$H^{*}\mathcal{P}(\phi)$ is an isomorphism.

Similarly for the other cases we found:

  1. $H^{*}\mathrm{Hom}_{\mathcal{A}/\mathcal{B}}(P_{2},P_{2})=\mathrm{Hom}_{\Lambda}(P_{2},P_{2})\bigoplus\langle\alpha\delta(\alpha\delta)^{n}|\ n\geq 0\rangle$,

  2. $H^{*}\mathrm{Hom}_{\mathcal{A}/\mathcal{B}}(P_{1},P_{2})=\mathrm{Hom}_{\Lambda}(P_{1},P_{2})\bigoplus\langle\alpha\delta(\alpha\delta)^{n}\alpha|\ n\geq 0\rangle$,

  3. $H^{*}\mathrm{Hom}_{\mathcal{A}/\mathcal{B}}(P_{2},P_{1})=\mathrm{Hom}_{\Lambda}(P_{2},P_{1})\bigoplus\langle\delta(\alpha\delta)^{n}|\ n\geq 0\rangle$,

which are isomorphic under $H^{*}\mathcal{P}(\phi)$ to the corresponding
spaces in $\mathcal{P}(\Lambda[\omega^{-1}])$.

$\square$

##  6\. Proof of the main results

We now prove Theorem 1.10 and Proposition 1.11. For this we will need two
lemmas.

######  Lemma 6.1.

Let $\Lambda$ be the quotient of a path algebra $KQ$ by a finitely generated
ideal $I=\langle\rho_{1},...,\rho_{r}\rangle$. Let $\overline{Q}_{0}\subseteq
Q_{0}$ be a subset of vertices and $e=\sum\limits_{i\in\overline{Q}_{0}}e_{i}$
the associated idempotent.

There is an isomorphism $[[-]]:eKQe\rightarrow K\overline{Q}$, where
$\overline{Q}$ is the quiver whose set of vertices is $\overline{Q}_{0}$, and
whose arrows from $i$ to $j$ are symbols $[\alpha_{r}...\alpha_{1}]$, where
$\alpha_{r}...\alpha_{1}$ is a path from $i$ to $j$ in $Q$ which passes
through no other vertices of $\overline{Q}_{0}$.

It induces an isomorphism $e\Lambda e\simeq K\overline{Q}/\overline{I}$, where
$\overline{I}=\langle[[\delta\rho_{i}\gamma]]\ |\ \forall i\in\\{1,...,r\\}\
\mathrm{and}\ \delta\in T,\ \gamma\in S\rangle$, with:

  1. -

$T=\\{\alpha_{r}...\alpha_{1}\ |\ r\geq 1,\ \forall i.\alpha_{i}\in Q_{1}\
\mathrm{and}\ \tau(\alpha_{r})\in\overline{Q}_{0}\ \mathrm{and}\ \forall
i.\sigma(\alpha_{i})\in Q_{0}\backslash\overline{Q}_{0}\\}\bigsqcup\\{e_{i}\
|\ i\in\overline{Q}_{0}\\}$, and dually

  2. -

$S=\\{\alpha_{r}...\alpha_{1}\ |\ r\geq 1,\ \forall i.\alpha_{i}\in Q_{1}\
\mathrm{and}\ \sigma(\alpha_{1})\in\overline{Q}_{0}\ \mathrm{and}\ \forall
i.\tau(\alpha_{i})\in Q_{0}\backslash\overline{Q}_{0}\\}\bigsqcup\\{e_{i}\ |\
i\in\overline{Q}_{0}\\}$.

Note that $\overline{Q}$ can have an infinite set of arrows, and that
$\overline{I}$ is not necessarily admissible.

######  Lemma 6.2.

Let $\Lambda=(KQ/\langle I\rangle,|\ .\ |)$ be a graded pinched gentle
algebra, $(\alpha,\beta)$ be a graded Kronecker in $Q$ as in Definition 1.9,
and $\omega=\alpha+\mu\beta$ for some $\mu\in K^{*}$. Suppose that the
characteristic of $K$ is different from 2. There are equivalences:

$\mathcal{D}(\Lambda[\omega^{-1}])\simeq\mathcal{D}(\Lambda_{(\alpha,\beta)})$
and $per(\Lambda[\omega^{-1}])\simeq per(\Lambda_{(\alpha,\beta)})$.

Proof: Since $P_{1}$ and $P_{2}$ become isomorphic in
$\Lambda[\omega^{-1}]=:K\widetilde{Q}/\langle\widetilde{I}\rangle$,
$\mathcal{D}(\Lambda[\omega^{-1}])\simeq\mathcal{D}(e\Lambda[\omega^{-1}]e)$
and $per(\Lambda[\omega^{-1}])\simeq per(e\Lambda[\omega^{-1}]e)$ where
$e=\sum\limits_{l\in Q_{0}\backslash\\{2\\}}e_{l}$.

One can compute $e\Lambda[\omega^{-1}]e=:K\overline{Q}/\overline{I}$ using
Lemma 6.1. First we have:

$\overline{Q}_{1}=\\{[\rho]\ |\ \rho\in\widetilde{Q}_{1}\ \mathrm{and}\
\widetilde{\sigma}(\rho)\neq
2\neq\widetilde{\tau}(\rho)\\}\bigsqcup\\{[\beta^{+}\beta],[\alpha^{+}\alpha],[\delta\alpha],[\delta\beta]\\}$,

and Lemma 6.1 gives the following description of $\overline{I}$:

