#  Quantitative analysis of interaction effects in generalized Aubry-André-
Harper models

Y.-T. Lin  Institut für Theorie der Statistischen Physik, RWTH Aachen
University and JARA—Fundamentals of Future Information Technology, 52056
Aachen, Germany  C. S. Weber  Institut für Theorie der Statistischen Physik,
RWTH Aachen University and JARA—Fundamentals of Future Information Technology,
52056 Aachen, Germany  D. M. Kennes  Institut für Theorie der Statistischen
Physik, RWTH Aachen University and JARA—Fundamentals of Future Information
Technology, 52056 Aachen, Germany  Max Planck Institute for the Structure and
Dynamics of Matter, Center for Free Electron Laser Science, 22761 Hamburg,
Germany  M. Pletyukhov  Institut für Theorie der Statistischen Physik, RWTH
Aachen University and JARA—Fundamentals of Future Information Technology,
52056 Aachen, Germany  H. Schoeller  Institut für Theorie der Statistischen
Physik, RWTH Aachen University and JARA—Fundamentals of Future Information
Technology, 52056 Aachen, Germany  V. Meden  Institut für Theorie der
Statistischen Physik, RWTH Aachen University and JARA—Fundamentals of Future
Information Technology, 52056 Aachen, Germany

###### Abstract

We present a quantitative analysis of two-particle interaction effects in
generalized, one-dimensional Aubry-André-Harper models with the Fermi energy
placed in one of the band gaps. We investigate systems with periodic as well
as open boundary conditions; for the latter focusing on the number of edge
states and the boundary charge. Both these observables are important for the
classification of noninteracting topological systems. In our first class of
models the unit cell structure stems from periodically modulated single-
particle parameters. In the second it results from the spatial modulation of
the two-particle interaction. For both types of models we find that the
single-particle band gaps are renormalized by the interaction in accordance
with expectations employing general field theoretical arguments. While
interaction induced effective edge states can be found in the local single-
particle spectral function close to a boundary, the characteristics of the
boundary charge are not modified by the interaction. This indicates that our
results for the Rice-Mele and Su-Schriefer-Heeger model [Phys. Rev. B 102,
085122 (2020)] are generic and can be found in lattice models with more
complex unit cells as well.

##  I Introduction

In the context of topological properties of fermionic systems, one-dimensional
(1D) lattice models with a rich structure of the unit cell have attracted a
tremendous interest in condensed matter physics [1, 2, 3, 4, 5, 6, 7, 8, 9,
10]. Besides edge states, i.e., bound states located at the boundary of the
system, also the investigation of the boundary charge accumulated at one end
of the system together with its fluctuations recently experienced a
significant revival [11, 12, 13, 14, 15, 16]. Corresponding models are
characterized by several bands which are separated by single-particle gaps.
The topological properties as well as the presence or absence of in-gap edge
states localized close to an open boundary can be tuned by changing the
single-particle parameters, i.e., the hopping matrix elements and the onsite
energies. Comparably simple models from this class are the Su-Schriefer-Heeger
(SSH) model [17, 18] and the Rice-Mele (RM) model [18, 19] with a unit cell
size of $Z=2$ lattice sites. Already shortly after these models were set up in
the early eighties effects of a complementary two-particle interaction were
investigated. This was mainly done using effective low-energy field theories
which are supposed to be applicable to more complex microscopic lattice models
as well [20, 21, 22, 23, 24].

We recently studied the spinless RM model and, as a special limit of this, the
SSH model with a homogeneous nearest-neighbor two-particle interaction
directly, i.e., without the approximate mapping to a low-energy continuum
field theory [25]. Employing an approximate functional renormalization group
(RG) [26, 27] approach and numerically exact density matrix renormalization
group (DMRG) [28, 29, 30, 31] we

  1. 1.

confirmed the low-energy power-law scaling of the renormalized gap as a
function of the bare one with an exponent which is interaction dependent.

This was revealed using general field theoretical arguments in the early
eighties [20, 21, 22] and recently reinvestigated employing continuum field
theories constructed as closely as possible to the microscopic model of
interest [32, 25, 15]. Based on these insights one expects that the power-law
renormalization of all band gaps is also found in microscopic models with more
complex unit cells and $Z\geq 2$.

With this sanity check passed, in Ref. 25 we went beyond established
interaction effects. We studied the decay of the inter unit cell density
oscillations of the interacting RM model away from an open boundary for a
Fermi energy placed within the gap. From this the associated boundary charge,
accumulated close to the boundary, can be computed and was investigated. We
finally computed the local spectral function. In Ref. [25] as well as here we
focus on the limit of vanishing temperature. We found

  1. 2.

that the density decay of the interacting gapped system remains dominantly
exponential (as it is the case for vanishing interaction) but that the
behavior of the pre-exponential function is altered by the two-particle
interaction and

  2. 3.

that the characteristics (see below) of the fractional part of the boundary
charge [33, 34, 35, 11, 12, 13, 15, 14, 36] is robust against interaction
effects and can fully be understood in terms of the (renormalized) bulk
properties, while

  3. 4.

the local spectral function close to the boundary might show an interaction
induced in-gap $\delta$-peak associated to an effective edge-state which does
not have a counterpart in the noninteracting limit.

It is remarkable that while the number of in-gap $\delta$ peaks of the
spectral function, indicating effective edge states, depends on the two-
particle interaction, the characteristics of the fractional part of the
boundary charge is robust. This is of interest in the context of topological
systems. It is known that in the noninteracting case topological bulk
invariants are related to the number or parity of zero-energy edge states
(bulk-boundary correspondence). These edge states are thus related to bulk
properties. We will refer to them as conventional edge-states. In the
interacting RM model the additional edge states instead originate from the
local modulation of the self-energy close to the open boundary which goes
beyond the intra unit cell structure. It follows from the interplay of the
boundary and the two-particle interaction as can conveniently be illustrated
using functional RG (see below). These edge states cannot be explained based
on (renormalized) bulk properties and we refer to them as unconventional edge
states.

The insights of Ref. [25] raise the question if the properties 2. to 4. are
specific to the interacting RM model or if they can be found in other, more
complex lattice models of the above class as well—we already commented on the
general expectation concerning the property 1.. To investigate this in a first
step we here study the generalized Aubry-André-Harper (AAH) model [15, 14, 11,
12, 13, 37, 38, 39, 40, 41, 42] with $Z=4$ and (homogeneous) nearest-neighbor
interaction at different fillings $f$ and the Fermi energy falling in one of
the $Z-1=3$ single-particle gaps. We confirm the properties 1. to 4. using
functional RG.

In a second step we pose another question: Is it conceivable to construct a
model with a homogeneous single-particle part—thus being gapless in the
noninteracting limit—which shows a similar phenomenology as described above
but rooted in a periodically modulated, local two-particle interaction?
Considering the cases of $Z=2$ and $Z=4$ periodicity and using functional RG
as well as DMRG we show that this is indeed possible. It can be understood as
follows. On the Hartree-Fock level the modulated two-particle interaction
generates a modulated onsite energy and/or modulated effective hopping, and
thus $Z-1$ (Hartree-Fock) gaps open. If the Fermi energy is chosen such that
it lies inside one of these gaps the RG procedure, which contains the Hartree-
Fock terms, but other diagrammatic contributions on top, leads to the effects
summarized in the above phenomenology. Within functional RG and for $Z=2$ this
can even be shown analytically.

However, there is an exception to this for $Z=4$. If one selects the Fermi
energy such that it lies in the gap associated to half filling the gap opening
is of second order in the interaction. As our approximate functional RG
procedure does not contain all terms to second order we cannot use it to study
this special case. Also DMRG is not the right tool to investigate the low-
energy physics in this case. For small interactions the gap, setting the low-
energy scale, is very small (of second order). It would thus require very
large systems inaccessible by DMRG to study the half-filled situation. We thus
leave the detailed study of this special case to future work.

The modulated interaction model was studied earlier in Ref. 43. Using DMRG and
field theory the authors provided evidence that due to the modulation of the
two-particle interaction a gap opens. However, their DMRG results for the
density of a finite system with two open boundaries were not obtained in the
ground state. The authors overlooked that with this method it is difficult to
discriminate between states which are almost degenerate, a situation naturally
arising in the presence of a zero-energy edge state (see below). In that case
the density the authors present is the one of a linear combination of the
ground and the first excited state.

Combining the results of the present paper with the ones of Ref. 25 provides
strong evidence that the phenomenological properties 1. to 4. hold quite
generally for 1D interacting lattice models with periodically modulated
single-particle parameters and, under certain conditions, similar effects can
also be found if the underlying noninteracting model has homogeneous
parameters but the local interaction is periodically modulated around a finite
average value.

We are not aware of any experimental realization of the 1d spinless modulated
interaction model so far. However, first steps towards the experimental
realization of extended, spinful models such as the Fermi-Hubbard model were
taken. One of the candidates is a semiconductor heterostructure leading to a
chain of gate-defined quantum dots, see Ref. [44]. Another possibility would
be a cold Fermi gas in an optical lattice. In the bosonic case a modulated
interaction model was e.g. realized in Ref. [45]. For possibilities to realize
noninteracting AAH models in various systems we refer to Refs. [40, 46, 39]

The rest of this paper is organized as follows. In the next section we present
the models we consider, give a brief account of the many-body methods
employed, and introduce the observables we study. Our results for the
interacting generalized AAH model are presented in Sect. III. Section IV is
devoted to the results obtained for the models with periodically modulated
interaction ($Z=2$ and $Z=4$). In Sect. V we summarize our results. The
functional RG flow equations are presented in Appendix A while Appendix B
contains the details of the analytical solution of the flow equations for the
modulated $U$ model with $Z=2$. In Appendix C we briefly discuss the model
with $Z=4$ at half filling and periodically modulated interaction.

We emphasize, that the present work should be viewed as a follow-up of Ref.
25, while still being self contained to an appropriate degree. We thus refrain
from presenting all the technical details and a comprehensive account of
earlier works; we here restrict ourselves to the ones which are of direct
relevance for our work. We furthermore only give a brief summary of the
behavior of the observables, i.e., the local single-particle spectral
function, the local density, and the boundary charge, in the noninteracting
limit; our focus is on the changes due to the two-particle interaction. For
further details we refer the reader to Refs. 25, 15, 14.

##  II Models, methods, and observables

###  II.1 Models

####  II.1.1 The interacting generalized Aubry-André-Harper model with $Z=4$

The first model we consider is the generalized 1D spinless AAH model with
uniform nearest-neighbor two-particle interaction. The noninteracting model is
given by the Hamiltonian

$\displaystyle H^{Z}_{\rm
AAH}=\sum_{j}\left(V_{j}n^{\phantom{}}_{j}-\left[t_{j}c_{j+1}^{\dagger}c_{j}^{\phantom{}}+\mbox{H.c.}\right]\right),$
(1)

where $c_{j}^{({\dagger})}$ is the second quantized annihilation (creation)
operator at the site $j$ and $n_{j}=c_{j}^{\dagger}c_{j}^{\phantom{}}$ the
local density operator. The on-site potential $V_{j}$ and the hopping
parameters $t_{j}$ are periodic with period $Z$, defining the number of
lattice sites of the unit cell,

$\displaystyle V_{j}=V\cos\left(\frac{2\pi j}{Z}+\varphi_{v}\right),$ (2)
$\displaystyle t_{j}=t+\delta t\cos\left(\frac{2\pi j}{Z}+\varphi_{t}\right).$
(3)

Here $V$ and $\delta t$ denote the amplitude of the modulation of the on-site
potential and hopping, respectively. Furthermore, $\varphi_{v}$ and
$\varphi_{t}$ are the phases of the corresponding modulation. For generic
parameters the noninteracting model has $Z-1$ band gaps of size
$2\Delta_{\nu}$, with $\nu=1,2,\ldots,Z-1$ [14].

