ we have anisotropy. Thus, we will skew
the coordinates of $\bm{r}$ to $\tilde{\bm{r}}$, such that

$\bm{r}\cdot\nabla_{R}\phi_{j}=x\partial_{R_{x}}\phi_{j}+y\partial_{R_{y}}\phi_{j}=\frac{x}{\sqrt{\tilde{v}_{j}}}\sqrt{\tilde{v}_{j}}\partial_{R_{x}}\phi_{j}+y\sqrt{\tilde{v}_{j}}\frac{\partial_{R_{y}}\phi_{j}}{\sqrt{\tilde{v}_{j}}}:=\tilde{x}\sqrt{\tilde{v}_{j}}\partial_{R_{x}}\phi_{j}+\tilde{y}\frac{\partial_{R_{y}}\phi_{j}}{\sqrt{\tilde{v}_{j}}}$
(94)

The measure is invariant under the change from $\bm{r}$ to $\tilde{\bm{r}}$ so
we have, for instance,

$\int d^{2}\bm{r}(\bm{r}\cdot\nabla_{R}\phi_{j})^{2}F_{j}(\bm{r})=\int
d^{2}\tilde{\bm{r}}(\tilde{\bm{r}}\cdot\tilde{\nabla_{R}\phi_{j}})^{2}\tilde{F}_{j}(\tilde{\bm{r}})$
(95)

where $\tilde{F}_{j}$ is given in Eq.(92). Introducing
$\tilde{g}_{i}=g_{i}/\bar{v}$ such that
$u_{i}=\tilde{g}_{i}/(\pi^{2}\alpha^{2})$, we have for the gradient terms
after a straightforward manipulation

$\displaystyle\frac{1}{2}\left[1+\left({\tilde{g}}_{2}^{2}(A_{1}\beta_{1}^{2}+A_{3}\beta_{3}^{2})+{\tilde{g}}_{3}^{2}(A_{1}\beta_{1}^{2}+A_{2}\beta_{2}^{2})\right)\frac{\beta_{1}^{2}}{8}dl\right]\int
d^{2}\bm{r}\left(\tilde{v}_{1}(\partial_{x}\phi_{i})^{2}+\frac{1}{\tilde{v}_{1}}(\partial_{y}\phi_{i})^{2}\right)$
(96) $\displaystyle+$
$\displaystyle\frac{1}{2}\left[1+\left({\tilde{g}}_{1}^{2}(A_{2}\beta_{2}^{2}+A_{3}\beta_{3}^{2})+{\tilde{g}}_{3}^{2}(A_{1}\beta_{1}^{2}+A_{2}\beta_{2}^{2})\right)\frac{\beta_{2}^{2}}{8}dl\right]\int
d^{2}\bm{r}\left(\tilde{v}_{2}(\partial_{x}\phi_{i})^{2}+\frac{1}{\tilde{v}_{2}}(\partial_{y}\phi_{i})^{2}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{2}\left[1+\left({\tilde{g}}_{2}^{2}(A_{1}\beta_{1}^{2}+A_{3}\beta_{3}^{2})+{\tilde{g}}_{1}^{2}(A_{2}\beta_{2}^{2}+A_{3}\beta_{3}^{2})\right)\frac{\beta_{3}^{2}}{8}dl\right]\int
d^{2}\bm{r}\left(\tilde{v}_{3}(\partial_{x}\phi_{i})^{2}+\frac{1}{\tilde{v}_{3}}(\partial_{y}\phi_{i})^{2}\right)$

Transforming this correction to the cosine terms (by rescaling the fields
$\phi_{j}$), we obtain the following RG equations

$\displaystyle\frac{d\beta_{1}^{2}}{dl}$
$\displaystyle=-\left({\tilde{g}}_{2}^{2}(A_{1}\beta_{1}^{2}+A_{3}\beta_{3}^{2})+{\tilde{g}}_{3}^{2}(A_{1}\beta_{1}^{2}+A_{2}\beta_{2}^{2})\right)\frac{\beta_{1}^{4}}{8}$
(97) $\displaystyle\frac{d\beta_{3}^{2}}{dl}$
$\displaystyle=-\left({\tilde{g}}_{1}^{2}(A_{2}\beta_{2}^{2}+A_{3}\beta_{3}^{2})+{\tilde{g}}_{3}^{2}(A_{1}\beta_{1}^{2}+A_{2}\beta_{2}^{2})\right)\frac{\beta_{2}^{4}}{8}$
$\displaystyle\frac{d\beta_{3}^{2}}{dl}$
$\displaystyle=-\left({\tilde{g}}_{2}^{2}(A_{1}\beta_{1}^{2}+A_{3}\beta_{3}^{2})+{\tilde{g}}_{1}^{2}(A_{2}\beta_{2}^{2}+A_{3}\beta_{3}^{2})\right)\frac{\beta_{3}^{4}}{8}$

