Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2309.11440 · Physical review. B./Physical review. B

Flat-band superconductivity in a system with a tunable quantum metric: The stub lattice

M. Thumin, G. Bouzerar

Published Jun 12, 2023• 22 citations
arXivHTMLPDF

Abstract

Over the past years, one witnesses a growing interest in flat band (FB) physics which has become a playground for exotic phenomena. In this study, we address the FB superconductivity in one-dimensional stub chain. In contrast to the sawtooth chain or the Creutz ladder, for a given strength of the attractive electron-electron interaction, the stub chain allows the tuning of the real space spreading of the FB eigenstates (quantum metric or QM). We study in detail the interplay between the interaction strength and the mean value of the QM $\ensuremath{\langle}g\ensuremath{\rangle}$ on the pairings and on the superfluid weight ${D}_{s}$. Our calculations reveal several interesting and intriguing features. For instance, in the weak coupling regime, ${D}_{s}$ with respect to $\ensuremath{\langle}g\ensuremath{\rangle},$ exhibits two different types of behavior. Despite the fact that the pairings differs drastically, ${D}_{s}$ scales linearly with the QM only when its $\ensuremath{\langle}g\ensuremath{\rangle}$ is large enough (small gap limit). On the other hand, when the QM is of small amplitude an unusual power law is found, more precisely ${D}_{s}\ensuremath{\propto}{\ensuremath{\langle}g\ensuremath{\rangle}}^{\ensuremath{\nu}}$, where $\ensuremath{\nu}\ensuremath{\rightarrow}2$ in the limit of large single particle gap. In addition to the numerical calculations, we have provided several analytical results which shed light on the physics in both the weak and strong coupling regime. Finally, we have addressed the impact of the thermal fluctuations on the superfluid weight.

DOI
10.1103/physrevb.107.214508

Related papers

Designer flat bands: Topology and enhancement of superconductivity

Si Min Chan, Benoît Grémaud, G. G. Batrouni

We construct quasi-one-dimensional topological and nontopological three-band lattices with a tunable band gap and winding number of the flat band. Using full multiband mean field (MF) and exact density matrix renormalization group (DMRG) calculations, we show explicitly how the band gap affects pairing and superconductivity (SC) in a Hubbard model with attractive interactions. We obtain excellent agreement between MF and DMRG calculations. The SC weight ${D}_{s}$ on the gapped topological, $W\ensuremath{\ne}0$, flat band increases linearly with interaction strength $U$ for low values and with a slope that depends on the details of the compact localized state at $U=0$. As $U\ensuremath{\rightarrow}0$ for the gapped nontopological flat band ($W=0$), ${D}_{s}$ decays with a power law faster than quadratically but slower than exponentially. In the gapless case (flat band touching the band above it), we find at low $U$ (for both $W=0$ and $W\ensuremath{\ne}0$) that ${D}_{s}\ensuremath{\propto}{U}^{\ensuremath{\varphi}}$, with $\ensuremath{\varphi}<1$. In other words, ${D}_{s}$ increases faster than linearly for low $U$, thus favoring SC at weak interaction more than the gapped case. Our results reestablish that the BCS mean field and quantum metric alone are insufficient to characterize SC at weak coupling.

arXiv:2206.10651•Physical review. B./Physical review. B•Sep 22, 2022•25 citations

Correlation functions and characteristic lengthscales in flat band superconductors