We take $t$ as our unit of energy and set $t=1$. Moreover, we use $W=2t$ to
denote half of the bandwidth of the (gapless) model with $\delta t=V=0$.

The homogeneous two-particle interaction is assumed to be of nearest neighbor
form and given by

$\displaystyle H_{\rm
int}=U\sum_{j}\left(n^{\phantom{}}_{j}-\frac{1}{2}\right)\left(n^{\phantom{}}_{j+1}-\frac{1}{2}\right),$
(4)

with the (repulsive) interaction of strength $U\geq 0$.

We take the total number of lattice sites $N$ to be an integer multiple of the
period $Z$ such that all of the unit cells remain intact. One can rewrite the
index $j$ of the Wannier basis into a unit cell index $n$ and intra-cell site
index $i$

$\displaystyle j=Z(n-1)+i.$ (5)

To study the bulk properties of the system we consider periodic boundary
conditions (PBCs). In this case the site index $j$ in the sum of equations (1)
and (4) runs from $1$ to $N$ and we identify sites $N+1$ and $1$. To study the
boundary physics, we take open boundary conditions (OBCs). In this case the
sum in the above equations runs from $1$ to $N$ for the diagonal addends and
from $1$ to $N-1$ for the off-diagonal ones.

For $Z=2$, Eq. (1) is the well-known Rice-Mele model, which is one of the
basic models considered in the field of topological Fermi systems. In
particular, it shows edge-state physics. In an earlier publication [25] we
have provided a comprehensive study of interaction effects in this model. Our
focus was on directly tackling the microscopic lattice model using the
functional RG and DMRG. In addition, we used the mapping to field theories and
methods applicable to those such as bosonization. As an extension beyond $Z=2$
we consider the interacting generalized AAH model with period $Z=4$ in the
present paper. However, we already now note that our methods are applicable
for arbitrary $Z$.

####  II.1.2 A tight-binding model with periodically modulated interaction

In a second extension of our RM model study we investigate a tight-binding
model with uniform hopping and vanishing single-particle potential but a
periodically modulated nearest-neighbor interaction. We thus leave the class
of models in which the gap, the edge-state physics, and the topological
properties are present already in the noninteracting limit. Throughout the
paper we consider interactions sufficiently small such that the homogeneous
part does not lead to the opening of a correlation-induced gap (as it happens
for our interaction Eq. (4) at half filling for sufficiently large repulsive
interactions [47]). If any of the above phenomena 1. to 4. is found for the
modulated interaction model, it thus originates from the inhomogeneous (but
periodic) part of the two-particle interaction. The noninteracting Hamiltonian
is

$\displaystyle
H_{0}=-t\sum_{j}\left(c_{j+1}^{\dagger}c_{j}^{\phantom{}}+\mbox{h.c.}\right).$
(6)

The interaction is given by

$\displaystyle H^{\rm Z}_{\rm
int}=\sum_{j}U_{j}\left(n^{\phantom{}}_{j}-\frac{1}{2}\right)\left(n^{\phantom{}}_{j+1}-\frac{1}{2}\right),$
(7)

with

$\displaystyle U_{j}=U+\delta U\cos\left(\frac{2\pi j}{Z}+\varphi_{U}\right)$
(8)

where $\delta U$ is the amplitude of the modulation, $U\geq 0$ is the average
value, and $\varphi_{U}$ is the phase. We, in particular, consider $Z=2$, with
$U_{j}=U_{j+2}$ and $Z=4$ with $U_{j}=U_{j+4}$ for all $j$. The same
statements on the boundary conditions and the related summation limits as made
for the generalized AAH model hold.

###  II.2 Many-body methods

To solve the interacting models we employ the same methods as used in our
interacting RM model study [25], i.e., functional RG [26] and DMRG [28]. In
addition, we refer to field theoretical considerations employed to effective
low-energy models [20, 21, 22, 23, 24, 32, 15]. To avoid any doubling we refer
the reader interested in an introduction to the application of these methods
to (single-particle) gapped systems to Sect. III of Ref. 25. We here only
summarize the individual advantages and shortcomings of the three approaches.

Functional RG and DMRG can directly be applied to the microscopic lattice
models while the use of field theoretical tools requires the mapping to a
continuum model. This only holds in the low-energy limit and for sufficiently
weak two-particle interactions. However, within the field theory one can
obtain analytical insights. In limiting cases this is also possible within
functional RG. As we use the lowest-order truncated approach the functional RG
results are only controlled for small to intermediate two-particle
interactions. If properly executed, DMRG can be considered as numerically
exact. When being interested in boundary effects, we have to employ finite
size DMRG. Due to the growth of the entanglement entropy it is, however,
limited to system sizes of the order of $10^{3}$ lattice sites. The inverse
system size sets a lower bound for the smallest accessible energy. In
contrast, functional RG can be applied for very large systems allowing to
access the asymptotic low-energy limit. Importantly, the lowest-order
truncated functional RG was shown to be capable to capture the entire series
of leading logarithms for ungapped [49, 48] as well as (single-particle)
gapped [25] many fermion systems. This was crucial for the RM model [25] and
will turn out to be equally important in our present extensions.

To derive a closed set of functional RG flow equations for the self-energy and
the local density of our models we employed the same approximations and cutoff
procedure as discussed in Ref. 25. Due to the enlarged size of the unit cell
as compared to the $Z=2$ RM model studied in this paper, they become more
complex. For completeness we present the RG equations in real space in
Appendix A.1 and in momentum space in A.2. Both can easily be solved on a
computer and in limiting cases even analytically; see Appendix B.

The functional RG approach truncated to lowest order in the two-particle
interaction leads to an approximate self-energy which is frequency
independent. At the end of the RG flow one thus has to deal with an effective
Hamiltonian with single-particle parameters (hoppings and onsite energies)
which are renormalized by the two particle interaction. For periodic boundary
conditions the periodicity with period $Z$ is preserved. In the presence of
open boundaries the effective single-particle parameters are modulated beyond
the intra unit cell structure. This additional structure decays from the
boundary towards the bulk and follows from the interplay of the boundary and
the two-particle interaction. For large distances the renormalized bulk values
obtained for periodic boundary conditions are reached. The approximate
effective single-particle picture of the interacting problem helps to
interpret the results (see Ref. [25] and below).

To obtain the DMRG data for the spectral gap and the local density we used the
same approach and benchmarking as described in Sect. III.C of Ref. 25.

In Ref. 25 we have shown that to capture the physics of the microscopic
lattice model beyond leading order (in the two-particle interaction) it is
advantageous to construct a field theory directly from the model under
consideration and use additional information available such as, e.g., the
exact Bethe ansatz solution of the interacting homogeneous model. Here we do
not aim at this level of accuracy and instead use general field theoretical
arguments which do not contain any information on the underlying microscopic
model beyond the leading order ones. For our present purposes this is
sufficient.

###  II.3 Observables

We briefly introduce the observables we compute to investigate the effects of
the two-particle interaction. Besides the band gaps we study the local
particle density, the boundary charge, and the single-particle spectral
function.

We note that for our purposes the details of the analytical results which can
be obtained for the noninteracting $Z=4$ generalized AAH model are not
crucial. We thus refer the interested reader to Ref. [14] for a discussion of
those.

####  II.3.1 The gap

In the noninteracting limit the three single-particle band gaps of size
$2\Delta_{\nu}$, with index $\nu=1,2,3$, of the $Z=4$ generalized AAH model
are determined by the single-particle parameters $V$, $\delta t$,
$\varphi_{v}$, and $\varphi_{t}$. Here we are not interested in the details of
this dependence [15, 14] but rather on how the gap is modified by the two-
particle interaction if the Fermi energy is placed in one of the gaps. From
field-theoretical considerations [20, 21, 22, 23, 24, 32, 15], which are
independent of the details of the underlying lattice model, and our study of
the $Z=2$ RM model [25] we expect power-law scaling of the renormalized gap as
a function of the bare one. The exponent will depend on $U$. In addition, we
expect a dependence on the band filling $f=\nu/Z$ associated to the
corresponding index $\nu$ via the dependence on the Fermi momentum $k_{F}=\pi
f$ where the gap opens; quarter filling for $\nu=1$, half filling for $\nu=2$,
and three-quarter filling for $\nu=3$.

For the modulated interaction model the noninteracting limit is gapless.
However, the periodic modulation of the two-particle interaction might lead to
gaps. This should not be confused with the gap which at half-filling opens
even for homogeneous interactions if $U$ becomes sufficiently large [47]. To
study the possibility of a gap opening by DMRG we consider the difference of
the energy of the first excited state and the ground state energy for systems
of up to ${\mathcal{O}}(10^{3})$ sites. Within lowest-order truncated
functional RG we can employ the effective single-particle picture. If the
modulated interaction will lead to modulated effective single-particle
parameters, a gap might open. We can study this using the noninteracting
formulas introducing the renormalized parameters. The same can be done for the
$Z=4$ interacting AAH model. Significantly larger systems than in DMRG or even
the thermodynamic limit can be studied within functional RG (see below).

####  II.3.2 The local spectral function

The single-particle spectral function $A_{j}(\omega)$ is of particular
interest if we consider a system with open boundaries. It, on the one hand,
shows the distribution of spectral weight
$\left|\psi_{k}^{(\alpha)}(j)\right|^{2}$, given by the delocalized single-
particle wave functions $\psi_{k}^{(\alpha)}(j)$ over the band energies
$\varepsilon^{\alpha}_{k}$, with band index $\alpha$ and quasi-momentum $k$
for given site $j$. On the other hand possible edge states show up as in-gap
$\delta$-peaks at their energy $\varepsilon_{\rm e}$ with a weight
$\left|\psi_{\rm e}(j)\right|^{2}$ and wave function $\psi_{\rm e}(j)$. In the
noninteracting limit the eigenenergies and eigenstates can be obtained
straightforwardly by numerically diagonalizing the Hamiltonian for a large but
finite system (for analytical results, see Ref. [14]). The same can be done
with the effective Hamiltonian at the end of the RG flow leading to an
approximation of the local spectral function for $U>0$. This holds for the
interacting $Z=4$ generalized AAH model as well as for the modulated
interaction model at $Z=2$ and $Z=4$.

To obtain the full spectral information from DMRG is computationally rather
challenging and suffers from certain shortcomings [29, 30, 31], mainly a
limited energy resolution due to finite size effects, a finite bond dimension,
and/or an artificial broadening. We thus refrain from using DMRG to obtain
$A_{j}(\omega)$.