In terms of the Luttinger parameters,

$\displaystyle\frac{dK_{1}}{dl}$
$\displaystyle=-2\pi^{2}K_{1}^{2}\left[{\tilde{g}}_{2}^{2}(A_{1}K_{1}+A_{3}K_{3})+{\tilde{g}}_{3}^{2}(A_{1}K_{1}+A_{2}K_{2})\right],$
(98) $\displaystyle\frac{dK_{2}}{dl}$
$\displaystyle=-2\pi^{2}K_{2}^{2}\left[{\tilde{g}}_{1}^{2}(A_{2}K_{2}+A_{3}K_{3})+{\tilde{g}}_{3}^{2}(A_{1}K_{1}+A_{2}K_{2})\right],$
$\displaystyle\frac{dK_{3}}{dl}$
$\displaystyle=-2\pi^{2}K_{3}^{2}\left[{\tilde{g}}_{2}^{2}(A_{1}K_{1}+A_{3}K_{3})+{\tilde{g}}_{1}^{2}(A_{2}K_{2}+A_{3}K_{3})\right].$

Also note that the cross terms in Eq.(93), give rise to additional
renormalization to the interactions in addition to Eq.(83).

$\displaystyle\frac{d{\tilde{g}}_{1}}{dl}$
$\displaystyle=\left[2-(K_{2}+K_{3})\right]{\tilde{g}}_{1}+4\pi
A_{1}^{\prime}{\tilde{g}}_{2}{\tilde{g}}_{3}K_{1},$ (99)
$\displaystyle\frac{d{\tilde{g}}_{2}}{dl}$
$\displaystyle=\left[2-(K_{1}+K_{3})\right]{\tilde{g}}_{2}+4\pi
A_{2}^{\prime}{\tilde{g}}_{1}{\tilde{g}}_{3}K_{2},$
$\displaystyle\frac{d{\tilde{g}}_{3}}{dl}$
$\displaystyle=\left[2-(K_{1}+K_{2})\right]{\tilde{g}}_{3}+4\pi
A_{3}^{\prime}{\tilde{g}}_{2}{\tilde{g}}_{1}K_{3}.$

where

$A_{j}\approx\frac{\pi}{\pi^{4}\alpha^{4}}\int_{0}^{1/\Lambda}drr^{3}\tilde{F}_{j}(r),~{}~{}~{}A_{j}^{\prime}\approx\frac{2\pi}{\pi^{2}\alpha^{2}}\int_{0}^{1/\Lambda}drr\tilde{F}_{j}(r).$
(100)

We can absorb the constants relating $\pi$ into the new definition of $A_{j}$
and $A_{j}^{\prime}$, so we end up with

$\boxed{\begin{aligned}
\frac{dK_{1}}{dl}&=-K_{1}^{2}\left[{\tilde{g}}_{2}^{2}(A_{1}K_{1}+A_{3}K_{3})+{\tilde{g}}_{3}^{2}(A_{1}K_{1}+A_{2}K_{2})\right],\\\
\frac{dK_{2}}{dl}&=-K_{2}^{2}\left[{\tilde{g}}_{1}^{2}(A_{2}K_{2}+A_{3}K_{3})+{\tilde{g}}_{3}^{2}(A_{1}K_{1}+A_{2}K_{2})\right],\\\
\frac{dK_{3}}{dl}&=-K_{3}^{2}\left[{\tilde{g}}_{2}^{2}(A_{1}K_{1}+A_{3}K_{3})+{\tilde{g}}_{1}^{2}(A_{2}K_{2}+A_{3}K_{3})\right],\\\
\frac{d{\tilde{g}}_{1}}{dl}&=\left[2-(K_{2}+K_{3})\right]{\tilde{g}}_{1}+A_{1}^{\prime}{\tilde{g}}_{2}{\tilde{g}}_{3}K_{1},\\\
\frac{d{\tilde{g}}_{2}}{dl}&=\left[2-(K_{1}+K_{3})\right]{\tilde{g}}_{2}+A_{2}^{\prime}{\tilde{g}}_{1}{\tilde{g}}_{3}K_{2},\\\
\frac{d{\tilde{g}}_{3}}{dl}&=\left[2-(K_{1}+K_{2})\right]{\tilde{g}}_{3}+A_{3}^{\prime}{\tilde{g}}_{2}{\tilde{g}}_{1}K_{3}.\end{aligned}}$
(101)