M. Thumin, G. Bouzerar

The possibility of an unconventional form of high temperature superconductivity in flat band (FB) material does not cease to challenge our understanding of the physics in correlated systems. Here, we calculate the normal and anomalous one-particle correlation functions in various one and two dimensional FB systems and systematically extract the characteristic lengthscales. When the Fermi energy is located in the FB, it is found that the coherence length ( \xi ξ ) is of the order of the lattice spacing and weakly sensitive to the strength of the electron-electron interaction. Recently, it has been argued that in FB compounds \xi ξ could be decomposed into a conventional part of BCS type ( \xi_{BCS} ξ B C S ) and a geometric contribution which characterises the FB eigenstates, the quantum metric ( \langle g \rangle ⟨ g ⟩ ). However, by calculating the coherence length in two possible ways, our calculations show that \xi ≠ \sqrt{\langle g \rangle.} ξ ≠ ⟨ g ⟩ . This may suggest that the link between QM and coherence length is more complex, and leaves us with the open question: what is the appropriate definition of the coherence length in flat-band systems?

arXiv:2405.06215•SciPost Physics•Jan 21, 2025•9 citations

Superconductivity in geometrically and topologically nontrivial lattice models

Aleksi Julku

In this thesis, superconductivity of fermions in two-dimensional Hubbard lattice models is theoretically studied. Each of the studied lattice structures features intriguing properties that provide valuable insight to understand superconductivity in general. While real condensed matter systems are too complicated to tackle with theoretical tools, the Hubbard model is one of the simplest quantum many-body models to possess a wide spectrum of physically relevant phenomena, from superconductivity to magnetism and topological order. Hubbard models are also interesting as they can be experimentally realized and accurately controlled by utilizing ultracold gases. The first part of the thesis explains the required theoretical background of superconductivity in the level of mean-field and linear response theories. Specifically, we primarily focus on the superfluid weight which implies the existence of the Meissner effect and dissipationless current, defining properties of superconductors. We further discuss how the Berezinskii-Kosterlitz-Thouless (BKT) transition, related to the loss of superconductivity due to the thermal fluctuations of the superconducting order parameter in high enough temperatures, can be accessed via the superfluid weight. In the second part, the main results are discussed. We show how the superconductivity of lattice models featuring dispersionless Bloch bands, so-called flat bands, can be large and originating from the quantum geometric properties of the Bloch states of the flat bands. This is in contrast to conventional dispersive bands where superconductivity is determined by the effective mass of the particles, i.e. by the dispersion relation of the Bloch bands alone and not by the Bloch states. This is well demonstrated in the Lieb lattice geometry, studied in Publication I, which features both a strictly flat and two dispersive Bloch bands, allowing direct comparison of superconductivity in dispersive and flat bands. We show how the superconductivity of the flat band is more robust than that of the dispersive bands and how the superfluid weight behaves non-monotonically as a function of the interaction strength. This non-monotonic behavior should be experimentally accessible. Flat band superconductivity is also the topic of Publication III, where the experimentally realized twisted bilayer graphene (TBG) is studied with local and non-local attractive interactions. We predict qualitative differences between local and non-local interaction schemes which could be distinguished experimentally and reveal the quantum geometric origin of superconductivity in TBG systems in case of both the assumed interaction schemes. Exotic Fulde-Ferrell (FF) superfluid phases, characterized by the non-zero momentum of Cooper pairs, were explored in Publication II. We show how the FF phases can be stabilized against thermal fluctuations in the presence of spin-orbit coupling (SOC) and that topologically non-trivial FF states, created with SOC and Zeeman fields, can be found at finite temperatures.; Tassa…

•Jan 1, 2019•0 citations

Pairing and superconductivity in the flat band: Creutz lattice

Rubem Mondaini, G. G. Batrouni, Benoît Grémaud

We use unbiased numerical methods to study the onset of pair superfluidity in a system that displays flat bands in the noninteracting regime. This is achieved by using a known example of flat band systems, namely, the Creutz lattice, where we investigate the role of local attractive interactions in the $U<0$ Hubbard model. Going beyond the standard approach used in these systems where weak interactions are considered, we map the superfluid behavior for a wide range of interaction strengths and exhibit a crossover between BCS and tightly bound bosonic fermion pairs. We further contrast these results with a standard two-leg fermionic ladder, showing that the pair correlations, although displaying algebraic decay in both cases, are longer ranged in the Creutz lattice, signifying the robustness of pairing in this system.