In Ref. [25] we have shown for the $Z=2$ RM model that besides the
conventional edge-states additional interaction induced unconventional edge
states appear in certain parts of the parameter space. They result from the
spatial modulation of the renormalized single-particle parameters beyond the
intra unit cell structure which decays from the boundaries towards the bulk.
We investigate if the same happens in the $Z=4$ generalized AAH model. For the
modulated interaction model both types of edge states, conventional and
unconventional ones, might be induced by the interaction. In the former case
the renormalized bulk parameters are such that the corresponding
noninteracting model with these parameters shows an edge state. For the
unconventional edge state this is not the case and they follow from the
interplay of the boundary and the interaction as described above.

####  II.3.3 The local density

For a noninteracting system it is straightforward to compute the local density
$\rho(j)$ for a large but finite system by exact diagonalization (for
analytical insights for the $Z=4$ generalized AAH model, see Ref. [14]). For
models of the present type it can have two contributions. One coming from the
filled bands $\rho_{\rm band}(j)$ and the other one coming from the filled
edge states $\rho_{\rm edge}(j)$. In the presence of an open boundary the
envelope modulating the intra unit cell density structure of $\rho_{\rm
band}(j)$ from the boundary towards the bulk decays exponentially if the Fermi
energy lies in one of the gaps. The decay length is given by the single-
particle parameters (for analytical results for the $Z=4$ generalized AAH
model, see Ref. [14]). The same holds for $\rho_{\rm edge}(j)$.

To obtain the density of interacting systems by functional RG we used two
approaches. One is to employ the effective single-particle picture and
diagonalize the effective Hamiltonian at the end of the RG flow. However, it
was earlier shown for metallic [49] as well as single-particle gapped models
[25] that writing down a flow equation for the density leads to improved
results. The corresponding flow equation is presented in Appendix A. We
therefore also here rely on this approach which can be used for systems of up
to $10^{6}$ lattice sites.

Within DMRG the ground state density is directly accessible and we can thus
use the numerically exact results for systems of the order of $10^{3}$ sites
to compare to the approximate functional RG ones.

For the $Z=2$ interacting RM model we have shown that the decay of the
envelope of the density remains exponential with a renormalized decay length
but that the pre-exponential function is altered by the interaction. We
investigate if the same holds for the interacting $Z=4$ generalized AAH model.

For the modulated interaction models the noninteracting density away from the
boundary decays generically as $1/j$. This is the decay of ordinary Friedel
oscillations of metallic systems in one spatial dimension [52]. However, at
half-filling the particle-hole symmetry prohibits Friedel oscillations and the
local density even in the presence of open boundaries is homogeneous [49] (and
equal to $1/2$). This also holds in the presence of the nearest-neighbor
interaction. We thus investigate the density of the $Z=4$ modulated
interaction model with the Fermi energy placed in the gaps with indices
$\nu=1,3$ corresponding to one- and three-quarter filling.

####  II.3.4 The boundary charge

In the presence of an open boundary, charge might be accumulated close to it.
This boundary charge $Q_{\rm B}$ can be computed as

$\displaystyle Q_{\rm
B}=\lim_{M\rightarrow\infty}\lim_{N\rightarrow\infty}\sum_{j=1}^{\infty}\left[\rho(j)-\bar{\rho}\right]f_{N,M}(j),$
(9)

where the bulk averaged particle density is given as

$\displaystyle\bar{\rho}=\frac{1}{Z}\sum^{Z}_{i=1}\rho_{\text{bulk}}(i),$ (10)

with $\rho_{\text{bulk}}(i)$ computed for periodic boundary conditions, and
$f_{N,M}=1-\theta_{MZ}(j-NZ)$ is an envelope function which defines the range
of the boundary on the scale $NZ$ and varies smoothly from unity to zero on
the scale $MZ$ (see also Fig. 3 of Ref. [15] for a sketch). Here,
$\theta_{\delta x}(x)$ denotes some representation of the $\theta$-function
with broadening $\delta x$. One needs to take $N\gg M\gg Z$ for the boundary
charge to become independent of $M$ and $N$. As described in the last
subsection $\rho(j)$ as well as $\rho_{\rm bulk}(i)$ are accessible by
functional RG and DMRG.

One of the most important universal properties of $Q_{\rm B}$ discussed in all
detail in Refs. [11, 12, 13, 14, 15, 16] is its transformation under a shift
of the lattice by one site towards the boundary, described by a change
$\varphi\rightarrow\varphi+2\pi/Z$ of all modulation phases $\varphi_{v}$,
$\varphi_{t}$, and $\varphi_{U}$. For generalized AAH models it was shown for
the noninteracting case in Refs. [13, 14] that the boundary charge can only
change by the average particle charge $f$ or the hole charge $f-1$ moved into
the boundary, with $f=\nu/Z$. Therefore, as function of
$\varphi=\varphi_{t}=\varphi_{v}$, one expects one of the two following
possibilities for the phase dependence of $Q_{\rm B}$

$\displaystyle Q_{\rm
B}=f(\varphi)+F(\varphi)+\frac{\varphi}{2\pi}\begin{cases}\nu\\\
\nu-Z\end{cases}\,,$ (11)

where $f(\varphi)=f(\varphi+2\pi/Z)$ is some smooth and periodic function, and
$F(\varphi)$ contains discrete jumps by unity when edge states cross the Fermi
energy. A similiar result is also expected in the presence of a modulated
interaction (with $\varphi=\varphi_{U}$) since, as shown below, the gap is
induced by an effective Hartree-Fock mechanism. For the generic case
(including interactions, several channels, and random disorder) it is expected
that $Q_{\rm B}$ will always change by $\nu/Z\,\text{mod}(1)$ when changing
the phase by $2\pi/Z$. This was motivated in Ref. [15] based on the
nearsightedness principle that charge correlations in insulators will decay
exponentially fast.

In addition, the universal properties of the boundary charge have also been
discussed within low-energy field-theoretical models for the case of small
gaps (including interactions via bosonization methods) [15, 16]. In this case
it was shown that the boundary charge can be written as

$\displaystyle Q_{\rm B}=\frac{\gamma_{\nu}}{2\pi}+\frac{\nu}{2Z}\,,$ (12)

where $\gamma_{\nu}$ denotes the phase of the gap parameter in gap $\nu$,
resulting from the resonant processes connecting the two Fermi points $\pm
k_{F}=\pm\pi f$. The relation of the phase $\gamma_{\nu}$ to the modulation
phase $\varphi$ has been analysed in all detail in Ref. [15] and is in general
quite non-trivial for half-filling $f=1/2$. Away from half-filling, one
obtains

$\displaystyle\gamma_{\nu}=\begin{cases}\nu\varphi+\text{const}&\text{for}\quad
f<1/2\\\ (\nu-Z)\varphi+\text{const}&\text{for}\quad f>1/2\end{cases}\,.$ (13)

Here, the constant part can often be fixed via Eq. (12) by special symmetry
points [15]. The universal result Eq. (12) shows that the boundary charge is
only sensitive to the phase variable $\gamma_{\nu}$ but not to the gap size in
the low-energy regime. For $Z=2$, this has been confirmed numerically via DMRG
and functional RG to hold in the interacting case as well [25, 16].

##  III Results for the interacting generalized AAH model with $Z=4$

We now present explicit results for the renormalized gap, the local spectral
function close to the boundary, the local density, and the boundary charge for
the interacting generalized AAH model with $Z=4$. As already emphasized, we
are only interested in the cases in which the Fermi energy lies within one of
the three gaps and the (noninteracting) system is a band insulator. Note that
the Fermi energy depends on the two-particle interaction $U$ and must thus be
chosen such that the target gap or the corresponding band filling is reached.
There is a unique relation between the band gap index $\nu$ and the band
filling $f$.

From effective low-energy field theories we expect that for all $\nu$ the
dependence of the renormalized gap $\Delta^{\rm ren}_{\nu}$ on the bare one
$\Delta_{\nu}$ is given by

$\displaystyle\frac{\Delta^{\text{ren}}_{\nu}}{\Delta_{\nu}}=\left(\frac{\Delta_{\nu}}{W}\right)^{\beta_{f}(U)}\text{
for }\Delta_{\nu}\ll W,$ (14)

with an interaction and filling dependent exponent $\beta_{f}(U)$.
Bosonization predicts that to leading order in the two-particle interaction
$\beta_{f}(U)=K(U,f)-1$ with the Tomonaga-Luttinger liquid parameter $K$ of
the underlying homogeneous model [50, 47, 51]. For the present case
$K(U,f)-1=-U[1-2\cos(2k_{\rm F})]/[2\pi\sin(k_{\rm F})]+{\mathcal{O}}(U^{2})$
and $k_{\rm F}=f\pi$ [49, 50, 47, 51].

We have earlier confirmed this for the microscopic $Z=2$ RM model at half-
filling (the single gap for $Z=2$ corresponds to half-filling) directly
without the approximate mapping to a field theory [25].

Figure 1: The $U$-dependence of the exponent of the power-law scaling of the
effective gap as a function of the bare one taken from functional RG data of
the interacting generalized AAH model with $Z=4$. Here $f$ is the filling. The
black dashed lines show the leading order exponent according to field theory.
Note that one-quarter and three-quarter filling lead to the same leading order
exponent. The lower dashed line is for $f=1/2$ the upper one for $f=1/4$ and
$3/4$. The phases of the modulated single-particle parameters are
$\varphi_{v}=\varphi_{t}=-0.45-0.25\pi$. For quarter filling, the other
parameters are $V=0.15,\delta t=0.2$, for half filling, $V=0.15,\delta
t=0.075$, and for three-quarter filling, $V=0.15,\delta t=0.1$.

Figure 1 shows the exponent $\beta_{f}$ as a function of $U$ for the
generalized interacting $Z=4$ AAH model at quarter, half, and three-quarter
filling. The values of the single-particle parameters are given in the
caption. For fixed $U$ and filling the exponent was determined as follows [see
Eq. (37) of Ref. 25]: $\Delta_{\nu}^{\text{ren}}$ was computed from numerical
diagonalization of the effective single-particle Hamiltonian in momentum space
(thermodynamic limit) at the end of the functional RG flow (see Appendix A.2).
After dividing by the bare gap $\Delta_{\nu}$ logarithmic centered differences
with respect to $\Delta_{\nu}$ were taken leading to a $U$ and filling
dependent constant in the limit of small $\Delta_{\nu}$. This value
corresponds to the exponent. In Fig. 1 the numerical data (symbols) are
compared to the leading order in $U$ prediction from field theory shown as
black dashed lines. Note that quarter and three-quarter fillings lead to the
same leading order expression for $\beta_{f}$. The data agree to the field
theoretical prediction to leading order in $U$. All this is as expected. To
save computational resources we therefore refrain from presenting DMRG data
for the gap scaling of the interacting generalized $Z=4$ AAH model. How
quickly the functional RG results for the exponent deviate from field theory
obviously when increasing $U$ depends on the filling (gap index). We, however,
emphasize, that employing the lowest-order truncated functional RG we do not
control the orders of the exponent beyond the leading one; for an extensive
discussion of this for the interacting RM model see Ref. [25].