$\displaystyle\overline{I}=$ $\displaystyle\langle[[\delta\rho_{i}\gamma]]\
|\rho_{i}\in\widetilde{I}=I^{g}\sqcup I^{p}\sqcup\\{\delta\omega-
e_{1},\omega\delta-e_{2}\\}\ \mathrm{and}\ \delta\in T,\ \gamma\in S\rangle$
$\displaystyle=$
$\displaystyle\langle\\{[\beta_{s}...\beta_{1}][\alpha_{r}...\alpha_{1}]\ |\
[\beta_{s}...\beta_{1}],[\alpha_{r}...\alpha_{1}]\in\overline{Q}_{1}\
\mathrm{and}\ \beta_{1}\alpha_{r}\in I^{g}\\}$ $\displaystyle\sqcup$
$\displaystyle\\{[\beta_{s}...\beta_{1}]([\gamma_{v}]+e_{v})\ |\
[\beta_{s}...\beta_{1}]\in\overline{Q}_{1}\ \mathrm{and}\
\beta_{1}(\gamma_{v}+e_{v})\in I^{p}\\}$ $\displaystyle\sqcup$
$\displaystyle\\{([\gamma_{v}]+e_{v})[\beta_{s}...\beta_{1}]\ |\
[\beta_{s}...\beta_{1}]\in\overline{Q}_{1}\ \mathrm{and}\
(\gamma_{v}+e_{v})\beta_{s}\in I^{p}\\}$ $\displaystyle\sqcup$
$\displaystyle\\{[\alpha_{r}...\alpha_{1}]([\gamma_{v}]-e_{v})\ |\
[\alpha_{r}...\alpha_{1}]\in\overline{Q}_{1}\ \mathrm{and}\
\alpha_{1}(\gamma_{v}-e_{v})\in I^{p}\\}$ $\displaystyle\sqcup$
$\displaystyle\\{([\gamma_{v}]-e_{v})[\alpha_{r}...\alpha_{1}]\ |\
[\alpha_{r}...\alpha_{1}]\in\overline{Q}_{1}\ \mathrm{and}\
(\gamma_{v}-e_{v})\alpha_{s}\in I^{p}\\}$ $\displaystyle\sqcup$
$\displaystyle\\{[[\delta\omega-
e_{1}]]=[\delta\alpha]+\mu[\delta\beta]-e_{1},$ $\displaystyle\
[[\alpha^{+}(\omega\delta-
e_{2})\alpha]]=[\alpha^{+}\alpha][\delta\alpha]-[\alpha^{+}\alpha],\
[[\alpha^{+}(\omega\delta-e_{2})\beta]]=[\alpha^{+}\alpha][\delta\beta],$
$\displaystyle\ [[\beta^{+}(\omega\delta-
e_{2})\alpha]]=\mu[\beta^{+}\beta][\delta\alpha],\ [[\beta^{+}(\omega\delta-
e_{2})\beta]]=\mu[\beta^{+}\beta][\delta\beta]-[\beta^{+}\beta]\\}\rangle.$

Let $\gamma:=[\delta\alpha]-\mu[\delta\beta]$. We have
$2[\delta\alpha]=e_{1}+[\delta\alpha]-\mu[\delta\beta]=e_{1}+\gamma$ and
$2\mu[\delta\beta]=e_{1}-([\delta\alpha]-\mu[\delta\beta])=e_{1}-\gamma$.
Renaming $[\beta^{+}\beta]=:\beta^{+}$, $[\alpha^{+}\alpha]=:\alpha^{+}$,
dropping the bracket notation and rewriting the relations with
$\gamma^{\prime}s$ gives the desired description.

$\square$

Proof of Theorem 1.10: Let $\mathcal{A}$ and $\mathcal{B}$ be as in the
Setting 2.13. Theorem 3.1 of [G22] gives the recollement:

${\mathcal{D}(\mathcal{A}/\mathcal{B})}$${\mathcal{D}(\mathcal{A})}$${\mathcal{D}(\mathcal{B})}$

First
$\mathcal{D}(\mathcal{A})\simeq\mathcal{D}(\mathcal{P}(\Lambda))\simeq\mathcal{D}(\Lambda)$.
Then one can show that $\mathrm{End}_{\mathcal{P}(\Lambda)^{pre-tr}}(B)$ is a
formal DG algebra, and thus
$\mathcal{D}(\mathcal{B})\simeq\mathcal{D}(\mathrm{End}_{\mathcal{P}(\Lambda)^{pre-
tr}}(B))\simeq\mathcal{D}(H^{*}\mathrm{End}_{\mathcal{P}(\Lambda)^{pre-
tr}}(B))\simeq\mathcal{D}(K[x]/(x^{2}))$, by Lemma 5.4.

Moreover
$\mathcal{D}(\mathcal{A}/\mathcal{B})\simeq\mathcal{D}((\mathcal{A}/\mathcal{B})_{\circ})\simeq\mathcal{D}(\Lambda[\omega^{-1}])$,
where the last equivalence is given by Proposition 5.3.

The second part of the theorem is given by Lemma 6.2.

$\square$

Proof of Proposition 1.11: By Theorems 2.10, 2.14 and Proposition 2.12,
$per(\Lambda)/\langle
B\rangle\simeq\mathcal{A}^{tr}/\mathcal{B}^{tr}\simeq(\mathcal{A}/\mathcal{B})^{tr}$,
and by Proposition 5.3,
$(\mathcal{A}/\mathcal{B})^{tr}\simeq(\mathcal{A}/\mathcal{B})_{\circ}^{tr}\simeq\mathcal{P}(\Lambda[\omega^{-1}])^{tr}\simeq
per(\Lambda[\omega^{-1}])$. We conclude using $per(\Lambda)/\langle
B\rangle\simeq per(\Lambda)/thick(B)$.

The second part of the proposition is given by Lemma 6.2.

