# Kohn-Luttinger-like mechanism for unconventional charge density waves

Hannes Braun  Max Planck Institute for Solid State Research, D-70569
Stuttgart, Germany  School of Natural Sciences, Technische Universität
München, D-85748 Garching, Germany  Michael M. Scherer  Institute for
Theoretical Physics III, Ruhr-University Bochum, D-44801 Bochum, Germany
Laura Classen  Max Planck Institute for Solid State Research, D-70569
Stuttgart, Germany  School of Natural Sciences, Technische Universität
München, D-85748 Garching, Germany

###### Abstract

Interaction-induced charge orders with electronic origin occur as states of
spontaneously broken symmetry in several materials platforms. An electronic
mechanism for charge order requires an attractive component in the effective
charge vertex. We put forward such a mechanism for the formation of
unconventional charge density waves in a metal. These states result from the
condensation of particle-hole pairs with finite wave vector and non-zero
angular momentum and correspond to bond or loop current order on a lattice.
The mechanism we describe can be viewed as Kohn Luttinger analysis in the
particle-hole channel with finite transferred momentum. It incorporates one-
loop spin and pairing correctionsn, which are then used as an input for a
summation in the charge channel triggering an instability. We extend our
analysis to a spin-fluctuation approach, where the effective charge
interaction is dressed by the particle-hole ladder with exchanged momentum. We
argue that this mechanism works for weakly-interacting metals with nested
Fermi surface and a large number of fermion flavors. We apply the Kohn-
Luttinger-like approach to square- and triangular-lattice Hubbard models with
SU($N_{f}$) flavour symmetry and show that it leads to different types of
$p$-wave charge density waves. We also study effects beyond weak coupling at
and away from Van Hove filling in terms of a phenomenological model with
additional exchange interaction. In the vicinity of Van Hove filling, we
obtain $d$-wave charge density waves with wave vectors determined by nesting
as leading instabilities. In addition, we find another charge density wave
with wave vector $K/4$ on the triangular lattice on both sides of Van Hove
filling. We demonstrate that this $K/4$ instability can win the competition
against pairing for $N_{f}=4$ via an unbiased functional renormalisation group
calculation.

##  I Introduction

Fascinating states of matter can arise from spontaneous symmetry breaking at
low temperatures in an interacting metal. They occur in different forms such
as spin, charge, or pairing orders. Studying the mechanisms behind their
formation provides insights about the fundamental processes in a quantum many-
body system. A famous example is superconductivity from repulsive
interactions, which relies on the renormalisation of an initially repulsive
electron-electroninteraction into an effective attraction between electrons
with non-zero relative angular momentum [1, 2, 3, 4]. In contrast, spin orders
can directly profit from the bare Coulomb interaction in the sense that an
instability is, in principle, possible if the interaction is large enough. The
Stoner criterion is a well-known example for this. For charge orders it is
less common to become the leading instability. One reason is that the standard
screening within a random phase approximation (RPA) does not lead to an
instability for a repulsive Coulomb interaction. However, Pomeranchuk
instabilities with higher angular momentum that distort the Fermi surface or
charge density waves (CDW) can occur due to additional renormalisation effects
[5, 6, 7].

Here, we are interested in a mechanism that can lead to a charge-ordered
ground state, in particular, one that breaks translation symmetry. Such a
charge density wave results from the condensation of particle-hole pairs with
a finite wave vector. Interestingly, the particle-hole pairs can also possess
a non-zero relative angular momentum [8]. We call the corresponding states
unconventional charge density waves (uCDW) in analogy to the case of
superconductivity from repulsive interactions. They need to be distinguished
from charge Pomeranchuk orders [5, 6], which are also based on particle-hole
pairs with different angular momenta but have zero wave vector, i.e., they do
not break translation symmetry. Consequently, a symmetry classification
differs between Pomeranchuk and CDW orders [9]. Examples for uCDWs include
bond or current orders, and were investigated in many systems. For example, a
Kekulé valence bond state can occur on the honeycomb lattice and was discussed
for graphene [10, 11, 12, 13, 14, 15, 16]. More recently, an incommensurate
more complex version, the incommensurate Kekulé spiral [17] was observed in
twisted bilayer graphene [18]. Current orders spontaneously break time-
reversal symmetry and they induce flux patterns if they form loop currents.
Loop current orders or staggered flux states, as well as bond orders, were
discussed in the context of the cuprates [19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31]. Furthermore, their topological properties were studied and it
was pointed out that topological density waves provide as a way to obtain
interaction-induced topological states [32, 33, 14, 34, 35, 36, 37]. Recently,
charge order that breaks translation and maybe also time-reversal symmetry was
observed in kagome metals [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50,
51, 52, 53], where an interesting interplay of multi-Q CDWs can occur [54, 55,
56].

A way to understand the formation of uCDWs out of a metallic state is to
consider them as interaction-induced instabilities of a Fermi liquid. In this
scenario, the effective interaction in the charge channel needs to develop a
singularity at a finite wave vector and higher angular momentum. However, an
RPA summation does not result in such a singularity for a repulsive density-
density interaction with a (screened) Coulomb-like profile. This indicates
that the feedback from spin and charge channels plays an important role for
the formation of uCDWs. An analogous mechanism can induce nematic order [57,
58] and assist in pairing near a nematic instability [59]. Inter-channel
feedback is taken into account in parquet or functional renormalisation group
(PRG/FRG) studies, which indeed reported charge bond order in the phase
diagram on the kagome lattice [60, 61, 62], or current loop orders on the
square and triangular lattice [63, 36, 37].

For strongly interacting systems, another route towards uCDWs is given via an
exchange interaction between neighboring spins, which can also induce
attractive components in the charge channel. This was analysed on the square
lattice with respect to Peierls or staggered flux states (a.k.a. bond and
current orders) [64, 65, 66, 67, 68, 69, 70]. In the examples above, uCDWs
compete with pairing or spin-density-wave (SDW) instabilities. Thus, for them
to win the competition the effective coupling for uCDWs needs to be larger
than for pairing or SDW. This can be achieved, for example, via an increased
number of fermion flavours which boost the uCDW coupling [63, 36, 37], or via
multi-orbital/sublattice interference effects which suppress competing SDW and
pairing couplings [71]. It is also possible to engineer such conditions, e.g,
in correlated fermion systems with SU($N_{f}$) flavour symmetry based on
ultra-cold atoms [72] or moiré heterostructures [73].

In this report, we present yet another mechanism to obtain uCDWs from
repulsive interactions inspired by the Kohn-Luttinger (KL) mechanism for
superconductivity [3, 4]. We start with a weakly-interacting metallic system
and discuss under which conditions an instability towards uCDW order can
arise. To this end, we assume that there is a large number of fermion flavours
$N_{f}$ and that the single-particle dispersion is at least approximately
nested. We argue that in this situation an attraction in the uCDW channel can
already occur when spin and pairing corrections are included on the one-loop
level and determine the instability condition for a general bare interaction.
As a next step, we consider higher-loop corrections in terms of a spin-
fluctuation approach, i.e., we use the effective interaction in the spin
channel as input for the charge channel.

Our analysis works out the minimal fluctuations that are required for an uCDW
instability and make the delicate interplay between spin and pairing
corrections clearly visible. In this sense, they complement PRG and FRG
approaches which take into account all channels on equal footing. We apply
this KL-like approach to the square- and triangular-lattice Hubbard model,
where we find that it results in $p$-wave CDW with wave vector given by the
nesting vectors. On the square lattice, $p_{x}$ and $p_{y}$ solutions are
degenerate and describe bond or current density waves depending on their
superposition. On the triangular lattice, there is a unique $p$-wave form
factor for each of the three non-equivalent nesting vectors. They separately
form bond density waves, while the exact ground state also depends on their
superposition.

Finally, we complement our KL-like analysis by studying a $t-U-J$ model with
Hubbard $U$ and SU($N_{f}$) nearest-neighbor exchange interaction as a
phenomenological description beyond weak coupling. For the square lattice, we
reproduce earlier results for the staggered flux state [64, 65]. For the
triangular lattice, we find a phase transition between two $d$-wave uCDW with
different wave vectors as function of the chemical potential. The first one
occurs at Van Hove filling and corresponds to the flux order with nesting wave
vectors $M_{i}$ that was discussed previously [9, 36, 37]. Moving away from
the Van Hove filling, the maximum of the susceptibility jumps from $M_{i}$ to
$K/4$, which is why the other $d$-wave uCDW with wave vector $K/4$ becomes the
leading instability. We confirm that the $K/4$ CDW can win against pairing in
an FRG calculation for the SU(4) case.

In the following, we first introduce the general formalism and describe the
mechanisms for uCDW instabilities in Sec. II. We also compare to the original
KL mechanism for pairing instabilities. In Sec. III, we present the
application to the square and triangular lattice model, as well as the
analysis of the phenomenological $t-U-J$ model. We summarise our findings and
comment on extensions of our approach in Sec. IV. In the appendix, we present
a KL-like analysis for uCDWs in a patch model for square and triangular
lattice and compare to the results of full model in the main text.

##  II Formalism and idea

###  II.1 Effective interaction and form-factor expansion

Figure 1: Effective interaction $V(k_{1},k_{2},k_{3})$ expanded up to one-loop
order. The first term on the left-hand side depicts the contribution from the
bare interaction $U(k_{1},k_{2},k_{3})$ (dashed line)and all other terms are
one-loop corrections. External solid lines determine the in-going and out-
going momenta of the interacting fermions, and internal solid lines in the
loops represent single-particle Green’s functions $\mathcal{G}$. Notably, the
closed internal loop in the first direct particle-hole diagram, which is also
known as “screening” diagram, contributes with the multiplicity of the flavour
number $N_{f}$.

In a many-body system, interactions receive corrections from multiple
scattering processes. Formally, this can be described by the vertex function
$\Gamma^{(2)}_{\alpha\beta\gamma\delta}(k_{1},\dots,k_{4})$ which is the one-
particle irreducible part of the two-particle Green’s function

$\displaystyle
G^{(2)}_{\alpha\beta\gamma\delta}(k_{1},\dots,{k}_{4})=\langle\hat{T}c_{{k}_{1}\alpha}^{\dagger}c_{{k}_{2}\beta}^{\dagger}c_{{k}_{3}\gamma}c_{{k}_{4}\delta}\rangle\,,$
(1)

where $\hat{T}$ is the time-ordering operator and the $c^{(\dagger)}_{\alpha
k}$ are fermion annihilation (creation) operators with spin $\alpha$ and
(2+1)d-momentum $k=(\omega,\mathbf{k})$ collecting Matsubara frequency and
wave vector. Flavour symmetry SU($N_{f}$), such as spin SU(2), dictates the
form

$\displaystyle\Gamma_{\alpha\beta\gamma\delta}(k_{1},\dots,k_{4})$
$\displaystyle=V(k_{1},k_{2},k_{3},k_{4})\delta_{\alpha\delta}\delta_{\beta\gamma}$
$\displaystyle\quad-V(k_{1},k_{2},k_{4},k_{3})\delta_{\alpha\gamma}\delta_{\beta\delta}\,,$
(2)

and we can view $V(k_{1},k_{2},k_{3},k_{4})$ as the effective interaction. It
appears in the effective action as

$\displaystyle\Gamma_{V}$
$\displaystyle=\frac{1}{2}\int_{{k}_{1},\dots,{k}_{4}}\sum_{\sigma,\sigma^{\prime}}V({k}_{1},\dots,{k}_{4})\delta({k}_{1}+{k}_{2}-{k}_{3}-{k}_{4})$
$\displaystyle\quad\times
c_{{k}_{1}\sigma}^{\dagger}c_{{k}_{2}\sigma^{\prime}}^{\dagger}c_{{k}_{3}\sigma^{\prime}}c_{{k}_{4}\sigma}\,,$
(3)

and we use the short-hand notation $T\sum_{\omega}\frac{1}{V}\int
d\mathbf{k}\equiv\int_{k}$.