where the constants $A_{j}$ and $A_{j}^{\prime}$ are now given by

$A_{j}\approx\frac{2\pi^{3}}{\pi^{4}\alpha^{4}}\int_{0}^{1/\Lambda}drr^{3}\tilde{F}_{j}(r),~{}~{}~{}A_{j}^{\prime}\approx\frac{8\pi^{2}}{\pi^{2}\alpha^{2}}\int_{0}^{1/\Lambda}drr\tilde{F}_{j}(r).$
(102)

In zero temperature limit, we can use Eq.(92) and obtain (setting
$\alpha\Lambda=1$)

$A_{j}=\frac{1}{\pi^{2}(\alpha\Lambda)^{4}}\int_{0}^{1}dxx^{3}\tilde{F}_{j}(x)\approx
0.02+0.002\delta_{j},~{}~{}A_{j}^{\prime}=\frac{4}{\pi(\alpha\Lambda)^{2}}\int_{0}^{1}dxx\tilde{F}_{j}(x)\approx
0.56+0.035\delta_{j}.$ (103)

where we remind that $\delta_{j}=\tilde{v}_{j}-1$.

In Fig.8 we show an example of the RG flows obtained by setting
$\tilde{g}_{2}(0)=\tilde{g}_{3}(0)=0$ and $K_{2}(0)=K_{3}(0)$. In this case,
$\tilde{g}_{2}$ and $\tilde{g}_{3}$ remain zero, $K_{1}$ does not flow,
$K_{2}$ and $K_{3}$ remain identical which we denote by $K$. In the vicinity
of $K\approx 1$ and $\tilde{g}_{1}\to 0$, this is the Kosterlitz-Thouless RG
flow.

Figure 8: RG flows in the $\tilde{g}_{1}$-$K$ plane with the condition
$K_{2}=K_{3}=K$ and $\tilde{g}_{2}=\tilde{g}_{3}=0$.

##  V V. Self-consistent Gaussian approximation for the massive phase

We now discuss when the three double-cosine terms in Eq.(73) are relevant, the
gap can be determined self-consistently in a variational approach, also known
as self-consistent Gaussian approximation. Let’s use the notations formulated
in Eq.(44) and(45).

Deep inside the massive phase, we can approximate the action as some Gaussian
form, expecting the soliton modes are excluded in the low energy limit. To
this end, we write the variational Gaussian action as (for $i=1,2,3$)

$S_{\text{var}}[\phi_{i}]=\frac{T}{2L}\sum_{n,q}\sum_{i,j}G^{-1}_{ij}(q,\omega_{n})\phi_{i}^{*}(q,\omega_{n})\phi_{j}(q,\omega_{n})$
(104)

where

$G^{-1}(\bm{q})=\begin{pmatrix}\frac{v_{1}}{\pi
K_{1}}\left(\frac{\omega_{n}^{2}}{v_{1}^{2}}+q^{2}+\frac{\Delta_{1}^{2}}{v_{1}^{2}}\right)&\Delta_{12}&\Delta_{13}\\\
\Delta_{21}&\frac{v_{2}}{\pi
K_{2}}\left(\frac{\omega_{n}^{2}}{v_{2}^{2}}+q^{2}+\frac{\Delta_{2}^{2}}{v_{2}^{2}}\right)&\Delta_{23}\\\
\Delta_{31}&\Delta_{32}&\frac{v_{3}}{\pi
K_{3}}\left(\frac{\omega_{n}^{2}}{v_{3}^{2}}+q^{2}+\frac{\Delta_{3}^{2}}{v_{3}^{2}}\right)\end{pmatrix}$
(105)

and $\Delta_{i}$ is the gap for the $i$-th sector. We will assume that
$\Delta_{ij}=\Delta_{ji}$ so $G^{-1}$ is a symmetric matrix which can be
diagonalize by an orthogonal matrix $U$:

$\phi^{\dagger}G^{-1}\phi=\tilde{\phi}^{\dagger}G_{d}^{-1}\tilde{\phi}$ (106)

where

$\phi=U\tilde{\phi},~{}~{}G^{-1}=UG_{d}^{-1}U^{-1}$ (107)

and $G_{d}$ is now a diagonal matrix.