$\square$

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Current address: ] Cooperative Institute for Research in the Atmosphere,
Colorado State University, Fort Collins, Colorado 80521, USA

#  Topological two-band electron-hole superconductors with d-wave symmetry:
a possible application to magic-angle twisted trilayer graphene

Senne Van Loon<EMAIL_ADDRESS>School of Physics, Georgia
Institute of Technology, Atlanta, Georgia 30332, USA  TQC, Universiteit
Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, België  [  C. A. R. Sá de
Melo  School of Physics, Georgia Institute of Technology, Atlanta, Georgia
30332, USA

###### Abstract

We discuss a two-band model for two-dimensional superconductors with electron
and hole bands separated by an energy gap and singlet $d$-wave pairing in each
band. This model exhibits a V-shaped to U-shaped transition in the density of
the states of the superconductor, which is similar to that found in recent
tunneling experiments in magic-angle twisted trilayer graphene Kim  _et al._
(2022). This qualitative difference in behavior occurs when the electron and
hole chemical potentials change, leading to topological quantum phase
transitions between gapless and gapful $d$-wave superconducting states, due to
the annihilation of chiral Dirac fermions at the critical points. Lastly, we
show that direct thermodynamic evidence of these topological quantum phase
transitions can be found in measurements of the compressibility, which
exhibits logarithmic singularities at the transition points.

In the search for high temperature superconductors, more and more evidence
points to two-dimensional (2D) structures as good candidates, but their
theoretical understanding remains limited, because many of these materials
have a complex structure. Specifically, multilayer graphene has been put
forward as a promising testbed for unconventional superconductivity Cao  _et
al._ (2018); Oh  _et al._ (2021); Park  _et al._ (2021); Kim  _et al._ (2022);
Zhang  _et al._ (2022); Park  _et al._ (2022). When the graphene layers are
twisted by a certain ‘magic’ angle, the emerging moiré superlattice allows for
a superconducting phase that can be tuned with electric fields. Experiments in
magic-angle twisted bilayer Cao  _et al._ (2018); Oh  _et al._ (2021) and
trilayer Hao  _et al._ (2021); Park  _et al._ (2021); Kim  _et al._ (2022)
graphene (MATBG and MATTG) point to the existence of the evolution of Bardeen-
Cooper-Schrieffer (BCS) to Bose superconductivity with a nodal order parameter
Lake  _et al._ (2022) as a function of carrier density.

The BCS-Bose evolution has been studied theoretically Nozières and Schmitt-
Rink (1985); Sá de Melo  _et al._ (1993) and experimentally Zwierlein  _et
al._ (2005); Sobirey  _et al._ (2021) in the context of $s$-wave ultracold
atoms, where single band systems are easily created and interactions are
continuously changed. Compared to the smooth crossover in $s$-wave systems,
interacting fermions with higher angular momentum pairing undergo topological
quantum phase transitions (TQPT) when interaction or density are tuned Duncan
and Sá de Melo (2000); Botelho and Sá de Melo (2005, 2005). Recent work has
suggested that high-$T_{c}$ cuprates may exhibit such evolution Harrison and
Chan (2022), while another indicates that experimental evidence is absent Sous
_et al._ (2022). Albeit the experimental success with s-wave systems,
observing tunable $p$\- or $d$-wave superfluidity in cold-atom experiments has
been a challenge Regal  _et al._ (2003); Beaufils  _et al._ (2009); Venu  _et
al._ (2023), so the advent of twisted graphene structures created an exciting
new avenue to study such TQPT in superconductors Lewandowski  _et al._ (2023).

In particular, recent experiments in MATTG Kim  _et al._ (2022) found evidence
of a phase transition between gapless and gapped superconducting phases, as
indicated by a dramatic change from a V-shaped to a U-shaped profile in the
tunneling conductance, which measures the density of states (DOS). As
suggested in the experimental work of the Caltech group Kim  _et al._ (2022),
a possible explanation for this is the presence of a nodal $d$-wave order
parameter in a system with an electron and a hole band. Here, we propose a
microscopic theory of such electron-hole superconductors and find a similar
behavior to that measured in conductance profiles.

Our work paves the way for a better understanding of two-band superconductors
with electron and hole bands and tunable chemical potentials or densities. Our
model presents a rich phase diagram with distinct superconducting phases that
are topological in nature. The $d$-wave symmetry of the pairing interaction
allows for multiple TQPT, distinguished by the presence of a gap in the
quasiparticle spectrum, which emerges via the annihilation of chiral Dirac
fermions. We show that these transitions are robust to the breaking of
particle-hole symmetry and describe qualitatively the evolution from V-shaped
to U-shaped density of states, similarly to tunneling conductance experiments
in twisted trilayer graphene Kim  _et al._ (2022). Furthermore, we show that a
direct thermodynamic signature of this transition is seen in the
compressibility, which exhibits a logarithmic singularity at the transition
point.

Effective action:  In general, the action for a two-band system of spin-1/2
fermions is

$\displaystyle S$
$\displaystyle=\beta\sum_{k,\sigma=\uparrow,\downarrow}\sum_{j=\pm}\bar{\psi}^{(j)}_{k,\sigma}(-\mathrm{i}\hbar\omega_{n}+\xi_{{\bf{k}}}^{(j)})\psi_{k,\sigma}^{(j)}$
(1)
$\displaystyle\kern-10.00002pt-\frac{\beta}{L^{2}}\\!\\!\\!\sum_{k,k^{\prime},q}\sum_{i,j=\pm}\\!\\!V_{ij}({\bf{k}},{\bf{k}^{\prime}})\bar{\psi}^{(i)}_{k+\frac{q}{2},\uparrow}\bar{\psi}^{(i)}_{-k+\frac{q}{2},\downarrow}\psi^{(j)}_{-k^{\prime}+\frac{q}{2},\downarrow}\psi^{(j)}_{k^{\prime}+\frac{q}{2},\uparrow},$