Due to momentum conservation, the interaction depends effectively only on
three momenta, which we use to shorten notation by omitting the fourth one in
the following. To first order, the effective interaction equals the bare
interaction $V(k_{1},k_{2},k_{3})=U(k_{1},k_{2},k_{3})$. Below we will
consider a constant Hubbard interaction $U(k_{1},k_{2},k_{3})=U$ and study the
repulsive case $U>0$. To second order, $V(k_{1},k_{2},k_{3})$ receives the
one-loop corrections shown diagrammatically in Fig. 1. They can be decomposed
into contributions of particle-particle type

$\displaystyle\tau_{\mathrm{pp}}(k_{1},k_{2},k_{3})=-\int_{p}\mathcal{G}({p}+Q_{\mathrm{pp}})\mathcal{G}(-{p})$
$\displaystyle\quad\quad\quad\quad\times
U({k}_{1},{k}_{2},Q_{\mathrm{pp}}-{p})U({p},Q_{\mathrm{pp}}-{p},{k}_{3})\,,$
(4)

of crossed particle-hole type

$\displaystyle\tau_{\mathrm{ph-c}}(k_{1},k_{2},k_{3})=-\int_{p}\mathcal{G}({p}+Q_{\mathrm{ph-c}})\mathcal{G}({p})$
$\displaystyle\quad\quad\quad\quad\times
U({k}_{1},{p},k_{3})U({p}+Q_{\mathrm{ph-c}},{k}_{2},p)\,,$ (5)

and of direct particle-hole type

$\displaystyle\tau_{\mathrm{ph-d}}(k_{1},k_{2},k_{3})=-\int_{p}\mathcal{G}({p}+Q_{\mathrm{ph-d}})\mathcal{G}({p})$
$\displaystyle\quad\quad\quad\quad\times\bigl{[}U({k}_{1},{p}+Q_{\mathrm{ph-d}},p)U(k_{2},p,k_{3})$
$\displaystyle\quad\quad\quad\quad+U({k}_{1},{p}+Q_{\mathrm{ph-d}},k_{4})U(p,k_{2},k_{3})$
$\displaystyle\quad\quad\quad\quad-
N_{f}U({k}_{1},{p}+Q_{\mathrm{ph-d}},{p})U({p},{k}_{2},{k}_{3})\bigr{]}\,.$
(6)

Here, $\mathcal{G}(k)=(i\omega-\xi_{\mathbf{k}})^{-1}$ is the single-particle
Green’s function with dispersion $\xi_{\mathbf{k}}$ and we introduced the
total, transferred, and exchanged momentum, $Q_{\mathrm{pp}}=k_{1}+k_{2}$,
$Q_{\mathrm{ph-c}}={k}_{1}-{k}_{3}$, and $Q_{\mathrm{ph-d}}={k}_{2}-{k}_{3}$,
respectively.

Generally, the effective interaction can be viewed as a function of either
total, transferred, or exchanged momentum according to these corrections. This
defines three channels $V_{X}$, $X\in\\{\text{pp, ph-d, ph-c}\\}$

$\displaystyle V_{\text{pp}}(k,k^{\prime},Q)$
$\displaystyle=V({k},{Q}-{k},{Q}-{k}^{\prime},{k}^{\prime})$ (7)
$\displaystyle V_{\text{ph-c}}(k,k^{\prime},Q)$
$\displaystyle=V({k},{k}^{\prime}+Q,{k}+{Q},{k}^{\prime})$ (8) $\displaystyle
V_{\text{ph-d}}(k,k^{\prime},Q)$
$\displaystyle=V({k},{k}^{\prime}+Q,{k}^{\prime},{k}+{Q})\,,$ (9)

where the effective interaction $V_{X}$ is two-particle reducible in the
particle-particle (pp), the direct particle-hole (ph-d), and the crossed
particle-hole (ph-c) channel, respectively.

This rewriting allows us to single out the decisive momentum ${Q}$ and two
less important momenta ${k},{k}^{\prime}$ for an instability in each channel.
At low temperatures, one or more of the $V_{X}$ can develop a singularity at a
specific $Q$, which signals the tendency to form an ordered phase with
particle-hole or particle-particle bound states. To study the most singular
correction, we set external frequencies to the lowest Matsubara frequency.

For example, the Cooper instability corresponds to a singularity in
$V_{\mathrm{pp}}$ for $Q=0$. The particle-hole channels are related to
instabilities towards charge and spin or more generally flavour order. This
can be seen when rewriting the effective action in Eq. 3 via charge and
flavour components

$\displaystyle\Gamma_{V}$
$\displaystyle=\frac{1}{2}\int_{k,k^{\prime},q}\sum_{\alpha\ldots\delta}c_{{k}\alpha}^{\dagger}c_{{k}^{\prime}+q\beta}^{\dagger}c_{k^{\prime}\gamma}c_{k+q\delta}$
$\displaystyle\left[V^{c}(k,k^{\prime},q)\delta_{\alpha\delta}\delta_{\beta\gamma}-V^{s}(k,k^{\prime},q)\sum_{i=1}^{N_{f}^{2}-1}T^{i}_{\alpha\delta}T^{i}_{\beta\gamma}\right]\,,$
(10)

where $T^{i}$ are the generators of SU($N_{f}$) in the defining representation
and we used the completeness relation
$2T^{i}_{\alpha\beta}T^{i}_{\gamma\delta}+\delta_{\alpha\beta}\delta_{\gamma\delta}/N_{f}=\delta_{\alpha\delta}\delta_{\beta\gamma}$
[74]. For SU(2), these are simply proportional to the Pauli matrices
$T^{i}=\sigma^{i}/2$. The effective interactions in charge and flavour channel
are given by
$V^{c}(k,k^{\prime},q)=[V_{\mathrm{ph-d}}(k,k^{\prime},q)-V_{\mathrm{ph-c}}(k,k^{\prime},q)/N_{f}]/2$
and $V^{c}(k,k^{\prime},q)=-V_{\mathrm{ph-c}}(k,k^{\prime},q)$. This means
that the effective interaction in the charge channel is well approximated by
the effective interaction in the ph-d channel for large $N_{f}$.

Figure 2: Kohn-Luttinger mechanism for pairing. Top: diagrammatic
representation of the self-consistent equation for the effective pairing
vertex $V_{pp}$, i.e., the effective interaction viewed as a function of total
incoming/outgoing momentum. The self-consistent equation amounts to summing up
the Cooper ladder. There is no pairing instability with only a bare Coulomb
interaction (dashed line). Using instead the bare interaction and one-loop
particle-hole corrections (bottom) as a basis in the Cooper ladder (blue
vertex), a pairing instability for pairs with non-zero angular momentum can
occur. Internal solid lines represent fermion propagators, external ones
denote the momentum dependence.

It is convenient to expand the momentum dependence of $V_{X}(k,k^{\prime},Q)$
in form factors according to the symmetry of the system. For example, we
choose the lattice harmonics of the square and triangular lattice in Sec. III
below. This can be thought of as an expansion in the angular momentum
generalised to a lattice system. For the expansion, we project the vertex into
form factor space

$\displaystyle\hat{V}_{X}^{l,l^{\prime}}(\mathbf{Q})=\int_{\mathbf{k},\mathbf{k}^{\prime}}f_{l}(\mathbf{k})f^{\ast}_{l^{\prime}}(\mathbf{k}^{\prime})V_{X}(\mathbf{k,\mathbf{k}^{\prime},\mathbf{Q})}$
(11)

where the form factors $f_{i}(\mathbf{k})$ form an orthonormal basis
$\delta(\mathbf{k}-\mathbf{k}^{\prime})=\sum_{l}f_{l}^{\ast}(\mathbf{k})f_{l}(\mathbf{k}^{\prime})$
and
$\delta_{l,l^{\prime}}=\int_{\mathbf{k}}f_{l}^{\ast}(\mathbf{k})f_{l^{\prime}}(\mathbf{k})$.
Analogously, we project the particle-particle (-) and particle-hole (+)
polarisations into the form-factor basis for later use

$\hat{\Pi}_{\pm}^{l,l^{\prime}}(\mathbf{Q})=\mp\int_{{p}}\mathcal{G}({p})\mathcal{G}(\pm{p}+Q)f^{\ast}_{l}(\mathbf{p})f_{l^{\prime}}(\mathbf{p})$
(12)

To a good approximation, it is usually sufficient to consider a finite number
of form factors. Here, we have chosen the signs of the particle-particle and
particle-hole polarisations such that $\Pi_{\pm}(\mathbf{Q})>0$.

###  II.2 Kohn Luttinger mechanism for pairing

Even in a weakly-interacting system of fermions with purely repulsive, bare
interactions, instabilities of the Fermi liquid arise at low temperatures. The
reason is that the effective interaction becomes attractive in a pairing
channel due to corrections from particle-hole channels, which then gives rise
to a Cooper instability. Kohn and Luttinger showed that the decisive
corrections come from a non-analytic contribution in the particle-hole
polarisation at a momentum transfer of twice the Fermi momentum $q=2k_{F}$,
which possesses attractive components in higher angular-momentum channels. To
see this, we consider the pairing channel, cf. Fig. 2,

$\displaystyle
V_{\text{pp}}(\mathbf{k},\mathbf{k}^{\prime},\mathbf{Q})=U(\mathbf{k},\mathbf{Q}-\mathbf{k},\mathbf{Q}-\mathbf{k}^{\prime})$
$\displaystyle-\\!\int_{p}U(\mathbf{k},\mathbf{Q}\\!-\\!\mathbf{k},\mathbf{Q}\\!-\\!\mathbf{p})\mathcal{G}({p})\mathcal{G}(Q\\!-\\!{p})V_{\text{pp}}(\mathbf{p},\mathbf{k}^{\prime},\mathbf{Q})\,,$
(13)

and project it into the form-factor basis

$\displaystyle\hat{V}^{l,l^{\prime}}_{\text{pp}}(\mathbf{Q})=\hat{U}_{\mathrm{pp}}^{l,l^{\prime}}(\mathbf{Q})-\hat{U}_{\mathrm{pp}}^{l,a}(\mathbf{Q})\hat{\Pi}_{-}^{a,b}(\mathbf{Q})\hat{V}_{\text{pp}}^{b,l^{\prime}}(\mathbf{Q})\,,$
(14)

where summation over repeated indices is implied and the bare interaction in
the particle-particle channel reads

$\displaystyle\hat{U}_{\mathrm{pp}}^{l,l^{\prime}}(\mathbf{Q})=\int_{\mathbf{k,^{\prime}k}}f_{l}(\mathbf{k})f^{\ast}_{l^{\prime}}(\mathbf{k}^{\prime})U(\mathbf{k},\mathbf{Q}-\mathbf{k},\mathbf{Q}-\mathbf{k}^{\prime})\,.$
(15)

The formal solution in matrix representation is given by

$\displaystyle\hat{V}_{\text{pp}}(\mathbf{Q})$
$\displaystyle=\frac{\hat{U}_{\mathrm{pp}}(\mathbf{Q})}{\mathbb{1}+\hat{U}_{\mathrm{pp}}(\mathbf{Q})\hat{\Pi}_{-}(\mathbf{Q})}\,.$
(16)

We note that for an isotropic system and $\bf{Q}=0$, as considered by KL, all
quantities become diagonal when expanding in angular harmonics, so that the
$l$th angular-momentum component reads
$\hat{V}^{l}_{\mathrm{pp}}=\hat{U}_{\mathrm{pp}}^{l}(1+\hat{U}_{\mathrm{pp}}^{l}\hat{\Pi}_{l})^{-1}$.

The bare Coulomb repulsion does not possess an attractive angular-momentum
component and with it alone no pairing instability can arise. However, taking
one-loop particle-hole corrections into account, we have to replace the bare
interaction in the pairing channel by

$\displaystyle U_{\mathrm{pp}}\rightarrow
V_{\mathrm{pp}}^{0}=U_{\mathrm{pp}}+\left.\tau_{\mathrm{ph-c}}\right|_{\mathrm{pp}}+\left.\tau_{\mathrm{ph-d}}\right|_{\mathrm{pp}}\,,$
(17)

where $\tau_{X}|_{\mathrm{pp}}=\tau_{X}(k,Q-k,Q-k^{\prime})$, see Fig. 2.
Explicitly, this reads

$\displaystyle\hat{V}_{\text{pp}}^{0,l,l^{\prime}}(\mathbf{Q})$
$\displaystyle=U\delta_{l,0}\delta_{l^{\prime},0}+U^{2}\int_{\mathbf{k},\mathbf{k}^{\prime}}f_{l}(\mathbf{k})f^{\ast}_{l^{\prime}}(\mathbf{k}^{\prime})$
$\displaystyle\quad\times\left[\Pi_{+}(\mathbf{k}+\mathbf{k}^{\prime})+(N_{f}-2)\cdot\Pi_{+}(\mathbf{k}-\mathbf{k}^{\prime})\right]\
.$ (18)

where the term with prefactor $N_{f}-2$ belongs to the ph-d fluctuations. The
other $\mathcal{O}(U^{2})$ term corresponds to the spin fluctuations.