Figure 2: Functional RG data of the local single-particle spectral function
$A_{j}(\omega)$ of the interacting, generalized, $Z=4$ AAH model with open
boundaries on site $j=1$ with $N=6000,V=0.27,\delta
t=0.15,\varphi_{v}=-0.05,\varphi_{t}=-0.17$. Data for three-quarter filling
and different $U$ are shown. The interaction induced effective edge states are
indicated by vertical arrows with the heights being proportional to the
corresponding spectral weight.

In Fig. 2 we show exemplary functional RG results for the local single-
particle spectral function on site $j=1$ of the interacting generalized $Z=4$
AAH model with open boundaries. Three-quarter filling is chosen. Energies
around the gap with index $\nu=3$ are shown corresponding to the low-energy
regime (the Fermi energy lies in the center of the gap shown). The parameters,
as given in the caption, are taken from a regime in which the $U=0$ function
shows a van-Hove singularity at the band edge of the gap but no in-gap edge
state (see the dark purple curve in Fig. 2). For the $Z=2$ RM model we have
shown that this is the parameter regime in which with increasing $U$ an
effective interaction-induced unconventional in-gap edge state forms [25]. As
Fig. 2 shows the same happens in the generalized AAH model. The edge state is
indicated by an in-gap $\delta$-peak displayed by a vertical arrow of a height
which is proportional to the spectral weight. With increasing $U$ the in-gap
state detaches from the van-Hove singularity and gains weight. It is not
located at the Fermi energy, i.e., it is not a zero energy edge state. The
appearance of this interaction induced edge state can be understood as
follows. During the RG flow the interplay of the open boundary and the
interaction leads to the build-up of a spatial modulation of the effective
single-particle parameters beyond the unit cell structure. The corresponding
envelope decays from the boundary towards the bulk and can lead to bound
states located close to the boundary (edge states). Deep in the bulk the same
effective single-particle parameters as obtained from a calculation with
periodic boundary conditions is reached. Crucially, the appearance of such
unconventional edge states cannot be understood from either the bare or the
renormalized single-particle parameters in the bulk and does thus not follow
the notion of the standard bulk-boundary correspondence which deals with
conventional edge states only.

The spectral function is computed for $N=6000$ lattice sites. To obtain a
continuous function for energies within the bands the spectral weight is
averaged over a few eigenenergies. Further increasing the system size does not
lead to any visible changes on the scale of the plot and the results can be
considered as being in the thermodynamic limit.

As a side remark we emphasize that Fig. 2 also illustrates that the gap
increases with increasing $U$ (see above).

Figure 3: Main panel: Functional RG data for of the local density relative to
the bulk value as a function of the unit cell index $n$ for $i=1$ and
different $U$. The single-particle parameters are $N=20000$, $V=0.03$, $\delta
t=0.08$, $\varphi_{v}=\varphi_{t}=\pi/4$. Half filling is considered. Note the
linear-log scale. The slope of the dashed line is computed plugging the
renormalized single-particle parameters into the analytical expression for
$-2\kappa^{\text{ren}}_{\text{bc}}$ at $U=0$, where
$\kappa^{\text{ren}}_{\text{bc}}$ is an inverse decay length calculated in
Ref. [14]. Inset: The logarithmic derivative of the pre-exponential function.
Solid lines are for $i=1$, dashed lines are for $i=2$, dotted lines are for
$i=3$ and the dashed-dotted ones are for $i=4$.

The bulk density on each intra-cell lattice site $i$ takes a value which is
independent of the unit cell index $n$. However, the density of the
interacting generalized AAH model becomes $n$-dependent if a system with an
open boundary is considered. The main part of Fig. 3 shows exemplary
functional RG results for the absolute value of the difference between the
bulk density and the one obtained in the presence of an open boundary for the
intra-cell index $i=1$ and different $U$. To be able to access large distances
from the boundary without any significant finite size effects a fairly large
system size of $N=20000$ sites is considered. Such cannot be reached by DMRG.
Without loss of generality we focus on half filling. The single-particle
parameters are given in the caption. For sufficiently large $n$ the decay
towards the bulk follows an exponential function with a $U$-dependent decay
length $(2\kappa_{\text{bp}})^{-1}$; note the linear-log scale of Fig. 3. This
length scale is independent of $i$ (not shown). For $U=0$ an analytic
expression for $2\kappa_{\text{bp}}$ in terms of the single-particle
parameters is known [14]. The details of this formula are irrelevant in the
present context and we do not reproduce it here. The figure indicates that
this expression can also be used in the interacting case if the bulk
renormalized single-particle parameters at the end of the RG flow are plugged
into the noninteracting formula; see the dashed lines. The decay length is
linked to the gap size and it is thus not surprising, that
$2\kappa_{\text{bp}}$ increases with increasing $U$.

After the exponential part of the $n$-dependence is known it can be divided
out. For $U=0$ the resulting pre-exponential function decays as $n^{-1/2}$.
This is shown in the inset of Fig. 3, in which we present the log-derivative
of the pre-exponential function as a function of $n$. For all $i$ the data
approach the asymptotic exponent $\zeta=-1/2$. This is known analytically
[14]. For increasing interaction the pre-exponential function changes. In
particular, it does no longer seem to decay as the inverse square root of the
unit cell index but instead shows a richer behavior. As for the gap and the
spectral function these findings are in full agreement with our results for
the interacting RM model [25].

Figure 4: Main panel: Functional RG data for the boundary charge of the
interacting three-quarter filled generalized AAH model as a function of
$\varphi=\varphi_{v}=\varphi_{t}$ for different $\delta t$ and $V$. The system
size is $N=4000$ and the interaction $U=0.08$. Inset: Derivative of the data
of the main panel with respect to $\varphi$. Black lines are the result
obtained from the noninteracting generalized AAH model but with the effective
bulk renormalized parameters.

Finally, we discuss the boundary charge of the interacting generalized AAH
model. In Fig. 4 we show $Q_{\rm B}$ (symbols) as a function of
$\varphi=\varphi_{v}=\varphi_{t}$ for $U=0.08$, different $V$ and $\delta t$
at three-quarter filling. The line shape is consistent with the second case of
Eq. (11) (involving $\nu-Z=-1$) and, for small gaps, with Eqs. (12) and (13).
The linear dependence of $Q_{\rm B}$ on $\varphi$ is known from noninteracting
models [11, 12, 13, 14], for small gaps from effective low-energy field
theories [15], and from our results for the microscopic RM model [25], see the
summary in Sect. II.3.4. As already emphasized in these works it is robust for
weak two-particle interactions. Moreover, the interaction enhanced corrections
to the linear behavior can be understood from the effective single particle
picture. In the inset of Fig. 4, we show the derivative of the data of the
corresponding main panel (symbols). Black lines are the results obtained from
the noninteracting AAH model but with the renormalized effective bulk
parameters. They coincide with the numerical data. Therefore, the interacting
boundary charge can be determined by the effective bulk parameters alone.

For large gaps (red diamonds in Fig. 4) plateaus at multiples of $1/4$
develop. These are also known from considering the atomic limit [15] and from
our discussion of the microscopic RM model [25]. We found similar results for
the one-quarter filled AAH model.

Figure 5: Functional RG data for the interaction correction of the boundary
charge for the generalized AAH $Z=4$ model at half filling as a function of
$U$. Different $V$ and $\delta t$ are considered and we choose
$\varphi_{v}=\varphi_{t}=5\pi/6$. The system size is $N=4000$. The black
dashed line indicates a power law $\sim U$, and the blue dashed line a power-
law $\sim U^{2}$. The weak $U$-dependence in the uniform hopping limit with
$\delta t=0$ indicates that the quantization of the fractional part of the
boundary charge is particularly robust against two-particle interactions. Note
the log-log scale.

For the half-filled noninteracting generalized AAH model, there is no simple
universal linear relation between the boundary charge and the phase of the
modulation in the small gap limit. Instead, Ref. 15 showed that the fractional
part of the boundary charge is quantized in the uniform hopping limit $\delta
t=0$ and takes values $\pm 1/2$ and $0$. In this work it was also shown, that
the quantization is robust towards two-particle interactions (and disorder).
In the following, we examine the robustness of the boundary charge for the
half-filled interacting generalized AAH model using our functional RG
approach. Away from the uniform hopping limit, the boundary charge at $U=0$ is
not quantized and shows a nonuiversal behavior we here do not further
investigate. We only studied the difference between the interacting and
noninteracting boundary charge for both the unifom hopping limit and generic
parameters to show that results for $Q_{\rm B}$ are in both cases rather
robust towards the two-particle interaction. In Fig. 5 we show the absolute
value of the interaction correction of the boundary charge for small
interactions as a function of $U$ on a log-log scale. On the one hand generic
exemplary single-particle parameters are taken (red and orange symbols), on
the other hand the subspace with $\delta t=0$ (blue symbols) is considered.
The black dashed line indicates a power law $\sim U$ and the blue one a power
law $\sim U^{2}$. The detailed values of the single-particle parameters are
given in the legend. We observe that for all single-particle parameters the
correction of the boundary charge is very small. Moreover, for the generic
cases, the correction of the boundary charge scales linearly in $U$ in the
small $U$ limit. In contrast, in the uniform hopping limit with $\delta t=0$,
the linear $U$ correction vanishes rendering the quantized boundary charge
particularly insensitive to the two-particle interaction. We note that our
approximate functional RG procedure does not contain all terms of order
$U^{2}$ thus we do not control the results in the uniform hopping limit
besides the insight that the linear term vanishes.

These results for the boundary charge again support our idea [25] that it
might be more appropriate and a more physical indicator of the relation
between boundary and bulk properties for interacting systems as compared to
the number of edge states.

The above insights on the interaction dependence of the gap, the single-
particle spectral function, the local density, as well as the charge
accumulated close to an open boundary of the generalized AAH model for $Z=4$
are in full accordance with the ones we gained for the $Z=2$ RM model. This
indicates that the phenomenology on the interaction dependence of the above
observables for gapped Fermi systems summarized by the properties 1. to 4. in
the Introduction and first formulated for the RM model seems to be
generalizable to models with more complex unit cell structure.

We now proceed and investigate if effects similar to the ones summarized in
this phenomenology are also found in models which are gapless in the
noninteracting limit. The gap is instead generated by a spatially modulated
interaction.

##  IV Results for the modulated interaction model

###  IV.1 The $Z=2$ model

In this section we discuss the modulated interaction model with $Z=2$. For
symmetry reasons a gap can only open around zero band energy. In the following
we thus consider a vanishing Fermi energy and half filling. Due to particle-
hole symmetry no effective onsite potential is generated during the RG flow.
As we have discussed in Ref. 25 this symmetry also implies that an open
boundary does not induce any density modulations beyond the intra unit cell
structure. For the present model we thus only consider the gap and the local
spectral function as observables.

To the best of our knowledge only the gap of the modulated interaction model
was so far studied in a reliable way [43]. We therefore have no expectations
from other models or field theory we can compare and benchmark our approximate
functional RG results for the ground state density and boundary charge
against. For this reason numerically exact DMRG results obtainable for small
to moderate system sizes will play a major role in this and the next
subsection.