where the fields $\psi_{k,\sigma}^{(j)}$ represent fermions of spin $\sigma$
in band $j$. We use the notation $k=({\bf{k}},\omega_{n})$, with ${\bf{k}}$
being momentum, $\omega_{n}=\pi(2n+1)/\beta$ fermionic Matsubara frequencies,
and $\beta=\hbar/k_{\rm B}T$. From this action, the partition sum is defined
as
$\mathcal{Z}=\int\mathcal{D}\bar{\psi}^{(+)}\mathcal{D}\psi^{(+)}\mathcal{D}\bar{\psi}^{(-)}\mathcal{D}\psi^{(-)}e^{-S/\hbar}$.
We study a system with an electron (particle) and a hole band separated by a
gap ${\epsilon_{\rm bg}}$ (see Fig. 1), and denote them by $j=+$ and $j=-$
respectively. The kinetic energy of the bands is given by
$\epsilon_{{\bf{k}}}^{(\pm)}=\pm(\hbar^{2}{\bf{k}}^{2}/2m^{(\pm)}+{\epsilon_{\rm
bg}}/2)$, and
$\xi_{\bf{k}}^{(\pm)}=\epsilon_{{\bf{k}}}^{(\pm)}\mp\mu^{(\pm)}$, where the
chemical potential $\mu^{(+)}$ $(\mu^{(-)})$ for the electron (hole) is
measured with respect to the bottom (top) of the electron (hole) band
$+{\epsilon_{\rm bg}}/2$ $(-{\epsilon_{\rm bg}}/2)$. We allow for both
intraband and interband coupling, and write the pairing interaction
$V_{ij}({\bf{k}},{\bf{k}^{\prime}})=\lambda_{ij}\Gamma_{\bf{k}}\Gamma_{\bf{k}^{\prime}}$
in separable form. The symmetry factors $\Gamma_{\bf{k}}$ characterize the
angular momentum $\ell$ of the pairing channel Duncan and Sá de Melo (2000);
Botelho and Sá de Melo (2005), which we take to be of the $d$-wave type
$(\ell=2)$ in two dimensions, such that $\Gamma_{\bf{k}}=\gamma_{{\bf{k}}}\cos
2\phi$ and
$\gamma_{{\bf{k}}}=(|{\bf{k}}|/k_{1})^{2}(1+|{\bf{k}}|/k_{0})^{-5/2}$. Here,
$k_{0}$ and $k_{1}$ set the scales at high and low $|{\bf k}|$ respectively,
with $R_{0}\sim k_{0}^{-1}$ being the effective interaction range. We present
our results in units of $\epsilon_{0}=\hbar^{2}k_{0}^{2}/2m^{(+)}$ and set
$k_{1}=k_{0}$ without loss of generality, as $k_{1}$ only scales the
interaction prefactors $\lambda_{ij}$.

Figure 1:  Energy bands of the free fermions with (right) and without (left)
mass imbalance.

Superconductivity emerges through the complex Hubbard-Stratonovich pair fields
$\Phi_{r}^{(\pm)}=|\Delta^{(\pm)}|(1+\phi_{r}^{(\pm)})\mathrm{e}^{\mathrm{i}\theta_{r}^{(\pm)}}$,
where $\Delta^{(\pm)}$ is order parameter, while $\phi_{r}^{(\pm)}$ is the
modulus and $\theta_{r}^{(\pm)}$ is the phase fluctuation in each band
$(\pm)$. For slowly varying bosonic fields with respect to fermionic degrees
of freedom, the action becomes

$\displaystyle S_{\rm HS}=$
$\displaystyle\int\frac{d^{3}r}{L^{2}}\bigg{[}\sum_{k}\bar{\eta}_{k}G^{-1}_{k}(r)\eta_{k}+\sum_{i,j=\pm}(\Phi^{(i)}_{r})^{\dagger}g_{ij}\Phi^{(j)}_{r}$
$\displaystyle\kern
30.00005pt+\sum_{{\bf{k}},j=\pm}j\Big{(}\xi_{{\bf{k}}}^{(j)}+\chi_{{\bf{k}}}^{(j)}(r)+\gamma_{{\bf{k}}}^{(j)}(r)\Big{)}\bigg{]}.$
(2)

We use the notation $r=({\bf{r}},\tau)$, $\int
d^{3}r=\int_{0}^{\beta}d\tau\int_{L^{2}}d^{2}{\bf{r}}$, define the effective
coupling
$g_{ij}=ijL^{2}\lambda_{-j,-i}/(\lambda_{++}\lambda_{--}-\lambda_{+-}\lambda_{-+})$,
and the functions
$\gamma_{\bf{k}}^{(\pm)}(r)=\mathrm{i}\hbar\partial_{\tau}\theta_{r}^{(\pm)}/2+(\hbar^{2}/8m)({\bm{\nabla}}\theta_{r}^{(\pm)})^{2}$
and
$\chi_{\bf{k}}^{(\pm)}(r)=(\hbar^{2}/4m)\mathrm{i}\nabla^{2}{\theta_{r}^{(\pm)}}-(\hbar^{2}/2m){\bm{\nabla}}{\theta_{r}^{(\pm)}}\cdot{\bf{k}}$.
The fermionic fields are represented by the Nambu spinors
$\bar{\eta}_{k}=(\bar{\psi}^{(+)}_{k,\uparrow},\psi^{(+)}_{-k,\downarrow},\bar{\psi}^{(-)}_{k,\uparrow},\psi^{(-)}_{-k,\downarrow})$,
and the inverse quasiparticle propagator is a blockdiagonal $4\times 4$ matrix
$G^{-1}={\rm diag}[(G^{(+)})^{-1},(G^{(-)})^{-1}]$ that can be separated into
saddle point and fluctuation parts
$(G^{(\pm)})^{-1}=(G_{0}^{(\pm)})^{-1}+F^{(\pm)}$, where