Figure 3: Kohn-Luttinger-like mechanism for uCDWs. Top: diagrammatic
representation of the large-$N_{f}$ self-consistent equation for the effective
interaction viewed as function of transferred momentum, i.e., in the direct
particle-hole channel $V_{\mathrm{ph-d}}$. In the large-$N_{f}$ limit
$V_{\mathrm{ph-d}}$ is a good approximation for the effective charge vertex.
The self-consistent equation is equivalent to summing the RPA screening
diagrams in the direct particle-hole channel. Bottom: similar to the Kohn-
Luttinger mechanism for pairing, an instability can arise in the RPA summation
if the bare interaction (dashed line) is supplemented by one-loop particle-
particle and remaining particle-hole diagrams (green vertex). These diagrams
are subleading in $N_{f}$. Internal solid lines represent fermion propagators,
external ones denote the momentum dependence.

KL showed that in an isotropic Fermi liquid, the bare repulsion
$\hat{U}_{\mathrm{pp}}^{l}$ is exponentially small for large $l$ and that
$\tau_{\mathrm{ph,c}}|_{\mathrm{pp}}+\tau_{\mathrm{ph,d}}|_{\mathrm{pp}}$
possesses attractive angular-momentum components due to the non-analyticity of
the particle-hole polarisation bubble at $\Pi_{-}^{l}(k\pm k^{\prime}\approx
2k_{F})$ so that $V_{\mathrm{pp}}^{0,l}<0$ for some $l>0$. As a result, a
pairing instability arises even in a weakly-interacting Fermi liquid based on
a purely electronic pairing mechanism.

Critical temperatures of this original KL mechanism remain very small, i.e.,
exponentially small in $l^{4}$, but they can be boosted through peaks in the
density of states in a lattice system when a Van Hove singularity is placed
near the Fermi level. Closely related in spirit is also the spin-fluctuation
mechanism: the one-loop particle-hole corrections are only the first terms of
the renormalisation in the particle-hole channel and the entire series should
be summed if they are large, as is the case close to a magnetic instability.

###  II.3 Mechanisms for unconventional CDW

Similar to the case of pairing, the couplings for charge ordering tendencies
are repulsive when starting with a (screened) Coulomb interaction. However,
following the idea of KL, corrections from other channels can introduce an
attraction and consequently induce instabilities towards charge order. We show
that such a KL-like mechanism also works for finite wave-vector transfer as
long as higher lattice harmonics are considered, i.e., the mechanism yields
instabilities towards “unconventional” charge density waves with order
parameters of the form $\Delta_{Q}(\mathbf{k})=f_{l}(\mathbf{k})\langle
c_{k+Q}^{\dagger}c_{k}\rangle$ where $l>0$ and $Q\neq 0$. Depending on $l$ and
$Q$, these orders describe, for example, bond density waves for real
$\Delta_{Q}$ or loop current/staggered flux phases for imaginary $\Delta_{Q}$.

For density-wave instabilities to become competitive at weak coupling, we
consider a system with (approximate) nesting, so that particle-particle and
particle-hole channels are parametrically of the same order. Specifically,
this means that they possess a (double-)logarithmic singularity away from (at)
Van Hove filling (and above an infrared cutoff scale associated with
deviations from perfect nesting). In addition, we consider a system with
SU($N_{f}$) symmetry where $N_{f}$ is large, which will give charge
instabilities a boost over pairing and spin instabilities.

In this situation, the leading corrections to the effective interaction are
based on the one-loop “screening” diagram, cf. Fig. 3, which is of the order
of $N_{f}\ln^{2}(T)$ (or $N_{f}\ln(T)$ without Van Hove singularity). Summing
up the corresponding ladder series shown in Fig. 3, we obtain the self-
consistent equation for the effective interaction in the direct particle-hole
channel

$\displaystyle
V_{\text{ph-d}}(\mathbf{k},\mathbf{k}^{\prime},\mathbf{Q})=U(\mathbf{k},\mathbf{k}^{\prime}+\mathbf{Q},\mathbf{k}^{\prime})$
$\displaystyle-\\!N_{f}\\!\\!\int_{\mathbf{p}}\\!U(\mathbf{k},\mathbf{p}\\!+\\!\mathbf{Q},\mathbf{p})\mathcal{G}({p})\mathcal{G}({p}\\!+\\!{Q})V_{\text{ph-d}}(\mathbf{p},\mathbf{k}^{\prime},\mathbf{Q})\,.$
(19)

We rewrite this in terms of the form factor basis

$\displaystyle\hat{V}_{\text{ph-d}}^{l,l^{\prime}}(\mathbf{Q})$
$\displaystyle=\hat{U}_{\mathrm{ph,d}}^{l,l^{\prime}}(\mathbf{Q})$
$\displaystyle\quad-
N_{f}\hat{U}^{l,a}_{\mathrm{ph,d}}(\mathbf{Q})\hat{\Pi}^{a,b}_{+}(\mathbf{Q})\hat{V}_{\text{ph-d}}^{b,l^{\prime}}(\mathbf{Q})\,,$
(20)

with
$\hat{U}_{\mathrm{ph,d}}^{l,l^{\prime}}(\mathbf{Q})\\!=\\!\int_{\mathbf{k,^{\prime}k}}U(\mathbf{k},\mathbf{k}^{\prime},\mathbf{Q})f_{l}(\mathbf{k})f^{\ast}_{l^{\prime}}(\mathbf{k}^{\prime})$,
and obtain the formal solution in matrix representation

$\displaystyle\hat{V}_{\text{ph-d}}(\mathbf{Q})=\frac{\hat{U}_{\text{ph-d}}(\mathbf{Q})}{\mathbb{1}+N_{f}\hat{U}_{\text{ph-d}}(\mathbf{Q})\hat{\Pi}_{+}(\mathbf{Q})}\,.$
(21)

Again, there cannot be any instability if the bare interaction does not
possess an attractive angular momentum component, which is usually not the
case. However, analogous to the KL scenario for pairing, interaction
corrections from other channels will introduce a non-trivial momentum
dependence into the renormalized interaction. Only now we need to project the
crossed particle-hole and particle-particle corrections into the direct
particle-hole momentum structure.

As a first step, we include the corresponding one-loop corrections, see Fig.
3. Then, we obtain that the bare interaction in the direct channel is replaced
by

$\displaystyle U_{\mathrm{ph-d}}\rightarrow
V_{\mathrm{ph-d}}^{0}=U^{(2)}_{\mathrm{ph-d}}\\!+\\!\left.\tau_{\mathrm{ph-c}}\right|_{\mathrm{ph-d}}\\!+\\!\left.\tau_{\mathrm{pp}}\right|_{\mathrm{ph-d}}\,,$
(22)

where $\tau_{X}|_{\mathrm{ph-d}}=\tau_{X}(k,k^{\prime}+Q,k^{\prime})$ and in
$U^{(2)}_{\mathrm{ph-d}}$ we incorporated the ph-d corrections that are of
subleading order $\mathcal{O}(1)$ in $N_{f}$ and not contained in Eq. (19),
i.e.,
$U^{(2)}_{\mathrm{ph-d}}=U_{\mathrm{ph-d}}-\int_{p}\mathcal{G}(p)\mathcal{G}({p}+Q)[U({k},{p}+Q,p)U(k^{\prime}+Q,p,k^{\prime})+U({k},{p}+Q,k+Q)U(p,k^{\prime}+Q,k^{\prime})]$,
cf. Eq. (6) or the last two diagrams in Fig. 3. In the case of a pure Hubbard
interaction $U(k_{1},k_{2},k_{3})=U$, we find

$\displaystyle
V_{\text{ph-d}}^{0}(\mathbf{k},\mathbf{k}^{\prime},\mathbf{Q})=U+2U^{2}\Pi_{+}(\mathbf{Q})$
$\displaystyle\quad\quad\quad\quad+U^{2}[\Pi_{+}(\mathbf{k}-\mathbf{k}^{\prime})-\Pi_{-}(\mathbf{k}+\mathbf{k}^{\prime}+\mathbf{Q})]\,.$
(23)

In this case, only the ph-c and pp corrections lead to an additional momentum
dependence, which can induce an attraction in a higher harmonic contribution.
The remaining ph-d corrections lead to a constant shift $2U^{2}\Pi_{+}(Q)$,
which changes the $s$-wave part of $V^{0}_{\text{ph-d}}$ to more repulsive.
Explicitly, we find after projecting into form factors

$\displaystyle\\!\\!\hat{V}_{\text{ph-d}}^{0,l,l^{\prime}}(\mathbf{Q})=[U+2U^{2}\Pi_{+}(\mathbf{Q})]\delta_{l,0}\delta_{l^{\prime},0}$
$\displaystyle\\!\\!+\\!U^{2}\\!\\!\int_{\mathbf{k}}\\!\int_{\mathbf{k}^{\prime}}\\!\\!f_{l}(\mathbf{k})f_{l^{\prime}}^{\ast}(\mathbf{k}^{\prime})[\Pi_{+}(\mathbf{k}\\!-\\!\mathbf{k}^{\prime})\\!-\\!\Pi_{-}(\mathbf{k}\\!+\\!\mathbf{k}^{\prime}\\!+\\!\mathbf{Q})]\,.$
(24)

Through the inclusion of ph-c and pp corrections, the sign of the effective
interaction in the ph-d channel for $l,l^{\prime}>0$ can be changed so that a
charge instability with a higher lattice harmonic becomes possible. Without
nesting, this could, in principle, lead to a Pomeranchuk instability
characterised by $Q=0$. But with nesting, the leading instability is towards
an unconventional charge density wave.

We note that generally neither
$\hat{V}_{\mathrm{ph-d}}^{0,ll^{\prime}}(\mathbf{Q})$ nor
$\hat{\Pi}_{+}^{ll^{\prime}}(\mathbf{Q})$ are diagonal when expanding in
lattice harmonics because of the lower symmetry at finite $Q\neq 0$. This
means that admixtures of different components within the same irreducible
representation of the extended point group are allowed by symmetry [9]. To
determine if there is an instability in the ph-d channel, we have to calculate
the eigenvalues of the matrix product
$\hat{V}_{\mathrm{ph-d}}^{0,ln}(\mathbf{Q})\hat{\Pi}_{+}^{nl^{\prime}}(\mathbf{Q})$.

###  II.4 Beyond KL

Going beyond the KL-like analysis to higher order in the bare interaction, the
question arises which diagrams to select for resummation to determine the
effective interaction in the ph-d channel. The diagrams leading in $N_{f}$ are
included in the ph-d channel. On the next level, ph-c and pp contributions are
of the same order. One choice is to only take the summed crossed particle-hole
series which is near-singular if the system is at a temperature close to the
magnetic instability that arises due to nesting. Such an approach is analogous
to a spin-fluctuation mechanism for pairing. For a constant, bare interaction,
it amounts to

$\displaystyle
V_{\mathrm{ph-c}}(\mathbf{k}_{1}-\mathbf{k}_{3})=\frac{U}{1-U\,\Pi_{+}(\mathbf{k}_{1}-\mathbf{k}_{3})}\,.$
(25)

Taking $V_{\mathrm{ph-c}}$ as effective interaction in the ph-d channel, we
obtain

$\displaystyle\hat{V}_{\mathrm{ph-d}}^{0,ll^{\prime}}$
$\displaystyle=\int_{\mathbf{k}}\int_{\mathbf{k}^{\prime}}f_{l}(\mathbf{k})f^{\ast}_{l^{\prime}}(\mathbf{k}^{\prime})V_{\mathrm{ph-c}}(\mathbf{k}-\mathbf{k}^{\prime})\,,$
(26)

and we note that
$V_{\mathrm{ph-d}}^{0,ll^{\prime}}=V_{\mathrm{ph-d}}^{0,l}\delta_{l,l^{\prime}}$
is diagonal in this spin-fluctuation approach because there are no particle-
particle contributions (see App. B).