The variational free energy to be minimized is,

$F_{\text{var}}=-T\ln Z_{\text{var}}+T\braket{S-S_{\text{var}}}_{\text{var}}$
(108)

where $Z_{\text{var}}$ is the partition function evaluated using
$S_{\text{var}}$ and the original $S$ is given in Eqs.(44) and (45). The first
term in Eq.(108) can be evaluated directly,

$-T\ln Z_{\text{var}}=-T\text{Tr}\ln G(q,\omega_{n}).$ (109)

Note in the second term, $\braket{S_{\text{var}}}_{\text{var}}$ does not
contain dependence on the variational parameter, so it can be dropped. We are
then left with evaluating $\braket{S}_{\text{var}}$, which involves in
evaluating the average of the product of two cosine terms
$g_{i}\cos(2\phi_{j})\cos(2\phi_{k})$ using the Gaussian distribution
function. This requires one first identify the classical ground state of these
cosine terms. In the case when there is no frustration, i.e. when
$\operatorname{sgn}(g_{1}g_{2}g_{3})>0$, the classical ground state is easy to
find. When all $g_{i}>0$, all $\phi_{i}$ should be around $\phi_{i}=0$. when
$g_{i}>0$ and $g_{j},g_{k}<0$, we should have $\phi_{j,k}=0$ and
$\phi_{i}=\pm\pi/2$. Since $\phi_{i}$ is around $\pm\pi/2$ we can shift
$\phi_{i}$ by $\pm\pi/2$ such that $\cos 2\phi_{i}$ picks up a minus sign,
which can be absorbed in $g_{j}$ and $g_{k}$. Therefore, in the case without
frustration, we can use the absolute value of $g_{i,j,k}$ and then average all
the cosine terms at $\phi_{i,j,k}$ around $0$.

$\displaystyle\braket{S}_{\text{var}}=\sum_{i=1}^{3}\frac{v_{i}}{2\pi
K_{i}}\sum_{n,q}\left(\frac{\omega_{n}^{2}}{v_{i}^{2}}+q^{2}\right)G_{i,i}(q,\omega_{n})$
(110)
$\displaystyle-\frac{1}{2\pi^{2}\alpha^{2}}\frac{L}{T}\left(|g_{1}|\braket{e^{i2(\phi_{2}+\phi_{3})}+e^{i2(\phi_{2}-\phi_{3})}}_{\text{var}}+|g_{2}|\braket{e^{i2(\phi_{3}+\phi_{1})}+e^{i2(\phi_{3}-\phi_{1})}}_{\text{var}}+|g_{3}|\braket{e^{i2(\phi_{1}+\phi_{2})}+e^{i2(\phi_{1}-\phi_{2})}}_{\text{var}}\right).$

Here we average the $\cos(2\phi_{i})$ around $\phi_{i}=0$. Going to the
$\tilde{\phi}$ basis, we have

$\braket{e^{i2(\phi_{2}\pm\phi_{3})}}_{\text{var}}=\braket{e^{i2\sum_{j}(U_{2j}\pm
U_{3j})\tilde{\phi}_{j}}}_{\text{var}}=e^{-\frac{2T}{L}\sum_{n,q}\sum_{j}(U_{2j}\pm
U_{3j})^{2}G_{d,j}(q,\omega_{n})}.$ (111)