$\displaystyle(G_{0}^{(\pm)})^{-1}=$
$\displaystyle\begin{pmatrix}-\mathrm{i}\hbar\omega_{n}+\xi_{{\bf{k}}}^{(\pm)}&-|\Delta_{{\bf{k}}}^{(\pm)}|\\\
-|\Delta_{{\bf{k}}}^{(\pm)}|&-\mathrm{i}\hbar\omega_{n}-\xi_{{\bf{k}}}^{(\pm)}\end{pmatrix},$
(3) $\displaystyle F^{(\pm)}=$
$\displaystyle\begin{pmatrix}\chi_{{\bf{k}}}^{(j)}(r)+\gamma_{{\bf{k}}}^{(j)}(r)&-|\Delta_{{\bf{k}}}^{(\pm)}|\phi_{r}^{(\pm)}\\\
-|\Delta_{{\bf{k}}}^{(\pm)}|\phi_{r}^{(\pm)}&\chi_{{\bf{k}}}^{(j)}(r)-\gamma_{{\bf{k}}}^{(j)}(r)\end{pmatrix},$
(4)

with $\Delta_{{\bf{k}}}^{(\pm)}\equiv\Delta^{(\pm)}\Gamma_{{\bf{k}}}$.

Integrating the fermions leads to the effective action

$\displaystyle S_{\rm eff}$ $\displaystyle=\sum_{j=\pm}\bigg{[}S_{\rm
sp}^{(j)}+\frac{1}{2}\int
d^{3}r\bigg{\\{}A^{(j)}(\hbar\partial_{\tau}\theta_{r}^{(j)})^{2}$
$\displaystyle+\rho_{s}^{(j)}(\nabla{\theta_{r}^{(j)}})^{2}+C^{(j)}(\phi_{r}^{(j)})^{2}+\mathrm{i}D^{(j)}\phi_{r}^{(j)}\hbar\partial_{\tau}\theta_{r}^{(j)}\bigg{\\}}\bigg{]}$
$\displaystyle+2g_{+-}\int\frac{d^{3}r}{L^{2}}|\Phi_{r}^{(+)}||\Phi_{r}^{(-)}|\cos(\theta_{r}^{(+)}-\theta_{r}^{(-)}),$
(5)

The first term in Eq. (5) describes the saddle point result

$\displaystyle\frac{1}{\beta}S_{\rm
sp}^{(\pm)}=g_{\pm\pm}|\Delta^{(\pm)}|^{2}$ (6)
$\displaystyle\quad+\sum_{{\bf{k}}}\bigg{[}\pm\xi_{{\bf{k}}}^{(\pm)}-E_{{\bf{k}}}^{(\pm)}-2k_{\rm
B}T\log\big{(}1+\mathrm{e}^{-E_{{\bf{k}}}^{(\pm)}/k_{\rm B}T}\big{)}\bigg{]},$

where
$E_{\bf{k}}^{(\pm)}=\sqrt{(\xi_{{\bf{k}}}^{(\pm)})^{2}+|\Delta_{{\bf{k}}}^{(\pm)}|^{2}}$
are the quasiparticle energies. The second term is the effective action for
each band describing phase $\theta_{r}^{(\pm)}$ and modulus fluctuations
$\phi_{r}^{(\pm)}$, where $A^{(\pm)}$ is related to the compressibility at
fixed temperature and order parameter; $\rho^{(\pm)}$ represents the
superfluid density; $C^{(\pm)}$ describes the modulus fluctuations; and
$D^{(\pm)}$ represents the coupling between modulus and phase fluctuations.
The last line in Eq. (5) describes the Josephson coupling between the bands.

We consider the saddle point approximation at $T=0$, where the thermodynamic
potential $\Omega=-k_{\rm B}T\ln\mathcal{Z}$ reduces to

$\displaystyle\Omega_{\rm sp}=$
$\displaystyle\sum_{j=\pm}\Big{[}g_{jj}|\Delta^{(j)}|^{2}+\sum_{{\bf{k}}}\Big{(}j\xi_{{\bf{k}}}^{(j)}-E_{{\bf{k}}}^{(j)}\Big{)}\Big{]}$
$\displaystyle+2g_{+-}|\Delta^{(+)}||\Delta^{(-)}|\cos\theta^{\prime},$ (7)

with $\theta^{\prime}$ being the relative phase difference between the order
parameters in each band. The order parameters are found from the condition
$\partial\Omega_{\rm sp}/\partial|\Delta^{(\pm)}|=0$, leading to two coupled
equations

$g_{\pm\pm}|\Delta^{(\pm)}|+g_{+-}|\Delta^{(\mp)}|\cos\theta^{\prime}=\sum_{\bf{k}}\frac{|\Delta^{(\pm)}|\Gamma_{{\bf{k}}}^{2}}{2E_{{\bf{k}}}^{(\pm)}}.$
(8)

The order parameters $\Delta^{(\pm)}$ are functions of $\mu^{(+)}$ and
$\mu^{(-)}$ or equivalently of the total chemical potential
$\mu=(\mu^{(+)}+\mu^{(-)})/2$ and the imbalance
$\zeta=(\mu^{(+)}-\mu^{(-)})/2$.

Figure 2:  Quasiparticle spectrum for different values of the total chemical
potential with $\widetilde{\epsilon}_{\rm b}=-4\epsilon_{0}$. Here,
$|\Delta^{(+)}|=|\Delta^{(-)}|$, $\zeta=0$ and $m^{(+)}=m^{(-)}$, such that
$E_{{\bf{k}}}^{(+)}=E_{{\bf{k}}}^{(-)}$.