It is instructive to compare this to the spin-fluctuation approach for
pairing, where the effective interaction in the pp channel becomes

$\displaystyle\hat{V}_{\mathrm{pp}}^{0,ll^{\prime}}(\mathbf{Q})$
$\displaystyle=\int_{\mathbf{k},\mathbf{k}^{\prime}}f_{l}(\mathbf{k})f^{\ast}_{l^{\prime}}(\mathbf{k}^{\prime})V_{\mathrm{ph-c}}(\mathbf{k}+\mathbf{k}^{\prime}-\mathbf{Q})\,.$
(27)

Typically, a pairing instability occurs for zero total momentum $Q=0$. In this
case, there is a direct analogy between the effective interaction in pp and
ph-d channels

$\displaystyle\hat{V}_{\mathrm{ph-d}}^{0,ll^{\prime}}=\pm\hat{V}_{\mathrm{pp}}^{0,ll^{\prime}}(0),$
(28)

where the upper (lower) sign is for $l,l^{\prime}$ even (odd).

Thus, we find that, if spin fluctuations induce an attraction in an even
(singlet) pairing channel, they also induce an attraction in the corresponding
unconventional CDW channel. For odd (triplet) channels, it is the other way
around: an attraction for pairing translates to a repulsion for CDW and vice
versa. We note, however, that in contrast to the instability analysis in the
pairing channel, the polarisation $\hat{\Pi}^{l,l^{\prime}}_{+}(\mathbf{Q})$
that enters the renormalisation of the ph-d vertex is not necessarily
diagonal, and one has to diagonalise
$\hat{V}_{\mathrm{ph,d}}^{0,l}\hat{\Pi}^{l,l^{\prime}}_{+}(\mathbf{Q})$ to
diagnose the instability.

In the spin-fluctuation approach, we focus on the ladder series in the ph-c
channel because it is almost singular. However, it is unclear whether we can
neglect pp diagrams because they are of the same size as the ph-c diagrams in
a situation with nesting. Another option to explore this issue is an FRG or
PRG analysis which takes these contribution into account on equal footing.

More generally, in the situation we consider, there is also a strong
competition of orders: SDW, CDW, and pairing channels all possess the same
logarithmic singularity. Usually, the coupling in the spin channel
$U_{s}=U(k_{1},k_{2},k_{4})$ is attractive on the bare level, giving SDW
instabilities a head start. In addition, if fluctuations can turn the coupling
in a CDW channel, there is also an attraction in a pairing channel. Assuming
$\Pi_{-}(0)\sim\Pi_{+}(Q)$, the condition for a CDW instability to win this
competition is

$\displaystyle N_{f}V_{\mathrm{ph,d}}^{0}>V_{\mathrm{pp}}^{0},U_{s}\,,$ (29)

which is always possible for large enough $N_{f}$. In addition, an
unconventional CDW can benefit from a suppression of the bare interaction in
the spin channel in systems with several orbitals/sublattices per unit cell.

In the next section, we show how the KL-like mechanism indeed facilitates
instabilities towards unconventional charge order in square and triangular
lattices. Furthermore, we discuss a spin-fluctuation approach and study a
mechanism based on a nearest-neighbor exchange coupling in a phenomenological
model with the FRG.

##  III Application to square and triangular lattice

###  III.1 Lattice models

We apply the general formalism developed in Sec. II.3 to the Hubbard model on
the square and triangular lattice with perfect nesting and SU($N_{f}$) flavour
symmetry. To this end, we consider the Hamiltonian
$\mathcal{H}=\mathcal{H}_{0}+\mathcal{V}$, where

$\displaystyle\mathcal{H}_{0}$
$\displaystyle=-t\sum_{\braket{i,j}}\sum_{\sigma}\left(c_{i,\sigma}^{\dagger}c_{j\sigma}^{\phantom{\dagger}}+\text{h.c.}\right)-\mu\sum_{i}\sum_{\sigma}n_{i,\sigma}$
(30) $\displaystyle\mathcal{V}$
$\displaystyle=\frac{U}{2}\sum_{i}\sum_{\sigma,\sigma^{\prime}}n_{i,\sigma}n_{i,\sigma^{\prime}}$
(31)

with nearest neighbor hopping $t$, chemical potential $\mu$, and on-site
Coulomb repulsion $U>0$. The density operator is given by
$n_{i,\sigma}=c_{i,\sigma}^{\dagger}c_{i\sigma}^{\phantom{\dagger}}$ and the
flavour index $\sigma$ runs through $1\leq\sigma\leq N_{f}$. The dispersion
relations from diagonalising $\mathcal{H}_{0}$ on the square and triangular
lattice are given by

$\displaystyle\epsilon_{\mathbf{k}}^{\square}$
$\displaystyle\\!=\\!-2t\left(\cos(k_{x})+\cos(k_{y})\right)-\mu\,,$ (32)
$\displaystyle\epsilon_{\mathbf{k}}^{\triangle}$
$\displaystyle\\!=\\!-2t\left(\cos(k_{y})\\!+\\!2\cos(k_{y}/2)\cos(\sqrt{3}kx/2)\right)\\!-\\!\mu\,,$
(33)

where we used nearest-neighbor vectors $\mathbf{a}_{1}=(1,0)$ and
$\mathbf{a}_{2}=(0,1)$ for the square and
$\mathbf{a}_{1}=(0,1),\mathbf{a}_{2}=(\sqrt{3},1)/2$, and
$\mathbf{a}_{3}=(\sqrt{3},-1)/2$ for the triangular lattice.

We focus on the regimes close to Van Hove filling for square and triangular
lattice, i.e., we set the chemical potential $\mu$ close to
$\mu_{\mathrm{VH}}^{\square}=0$ and $\mu_{\mathrm{VH}}^{\triangle}=2t$,
respectively. At these fillings, the density of states possesses a logarithmic
singularity and the Fermi surface is nested
$\epsilon_{\mathbf{k}+\mathbf{Q}}=-\epsilon_{\mathbf{k}}$. Nesting is perfect
on the square lattice with $\mathbf{Q}^{\square}=(\pm\pi,\pm\pi)$. On the
triangular lattice it is approximate around the Fermi surface with
$\mathbf{Q}^{\triangle}\in\\{M_{1},M_{2},M_{3}\\}$ and $M_{1}=(\pm
2\pi/\sqrt{3},0),M_{2}=\pm(\pi/\sqrt{3},\pi),M_{3}=\pm(\pi/\sqrt{3},-\pi)$.
Under these conditions, there is a double logarithmic singularity in both the
particle-particle and the particle-hole polarisation
$\Pi_{-}(0)\sim\Pi_{+}(\mathbf{Q})\sim\ln^{2}(W/\text{max}(T,\delta\mu))$ with
bandwidth $W$, temperature $T$, and distance to Van Hove filling
$\delta\mu=|\mu-\mu_{\mathrm{VH}}|$. Thus, the prerequisites outlined in Sec.
II.3 for an instability towards an unconventional CDW are satisfied for
sufficiently large $N_{f}$.

###  III.2 Kohn-Luttinger-like mechanisms applied

We first consider CDW instabilities with wave vector
$\mathbf{Q}=\mathbf{Q}^{\square}$ or $\mathbf{Q}=\mathbf{Q}^{\triangle}$ on
the square and triangular lattice, respectively. In each case, we numerically
calculate the form-factor expansion of the effective interaction that we use
as input for the RPA summation of the ph-d channel according to Eq. (24). This
means we include one-loop pp and ph-c corrections, as well as the constant
ph-d one-loop corrections which are a factor of $N_{f}$ smaller. Similarly, we
numerically calculate and expand the polarisation operator
$\hat{\Pi}^{l,l^{\prime}}_{+}(\mathbf{Q})$. We use the numerical error
estimates to round the results.

For our calculation, we use the lattice harmonics for the square and
triangular lattices as a form-factor basis. We truncate the expansion after
nearest neighbors and check that including next-nearest neighbors does not
qualitatively change the leading CDW instability (see App. A for the explicit
expressions.) For the square lattice this amounts to the inclusion of $s$-,
$s^{\prime}$-, $d_{x^{2}-y^{2}}$-, $p_{x}$, and $p_{y}$-wave nearest-neighbor
form factors. For the triangular lattice it corresponds to the inclusion of
$s$-, $s^{\prime}$-, $f_{y(y^{2}-3x^{2})}$-, $d_{x^{2}-y^{2}}$-, $d_{xy}$-,
$p_{x}$, and $p_{y}$-wave form factors. Interestingly, we find that in the
considered cases pp and ph-c fluctuations work against each other for even
harmonics, while they work together for odd harmonics. To show this, we
rewrite for $l,l^{\prime}>0$

$\displaystyle\hat{V}_{\text{ph-d}}^{0,l,l^{\prime}}(\mathbf{Q})=U^{2}\int_{\mathbf{k}}\int_{\mathbf{k}^{\prime}}f_{l}(\mathbf{k})f_{l^{\prime}}^{\ast}(\mathbf{k}^{\prime})$
$\displaystyle\quad\quad\quad\times[\Pi_{+}(\mathbf{k}-\mathbf{k}^{\prime})-(-1)^{l^{\prime}}\Pi_{-}(\mathbf{k}-\mathbf{k}^{\prime}+\mathbf{Q})]\,.$
(34)

using that $f_{l}(-\mathbf{k})=(-1)^{l}f_{l}(\mathbf{k})$. On the square
lattice,
$\Pi_{-}(\mathbf{k}-\mathbf{k}^{\prime}+\mathbf{Q}^{\square})=\Pi_{+}(\mathbf{k}-\mathbf{k}^{\prime})$
due to perfect nesting. It follows that
$\hat{V}^{0,l,l^{\prime}}_{\mathrm{ph-d}}(\mathbf{Q}^{\square})=0$ for even
$l^{\prime}$, and
$\hat{V}^{0,l,l^{\prime}}_{\mathrm{ph-d}}(\mathbf{Q}^{\square})=2U^{2}\int_{\mathbf{k}}\int_{\mathbf{k}^{\prime}}f_{l}(\mathbf{k})f_{l^{\prime}}^{\ast}(\mathbf{k}^{\prime})\Pi_{+}(\mathbf{k}-\mathbf{k}^{\prime})$
for odd $l^{\prime}$. On the triangular lattice, nesting is not perfect,
however the leading logarithmic behavior still obeys
$\Pi_{-}(\mathbf{k}-\mathbf{k}^{\prime}+\mathbf{Q}^{\triangle})\approx\Pi_{+}(\mathbf{k}-\mathbf{k}^{\prime})$,
i.e., to a good approximation these contributions also cancel for even
$l^{\prime}$ and add up for odd $l^{\prime}$.

Figure 4: Square lattice. Top: effective interaction in the ph-d channel
$V_{\text{ph-d}}^{0,l}(\mathbf{Q}^{\square})$ including Kohn-Luttinger-like
corrections, expanded in nearest-neighbor lattice harmonics
$l\in\\{s,s^{\prime},p_{x},p_{y},d_{x^{2}-y^{2}}\\}$ as function of the
chemical potential $\mu$ at temperature $T=0.05/t$. Middle: Product of
effective interaction and polarization
$V_{\text{ph-d}}^{0,l}(\mathbf{Q}^{\square})\Pi_{+}^{l}(\mathbf{Q}^{\square})$
as it enters the direct particle-hole channel, cf. Eq. (21), i.e., the most
negative one determines the CDW instability. Bottom: eigenvalues of
$V_{\text{ph-d}}^{0,l}(\mathbf{Q}^{\square})\Pi_{+}^{l}(\mathbf{Q}^{\square})$
as a function of temperature $T/t$ at $\mu=\mu_{VH}$.

####  III.2.1 Square lattice

In the square-lattice case, both the effective interaction and the
polarisation operator decouple between the different irreducible
representations of the lattice point group for
$\mathbf{Q}=\mathbf{Q}^{\square}$, see App. B. This means that they are
diagonal in our truncated form-factor basis
$\hat{V}_{\text{ph-d}}^{0,l,l^{\prime}}(\mathbf{Q}^{\square})=V_{\text{ph-d}}^{0,l}(\mathbf{Q}^{\square})\delta_{l,l^{\prime}}$
and
$\hat{\Pi}_{+}^{l,l^{\prime}}(\mathbf{Q}^{\square})=\Pi_{+}^{l}(\mathbf{Q}^{\square})\delta_{l,l^{\prime}}$,
except for $A_{1}$ representatives, i.e., these relations hold for
$l,l^{\prime}\notin A_{1}$. Furthermore, all components of the polarisation
operator are positive so that it suffices to analyse the sign of
$V_{\text{ph-d}}^{0,l}(\mathbf{Q}^{\square})$ to determine if there is an
instability towards an unconventional CDW.