However, we first discuss approximate analytical insights which can be gained
by functional RG. The RG flow equations can be found in Appendix A. For
periodic boundary conditions, the number of coupled flow equations is reduced
to two: the ones for $t^{\Lambda}_{1}$ and $t^{\Lambda}_{2}$. This shows that
an effective SSH model is generated by the modulation of the interaction. To
the given approximation, the physics of this effective single-particle model
corresponds to the one of the modulated interaction model.

The effective flowing gap is determined by

$\displaystyle
2\Delta^{\Lambda}=2\left|t^{\Lambda}_{\text{1}}-t^{\Lambda}_{\text{2}}\right|.$
(15)

In Appendix B, we present the details of an analytical derivation of
functional RG results for the effective gap in the small interaction and the
small gap limit. In a first step we prove that the gap in first order
perturbation theory for the self-energy, i.e. on Hartree-Fock level, is given
by

$\displaystyle 2\Delta^{\text{HF}}=\frac{4\delta U}{\pi}.$ (16)

In a second step we show that for $\Lambda_{\text{IR}}\ll W$, the solution of
the flow equation of the effective gap is given by

$\displaystyle
2\Delta(\Lambda_{\text{IR}})=2\Delta^{\text{HF}}\left(\frac{\Lambda_{\rm
IR}}{4t}\right)^{-U/\pi t}(2-2^{U/\pi t}),$ (17)

where $\Lambda_{\text{IR}}$ is the infrared (IR) scale, cutting off the RG
flow. It needs to be determined numerically. In Fig. 12, the momentum space RG
flow for the effective gap divided by $\delta U$ is shown for very small
$\delta U$. From this figure (and related ones for other parameter sets) we
conclude in a third step that the IR cutoff is given by the Hartree-Fock gap
$\Lambda^{\text{IR}}=2\Delta^{\text{HF}}$. This appears to be reasonable on
general grounds and could have been guessed even without the numerical
results. The final result for the renormalized gap

$\displaystyle
2\Delta^{\text{ren}}=2\Delta^{\text{HF}}\left(\frac{\Delta^{\text{HF}}}{2t}\right)^{-U/(\pi
t)}(2-2^{U/\pi t})$ (18)

shows that in the RG procedure logarithms are resummed to a power law as it is
the case in the interacting RM model and the interacting generalized AAH
model.

We, however, emphasize that the appearance of the interaction in the basis [in
form of the modulation amplitude $\delta U$; see Eq. (16)] and the exponent
(in the form of the unit cell average $U$) makes it less obvious that our
approximate functional RG indeed captures the physics for small to moderate
two-particle interactions.

Figure 6: Main plot: Ratio of the renormalized gap and the Hartree-Fock gap
$2\Delta^{\mathrm{HF}}$ as a function of $2\Delta^{\mathrm{HF}}$. Data of DMRG
(open symbols), real-space FRG (solid lines with filled diamonds), and
$k$-space functional RG (black dotted lines) are shown. Inset: The log-
derivative and thus the apparent exponent $\beta$ of the data in the main
plot. The system size is $N=1000$. The corresponding $\delta U$ are
$0.01,0.05,0.1,0.15,0.2,0.25,0.5,1.0$. Three different $U$ are considered.
Note the log-log scale of the main plot.

In Fig. 6 we therefore compare numerical results for the gap at the end of the
RG flow obtained by real-space as well as $k$-space functional RG with DMRG
results. We note that besides the lowest order trunction the numerical
solution of the functional RG equations does not contain any further
approximations. This has to be contrasted to the analytical functional RG
analysis of Appendix B leading to Eq. (18) which requires such steps. The main
panel shows the comparison of the renormalized gap divided by
$2\Delta^{\text{HF}}$ as a function of the Hartree-Fock gap. The real-space
functional RG data are shown as color-coded filled diamonds. In comparison
DMRG data are indicated by open symbols, and the momentum space functional RG
results are displayed as dotted lines.

For not too large interactions, that is $U=0.1$ and $U=0.5$, the functional RG
data and the DMRG data agree nicely when $2\Delta^{\text{HF}}$ is not too
small. The upturn of the DMRG data at small Hartree-Fock gaps and the
resulting deviation from the functional RG data can be understood as follows.
For very small gaps, very low energy scales must be accessible. In DMRG the
low-energy scale is set by the inverse system size. If the gap becomes smaller
than this scale, finite size effects prevail and in the present case lead to
the upturn. Due to limitations in computing resources it is not possible to
produce DMRG data which are converged with respect to the bond dimension—as it
is the case in Fig. 6—for systems significantly larger than the $N=1000$
lattice sites considered here. The $k$-space functional RG results were
obtained in the thermodynamic limit instead and do not suffer from finite size
effects. The real-space functional RG data were, however, obtained for the
same system size as studied by DMRG. Thus finite size effects matter for these
as well but apparently do not show up in Fig. 6. Two reasons for this appear
to be reasonable. Firstly, the way the gap is extracted in finite size DMRG
and real-space functional RG (gap in the many-body spectrum versus gap in the
effective single-particle spectrum) differ. Secondly, truncated functional RG
is an approximate tool, while DMRG (for finite systems) is numerically exact.
Both might affect the details of the finite size corrections.

As the renormalized gap increases with increasing $U$ and thus the low-energy
scale increases, this finite size effect becomes less severe for larger $U$.
Accordingly, the $U=1$ DMRG data show barely any upturn at small
$2\Delta^{\text{HF}}$. However, for this interaction higher-order corrections
missed in the functional RG approach become important and DMRG and functional
RG data deviate for all $2\Delta^{\text{HF}}$ (see the green data).

Most notably all data sets of Fig. 6 show linear behavior for intermediate
$2\Delta^{\text{HF}}$ on a log-log scale. In the inset the log-derivative of
the data is shown. Up to the finite-size issues just discussed and higher
order (in $U$) corrections DMRG confirms the functional RG result Eq. (18) of
power-law scaling. We observe that the DMRG and functional RG exponents
deviate from the leading order expression $-U/\pi$ (black dashed lines in the
inset of Fig. 6) in similar ways. We have already observed this for the $Z=2$
interacting RM model. For a discussion of this, see Ref. 25. As the full
numerical solution of the RG equations contains terms beyond the leading
order, the deviation of the numerical exponent for $U$ of order 1 from the
analytical exponent $-U/\pi$ is not surprising.

All this confirms that our functional RG analysis based on the lowest-order
truncation indeed provides the correct answer. A gap opens in the modulated
interaction model which is triggered by the modulation amplitude $\delta U$
and scales as a power-law in the Hartree-Fock gap, with the interaction $U$
averaged over the unit cell entering the exponent to leading order. The
leading order exponent $-U/\pi$ is the same as found in the $Z=2$ half-filled
interacting RM model [25] as well as the $Z=4$ half-filled interacting
generalized AAH model (see Sect. III) and is consistent with field theory.
Within truncated functional RG this can be understood from the effective
single-particle SSH model generated during the RG flow. From a perturbative
perspective it, in contrast, can be understood as follows. Lowest order
perturbation theory for the self-energy, i.e. the (non-self-consistent)
Hartree-Fock approximation, leads to a gap $\sim\delta U$. In the RG process,
which includes more than the Hartree-Fock diagrams, this gap is elevated to a
power-law with a $U$-dependent exponent.

We numerically verified that for vanishing average interaction $U=0$ but
$\delta U\neq 0$ a gap opens but does not show power-law scaling (in the
Hartree-Fock gap) consistent with Eq. (18). In short, for $U=0$ the $\delta U$
term alone generates an effective SSH model which shows noninteracting
physics.

Figure 7: Functional RG data for the local single-particle spectral function
$A_{j}(\omega)$ of the modulated interaction model with $Z=2$, $\delta
U=0.19$, $U=0.5$, and $\varphi_{U}=\pi/4$. Modulated interaction induced in
gap zero energy $\delta$-peak are shown as the vertical arrows and the height
is scaled up.

After gaining confidence that also the modulated interaction model can
reliably be analyzed using truncated functional RG we proceed and study the
local spectral function employing this approach (see our comment of Sect.
II.3.2 on the use of DMRG for this observable).

In Fig. 7 we show functional RG results for the local spectral function
$A_{j}(\omega)$ of the modulated interaction model with $Z=2$ and open
boundary conditions as a function of $\omega$ and the lattice site index $j$.
The average interaction is $U=0.5$ with a modulation of $\delta U=0.19$. The
phase $\varphi_{U}=\pi/4$ and the system size is given as $N=6000$. Obviously,
an effective edge state forms which shows as an in-gap $\delta$-peak. As the
figure indicates the effective edge state has a nontrivial spatial structure.
When going from the boundary towards the bulk the spectral weight first
increases before its starts to decrease (asymptotically is decreases
exponentially; not shown), the latter as it is supposed to be the case for an
edge state. Due to symmetry the spectral weight of the edge state is only
nonvanishing on odd sites (or, equivalently, for the unit cell index $i=1$).

We now have to ask if this edge state appears because the effective SSH model
at the end of the RG flow has bulk single-particle parameters which imply a
conventional edge state or if the edge state is of unconventional type. For
the effective bulk SSH model an (topological) edge state appears if
$t_{2}^{\rm ren}<t_{1}^{\rm ren}$ [17, 18, 25]. For the parameters of Fig. 7
this indeed holds and the edge state is of conventional type. It is the well
known zero energy edge state of the effective SSH model in its topological
phase. The van-Hove-singularity discussed in connection with Fig. 2 as a
necessary requirement for the appearance of unconventional edge states, is
associated to a modulated onsite energy. For $Z=2$ and half-filling no
(modulated) onsite energy is generated during the RG flow and thus only
conventional edge states can be realized. Note that in accordance with
property 3. of the phenomenology described in the Introduction one can still
say that the (conventional) edge state is induced by the interaction as the
noninteracting model does not have any edge states.

To summarize, this shows that the $Z=2$ modulated interaction model shows
effects similar to the ones summarized in the Introduction. However, due to
particle-hole symmetry the density takes its bulk value regardless of the
boundary condition. To overcome this limitation we next study the $Z=4$ case.

###  IV.2 The $Z=4$ model

From what we just observed for $Z=2$ and our results for the interacting
generalized AAH model we expect that the $Z=4$ modulated interaction model
will show three gaps corresponding to one-quarter, one-half, and three-quarter
filling. Again the Fermi energy is selected such that it lies in the middle of
one of the developing gaps. Functional RG but also the Hartree-Fock
approximation confirms this.