Topological quantum phase transition:  The chemical potentials
${\widetilde{\mu}}^{(\pm)}=\mu^{(\pm)}-\epsilon_{\rm bg}/2$ play an important
role in determining the phase diagrams of our system, as can be seen in the
quasiparticle spectrum. In the most general case, four quasiparticle/quasihole
(qp-qh) bands are present, two for the electrons $\pm E_{{\bf{k}}}^{(+)}$ and
two for the holes $\pm E_{{\bf{k}}}^{(-)}$. Due to the nature of the $d$-wave
interactions, TQPT of the Lifshitz type Duncan and Sá de Melo (2000); Botelho
and Sá de Melo (2005) occur at ${\widetilde{\mu}}^{(\pm)}=0$, when the
quasiparticle energies go to zero at $|{\bf{k}}|=0$. For
${\widetilde{\mu}}^{(\pm)}>0$, the quasiparticle spectrum is gapless, with
$E_{{\bf{k}}}^{(\pm)}=0$ at
$\hbar|{\bf{k}}|=\sqrt{2m^{(\pm)}\widetilde{\mu}^{(\pm)}}\equiv\hbar
k_{\mu}^{(\pm)}$ and $\phi=\pi/4+n\pi/2$. There are four chiral Dirac
quasiparticles in the electron band and four in the hole band with topological
indices (winding numbers)

$N_{w}^{(\pm)}=\frac{1}{2\pi}\oint_{C}d\ell~{}{\hat{\bf z}}\cdot{\bf
m}_{\pm}\times\frac{d{\bf m}_{\pm}}{d\ell},$ (9)

where ${\bf m}_{\pm}=(\xi_{\bf k}^{(\pm)},|\Delta_{\bf k}^{(\pm)}|)/E_{\bf
k}^{(\pm)}$, and the path $C$ encircles each Dirac point separately. Notice
that $N_{w}^{(\pm)}$ alternates between $\mp 1$ $(\phi=\pi/4,5\pi/4)$ and $\pm
1$ $(\phi=3\pi/4,7\pi/4)$ as the Dirac points are circled counterclockwise
with increasing values of $\phi\in[0,2\pi]$. The net chirality of the system
is zero, as there is no global chiral symmetry breaking. However, the
annihilation of these chiral Dirac quasiparticles at
${\widetilde{\mu}}^{(\pm)}=0$ defines TQPT in momentum space. For
${\widetilde{\mu}}^{(\pm)}<0$, the spectrum is fully gapped, with
$\min_{{\bf{k}}}\\{E_{\bf{k}}^{(\pm)}\\}=|{\widetilde{\mu}}^{(\pm)}|$ at
$|{\bf{k}}|=0$, and $N_{w}^{(\pm)}$ is always zero, as there are no chiral
Dirac quasiparticles. An example of the TQPT is visible in Fig. 2, where the
quasiparticle energies are shown in the simplest case where
$E_{{\bf{k}}}^{(+)}=E_{{\bf{k}}}^{(-)}$.

Measurements of the density of states (DOS) via tunelling conductance in MATTG
Kim  _et al._ (2022) revealed a remarkable V-shaped to U-shaped transition for
sufficiently low-carrier densities as the chemical potential is tuned. Below,
we show that such transition naturally occurs in our model and is a direct
consequence of the chemical potentials
${\widetilde{\mu}}^{(\pm)}=\mu^{(\pm)}-\epsilon_{\rm bg}/2$ being the relevant
small energy scale in the problem. For sufficiently weak interactions or high
carrier densities and $\widetilde{\mu}^{(\pm)}>0$, $E_{\bf k}^{(\pm)}$ has a
local maximum at ${\bf k=0}$ and saddle points at $\phi=n\pi/2$ and
$|{\bf{k}}|<k_{\mu}^{(\pm)}$ leading to four peaks in the DOS at low energies,
two on the positive and two on the negative side, when
${\widetilde{\mu}}^{(\pm)}\sim|\Delta^{(\pm)}|\gamma_{k_{\mu}^{(\pm)}}$. For
strong interactions or low carrier densities and $\widetilde{\mu}^{(\pm)}>0$,
the saddle points at nonzero ${\bf k}$ become indistinguishable from the
maximum at ${\bf k}={\bf 0}$ producing two peaks in the DOS at low energies,
one on the positive and one on the negative side, when
${\widetilde{\mu}}^{(\pm)}\ll|\Delta^{(\pm)}|\gamma_{k_{\mu}^{(\pm)}}$. A two-
peak structure in the V-shaped phase is seen in the MATTG DOS experiments Kim
_et al._ (2022), suggesting that this system is in the strong interaction
regime.

The TQPT described above are controlled only by the chemical potentials
$\mu^{(\pm)}$, such that specific values of the masses $m^{(\pm)}$,
interactions $g_{++},g_{--},g_{+-}$, or order parameters $|\Delta^{(\pm)}|$ do
not matter. However, the transition from V-shaped to U-shaped in the low-
energy DOS, with two peaks only, requires that the interactions are
sufficiently strong to eliminate saddle points at nonzero ${\bf k}$ in $E_{\bf
k}^{(\pm)}$, that is, it requires that
${\widetilde{\mu}}^{(\pm)}\ll|\Delta^{(\pm)}|\gamma_{k_{\mu}^{(\pm)}}$. To
describe TQPT controlled by $\mu^{(\pm)}$, the simplest choices of parameters
are intraband interactions $g_{++}=g_{--}$ $(\lambda_{++}=\lambda_{--})$, band
masses $m^{(+)}=m^{(-)}$, and small Josephson coupling $g_{+-}$. Thus, in Eq.
(8), we set $|\Delta^{(+)}|=|\Delta^{(-)}|\equiv|\Delta|$ and $g_{+-}\to 0$.
Finally, we fix the interaction by using the relation between the two-body
binding energy ${\widetilde{\epsilon}}_{\rm b}^{(\pm)}=\epsilon_{\rm
b}^{(\pm)}-\epsilon_{\rm bg}$ with respect to $\epsilon_{\rm bg}$ and the
interaction parameters
$L^{2}/\lambda_{\pm\pm}=\sum_{\bf{k}}|\Gamma_{\bf{k}}|^{2}/(\hbar^{2}{\bf{k}}^{2}/m^{(\pm)}-{\widetilde{\epsilon}}_{\mathrm{b}}^{(\pm)})$.
When the electron and hole masses and the electron-electron and hole-hole
interactions are the same, that is, $m^{(+)}=m^{(-)}$ and
$\lambda_{++}=\lambda_{--}$, the binding energies ${\widetilde{\epsilon}}_{\rm
b}^{(+)}={\widetilde{\epsilon}}_{\rm b}^{(-)}={\widetilde{\epsilon}}_{\rm b}$
are also identical.