We show the result in Fig. 4. We find that the $p_{x}$ and $p_{y}$ components
become attractive so that an instability in the $p$-wave CDW channel arises.
This means that the singular part of the effective interaction is of the form

$\displaystyle
V=\sum_{k,\sigma}\sum_{\alpha=x,y}p_{\alpha}(\mathbf{k})p_{\alpha}(\mathbf{k}^{\prime})c_{\mathbf{k}\sigma}^{\dagger}c_{\mathbf{k}^{\prime}+\mathbf{Q}\sigma^{\prime}}^{\dagger}c_{\mathbf{k}^{\prime}\sigma^{\prime}}c_{\mathbf{k}+\mathbf{Q}\sigma}\,,$
(35)

suggesting a mean-field decoupling with order parameter

$\displaystyle\Delta_{\alpha}(\mathbf{Q})=\sum_{k,\sigma}p_{\alpha}(\mathbf{k})c_{\mathbf{k}+\mathbf{Q}\sigma}^{\dagger}c_{\mathbf{k}\sigma^{\prime}}\,.$
(36)

The resulting state of broken symmetry corresponds to a bond density wave with
alternating strong and weak bonds in $x$\- and $y$-direction, breaking
inversion symmetry. The exact pattern of the bond density wave depends on the
linear combination of $p_{x}$ and $p_{y}$ in the ground state. It can be
alternating in just one of the directions or, if both contributions are of the
same order, it yields a zig-zag bond pattern. An imaginary $p_{x}+ip_{y}$
combination can also introduce a current pattern.

We also find that at Van Hove filling, pp and ph-c contributions exactly
cancel for $d$-wave and extended $s$-wave components
$V_{\text{ph-d}}^{0,d}(\mathbf{Q}^{\square})=V_{\text{ph-d}}^{0,s^{\prime}}(\mathbf{Q}^{\square})=0$
as expected from Eq. (34) due to perfect nesting. We confirm this observation
for the full model also by an effective low-energy patch model that only keeps
states around Van Hove points, see App. C. Away from Van Hove filling
$\Pi_{-}(0)>\Pi_{+}(\mathbf{Q}^{\square})$ and the coupling in $s^{\prime}$\-
and $d$-wave CDW channels becomes repulsive, see Fig. 4.

####  III.2.2 Triangular lattice

For the triangular lattice, the analysis is slightly more involved. In this
case, the KL-like interaction
$\hat{V}_{\text{ph-d}}^{0,l,l^{\prime}}(\mathbf{Q}^{\triangle})$ and the
polarisation $\hat{\Pi}_{+}^{l,l^{\prime}}(\mathbf{Q}^{\triangle})$ are not
diagonal because of the lower two-fold symmetry at $\mathbf{Q}^{\triangle}$.
Mixing between different nearest-neighbor form factors is now allowed because
they fall into the same irreducible representations of the little group at
$\mathbf{Q}^{\triangle}$. For example, if $Q^{\triangle}=M_{1}$ mixing can
occur between $s$, $s^{\prime}$\- and $d_{x^{2}-y^{2}}$\- and between
$f_{y(y^{2}-3x^{2})}$\- and $p_{y}$-basis functions. To analyse if an
instability occurs in the ph-d channel, we determine the eigenvalues and
-functions of $\hat{V}_{\text{ph-d}}^{0}\cdot\hat{\Pi}_{+}$. We find three
attractive channels with negative eigenvalues as shown in Fig. 5. They
correspond to $p$\- and extended $s$-wave form factors for the little group,
which we obtain through linear combinations of the nearest-neighbor harmonics
of the triangular lattice (see App. A). The strongest attraction occurs in the
$\tilde{p}_{y-}$ channel

$\displaystyle\tilde{p}_{y-}$
$\displaystyle=\frac{1}{\sqrt{3}}(p_{y}-\sqrt{2}f)$
$\displaystyle=\sin(\mathbf{a}_{2}\cdot\mathbf{k})-\sin(\mathbf{a}_{3}\cdot\mathbf{k}$
(37)

An analogous analysis applies to $Q^{\triangle}\in\\{M_{2},M_{3}\\}$, which
can be obtained from the expressions for $M_{1}$ by rotation of $\pm\pi/3$.
This suggests an instability towards a bond density wave described by three
possible order parameters $\Delta(M_{1}),\Delta(M_{2}),\Delta(M_{3})$ with

$\displaystyle\Delta(M_{1})=\sum_{k,\sigma}[\sin(\mathbf{a}_{2}\cdot\mathbf{k})-\sin(\mathbf{a}_{3}\cdot\mathbf{k})]c_{\mathbf{k}+\mathbf{Q}\sigma}^{\dagger}c_{\mathbf{k}\sigma^{\prime}}\,$
(38)

and analogously for $\Delta(M_{2}),\Delta(M_{3})$. The resulting bond density
wave for $\Delta(M_{1})\neq 0$ has a zig-zag pattern that breaks inversion
with alternating bonds along $\mathbf{a}_{2}$ and $\mathbf{a}_{3}$. The
pattern in the ground state depends on the linear combination of all three
$\Delta(M_{i})$.

Figure 5: Triangular lattice. Top: Eigenvalues of
$\hat{V}_{\text{ph-d}}^{0}(M_{1})$ including Kohn-Luttinger-like corrections
expanded in nearest-neighbor lattice harmonics
$l\in\\{s,s^{\prime},p_{x},p_{y},d_{x,y},d_{x^{2}-y^{2}},f_{y(y^{2}-3x^{2})}\\}$
as function of the chemical potential $\mu$ at temperature $T=0.05/t$. The
eigenvectors are linear combinations of the lattice harmonics and labeled
according to the lower symmetry at $\mathbf{Q}^{\triangle}$ (see App. A for
explicit expressions). Middle: Eigenvalues of
$\hat{V}_{\text{ph-d}}^{0}(M_{1})\hat{\Pi}_{+}(M_{1})$ which enter the direct
particle-hole channel, cf. Eq. (21), i.e., the most negative one determines
the CDW instability. Bottom: eigenvalues of
$\hat{V}_{\text{ph-d}}^{0}(M_{1})\hat{\Pi}_{+}(M_{1})$ as a function of
temperature $T/t$ at $\mu=\mu_{VH}$.

###  III.3 Spin-fluctuation approach

Away from large $N_{f}$, diagrams beyond the KL-like approach need to be
included. We discussed two options to do so in Sec. II.3, a spin-fluctuation
and a functional renormalisation group (FRG) approach. In the spin-fluctuation
approach, the effective interaction in pp and ph-d channels are related via
$\hat{V}_{\mathrm{ph,d}}^{0,l}=\pm\hat{V}_{\mathrm{pp}}^{0,l}(0)$ (both are
diagonal in $l$). Thus, we can utilise earlier studies of the spin-fluctuation
mechanism for pairing in the square and triangular lattice [75, 76, 77] to
extract the effective interaction in the ph-d channel. We conclude that an
instability towards a $d_{x^{2}-y^{2}}$ CDW occurs on the square lattice as
$d_{x^{2}-y^{2}}$ superconductivity has been established as the leading
pairing instability there [2].

The resulting state of broken symmetry describes a loop-current order with a
staggered-flux pattern on neighboring plaquettes. This is also consistent with
PRG and FRG studies, which find the staggered-flux state for $N_{f}\geq 4$
[36] and $N_{f}\geq 6$ [63], respectively. On the triangular lattice, we can
infer an attraction in the $d$-wave CDWs channel based on earlier analyses
[77]. We note that the $f$-wave pairing state which is competitive on the
triangular lattice, becomes repulsive in the CDW channel because of the
additional minus sign for odd harmonics in the relation between the couplings
in pairing and ph-d channel. This suggests that an instability towards a
$d$-wave CDW can occur. However, for a proper treatment, the eigenvalues of
$\hat{V}_{\mathrm{ph-d}}^{0}\hat{\Pi}_{+}$ need to be determined (see next
Sec. III.4).

The $d$-wave CDW state can leads to various loop current and flux patterns
depending on the combination of the three degenerate CDW with wave vectors
$M_{1},M_{2},M_{3}$. For a single $M$ point, an alternating current pattern
along two directions of the triangular lattice is formed, and for triple-M
order a quantum anomalous Hall state that quadruples the unit cell appears
[9].

###  III.4 Instabilities without nesting

If we depart from a weak-coupling analysis, unconventional CDWs can arise as
Stoner instabilities for large $N_{f}$ and appropriate bare momentum-dependent
interactions even without nesting. As an example, we consider a
phenomenological $t-U-J$ model on the square and triangular lattice
$\mathcal{H}\rightarrow\mathcal{H}_{0}+\mathcal{V}+\mathcal{H}_{J}$, where

$\displaystyle\mathcal{H}_{J}=J\sum_{\langle
i,j\rangle}\sum_{\alpha\ldots\delta}\sum_{n}(c_{i\alpha}^{\dagger}T^{n}_{\alpha\beta}c_{i\beta})(c_{j\gamma}^{\dagger}T^{n}_{\gamma\delta}c_{j\delta})\,,$
(39)

with the SU($N_{f}$) generators $T^{n}$ we used previously in Eq. (10) and
$\mathcal{H}_{0}$, $\mathcal{V}$ are given in Eqs. (30) and (31).

After a Fourier transform, we expand the exchange interaction into form
factors in the pairing $U_{\mathrm{pp}}$ ($k_{1}=-k_{2}=k,k_{3}=k^{\prime}$),
ph-c $U_{\mathrm{ph-c}}$ ($k_{1}=k,k_{2}=k^{\prime}+Q,k_{3}=k+Q$) and ph-d
$U_{\mathrm{ph-d}}$ ($k_{1}=k,k_{2}=k^{\prime}+Q,k_{3}=k^{\prime}$) channels.
Explicitly, on the square lattice, we then obtain

$\displaystyle U_{\mathrm{pp}}(\mathbf{k},\mathbf{k}^{\prime},0)=U$
$\displaystyle-\frac{J}{2}\left(1+\frac{1}{N_{f}}\right)[s^{\prime}(\mathbf{k})s^{\prime}(\mathbf{k^{\prime}})+d_{x^{2}-y^{2}}(\mathbf{k})d_{x^{2}-y^{2}}(\mathbf{k^{\prime}})]$
$\displaystyle-\frac{J}{2}\left(1-\frac{1}{N_{f}}\right)[p_{x}(\mathbf{k})p_{x}(\mathbf{k^{\prime}})+p_{y}(\mathbf{k})p_{y}(\mathbf{k^{\prime}})]\,,$
(40) $\displaystyle
U_{\mathrm{ph-c}}(\mathbf{k},\mathbf{k}^{\prime},\mathbf{Q})=U-\frac{J}{4}s^{\prime}(\mathbf{Q})$
$\displaystyle-\frac{J}{2N_{f}}[s^{\prime}(\mathbf{k})s^{\prime}(\mathbf{k^{\prime}})+p_{x}(\mathbf{k})p_{x}(\mathbf{k^{\prime}})+p_{y}(\mathbf{k})p_{y}(\mathbf{k^{\prime}})$
$\displaystyle\qquad\quad+d_{x^{2}-y^{2}}(\mathbf{k})d_{x^{2}-y^{2}}(\mathbf{k^{\prime}})]\,,$
(41) $\displaystyle
U_{\mathrm{ph-d}}(\mathbf{k},\mathbf{k}^{\prime},\mathbf{Q})=U-\frac{J}{4N_{f}}s^{\prime}(\mathbf{Q})$
$\displaystyle-\frac{J}{2}[s^{\prime}(\mathbf{k})s^{\prime}(\mathbf{k^{\prime}})+p_{x}(\mathbf{k})p_{x}(\mathbf{k^{\prime}})+p_{y}(\mathbf{k})p_{y}(\mathbf{k^{\prime}})$
$\displaystyle\qquad+d_{x^{2}-y^{2}}(\mathbf{k})d_{x^{2}-y^{2}}(\mathbf{k^{\prime}})]\,,$
(42)