For $Z=4$ it is less straight forward to determine an analytic expression for
the gap at Hartree-Fock level [see Eq. (16) for $Z=2$]. However, one can
easily obtain the Hartree-Fock gap numerically. This shows that $\Delta^{\rm
HF}_{\nu}\sim\delta U$ for one- and three-quarter filling ($\nu=1,3$), that is
for the first and the third gap. At half-filling the numerics instead
indicates $\Delta^{\rm HF}_{2}\sim\delta U^{2}$, which is an inconsistent
result, as the Hartree-Fock approximation does only include some, but not all,
second order contributions for the gap. As discussed in Appendix C the $\delta
U^{2}$ dependence of the Hartree-Fock gap can be confirmed analytically. We
conclude that the special symmetry at half-filling leads to a nongeneric
behavior of the corresponding gap. As also in lowest-order truncated
functional RG only parts of the second order diagrams for the gap are included
this special case can also not be treated in a meaningful way by this method.
Half-filling is furthermore difficult to be treated by DMRG. For small $\delta
U$, the interesting case when it comes to the power-law scaling, the gap is
exceedingly small as indicated by $\Delta^{\rm HF}_{2}\sim\delta U^{2}$. To
access the corresponding low energy regime by DMRG would require huge system
sizes and very large bond dimensions both beyond reasonable computing
resources. We thus refrain from investigating the nongeneric half-filled case
any further. It might be a topic for a future study.

Figure 8: Main panel: The ratio of the functional RG effective gap
$2\Delta_{\nu}$ and the effective gap in to first order perturbation theory
$2\Delta^{\mathrm{HF}}_{\nu}$ as a function of $2\Delta^{\mathrm{HF}}_{\nu}$.
A comparison of quarter filling (filled symbols) and three-quarter (open
symbols) for different $U$ is shown. The parameters are $U=0.0,0.1,0.2,0.45$,
$\varphi_{U}=-0.45-\pi/4$, the system size is $N=4000$ and $\delta
U=0.01,0.015,\ldots,0.055$. Note the log-log scale. Inset: The log-derivative
of the data in the main plot. The dashed line indicates the leading order
exponent from field theory.

For one- and three-quarter filling we first investigate if Eq. (18) for the
gap also holds at $Z=4$. Based on the finding that $\Delta^{\rm
HF}_{\nu}\sim\delta U$ (for $\nu=1,3$) and the mechanism leading to the power
law as described in the last subsection this appears to be reasonable. In the
main panel of Fig. 8, we show functional RG data for
$2\Delta_{\nu}/2\Delta^{\text{HF}}_{\nu}$ as a function of
$2\Delta^{\text{HF}}_{\nu}$ for different $U$. Data for quarter filling
($\nu=1$) is labeled by filled symbol and for three-quarter filling ($\nu=3$)
by open symbols. The values of the parameters are given in the caption. For
fixed $U$ the two data sets agree and show linear behavior on a log-log scale.
This indicates again the power-law scaling of the effective gap with respect
to $2\Delta^{\text{HF}}_{\nu}$ and the same exponent for $\nu=1$ and $\nu=3$.
In the inset of Fig. 8, we show the centered logarithmic differences of the
data in the main panel. For small $2\Delta^{\text{HF}}_{\nu}$ (small $\delta
U$), both results approach the leading order exponent
$\beta_{f}=-U[1-2\cos(2k_{\rm F})]/[2\pi\sin(k_{\rm F})]$ from field theory
which is indicated as the dashed horizontal lines. As we emphasized in Sec.
III and our paper on the interacting RM model [25], we do not control the
orders of the exponent beyond the leading one.

Based on these numerical results, we conclude that concerning the gap induced
by the modulated interaction the $Z=4$ model at one- and three-quarter filling
shows the same behavior as the $Z=4$ interacting generalized AAH model [see
Eq. (14)]. In fact, the effective single-particle model at the end of the RG
flow of the $Z=4$ modulated interaction model is a AAH model and it appears to
be reasonable that both models share the same low energy effective theory.

As for $Z=2$ we verified numerically that even for $U=0$ but $\delta U\neq 0$
gaps open but that they do not scale as a power law in the Hartree-Fock gap.
The physics of the effective AAH model is that of a noninteracting one.

Figure 9: The local single particle spectrum function $A_{j}(\omega)$ of the
three-quarter filling $Z=4$ modulated interaction model as a function of
$\omega$ for different lattice sites $j$. The interaction induced effective
edge states are indicated by vertical arrows. The parameters are
$U=0.25,\delta U=0.15,N=6000,\varphi_{U}=-\pi/4$.

Next, we discuss the local spectral function computed by functional RG
considering open boundary conditions. In Fig. 9 we show $A_{j}(\omega)$ at
three-quarter filling as a function of $\omega$ and the lattice site index
$j$. The parameters are given in the caption. The low-energy regime around the
Fermi energy which is located in the gap centered around $\omega\approx 1.6$
is shown. The interaction obviously induces an edge state with spectral weight
which has a rich spatial structure inherited from the $Z=4$ spatial modulation
of the interaction.

As for $Z=2$ the question arises whether the edge state is a conventional or
an unconventional one. We verified that for the parameters of Fig. 9 the
former holds. In contrast to the $Z=2$ case with half-filling the state is,
however, not a zero energy edge state. Compared to the interacting generalized
AAH model, in which in addition to the interaction $U$, the modulation
amplitude and phase of the hopping as well as the onsite energy can be chosen
independently the parameter space of the modulated interaction model is
significantly smaller. For this only $U$, $\delta U$ and $\varphi_{U}$ can be
chosen. Despite an extensive search of the parameter space we did not succeed
to identify a regime in which an unconventional edge state is realized. We,
however, emphasize that this does not exclude that such a regime exists. In
any case, the (conventional) edge state is induced by the two-particle
interaction in accordance with the property 3. of the Introduction.

Figure 10: Main plot: Total density $\rho$ as a function of the site index $j$
for quarter filling modulated interaction model with $Z=4,U=-5\delta U=0.375$,
and $N=1000$. The resulting functional RG local density which is labeled by
“fRG” are compared with the DMRG result. Inset: The largest absolute value of
the difference between FRG and DMRG data taken over all of the lattices sites
as a function of $\tilde{U}$, where $\tilde{U}=1,2,...,11$ and $(U,\delta
U)=\tilde{U}(0.025,-0.005)$. A log-log scale is used. The dashed line
indicated the power law $\tilde{U}^{2}$.

We now study the density modulation induced by the modulated interaction for
$Z=4$. The main panel of Fig. 10 shows a comparison of the local density as a
function of lattice site index $j$ computed by functional RG and by DMRG at
quarter filling. The parameters can be found in the caption. As expected the
modulation of the interaction leads to a modulated density around the average
value $1/4$. The bulk density, which on the $y$-axis scale of the plot is
approximately reached already for $j\approx 50$, reflects the $Z=4$
periodicity. Close to the boundary this periodicity is disturbed by the
interplay of the boundary and the (average) interaction. This modulation
beyond the unit cell structure decays quickly towards the bulk.

Based on our results for the density of the $Z=4$ interacting generalized AAH
model we expect that the decay towards the bulk value is exponential. We
verified this (not shown). The decay rate can again be computed by plugging
the renormalized bulk parameters at the end of the RG flow into the analytical
expression for the noninteracting AAH model. This way the exponential part can
be divided out very precisely. Based on our insights on the different roles of
$\delta U$ and $U$ we anticipate that the pre-exponential function for $U=0$
decays as $1/\sqrt{n}$ (independent of the intra unit cell index $i$) while a
more complex, $i$-dependent behavior is found for $U>0$. We also verified this
(not shown). The corresponding data look similar to the ones shown for the
interacting AAH model in Fig. 3.

On the scale of Fig. 10 the approximate functional RG data cannot be
distinguished from the DMRG ones. In the inset, we show the maximal absolute
difference over all lattices sites as a function of $\tilde{U}$, where we
define $\tilde{U}$ as $(U,\delta U)=\tilde{U}(0.025,-0.005)$ in order to keep
the ratio between $U$ and $\delta U$ constant during the scaling of the
interaction. The difference scales as a power-law $\sim\tilde{U}^{2}$ (dashed
line) in full accordance with the approximate nature of the truncated
functional RG which does not capture all diagrammatic contributions to second
and higher order.

We found similar results for three-quarter filling corresponding to the gap
with index $\nu=3$.

Figure 11: The boundary charge $Q_{\rm B}$ of the three-quarter filled
modulated interaction model with $Z=4$ as a function of $\varphi=\varphi_{U}$
(in the azimuthal direction) and $\delta U$ (in the radial direction), see Eq.
(19). The system size is given as $N=4000$ and the interaction is $U=0.08$.

.

Finally, we investigate the boundary charge of the $Z=4$ modulated interaction
model at three-quarter filling. In Fig. 11 we show functional RG data for the
boundary charge as a two-dimensional color-coded plot in polar coordinates.
The azimuthal direction displays the angle $\varphi=\varphi_{U}$ and the
radial direction the amplitude of the modulation $\delta U$, which is
restricted to small values corresponding to small gaps. The other parameters
are given in the caption. We find that $Q_{\rm B}$ depends linearly on
$\varphi$ and has a rather weak dependence on $\delta U$. Numerically, we find
that the boundary charge is approximately given by

$\displaystyle Q_{\rm B}\approx-\frac{\varphi}{2\pi}+\frac{1}{2}.$ (19)

This result is consistent with Eqs. (12) and (13) as derived in Refs. [15,
16]. The constant part $1/2$ can be obtained from the special symmetry point
$\varphi=\varphi_{U}=\pi$, where the model has local inversion symmetry such
that $Q_{\rm B}=0$. Our result indicates that $Q_{\rm B}$ shows the same
behavior as for the interacting AAH model for small gaps; see Sec. III. This
is consistent with the observation from the other observables that the
modulated interaction and the AAH model share the same effective low-energy
theory. We, moreover, note (without showing data) that one can use the
noninteracting AAH model with the bare bulk parameters replaced by the
renormalized effective bulk parameters to generate the same result. This
supports the idea that the characteristics of the boundary charge can fully be
determined by the bulk properties with effective single-particle parameters.
This is summarized by the property 4. of the introduction. We found similar
results for one-quarter filling.

##  V Summary

This work generalizes the results of Ref. [25] on interaction effects in the
$Z=2$ RM model to microscopic lattice models with more complex unit cell
structure. A posteriori the success of this generalization also lends credence
to similar arguments obtained from a purely field theoretical approach.

We first investigated a direct extension of the $Z=2$ RM model, namely, an
interacting generalized 1d AAH model with a unit cell of size $Z=4$. We
employed functional RG, which proved to be a valuable tool in the prior study
of the RM model. We were able to confirm the phenomenology already identified
for the interacting RM model also in the case $Z=4$. The gap is renormalized
by the two-particle interaction. For small bare gaps the renormalized one
scales as a power law with the two-particle interaction entering the exponent.
The local spectral function close to an open boundary can show (depending on
the single-particle parameters) an interaction induced in-gap $\delta$-peak
associated to an effective edge state. Its appearance cannot be explained
based on the bare or renormalized bulk single-particle parameters. In
contrast, the characteristics of the boundary charge accumulated close to an
open boundary are robust towards the two-particle interaction. This
consolidates our idea that the boundary charge might provide a more robust
relation between bulk and boundary properties than the number of edge states
(conventional bulk-boundary correspondence).

In a second step we investigated a lattice model which in the noninteracting
limit is translationally invariant by a single-lattice site (for PBCs). In
this the non-trivial unit cell structure is induced by a periodically
modulated two-particle interaction. Up to singular situations with a special
symmetry this model also followed the above discussed phenomenology, with the
difference being that the interaction induced effective edge states can be
predicted based on the renormalized single-particle parameters. For the
parameter space we explored, we did not find any interaction induced edge
states which cannot be explained this way.

## Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) via RTG 1995 and under Germany’s Excellence Strategy -
Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1
- 390534769. DMK acknowledges support from the Max Planck-New York City Center
for Non-Equilibrium Quantum Phenomena. Simulations were performed with
computing resources granted by RWTH Aachen University.

##  Appendix A The flow equations

In this Appendix we present the truncated functional RG equations for the
self-energy and the density for arbitrary $Z$. They hold for both the
interacting generalized AAH model as well as the modulated interaction models.
We start out with the real space equations which can be employed for open as
well as periodic boundary conditions.

###  A.1 Real space

We focus on the lowest-order truncated functional RG scheme featuring a static
flowing self-energy $\Sigma^{\Lambda}$ and consider a sharp frequency cutoff
in Matsubara space [25, 49]. The two-particle vertex function in real space is
given as

$\displaystyle\Gamma_{j^{\prime}_{1},j^{\prime}_{2};j_{1},j_{2}}=U_{j_{1},j_{2}}\left(\delta_{j_{1},j^{\prime}_{1}}\delta_{j_{2},j^{\prime}_{2}}-\delta_{j_{1},j^{\prime}_{2}}\delta_{j_{2},j^{\prime}_{1}}\right),$
(20) $\displaystyle
U_{j_{1},j_{2}}=U_{j_{1}}\delta_{j_{1},{j_{2}-1}}+U_{j_{1}-1}\delta_{j_{1},j_{2}+1},$
(21)

and the lowest-order flow equation of the self-energy reads

$\displaystyle\partial_{\Lambda}\Sigma^{\Lambda}_{j^{\prime}_{1},j_{1}}$
$\displaystyle=-\frac{1}{2\pi}\sum_{\omega=\pm\Lambda}e^{i\omega
0^{+}}\mathcal{G}^{\Lambda}_{j_{2},j^{\prime}_{2}}(i\omega)\Gamma_{j^{\prime}_{1},j^{\prime}_{2};j_{1},j_{2}}$
(22)
$\displaystyle=-\frac{1}{\pi}\text{Re}\left[\mathcal{G}^{\Lambda}_{j_{2},j^{\prime}_{2}}(i\Lambda)U_{j_{1},j_{2}}\delta_{j_{1},j^{\prime}_{1}}\delta_{j_{2},j^{\prime}_{2}}\right]$
(23) $\displaystyle\hskip
10.00002pt+\frac{1}{\pi}\text{Re}\left[\mathcal{G}^{\Lambda}_{j_{2},j^{\prime}_{2}}(i\Lambda)U_{j_{1},j_{2}}\delta_{j_{1},j^{\prime}_{2}}\delta_{j_{2},j^{\prime}_{1}}\right]$
(24)

with a cutoff dependent propagator

$\displaystyle{\mathcal{G}}^{\Lambda}({\rm
i}\omega)=\left\\{\left[{\mathcal{G}}_{0}({\rm
i}\omega)\right]^{-1}-\Sigma^{\Lambda}\right\\}^{-1}.$ (25)

Equations (23) and (24) are the flow equations for the diagonal part of the
self-energy and the off-diagonal one respectively. They can be brought in the
more explicit forms

$\displaystyle\partial_{\Lambda}\Sigma^{\Lambda}_{j,j}=-\frac{1}{\pi}\text{Re}\left[\mathcal{G}^{\Lambda}_{j+1,j+1}(i\Lambda)U_{j}+\mathcal{G}^{\Lambda}_{j-1,j-1}(i\Lambda)U_{j-1}\right].$
(26) $\displaystyle\partial_{\Lambda}\Sigma^{\Lambda}_{j,j\pm
1}=\frac{1}{\pi}\text{Re}\left[\mathcal{G}^{\Lambda}_{j,j+1}(i\Lambda)U_{j/j-1}\right]$
(27)

To consistently compute the local density $\rho(j)$ we set up according flow
equations for this observable

$\displaystyle\frac{\partial}{\partial\Lambda}\rho^{\Lambda}(j)=-\frac{1}{2\pi}\sum_{\omega=\pm\Lambda}\mbox{tr}\left[e^{\mbox{i}\omega
0^{+}}{{\mathcal{G}}}^{\Lambda}({{\rm i}\omega})R^{\Lambda}_{j}({{\rm
i}\omega})\right].$ (28)

They involve a density response vertex $R^{\Lambda}_{j}$ which obeys the
equations

$\displaystyle\frac{\partial}{\partial\Lambda}R^{\Lambda}_{j;l,l}$
$\displaystyle=-\frac{1}{2\pi}\sum_{\omega=\pm\Lambda}\sum_{l^{\prime}}\sum_{r^{{}^{\prime}}=0,\pm
1}U_{l}{{\mathcal{G}}}^{\Lambda}_{l+1,l^{\prime}}({{\rm
i}\omega})R^{\Lambda}_{j;l^{\prime},l^{\prime}+r^{\prime}}{{\mathcal{G}}}_{l^{\prime}+r^{\prime},l+1}^{\Lambda}({{\rm
i}\omega})+U_{l-1}{{\mathcal{G}}}^{\Lambda}_{l-1,l^{\prime}}({{\rm
i}\omega})R^{\Lambda}_{j;l^{\prime},l^{\prime}+r^{\prime}}{{\mathcal{G}}}_{l^{\prime}+r^{\prime},l-1}^{\Lambda}({{\rm
i}\omega}),$ (29)
$\displaystyle\frac{\partial}{\partial\Lambda}R^{\Lambda}_{j;l,l\pm 1}$
$\displaystyle=-\frac{1}{2\pi}\sum_{\omega=\pm\Lambda}\sum_{mm^{\prime}}U_{l/l-1}{{\mathcal{G}}}^{\Lambda}_{l,m^{\prime}}({{\rm
i}\omega})R^{\Lambda}_{j;m^{\prime},m}{{\mathcal{G}}}_{m,l\pm
1}^{\Lambda}({{\rm i}\omega}).$ (30)

Details on the advantage of computing $\rho(j)$ via its own flow equation as
compared to an approach employing the Green function (and thus the self-
energy) can be found in Ref. [25].

Similar to what is done in Ref. [25], one can decompose the self-energy into
unit cell index independent and dependent parts, and the renormalized
parameters at the end of the RG prodedure are given as

$\displaystyle V^{\rm ren}_{j=Z(n-1)+i}=V_{j}+\Sigma^{\Lambda=0}_{j,j}=V^{\rm
ren}_{i}+V^{\rm F}_{i}(n),$ (31) $\displaystyle t^{\rm
ren}_{j=Z(n-1)+i}=t_{j}+\Sigma^{\Lambda=0}_{j,j+1}=t^{\rm ren}_{i}+t^{\rm
F}_{i}(n).$ (32)

where “ren” and “F” denotes the unit cell index $n$ independent and dependent
parts of the effective onsite potential and the hopping parameters.

The above sets of equations can for large but finite systems easily be solved
on a computer.

###  A.2 Momentum space

For underlying periodic boundary conditions on can derive flow equations for
the effective single particle parameters in $k$-space. They are set up
directly in the thermodynamic limit. For the hopping parameters one obtains

$\displaystyle\partial_{\Lambda}t^{\Lambda}_{i}=-\frac{U_{i}}{2\pi}\sum_{\omega=\pm\Lambda}\int^{\pi}_{-\pi}\frac{dk}{2\pi}\mathcal{G}^{\Lambda}_{i,i+1}(k,i\omega)\text{
for $i\neq Z$},$
$\displaystyle\partial_{\Lambda}t^{\Lambda}_{i=Z}=-\frac{U_{Z}}{2\pi}\sum_{\omega=\pm\Lambda}\int^{\pi}_{-\pi}\frac{dk}{2\pi}e^{ik}\mathcal{G}^{\Lambda}_{Z,1}(k,i\omega)\text{
for $i=Z$}.$ (33)

For the onsite potential they read

$\displaystyle\partial_{\Lambda}V^{\Lambda}_{i=1}=-\frac{1}{2\pi}\sum_{\omega=\pm\Lambda}\int^{\pi}_{-\pi}\frac{dk}{2\pi}\left\\{U_{1}\mathcal{G}^{\Lambda}_{2,2}(k,i\omega)+U_{Z}\mathcal{G}^{\Lambda}_{Z,Z}(k,i\omega)\right\\}\text{
for $i=1$},$
$\displaystyle\partial_{\Lambda}V^{\Lambda}_{i=Z}=-\frac{1}{2\pi}\sum_{\omega=\pm\Lambda}\int^{\pi}_{-\pi}\frac{dk}{2\pi}\left\\{U_{Z}\mathcal{G}^{\Lambda}_{1,1}(k,i\omega)+U_{Z-1}\mathcal{G}^{\Lambda}_{Z-1,Z-1}(k,i\omega)\right\\}\text{
for $i=Z$},$
$\displaystyle\partial_{\Lambda}V^{\Lambda}_{i}=-\frac{1}{2\pi}\sum_{\omega=\pm\Lambda}\int^{\pi}_{-\pi}\frac{dk}{2\pi}\left\\{U_{i}\mathcal{G}^{\Lambda}_{i+1,i+1}(k,i\omega)+U_{i-1}\mathcal{G}^{\Lambda}_{i-1,i-1}(k,i\omega)\right\\}\text{
else}.$

and the single scale propagator in the right hand side of the RG equations is
given as

$\displaystyle\\{\mathcal{G}^{\Lambda}(k,i\omega)\\}^{-1}=\left(\begin{array}[]{rrrr}i\Lambda+\mu-V^{\Lambda}_{1}&t^{\Lambda}_{1}\phantom{--}&&t^{\Lambda}_{Z}e^{-ik}\phantom{-}\\\
t^{\Lambda}_{1}\phantom{--}&i\Lambda+\mu-V^{\Lambda}_{2}&\ddots&\phantom{--}\\\
\phantom{--}&\ddots\phantom{--}&\ddots&t^{\Lambda}_{Z-1}\phantom{--}\\\
t^{\Lambda}_{Z}e^{ik}\phantom{-}&&t^{\Lambda}_{Z-1}&i\Lambda+\mu-V^{\Lambda}_{Z}\\\
\end{array}\right).$ (39)

The above set can be easily be solved numerically. However, it can also be
used as the starting point for an analytical analysis; see Appendix B for the
$Z=2$ modulated interaction model.