Since $G_{d}$ is diagonal, and $U^{T}=U^{-1}$, we have

$\braket{e^{i2(\phi_{2}\pm\phi_{3})}}_{\text{var}}=e^{-\frac{2T}{L}\sum_{n,q}(G_{22}+G_{33}\pm
2G_{23})}$ (112)

and similarly for other two terms in Eq.(110). Therefore

$\displaystyle\braket{S}_{\text{var}}=\sum_{i=1}^{3}\frac{v_{i}}{2\pi
K_{i}}\sum_{n,q}\left(\frac{\omega_{n}^{2}}{v_{i}^{2}}+q^{2}\right)G_{i,i}(q,\omega_{n})-\frac{1}{\pi^{2}\alpha^{2}}\frac{L}{T}\times$
(113)
$\displaystyle\times\left(|g_{1}|e^{-\frac{2T}{L}\sum_{n,q}(G_{22}+G_{33})}\cosh[4\text{Tr}G_{23}]+|g_{2}|e^{-\frac{2T}{L}\sum_{n,q}(G_{11}+G_{33})}\cosh[4\text{Tr}G_{13}]+|g_{3}|e^{-\frac{2T}{L}\sum_{n,q}(G_{11}+G_{22})}\cosh[4\text{Tr}G_{12}]\right).$

Here $\text{Tr}G_{ij}=\frac{T}{L}\sum_{q,n}G_{ij}(q,\omega_{n})$. To obtain
$G$, we need to inverse the matrix $G^{-1}$. In the case when all
$\Delta_{ij}$ are close to zero, we can approximately have

$G(\bm{q})\approx\begin{pmatrix}\frac{\pi
K_{1}}{v_{1}}\left(\frac{\omega_{n}^{2}}{v_{1}^{2}}+q^{2}+\frac{\Delta_{1}^{2}}{v_{1}^{2}}\right)^{-1}&-\frac{\pi^{2}K_{1}K_{2}}{v_{1}v_{2}}\frac{\Delta_{12}}{(\frac{\omega_{n}^{2}}{v_{1}^{2}}+q^{2}+\frac{\Delta_{1}^{2}}{v_{1}^{2}})(\frac{\omega_{n}^{2}}{v_{2}^{2}}+q^{2}+\frac{\Delta_{2}^{2}}{v_{2}^{2}})}&-\frac{\pi^{2}K_{1}K_{3}}{v_{1}v_{3}}\frac{\Delta_{13}}{(\frac{\omega_{n}^{2}}{v_{1}^{2}}+q^{2}+\frac{\Delta_{1}^{2}}{v_{1}^{2}})(\frac{\omega_{n}^{2}}{v_{3}^{2}}+q^{2}+\frac{\Delta_{3}^{2}}{v_{3}^{2}})}\\\
-\frac{\pi^{2}K_{1}K_{2}}{v_{1}v_{2}}\frac{\Delta_{12}}{(\frac{\omega_{n}^{2}}{v_{1}^{2}}+q^{2}+\frac{\Delta_{1}^{2}}{v_{1}^{2}})(\frac{\omega_{n}^{2}}{v_{2}^{2}}+q^{2}+\frac{\Delta_{2}^{2}}{v_{2}^{2}})}&\frac{\pi
K_{2}}{v_{2}}\left(\frac{\omega_{n}^{2}}{v_{2}^{2}}+q^{2}+\frac{\Delta_{2}^{2}}{v_{2}^{2}}\right)^{-1}&-\frac{\pi^{2}K_{2}K_{3}}{v_{2}v_{3}}\frac{\Delta_{23}}{(\frac{\omega_{n}^{2}}{v_{2}^{2}}+q^{2}+\frac{\Delta_{2}^{2}}{v_{2}^{2}})(\frac{\omega_{n}^{2}}{v_{3}^{2}}+q^{2}+\frac{\Delta_{3}^{2}}{v_{3}^{2}})}\\\
-\frac{\pi^{2}K_{1}K_{3}}{v_{1}v_{3}}\frac{\Delta_{13}}{(\frac{\omega_{n}^{2}}{v_{1}^{2}}+q^{2}+\frac{\Delta_{1}^{2}}{v_{1}^{2}})(\frac{\omega_{n}^{2}}{v_{3}^{2}}+q^{2}+\frac{\Delta_{3}^{2}}{v_{3}^{2}})}&-\frac{\pi^{2}K_{2}K_{3}}{v_{2}v_{3}}\frac{\Delta_{23}}{(\frac{\omega_{n}^{2}}{v_{2}^{2}}+q^{2}+\frac{\Delta_{2}^{2}}{v_{2}^{2}})(\frac{\omega_{n}^{2}}{v_{3}^{2}}+q^{2}+\frac{\Delta_{3}^{2}}{v_{3}^{2}})}&\frac{\pi