and on the triangular lattice, we find

$\displaystyle U_{\mathrm{pp}}(\mathbf{k},\mathbf{k}^{\prime},0)=U$
$\displaystyle-\frac{J}{2}\left(1+\frac{1}{N_{f}}\right)[s^{\prime}(\mathbf{k})s^{\prime}(\mathbf{k^{\prime}})+d_{xy}(\mathbf{k})d_{xy}(\mathbf{k^{\prime}})$
$\displaystyle\hskip
71.13188pt+d_{x^{2}-y^{2}}(\mathbf{k})d_{x^{2}-y^{2}}(\mathbf{k^{\prime}})]$
$\displaystyle-\frac{J}{2}\left(1-\frac{1}{N_{f}}\right)[p_{x}(\mathbf{k})p_{x}(\mathbf{k^{\prime}})+p_{y}(\mathbf{k})p_{y}(\mathbf{k^{\prime}})$
$\displaystyle\hskip
71.13188pt+f_{y(y^{2}-3x^{2})}(\mathbf{k})f_{y(y^{2}-3x^{2})}(\mathbf{k^{\prime}})]\,,$
(43) $\displaystyle
U_{\mathrm{ph-c}}(\mathbf{k},\mathbf{k}^{\prime},\mathbf{Q})=U-\sqrt{\frac{3}{2}}Js^{\prime}(\mathbf{Q})$
$\displaystyle-\frac{J}{2N_{f}}[s^{\prime}(\mathbf{k})s^{\prime}(\mathbf{k^{\prime}})+p_{x}(\mathbf{k})p_{x}(\mathbf{k^{\prime}})+p_{y}(\mathbf{k})p_{y}(\mathbf{k^{\prime}})$
$\displaystyle\qquad\quad+d_{x^{2}-y^{2}}(\mathbf{k})d_{x^{2}-y^{2}}(\mathbf{k^{\prime}})+d_{xy}(\mathbf{k})d_{xy}(\mathbf{k^{\prime}})$
$\displaystyle\qquad\quad+f_{y(y^{2}-3x^{2})}(\mathbf{k})f_{y(y^{2}-3x^{2})}(\mathbf{k^{\prime}})]\,,$
(44) $\displaystyle
U_{\mathrm{ph-d}}(\mathbf{k},\mathbf{k}^{\prime},\mathbf{Q})=U-\sqrt{\frac{3}{2}}\frac{J}{N_{f}}s^{\prime}(\mathbf{Q})$
$\displaystyle-\frac{J}{2}[s^{\prime}(\mathbf{k})s^{\prime}(\mathbf{k^{\prime}})+p_{x}(\mathbf{k})p_{x}(\mathbf{k^{\prime}})+p_{y}(\mathbf{k})p_{y}(\mathbf{k^{\prime}})$
$\displaystyle\qquad+d_{x^{2}-y^{2}}(\mathbf{k})d_{x^{2}-y^{2}}(\mathbf{k^{\prime}})+d_{xy}(\mathbf{k})d_{xy}(\mathbf{k^{\prime}})$
$\displaystyle\qquad+f_{y(y^{2}-3x^{2})}(\mathbf{k})f_{y(y^{2}-3x^{2})}(\mathbf{k^{\prime}})]\,.$
(45)

At these expressions, we can read off the components of the bare interaction
in the form-factor expansion, which are all diagonal
$\hat{U}^{l,l^{\prime}}_{X}=\hat{U}^{l}_{X}\delta_{l,l^{\prime}}$, $X\in$
{pp,ph-c,ph-d}. We see that the bare interaction is attractive in the
($s$-wave) spin channel, the pairing channel, and the charge channel for all
nearest-neighbor harmonics. We note in passing that the instability in the
spin channel is determined by the maximum value of
$[U-c_{i}Js^{\prime}(\mathbf{Q})]\Pi_{+}(\mathbf{Q})$ ($c_{\square}=1/4$,
$c_{\triangle}=\sqrt{3/2}$), which leads to the expected SDWs with
$\mathbf{Q}^{\square}$ or $\mathbf{Q}^{\triangle}$. Similarly, the pairing
instability with the largest critical temperature is a $d$-wave, because it
has a larger interaction and a larger $\Pi_{-}^{l}(0)$ than other harmonics.
It generally also depends on the filling, which of the spin, pairing, or
charge instabilities wins. However, for a large enough value of $N_{f}$, we
can ensure that a charge instability wins the competition. Its type is
determined by the singularities of

$\displaystyle\hat{V}^{l,l^{\prime}}_{\text{ph-d}}(\mathbf{Q})=-2\left[\frac{1}{\mathbb{1}+N_{f}\hat{U}_{\mathrm{ph-d}}\hat{\Pi}_{+}(\mathbf{Q})}\right]_{ll^{\prime}}\hat{U}_{\mathrm{ph-d}}^{l^{\prime}}\,$
(46)

Figure 6: Electronic polarisation at various fillings for the triangular
lattice. Top: bare particle-hole polarisation $\Pi_{+}(\mathbf{k})$ at
$\mu=1.6t$ (left), $\mu_{\mathrm{VH}}$ (middle), and $\mu=2.4t$ (right). The
position of the largest peak shifts from $M$ to $K/4$. Bottom left: The
position of $Q$ which maximises the polarisation $\Pi_{+}$.  Figure 7: Most
negative eigenvalue of
$\hat{V}_{\mathrm{ph-d}}^{ln}(Q)\hat{\Pi}_{+}^{nl^{\prime}}(Q)$ for $Q=M$
(blue) and $Q=K/4$ (red), which determines the uCDW instability. The
eigenfunctions have different $d$-wave forms as described in the Sec. III.4.
The crossing implies a phase transition where the wave vector changes from $M$
to $K/4$.

On the square lattice, we find that the leading instability at Van Hove
filling occurs for $\mathbf{Q}=\mathbf{Q}^{\square}$, where
$\Pi^{l,l^{\prime}}_{+}(\mathbf{Q})$ is diagonal and largest for
$l=d_{x^{2}-y^{2}}$. This is again the staggered flux state and reproduces
earlier analyses for large $N_{f}$ [64, 65].

Similarly, the leading instability on the triangular lattice at Van Hove
filling is a $d$-wave CDW at $\mathbf{Q}=\mathbf{Q}^{\triangle}$, where
$\mathbf{Q}^{\triangle}$ is one of the $M$ points. For a given $M$ point, the
unconventional CDW has a unique $d$-wave form factor. To determine it, we
diagonalise $\hat{U}_{\mathrm{ph}}\hat{\Pi}_{+}(\mathbf{Q})$, which yields a
pure $d_{xy}$ at $M_{1}$, and the appropriately rotated superposition
$\cos(\pi/3)d_{x^{2}-y^{2}}\pm\sin(\pi/3)d_{xy}$ at $M_{2,3}$. This is the
alternating or loop current state for single-/triple-M order we mentioned in
the previous section.

Changing the chemical potential slightly away from Van Hove doping, the wave
vector where the polarisation $\Pi_{+}(\mathbf{Q})$ peaks moves away from
$\mathbf{Q}^{\square}$ or $\mathbf{Q}^{\triangle}$ and becomes incommensurate.
Interestingly, in the triangular lattice, we find that not too far away from
Van Hove filling when $\mu<\mu_{c1}\approx 1.8t$ and $\mu>\mu_{c2}\approx
2.1$, the global maximum of $\Pi_{+}(\mathbf{Q})$ abruptly changes from
$\mathbf{Q}\approx\mathbf{Q}^{\triangle}$ to $\mathbf{Q}\approx K/4$ and
symmetry equivalent points, see Fig. 6. An analogous change does not occur on
the square lattice. The wave vector of $K/4$ approximately connects the locus
of Fermi surface pockets for $\mu>\mu_{c2}$. For $\mu<\mu_{c1}$, $K/4$
corresponds to the wave vector that connects points on the Fermi surface
closest to the Van Hove points back-folded into the first Brillouin zone.

The change of the peak position of the polarisation $\Pi_{+}$ implies that
there is a phase transition as function of chemical potential where the type
of the uCDW changes. We determine the instabilities of
$\hat{V}_{\mathrm{ph}}^{l,l^{\prime}}(K/4)$ for this case. It depends on the
chemical potential $\mu$, in which angular-momentum channel the leading
instability occurs. We find that it is predominantly of $d$-wave form for
$\mu<\mu_{c1}$ and of $f$-wave form for $\mu>\mu_{c2}$. Similar to the
$M$-point instability, the $d$-wave form factors at the six different K/4
points are related by $\pi/3$ rotation. For $K_{1}/4=(0,\pi/3)$ and
$\mu<\mu_{c1}$, we find a predominant $d_{xy}$ symmetry. This $d$-wave CDW
with wave-vector K/4 overtakes the $d$-wave CDW with wave-vector M at
$\mu\lesssim\mu_{{}_{c}1}$ whereas the $f$-wave CDW with wave vector K/4 wins
for $\mu\gtrsim\mu_{c2}$ as we show in Fig. 7.

To show that the uCDW with wave vector K/4 can win against pairing for
realistic $N_{f}$ and upon inclusion of all channels, we perform an FRG
calculation for the $t-U-J$ model on the triangular lattice with $N_{f}=4$.
Within the FRG truncation we employ here, the effective vertex is obtained as
the solution of the differential equation

$\displaystyle\partial_{T}V=\tau^{T}_{\mathrm{pp}}+\tau^{T}_{\mathrm{ph-c}}+\tau^{T}_{\mathrm{ph-d}}\,,$
(47)

where temperature is used as the flow parameter and $\tau^{T}_{X}$ are given
by the expressions in Eqs. (4)-(6) with the replacement
$\mathcal{G}(k)\rightarrow\mathcal{G}^{T}(k)=\sqrt{T}/(i\omega-\xi_{\mathbf{k}})$
[78]. The differential form incorporates an all-channel-to-all-channel
feedback, i.e., this analysis goes beyond the single channel resummation of
Eq. (46). The momentum dependence is resolved via a so-called truncated-unity
scheme, which also relies on a form-factor expansion of the effective vertex
in the pp, ph-c, and ph-d channels [79]. The details of our implementation are
described in Ref. [80].

We show the results of the FRG analysis for $U=3t$ and $J=0.4t$ in Fig. 8. We
note in passing that we do not obtain a uCDW instability for $N_{f}=4$ and
$J=0$ on the triangular lattice. For non-zero $J$ and varying chemical
potential, we find three different instabilities. At and around the immediate
vicinity of Van Hove filling $\mu_{\mathrm{VH}}=2t$, the leading instability
occurs in the ph-d channel with wave vector $M_{i}$ signaling a CDW formation.
To determine the form factor, we diagonalise the effective ph-d vertex
$V_{\mathrm{ph-d}}(k,k^{\prime},M_{i})$ (see Fig. 8), which yields the same
$d$-wave form factors that rotate with the $M_{i}$ as described above. When
the chemical potential deviates further from $\mu_{\mathrm{VH}}$, we find a
transition from the $d$-wave uCDW towards a pairing instability.
Diagonalisation of the FRG pairing vertex gives degenerate pairing solutions
with $d_{xy}$ and $d_{x^{2}-y^{2}}$ symmetry. For $1.75\leq\mu\leq 1.77$, the
pairing instability is interrupted, and instead we obtain another instability
in the ph-d channel, but with wave vector $K/4$. The form factors we extract
from $V_{\mathrm{ph-d}}(k,k^{\prime},K/4)$ in this case (see Fig. 8) also have
a $d$-wave form that matches with the RPA analysis and rotate with the six
different choices for $K/4$.

Figure 8: FRG results for SU(4) t-U-J model on the triangular lattice. Top:
critical temperature $T_{c}$ of the instability as function of $\mu$ for
$U/t=3.0$ and $J/t=0.4$. There are three different instabilities towards
charge (orange) and pairing order (blue). The charge order corresponds to
$d$-wave CDWs with wave vector $M$ around $\mu=2t$ and wave vector $K/4$
around $\mu=1.75t$. Bottom: symmetry of the uCDW (left and middle) and pairing
eigenfunctions (right) extracted from the singular FRG vertex.

##  IV Conclusion

In summary, we have put forward a mechanism for the formation of interaction-
induced unconventional CDWs in metals, resulting from the condensation of
particle-hole pairs with non-zero relative angular momentum. The generic
electron-based mechanism works for repulsive Hubbard and (screened) Coulomb
interactions and is reminiscent of the Kohn-Luttinger mechanism for
superconductivity. More specifically, it takes into account one-loop pp and
ph-c corrections in the bare interaction for the direct ph channel. This can
generate attractive components in the effective interaction of the ph-d
channel, leading to a charge-ordering instability with finite wave vector when
(approximate) nesting is operative.