Figure 3:  (Top) Density plot of $|\Delta|$ vs $\zeta/\epsilon_{0}$ and
$\widetilde{\mu}/\epsilon_{0}=(\mu-{\epsilon_{\rm bg}}/2)/\epsilon_{0}$ for
$m^{(-)}=m^{(+)}$ and ${\widetilde{\epsilon}}_{\rm b}=-4\epsilon_{0}$. The red
lines separate the different superfluid phases, while the blue line shows the
transition to an insulating phase where $|\Delta|=0$. The green dots show
critical values for $|\zeta_{c}|$ above which no fully gapped
superconductivity is possible. In (a) both the electron and hole sectors are
gapped (Bose-like), in (b) the electron sector is gapless (BCS-like), while
the hole sector is gapped (Bose-like), in (c) the electron sector is gapped
(Bose-like), while the hole sector is gapless (BCS-like), and in (d) both the
electron and hole sectors are gapless (BCS-like). The white dots show the
values of $\mu$ and $\zeta$ used in the bottom panel. (Bottom) The
dimensionless density of states
$\widetilde{D}(\varepsilon)=\epsilon_{0}D{(\varepsilon)}$ vs.
$\varepsilon/\epsilon_{0}$ for $\zeta=0$, and various $\widetilde{\mu}$ with
${\widetilde{\epsilon}}_{\rm b}=-4\epsilon_{0}$.

Phase diagram:  On the top panel of Fig. 3, the phase diagram and $|\Delta|$
are shown in the ${\widetilde{\mu}}$-$\zeta$ plane for
${\widetilde{\epsilon}}_{\rm b}=-4\epsilon_{0}$ ($\lambda_{\pm\pm}\simeq
121.9\epsilon_{0}/k_{0}^{2}$), which guarantees strong coupling in the sense
of ${\widetilde{\mu}}^{(\pm)}\ll|\Delta^{(\pm)}|\gamma_{k_{\mu}^{(\pm)}}$. As
seen below, this choice of $\widetilde{\epsilon}_{\rm b}$ gives a two-peaked
DOS at low energies with a clear V-shaped to U-shaped transition. There are
five different phases depending on the chemical potentials of the electrons
and holes. Each quasiparticle/quasihole band undergoes a Lifshitz transition
when ${\widetilde{\mu}}^{(\pm)}=0$, or equivalently when
${\widetilde{\mu}}=\pm\zeta$, shown as red lines in the phase diagram. When
both ${\widetilde{\mu}}^{(\pm)}<0$, the quasiparticle bands are fully gapped,
with both electron and hole sectors behaving as Bose-like superconductors
[phase (a)]. Conversely, when ${\widetilde{\mu}}^{(\pm)}>0$, the quasiparticle
bands are gapless, with both electron and hole sectors behaving as BCS-like
superconductors [phase (d)]. When ${\widetilde{\mu}}^{(+)}>0$ and
${\widetilde{\mu}}^{(-)}<0$, the quasiparticles for the electron sector are
gapless (BCS-like), while the quasiparticles for the hole sector are fully
gapped (Bose-like) [phase (b)]. When ${\widetilde{\mu}}^{(+)}<0$ and
${\widetilde{\mu}}^{(-)}>0$, the quasiparticles for the electron sector are
fully gapped (Bose-like), while the quasiparticles for the hole sector are
gapless (BCS-like) [phase (c)]. The transition line into a normal (insulating)
state with $|\Delta|=0$, where no charge carriers are available, is shown in
blue on the top panel of Fig. 3. Note that an insulating phase is only present
when both ${\widetilde{\mu}}^{(\pm)}<0$, or ${\widetilde{\mu}}<-|\zeta|$.
Outside of this region, the right hand side of Eq. (8) with $|\Delta|=0$
contains an imaginary part and a solution can never be found. Thus, there is a
critical value of the imbalance $|\zeta_{c}|$ (shown as green dots in Fig. 3)
indicating the largest value of $|\zeta|$ where fully gapped superfluidity is
possible. For $|\zeta|>|\zeta_{c}|$ and ${\widetilde{\mu}}<-|\zeta|$, the
system is always in the insulating state.

For $m^{(+)}=m^{(-)}$, the phase diagram is completely symmetric around
$\zeta=0$, reflecting the particle-hole symmetry in the energy spectrum.
However, for $m^{(+)}\neq m^{(-)}$ or $\lambda_{++}\neq\lambda_{--}$,
particle-hole symmetry is broken, but the red phase boundaries remain the
same, while the blue boundary becomes asymmetric with respect to $\zeta$.