##  Appendix B Analytical insights for the modulated interaction model with
$Z=2$

Here we consider the $Z=2$ modulated interaction model with vanishing Fermi
energy, i.e. at half filling. After plugging in the propagator and performing
the $k$-integration in the right hand side of the RG equation (33), one
obtains

$\displaystyle\partial_{\Lambda}2\Delta^{\Lambda}=2\Delta^{\Lambda}\frac{U}{\pi}\frac{1}{b^{\Lambda}}\left\\{1-\frac{a^{\Lambda}+b^{\Lambda}}{\sqrt{(a^{\Lambda})^{2}-(b^{\Lambda})^{2}}}\right\\}$
$\displaystyle\hskip 45.00006pt-2W^{\Lambda}\frac{\delta
U}{\pi}\frac{1}{b^{\Lambda}}\left\\{1-\frac{a^{\Lambda}-b^{\Lambda}}{\sqrt{(a^{\Lambda})^{2}-(b^{\Lambda})^{2}}}\right\\},$
(40)
$\displaystyle\partial_{\Lambda}2W^{\Lambda}=-2W^{\Lambda}\frac{U}{\pi}\frac{1}{b^{\Lambda}}\left\\{1-\frac{a^{\Lambda}-b^{\Lambda}}{\sqrt{(a^{\Lambda})^{2}-(b^{\Lambda})^{2}}}\right\\}$
$\displaystyle\hskip 45.00006pt+2\Delta^{\Lambda}\frac{\delta
U}{\pi}\frac{1}{b^{\Lambda}}\left\\{1-\frac{a^{\Lambda}+b^{\Lambda}}{\sqrt{(a^{\Lambda})^{2}-(b^{\Lambda})^{2}}}\right\\}$
(41)

for the difference and the sum of the two hoppings $t_{1}^{\Lambda}$ and
$t_{2}^{\Lambda}$, respectively. Here
$a^{\Lambda}=\Lambda^{2}+(t^{\Lambda}_{1})^{2}+(t^{\Lambda}_{2})^{2}$, and
$b^{\Lambda}=2t^{\Lambda}_{1}t^{\Lambda}_{2}$.

From these flow equations one can obtain the effective gap and bandwidth in
first order perturbation theory, i.e. on Hartree-Fock level, by turning off
the feedback of self-energy in the right hand side of Eq. (40) and (41) [25].
The $\Lambda$-integral can then be performed leading to

$\displaystyle 2\Delta^{\Lambda=0}=2\Delta^{\text{HF}}=\frac{4\delta U}{\pi},$
(42) $\displaystyle 2W^{\Lambda=0}=2W^{\text{HF}}=4t\left(1+\frac{U}{\pi
t}\right),$ (43)

for the Hartree-Fock gap $2\Delta^{\text{HF}}$ and bandwidth $2W^{\text{HF}}$.
For $\Delta^{\Lambda}\ll\Lambda,W^{\Lambda}$, that is in the small gap limit,
and keeping terms to linear order in $U$ and $\delta U$ on the right hand side
of Eq. (40), one finds

$\displaystyle\partial_{\Lambda}\Delta^{\Lambda}=\frac{1}{2t^{2}}\left\\{\Delta^{\Lambda}\frac{U}{\pi}\left(1-\frac{\sqrt{\Lambda^{2}+4t^{2}}}{\Lambda}\right)\right.$
$\displaystyle\left.-2t\frac{\delta
U}{\pi}\left(1-\frac{\Lambda}{\sqrt{\Lambda^{2}+4t^{2}}}\right)\right\\}.$

Figure 12: Exemplary numerical data for the $k$-space RG flow (color dashed
lines) and the analytical result [Eq. (47) with Eq. (48) substituted; black
dotted lines] of the ratio of effective gap and $\delta U$ are shown. The
model parameters are given as $U=0.001$, $t=1$ and different $\delta U$ are
shown in the plot. The infrared cutoff scale is clearly visible.

One can rescale all of the parameters $\Delta^{\Lambda}\rightarrow
y=\Delta^{\Lambda}/2t$, $\Lambda\rightarrow x=\Lambda/2t$, $U\rightarrow
a=U/{\pi t}$ and $\delta U\rightarrow c=\delta U/{\pi t}$, to obtain the
dimensionless equation

$\displaystyle\frac{dy}{dx}=ay\left(1-\frac{\sqrt{1+x^{2}}}{x}\right)-c\left(1-\frac{1}{\sqrt{1+x^{2}}}\right),$
(44)

with the initial condition $y(x=\infty)=0$. The solution of Eq. (44) is given
as

$\displaystyle y(x)=c$
$\displaystyle\exp\left[a(x-\sqrt{x^{2}+1})\right]\left(\frac{x}{1+\sqrt{1+x^{2}}}\right)^{-a}\times$
$\displaystyle\int^{x}_{\infty}d\xi\left\\{\exp\left[a(\xi-\sqrt{\xi^{2}+1})\right]\left(\frac{\xi}{1+\sqrt{1+\xi^{2}}}\right)^{a}\right.$
$\displaystyle\left.\hskip
15.00002pt\left(\frac{\xi-\sqrt{1+\xi^{2}}}{\sqrt{1+\xi^{2}}}\right)\right\\}$
(45)

In order to obtain the result in the small $a$ and $c$ limit (small $\Delta U$
and $U$ limit), one keeps the linear in $c$ term in front of the expression
and the linear in $a$ term in the exponent. Moreover, in this limit one finds
$\exp\left[a(x-\sqrt{x^{2}+1})\right]\cong 1$ and
$\exp\left[a(\xi-\sqrt{\xi^{2}+1})\right]\cong 1$. Therefore,

$\displaystyle y(x)\cong$ $\displaystyle
c\left(\frac{x}{1+\sqrt{1+x^{2}}}\right)^{-a}\frac{2\left(\tan\frac{\xi}{2}+1\right)^{a-1}}{(a-1)}\times$
$\displaystyle\hskip
15.00002pt\left.{}_{2}F_{1}\left(1-a,-a;2-a;\frac{1}{1+\tan(\frac{\xi}{2})}\right)\right|^{\xi=\frac{\pi}{2}}_{\xi=\tan^{-1}x}$
$\displaystyle\cong$ $\displaystyle
c\left(\frac{x}{2}\right)^{-a}\left(2-2^{a}\right)\hskip 15.00002pt\mbox{for
$x=\frac{\Lambda}{2t}\ll 1$}$ (46)

Finally, the effective gap for small two-particle interactions is given as

$\displaystyle 2\Delta^{\text{eff}}=\frac{4\delta
U}{\pi}\left(\frac{\Lambda_{\text{IR}}}{4t}\right)^{-U/\pi t}(2-2^{U/\pi t}),$
(47)

where $\Lambda_{\text{IR}}$ is the IR cutoff of the flow. From numerical
observation, we find

$\displaystyle\Lambda_{\text{IR}}=2\Delta^{\text{HF}}=\frac{4\delta U}{\pi},$
(48)

which is the effective gap in first order perturbation theory, see Eq. (42).
For an exemplary parameter set this is illustrated in Fig. 12.

##  Appendix C The modulated interaction model with $Z=4$ at half filling

In this Appendix, we investigate the effective gap in lowest order
perturbation theory (for the self-energy) for the modulated interaction model
with $Z=4$ at half filling. For this we can use the flow equations in momentum
space. Due to the particle hole symmetry, the flow of the on-site potential
vanishes. The flow of the effective hopping parameters are given as

$\displaystyle\partial_{\Lambda}t^{\Lambda}_{1}=-\frac{U_{1}}{2\pi}\sum_{\omega=\pm\Lambda}\int^{\pi}_{-\pi}\frac{dk}{2\pi}\mathcal{G}^{\Lambda}_{1,2}(k,i\omega),$
$\displaystyle\partial_{\Lambda}t^{\Lambda}_{2}=-\frac{U_{2}}{2\pi}\sum_{\omega=\pm\Lambda}\int^{\pi}_{-\pi}\frac{dk}{2\pi}\mathcal{G}^{\Lambda}_{2,3}(k,i\omega),$
$\displaystyle\partial_{\Lambda}t^{\Lambda}_{3}=-\frac{U_{2}}{2\pi}\sum_{\omega=\pm\Lambda}\int^{\pi}_{-\pi}\frac{dk}{2\pi}\mathcal{G}^{\Lambda}_{3,4}(k,i\omega),$
$\displaystyle\partial_{\Lambda}t^{\Lambda}_{4}=-\frac{U_{4}}{2\pi}\sum_{\omega=\pm\Lambda}\int^{\pi}_{-\pi}\frac{dk}{2\pi}e^{-ik}\mathcal{G}^{\Lambda}_{4,1}(k,i\omega),$
(49)

Without the feedback of the self-energy in the single scale propagator, that
is on Hartree-Fock level, the effective hopping parameters at the end of the
flow are given as

$\displaystyle t^{\text{HF}}_{i}$ $\displaystyle=t^{\Lambda=0}_{i}$
$\displaystyle=t-\frac{U_{i}}{\pi}\int^{0}_{\infty}d\Lambda\int^{\pi}_{-\pi}\frac{dk}{2\pi}\frac{t\left[\Lambda^{2}+t^{2}(1-\cos(k))\right]}{\Lambda^{4}+4t^{2}\Lambda^{2}+2t^{4}(1-\cos(k))}$
$\displaystyle=t-\frac{U_{i}}{\pi}\int^{0}_{\infty}d\Lambda\frac{1}{2t}\left\\{1-\frac{\Lambda}{\sqrt{\Lambda^{2}+(2t)^{2}}}\right\\}$
$\displaystyle=t+\frac{U_{i}}{\pi}$ (50)

Without loss of generality, we choose $\varphi_{U}=\pi/2$ and set

$\displaystyle t^{\text{HF}}_{1}=t^{\text{HF}}_{3}=t+\frac{U}{\pi}=\bar{t},$
(51) $\displaystyle t^{\text{HF}}_{2/4}=t+\frac{U\pm\delta
U}{\pi}=\bar{t}\pm\delta\bar{U}.$ (52)

With the effective single-particle Hamiltonian in momentum space $h_{k}$ and
the dispersion relation $E(k)$, one can rewrite the eigenvalue equation
$\det[h_{k}-E(k)]=0$ into

$\displaystyle 2\bar{t}^{2}(\bar{t}^{2}-\delta\bar{U}^{2})\cos
k=E^{4}(k)-AE^{2}(k)+B,$ (53)

where $A=4\bar{t}^{2}+2\delta\bar{U}^{2}$ and $B=\delta\bar{U}^{4}$.
Therefore, the band energy is given as

$\displaystyle
E(k)=\pm\sqrt{\frac{A\pm\sqrt{{A^{2}-4(B-2\bar{t}^{2}(\bar{t}^{2}-\delta\bar{U}^{2})\cos
k)}}}{2}}$ (54)

The gap is located at $k=0$ and

$\displaystyle 2\Delta^{\text{HF}}_{\nu=2}=$ $\displaystyle
2\sqrt{2\bar{t}^{2}+\delta\bar{U}^{2}-2\bar{t}^{2}\sqrt{1+\frac{\delta\bar{U}^{2}}{\bar{t}^{2}}}}$
(55) $\displaystyle\cong$ $\displaystyle\frac{\delta U^{2}}{t\pi^{2}}.$ (56)

We conclude, that even with the linear $U$ and $\delta U$ terms in the
effective hopping parameters, the linear term in the effective gap is
cancelled out. This shows that the effective gap of the modulated interaction
model with $Z=4$ at half filling is at least of second order in the
interaction. In our lowest order truncated functional RG, not all of the
second order terms are taken into account. Therefore, we can not obtain
controlled results within lowest-order truncated functional RG.

## References

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