Interestingly, this mechanism typically occurs in a regime with a strong
competition of various instabilities, i.e., SDW, CDW, and pairing. Therefore,
it is generally quite challenging to identify the leading ordering tendency
and, in principle, all these channels need to be taken into account on equal
footing, using an unbiased many-body approach, e.g., FRG or PRG. However, the
dominance of the Kohn-Luttinger-like uCDW formation can be controlled by going
to a large number of fermion flavors, which can be realized in materials
platforms with, e.g., spin, orbital, and/or valley degrees of freedom.

For an application of the Kohn-Luttinger-like mechanism, we discuss the
examples of square- and triangular-lattice Hubbard models at (approximate)
nesting. We numerically evaluate the effective interaction in an expansion of
lattice harmonics and show the emergence of attractive uCDW components for
both, the square and the triangular lattice. Our analysis suggests the
formation of different types of $p$-wave charge density waves, classifying as
loop-current or bond-ordered states.

Further, we phenomenologically explore effects beyond weak coupling at and
away from Van Hove filling on both lattices by adding exchange interactions.
We find $d$-wave charge density waves with wave vector corresponding to the
nesting vector as leading instabilities at Van Hove filling. On the square
lattice these correspond to the staggered flux state and on the triangular
lattice they are loop current states, which can form an interaction-induced
topological CDW phase. We also argue that phase transitions towards other CDWs
with wave vector K/4 occur on the triangular lattice on both sides of Van Hove
filling. They are towards another $d$-wave CDW for smaller $\mu$ and towards
an $f$-wave CDW for larger $\mu$. We employ an FRG approach as an independent
method to confirm this result for $N_{f}=4$, demonstrating that the $K/4$ CDW
can win the competition of orders against pairing.

In future studies, it will be interesting to extend our formalism to multi-
flavour systems relaxing SU($N_{f}$) symmetry for broader applications. We
expect that charge or spin density waves with wave vector $K/4$ can be
possible on other hexagonal lattices. Since the effect is based on the
behavior of the electronic polarisation, an ordinary charge density wave with
wave vector $K/4$ can also occur if the coupling in the $s$-wave charge
channel becomes attractive.

## Acknowledgments

We thank Andrey Chubukov, Lorenzo Del Re, Nico Gneist, and Nikolaos Parthenios
for helpful discussions. MMS acknowledges funding from the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) within Project-ID
277146847, SFB 1238 (project C02), and the DFG Heisenberg programme (Project-
ID 452976698).

##  Appendix A Form factors

For concreteness, we list the form factors we used explicitly. For the square
lattice, they are

  * •

The on-site form factor $f_{s}(\mathbf{k})=1$

  * •

The nearest neighbor components:

$\displaystyle f_{s^{\prime}}(\mathbf{k})$
$\displaystyle=\cos(k_{x})+\cos(k_{y})$ $\displaystyle f_{p_{x}}(\mathbf{k})$
$\displaystyle=\sqrt{2}\sin(k_{x})$ $\displaystyle f_{p_{y}}(\mathbf{k})$
$\displaystyle=\sqrt{2}\sin(k_{y})$ $\displaystyle
f_{d_{x^{2}-y^{2}}}(\mathbf{k})$ $\displaystyle=\cos(k_{x})-\cos(k_{y})$

  * •

The next-to-nearest neighbor components:

$\displaystyle f_{2s^{\prime}}(\mathbf{k})$
$\displaystyle=\cos(k_{x}+k_{y})+\cos(k_{x}-k_{y})$ $\displaystyle
f_{2p_{x+y}}(\mathbf{k})$ $\displaystyle=\sqrt{2}\sin(k_{x}+k_{y})$
$\displaystyle f_{2p_{x-y}}(\mathbf{k})$
$\displaystyle=\sqrt{2}\sin(k_{x}-k_{y})$ $\displaystyle
f_{2d_{xy}}(\mathbf{k})$ $\displaystyle=\cos(k_{x}+k_{y})-\cos(k_{x}-k_{y})$

And for the triangular lattice

  * •

The on-site form factor $f_{s}(\mathbf{k})=1$

  * •

The nearest neighbor components:

$\displaystyle f_{s^{\prime}}(\mathbf{k})$
$\displaystyle=\sqrt{\frac{2}{3}}(\cos(k_{y})+2\cos(\frac{k_{y}}{2})\cos(\frac{\sqrt{3}k_{x}}{2}))$
$\displaystyle f_{f_{1}}(\mathbf{k})$
$\displaystyle=\sqrt{\frac{8}{3}}(\cos(\frac{k_{y}}{2})-\cos(\frac{\sqrt{3}k_{x}}{2}))\sin(\frac{k_{y}}{2})$
$\displaystyle f_{p_{x}}(\mathbf{k})$
$\displaystyle=\sqrt{\frac{4}{3}}(2\cos(\frac{k_{y}}{2})+\cos(\frac{\sqrt{3}k_{x}}{2}))\sin(\frac{k_{y}}{2})$
$\displaystyle f_{p_{y}}(\mathbf{k})$
$\displaystyle=2\cos(\frac{k_{y}}{2})\sin(\frac{\sqrt{3}k_{x}}{2})$
$\displaystyle f_{d_{x^{2}-y^{2}}}(\mathbf{k})$
$\displaystyle=\sqrt{\frac{4}{3}}(\cos(k_{y})-\cos(\frac{k_{y}}{2})\cos(\frac{\sqrt{3}k_{x}}{2}))$
$\displaystyle f_{d_{xy}}(\mathbf{k})$
$\displaystyle=2\sin(\frac{k_{y}}{2})\sin(\frac{\sqrt{3}k_{x}}{2})$

  * •

The next-to-nearest neighbor components:

$\displaystyle f_{2s^{\prime}}(\mathbf{k})$
$\displaystyle=\sqrt{\frac{2}{3}}(2\cos(\frac{3k_{y}}{2})\cos(\frac{\sqrt{3}k_{x}}{2})+\cos(\sqrt{3}k_{x}))$
$\displaystyle f_{f_{2}}(\mathbf{k})$
$\displaystyle=\sqrt{\frac{8}{3}}(\cos(\frac{3k_{y}}{2})-\cos(\frac{\sqrt{3}k_{x}}{2})\sin(\frac{\sqrt{3}k_{x}}{2}))$
$\displaystyle f_{2p_{x+y}}(\mathbf{k})$
$\displaystyle=\frac{1}{\sqrt{3}}(\sin(\sqrt{3}k_{x})+\sin(\frac{\sqrt{3}(k_{y}-k_{x})}{2})$
$\displaystyle\quad\quad+2\sin(\frac{\sqrt{3}(k_{y}+k_{x})}{2}))$
$\displaystyle f_{2p_{x+y}}(\mathbf{k})$
$\displaystyle=\sqrt{\frac{8}{3}}(\cos(\frac{3k_{y}}{2})+\cos(\frac{\sqrt{3}k_{x}}{2})\sin(\frac{\sqrt{3}k_{x}}{2}))$
$\displaystyle f_{2d_{x^{2}-y^{2}}}(\mathbf{k})$
$\displaystyle=\sqrt{\frac{4}{3}}(\cos(\sqrt{3}k_{x})-\cos(\frac{3k_{y}}{2})\cos(\frac{\sqrt{3}k_{x}}{2}))$
$\displaystyle f_{2d_{xy}}(\mathbf{k})$
$\displaystyle=2\sin(\frac{3k_{y}}{2})\sin(\frac{\sqrt{3}k_{x}}{2})$ (48)

On the triangular lattice at $\mathbf{Q}^{\triangle}$, we need linear
combinations of these form factors which can be classified according to the
lower symmetry. The eigenfunctions of $\hat{V}(\mathbf{Q}^{\triangle})$ in
terms of the nearest-neighbor lattice harmonics are given by

$\displaystyle\tilde{s}=s$ (49)
$\displaystyle\tilde{s}_{-}^{\prime}=\frac{1}{\sqrt{3}}(\sqrt{2}s^{\prime}-d_{x^{2}-y^{2}})=\cos(\mathbf{a}_{2}\cdot\mathbf{k})-\cos(\mathbf{a}_{3}\cdot\mathbf{k})$
(50)
$\displaystyle\tilde{s}_{+}^{\prime}=\frac{1}{\sqrt{3}}(s^{\prime}+\sqrt{2}d_{x^{2}-y^{2}}))=\sqrt{2}\cos(\mathbf{a}_{1}\cdot\mathbf{k})$
(51)
$\displaystyle\tilde{p}_{y-}=\frac{1}{\sqrt{3}}(p_{y}-\sqrt{2}f)=\sin(\mathbf{a}_{2}\cdot\mathbf{k})-\sin(\mathbf{a}_{3}\cdot\mathbf{k})$
(52)
$\displaystyle\tilde{p}_{y+}=\frac{1}{\sqrt{3}}(\sqrt{2}p_{y}+f)=\sqrt{2}\sin(\mathbf{a}_{1}\cdot\mathbf{k})$
(53) $\displaystyle\tilde{p}_{x}=p_{x}$ (54)
$\displaystyle\tilde{d}_{xy}=d_{xy}\,.$ (55)

These also describe the eigenvectors of
$\hat{V}(\mathbf{Q}^{\triangle})\hat{\Pi}(\mathbf{Q}^{\triangle})$, only
$s\tilde{s}_{+}^{\prime}\rightarrow\tilde{s}_{+}^{\prime}=(3s+\sqrt{2}s^{\prime}+2d_{x^{2}-y^{2}}))/\sqrt{15}$.

##  Appendix B Decoupling according to lattice symmetry

In this section we show under which conditions the corrections in both, the
Kohn Luttinger mechanism for pairing Eq. (18) and the mechanism for uCDW Eq.
(24) become (block-)diagonal in the irreducible representations of the lattice
symmetry group. We start with the corrections originating in the ph channels.
We define

$\displaystyle{\Xi}^{l,l^{\prime}}_{\pm}\equiv\int_{\mathbf{k},\mathbf{k}^{\prime}}f_{l}(\mathbf{k}^{\prime}){\Pi}_{+}(\mathbf{k}\pm\mathbf{k}^{\prime})f_{l^{\prime}}^{\ast}(\mathbf{k})\,.$
(56)

Comparing the mechanisms, we find that
$\tau_{\mathrm{ph-c}}|_{\mathrm{ph-d}}\sim\tau_{\mathrm{ph-d}}|_{\mathrm{pp}}\sim\Xi^{l,l^{\prime}}_{-}$,
and that $\tau_{\mathrm{ph-c}}|_{\mathrm{pp}}\sim\Xi^{l,l^{\prime}}_{+}$
differs only by a sign between the arguments of the bubble $\hat{\Pi}_{+}$. We
can make use of the symmetry properties of the lattice harmonics we choose as
form factor basis, which can be distributed into the irreducible
representations of the lattice symmetry group. For clarity we change the way
we label the form factors $l\rightarrow(\Gamma_{l},\alpha)$ where $\Gamma$
labels the irreducible representation and $\alpha$ labels the basis function
within that irreducible representation, i.e., it runs from 1 to the dimension
$l_{n}$ of $\Gamma$. We now perform a group operation $\mathcal{R}$ which
belongs to the lattice group of our system

$\displaystyle{\Xi}^{l,l^{\prime}}_{\pm}(0)=\int_{\mathbf{k},\mathbf{k}^{\prime}}f_{\Gamma_{n},\alpha}(\mathcal{R}\mathbf{k}^{\prime}){\Pi}_{+}(\mathcal{R}(\mathbf{k}\pm\mathbf{k}^{\prime}))f_{{\Gamma_{n^{\prime}},\alpha^{\prime}}}^{\ast}(\mathcal{R}\mathbf{k})\,.$
(57)

The form factor $f_{\Gamma_{n},\alpha}(\mathcal{R}\mathbf{k})$ transforms like

$\displaystyle
f_{\Gamma_{n},\alpha}(\mathcal{R}\mathbf{k})=\sum_{j}g_{j\alpha}^{\Gamma_{n}}(\mathcal{R})f_{\Gamma_{n},j}(\mathbf{k}).$
(58)

We make use of this and since the particle-hole bubble $\Pi_{+}(\mathbf{k})$
is symmetric under such operations $\mathcal{R}$ we obtain