Density of states:  The DOS for electrons or holes
$D^{(\pm)}(\varepsilon)=-\sum_{{\bf{k}}}{\rm
Im}[(G_{0}^{(\pm)}({\bf{k}},\varepsilon+\mathrm{i}0^{+}))_{11}]/\pi$ is found
by analytically continuing
$\mathrm{i}\hbar\omega_{n}\rightarrow\varepsilon+\mathrm{i}0^{+}$ in the
quasiparticle propagator defined in Eq. (3). The total density of states is
the sum

$\displaystyle D(\varepsilon)=$
$\displaystyle\\!\\!\sum_{{\bf{k}},j=\pm}\Big{[}\big{(}U_{{\bf{k}}}^{(j)}\big{)}^{2}\delta\big{(}\varepsilon-
E_{{\bf{k}}}^{(j)}\big{)}+\big{(}V_{{\bf{k}}}^{(j)}\big{)}^{2}\delta\big{(}\varepsilon+E_{{\bf{k}}}^{(j)}\big{)}\Big{]},$
(10)

where
$U_{\bf{k}}^{(\pm)}=\sqrt{(1+\xi_{\bf{k}}^{(\pm)}/E_{\bf{k}}^{(\pm)})/2}$ and
$V_{\bf{k}}^{(\pm)}=\sqrt{(1-\xi_{\bf{k}}^{(\pm)}/E_{\bf{k}}^{(\pm)})/2}$ are
the Bogoliubov coefficients.

The black lines in the bottom panel of Fig. 3 show the DOS for
$m^{(+)}=m^{(-)}$. If also the chemical potentials of electrons and holes are
equal, $\mu^{(+)}=\mu^{(-)}$, the DOS is particle-hole symmetric. Only when
${\widetilde{\mu}}<-|\zeta|$ is the system fully gapped, and a U-shape can be
discerned in the DOS, with $D(\varepsilon)$ being zero within the gap between
the quasiparticle and quasihole energies. In this case, the electron and hole
sectors exhibit Bose-like superconductivity. When one or both of the
quasiparticle bands undergo a Lifshitz phase transition to the gapless state,
either the electron or hole sector or both exhibit BCS-like superconductivity,
and the DOS acquires a V-shape, only going to zero at $\varepsilon=0$. For
$\zeta=0$, the peaks at the gap edges for the U-shaped regime occur at
$\varepsilon=\pm|{\widetilde{\mu}}|$, and the two peaks in the V-shaped region
also occur at $\varepsilon=\pm|{\widetilde{\mu}}|$, since the system is in the
strongly interacting regime. A similar transition was observed in the
tunneling conductance of MATTG Kim  _et al._ (2022).

Particle-hole symmetry can be broken in multiple ways. When $\zeta\neq 0$,
three different phases emerge with changing ${\widetilde{\mu}}$, and two
asymmetric peaks are present at $\varepsilon=\mp\widetilde{\mu}^{(\pm)}$. When
$m^{(+)}\neq m^{(-)}$, particle-hole symmetry is broken even at $\zeta=0$, see
the dashed red lines in the bottom panel of Fig. 3. The DOS becomes larger on
the hole side, because the mass of the holes is larger $m^{(-)}=2m^{(+)}$.
However, the largest Van Hove singularity switches from positive to negative
energies at the TPQT when crossing the phase boundary from BCS-like [phase
(d)] to Bose-like [phase (a)], a feature that is controlled by the Bogoliubov
coefficients, which act as the residues of the poles at $\varepsilon=\pm
E_{\bf k}^{(\pm)}$.

Figure 4:  The dimensionless compressibility
$\chi=n^{2}\kappa(2\pi\epsilon_{0}/k_{0}^{2})$ as a function of the total
chemical potential $\widetilde{\mu}$ for different values of the imbalance
$\zeta$, with $\widetilde{\epsilon}_{\rm b}=-4\epsilon_{0}$ and
$m^{(+)}=m^{(-)}$.

Compressibility:  In Fig. 4, we show the compressibility
$n^{2}\kappa=dn/d\mu|_{T}$ as a thermodynamic signature of TQPT. For $d$-wave
pairing in 2D, logarithmic singularities Duncan and Sá de Melo (2000); Botelho
and Sá de Melo (2005) exist at the phase boundaries where chiral Dirac
fermions annihilate and gaps in the quasiparticle excitation spectrum of the
electron and/or hole sectors emerge. When $\mu^{(+)}=\mu^{(-)}$ or $\zeta=0$,
there is one peak, while when $\mu^{(+)}\neq\mu^{(-)}$ or $\zeta\neq 0$ there
are two. The number of peaks represents the number of TQPT of the Lifshitz-
type that are crossed. This TQPT is one possible interpretation for the change
from V-shaped to U-shaped in the tunnelling data of MATTG Kim  _et al._
(2022), where the symmetry of the order parameter does not change. We
emphasize, however, that such TQPT from BCS-like to Bose-like $d$-wave
superconductivity, must be accompanied by logarithmic singularities in the
compressibility.

Conclusion:  We presented a simple two-band model for two-dimensional
superconductors with electron and hole bands and singlet $d$-wave pairing in
each band. We showed that this model exhibits a V-shaped to U-shaped
transition in the density of states of the superconductor similar to that
found in tunneling experiments on magic-angle twisted trilayer graphene Kim
_et al._ (2022). This transition is topological in nature and is associated
with the annihilation of chiral Dirac fermions, as the electron and hole
sectors evolve from BCS-like to Bose-like superconductivity. Furthermore, we
emphasized that the V-shaped to U-shaped transition also has an important
thermodynamic signature in the compressibility, where a logarithmic
singularity must be observed whenever topological quantum phase transition
boundaries are crossed. As an important outlook, we note that it is necessary
to investigate the effects of disorder and of attractive interactions between
electrons and holes to test the robustness of the results presented here.

###### Acknowledgements.

Acknowledgments: We thank the Belgian American Educational Foundation for
support.

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