$\displaystyle{\Xi}^{l,l^{\prime}}_{\pm}(0)=\int_{\mathbf{k},\mathbf{k}^{\prime}}\sum_{j,j^{\prime}}$
$\displaystyle\ \
g_{j\alpha}^{\Gamma_{n}}(\mathcal{R})f_{\Gamma_{n},j}(\mathbf{k}^{\prime}){\Pi}_{+}(\mathbf{k}\pm\mathbf{k}^{\prime})g_{j^{\prime}\alpha^{\prime}}^{\Gamma_{n^{\prime}}}(\mathcal{R}^{-1})f^{\ast}_{\Gamma_{n^{\prime}},j^{\prime}}(\mathbf{k})$
(59)

Now we sum over all $N_{\Gamma}$ symmetry operations and use the
orthonormality theorem [81]

$\displaystyle{\Xi}^{l,l^{\prime}}_{\pm}(0)$
$\displaystyle=\int_{\mathbf{k},\mathbf{k}^{\prime}}\sum_{j,j^{\prime}}\frac{1}{N_{\Gamma}}\sum_{\mathcal{R}}g_{j\alpha}^{\Gamma_{n}}(\mathcal{R})g_{j^{\prime}\alpha^{\prime}}^{\Gamma_{n^{\prime}}}(\mathcal{R}^{-1})$
$\displaystyle\ \
f_{\Gamma_{n},j}(\mathbf{k}^{\prime}){\Pi}_{+}(\mathbf{k}\pm\mathbf{k}^{\prime})f^{\ast}_{\Gamma_{n^{\prime}},j^{\prime}}(\mathbf{k})$
$\displaystyle=\int_{\mathbf{k},\mathbf{k}^{\prime}}\sum_{j,j^{\prime}}\frac{1}{l_{n}}\delta_{\Gamma_{n},\Gamma_{n^{\prime}}}\delta_{j,j^{\prime}}\delta_{\alpha,\alpha^{\prime}}$
$\displaystyle\ \
f_{\Gamma_{n},j}(\mathbf{k}^{\prime}){\Pi}_{+}(\mathbf{k}\pm\mathbf{k}^{\prime})f^{\ast}_{\Gamma_{n^{\prime}},j^{\prime}}(\mathbf{k})$
(60)

At this expression, we see that $\hat{\Xi}^{l,l^{\prime}}_{\pm}(0)$ is of
diagonal structure. Now we look at the remaining correction

$\tau_{\mathrm{pp}}|_{\mathrm{ph-d}}^{l,l^{\prime}}(\mathbf{Q})=\int_{\mathbf{k},\mathbf{k}^{\prime}}f_{l}(\mathbf{k})f_{l^{\prime}}(\mathbf{k}^{\prime})\Pi_{-}(\mathbf{k}-\mathbf{k}^{\prime}+\mathbf{Q})$
(61)

with $\mathbf{Q}^{\square}=(\pm\pi,\pm\pi)^{t}$ and
$\mathbf{Q}^{\triangle}\in\\{M_{1},M_{2},M_{3}\\}$. Performing again a group
operation, we obtain

$\displaystyle\Pi_{-}(\mathcal{R}(\mathbf{k}-\mathbf{k}^{\prime})+\mathbf{Q})$
$\displaystyle=\Pi_{-}(\mathcal{R}(\mathbf{k}-\mathbf{k}^{\prime}+\mathcal{R}^{-1}\mathbf{Q}))$
$\displaystyle=\Pi_{-}(\mathbf{k}-\mathbf{k}^{\prime}+\mathcal{R}^{-1}\mathbf{Q})$
(62)

Here we have to distinguish the two lattices. In the square lattice,
$\mathbf{Q}^{\square}$ is invariant under the respective operations
$\mathcal{R}\in C_{4v}$. For the triangular lattice, this is not the case.
This means that we can apply the orthogonality theorem after analogous steps
as before for the square lattice, i.e.,
$V^{0,l,l^{\prime}}_{\mathrm{ph-d}}(\mathbf{Q}^{\square})$ is diagonal. For
the triangular lattice this does not work due to the pp fluctuations. In this
case, we can only state that even and odd form factors have no overlap and
their respective components in
$V^{0,l,l^{\prime}}_{\mathrm{ph-d}}(\mathbf{Q}^{\triangle})$ vanishes. This
argumentation holds analogously for $\hat{\Pi}^{l,l^{\prime}}_{\pm}$.

##  Appendix C Patch models

Figure 9: Patch models for square lattice (left) and tetragonal or hexagonal
systems (right) at Van Hove filling with nested Fermi surface. Van Hove points
are located inside the patches.

As a general analysis, we consider effective low-energy models for tetragonal
and hexagonal systems with a Van Hove singularity at the Fermi level and a
nested Fermi surface. The low-energy theory is restricted to momentum states
in patches around $N_{p}$ inequivalent Van Hove points $P_{i}$. For example,
this situation arises on the square ($N_{p}=2$), triangular, honeycomb, or
kagome ($N_{p}=3$) lattice with nearest-neighbor hopping, see Fig. 9. The non-
interacting part of the Hamiltonian is then given by

$\displaystyle
H_{0}=\sum_{p=1}^{N_{p}}\sum_{\sigma=1}^{N_{f}}\sum_{\mathbf{k}}(\epsilon_{p,\mathbf{k}}-\mu)c_{\mathbf{k},p,\sigma}^{\dagger}c_{\mathbf{k},p,\sigma}\,,$
(63)

with chemical potential $\mu$ and fermion annihilation (creation) operator
$c^{(\dagger)}_{\mathbf{k},p,\sigma}$ for patch $p$, flavour $\sigma$, and
momentum $\mathbf{k}$. The dispersion describes saddle points
$\epsilon_{1,\mathbf{k}}=ak_{x}^{2}-bk_{y}^{2}$ with $a,b>0$ and
$\epsilon_{p,\mathbf{k}}$ with $p>1$ is obtained from
$\epsilon_{1,\mathbf{k}}$ through rotation by $\pi/2$ for $N_{p}=2$ or
$\pm\pi/3$ for $N_{p}=3$. The theory is further equipped with all symmetry-
allowed, constant interactions between patches

$\displaystyle H_{V}$
$\displaystyle=\frac{1}{2}\sum_{\mathbf{k}_{1},\dots,\mathbf{k}_{4}}\sum_{\sigma,\sigma^{\prime}}\delta(\mathbf{k}_{1}+\mathbf{k}_{2}-\mathbf{k}_{3}-\mathbf{k}_{4})$
$\displaystyle\times\left(\sum_{p\neq
p^{\prime}}\left[\frac{g_{1}}{N_{p}^{2}}c^{\dagger}_{p\sigma}(\mathbf{k}_{1})c^{\dagger}_{p^{\prime}\sigma^{\prime}}(\mathbf{k}_{2})c_{p^{\prime}\sigma^{\prime}}(\mathbf{k}_{3})c_{p\sigma}(\mathbf{k}_{4})\right.\right.$
$\displaystyle+\frac{g_{2}}{N_{p}^{2}}c^{\dagger}_{p\sigma}(\mathbf{k}_{1})c^{\dagger}_{p^{\prime}\sigma^{\prime}}(\mathbf{k}_{2})c_{p\sigma^{\prime}}(\mathbf{k}_{3})c_{p^{\prime}\sigma}(\mathbf{k}_{4})$
$\displaystyle\left.+\frac{g_{3}}{N_{p}^{2}}c^{\dagger}_{p\sigma}(\mathbf{k}_{1})c^{\dagger}_{p\sigma^{\prime}}(\mathbf{k}_{2})c_{p^{\prime}\sigma^{\prime}}(\mathbf{k}_{3})c_{p^{\prime}\sigma}(\mathbf{k}_{4})\right]$
$\displaystyle\left.+\frac{g_{4}}{N_{p}}\sum_{p}c^{\dagger}_{p\sigma}(\mathbf{k}_{1})c^{\dagger}_{p\sigma^{\prime}}(\mathbf{k}_{2})c_{p\sigma^{\prime}}(\mathbf{k}_{3})c_{p\sigma}(\mathbf{k}_{4})\right)$
(64)

Low-energy effective theories based on $H_{0}+H_{V}$ were extensively studied
to analyse competing orders [82, 83, 84]. In particular, it was shown in a
parquet RG analysis that there is an instability towards imaginary CDW for
$N_{f}\geq 4$ [36]. Here, we apply the KL-like analysis of the main text to
the patch mode. We caution that the patch model with constant $g_{i}$ cannot
analyse odd-$l$ harmonics because the the momentum dependence around Van Hove
point at $P_{i}$ and $-P_{i}$ must be distinguished for this (this is
analogous to triplet pairing). We analyse the renormalisation of the effective
interaction
$V_{\text{ph,d}}(k,k^{\prime},Q)=V(\mathbf{k},\mathbf{k}^{\prime}+\mathbf{Q},\mathbf{k}^{\prime},\mathbf{k}+\mathbf{Q})$,
which drives the CDW instability for large $N_{f}$. In the patch model, there
are two such effective couplings $V_{2}=V_{\text{ph,d}}(P,P,Q)$ and
$V_{3}=V_{\text{ph,d}}(P,P^{\prime},Q)$, where $P\neq P^{\prime}$ are Van Hove
points and $Q=P-P^{\prime}$. On the bare level, $V^{0}_{2}=g_{2}$ and
$V^{0}_{3}=g_{3}$. Including pp, ph-c, and subleading ph-d corrections, we
obtain

$\displaystyle V^{0}_{2}$
$\displaystyle=g_{2}-(2g_{2}g_{4}+(N_{p}-2)g_{2}^{2})\Pi_{+}(0)$
$\displaystyle-2g_{1}g_{2}\Pi_{-}(\mathbf{Q})+2(g_{1}g_{2}+g_{3}^{2})\Pi_{+}(\mathbf{Q})$
$\displaystyle V^{0}_{3}$
$\displaystyle=g_{3}+(4g_{1}g_{3}+2g_{2}g_{3})\Pi_{+}(\mathbf{Q})$
$\displaystyle-(2g_{3}g_{4}+(N_{p}-2)g_{3}^{2})\Pi_{-}(0)\,.$ (65)

We use this as the initial interaction in the direct channel and sum the
corresponding RPA series. As a result, we obtain

$\displaystyle V_{\pm}=\frac{V_{\pm}^{0}}{1+N_{f}V_{\pm}^{0}\Pi_{+}(Q)}\,,$
(66)

where $V_{\pm}^{(0)}=V_{2}^{(0)}\pm V_{3}^{(0)}$. There can be an instability
if $V_{\pm}^{0}<0$. For a Hubbard-like interaction with all $g_{i}=U$, we find
$V_{+}^{0}=2U+U^{2}[N_{p}(\Pi_{+}(0)-\Pi_{-}(0))-2\Pi_{-}(Q)+10\Pi_{+}(Q)]$,
and $V_{-}=U^{2}[N_{p}(\Pi_{+}(0)+\Pi_{-}(0))-2\Pi_{-}(Q)-2\Pi_{+}(Q)]$. For
low temperatures and approximate nesting $\Pi_{-}(0)\gtrsim\Pi_{+}(Q)$, and we
can neglect other contributions in comparison. In that case, $V_{+}^{0}>0$.
For $V_{-}$, we observe a competition between ph and pp fluctuations in
$V_{-}^{0}\approx U^{2}[N_{p}\Pi_{-}(0)-2\Pi_{+}(Q)]$ and that larger numbers
of patches like in the hexagonal lattices $N_{p}\geq 3$ prevent an attractive
interaction on the one-loop level. The coupling $V_{-}^{0}$ can at best become
zero for perfect nesting $\Pi_{-}(0)=\Pi_{+}(Q)$ on the square lattice
$N_{p}=2$. However, including higher orders of the pp and ph-c ladders would
turn the sign of $V_{-}^{0}$ in favor of a CDW instability because the sign in
the pp channel alternates between even and odd-loop orders. Thus, while pp and
ph-c diagrams cancel for odd-loop orders they add for even-loop orders making
$V_{-}^{0}<0$. We comment on ways to go beyond a KL approach in the main text,
either via a spin fluctuation approach based on the proximity to an
instability in the ph-c channel due to perfect nesting, or via the FRG which
takes into account all channels on equal footing. We also emphasise again that
we cannot make a statement on odd-$l$ harmonics within this patch model using
momentum-independent interactions $g_{i}$. As discussed in the main text, the
KL mechanism gives an attractive coupling in the $p$-wave CDW channel on the
square and triangular lattice.

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