arXiv:1805.09359•Physical review. B./Physical review. B•Oct 25, 2018•69 citations

Disorder and robustness of superconductivity on the quasi-one-dimensional flat-band Creutz lattice

Si Min Chan, Benoît Grémaud, G. G. Batrouni

We study the interplay between on-site disorder and fermion pairing on the quasi-one-dimensional flat-band Creutz lattice. Both disorder and flat bands localize particles, but an attractive interaction results in pair formation and delocalization giving rise to superconductivity. In this work, we examine the attractive Hubbard model on the Creutz lattice to study the competition between these two effects and elucidate the properties of the superconducting phase and the localization quantum phase transition as the disorder strength is increased. Our main result is that flat-band superconductivity is robust against disorder: The critical disorder strength ${W}_{c}$ required to localize the fermion pairs and destroy superconductivity is finite at any interaction strength $U$ and is proportional to the superconducting weight ${D}_{s}$ of the clean system. Using large-scale density matrix renormalization group computations, we show that this transition is of the Berezinskii-Kosterlitz-Thouless form. In addition, even at very small interaction strength, the localization is not due to single-fermion localization but to pair localization. For completeness, we briefly study this disorder-induced localization with mean field theory and show that ${W}_{c}$ can be accurately determined by using an appropriate scaling function.

arXiv:2506.07095•Physical review. B./Physical review. B•Aug 19, 2025•2 citations

Quantum Geometry, Anomalous Scaling, and Strong Pseudogap Superfluidity in a Flat-Band Lieb Lattice

Lin Sun, Hao Deng, Yuxuan Wu +6 more

Flat-band systems such as magic-angle twisted bilayer graphene host strong-correlation superconductivity at vanishingly weak coupling, yet how quantum geometry and pairing fluctuations conspire to drive this phenomenon remains an open question. We investigate finite-temperature superfluidity in a quasi-two-dimensional Lieb lattice using a pairing fluctuation theory with a band-uniform attractive interaction $g<0$ that isolates the intrinsic quantum geometric contributions. Quantum geometry significantly amplifies superfluidity; the geometric pair hopping integral surpasses its conventional counterpart, and the geometric superfluid density becomes the dominant in-plane transport component. When the Fermi level enters the flat band, the BCS paradigm breaks down entirely; the pairing gap and $T_\text{c}$ shift from exponential to anomalous power-law scaling $Δ, T_\text{c} \propto |g|^ν$ ($ν>1$), and the superfluid density inherits an unconventional power-law temperature dependence at low temperatures. In the 2D limit ($t_z=0$), the pseudogap at $T_\text{c}$ nearly saturates the zero-temperature gap even at $|g|/t=0.001$, placing the system in a strong-pseudogap regime that would otherwise require unitary or BEC-scale interactions. These findings establish a microscopic mechanism for flat-band enhanced superfluidity and offer testable predictions for ultracold atom experiments.

arXiv:2401.02990•Jan 3, 2024•0 citations
Over the past years, one witnesses a growing interest in flat band (FB) physics which has become a playground for exotic phenomena. In this study, we address the FB superconductivity in one-dimensional stub chain. In contrast to the sawtooth chain or the Creutz ladder, for a given strength of the attractive electron-electron interaction, the stub chain allows the tuning of the real space spreading of the FB eigenstates (quantum metric or QM). We study in detail the interplay between the interaction strength and the mean value of the QM \ensuremath⟨g\ensuremath⟩\ensuremath{\langle}g\ensuremath{\rangle}\ensuremath⟨g\ensuremath⟩ on the pairings and on the superfluid weight Ds{D}_{s}Ds​. Our calculations reveal several interesting and intriguing features. For instance, in the weak coupling regime, Ds{D}_{s}Ds​ with respect to \ensuremath⟨g\ensuremath⟩,\ensuremath{\langle}g\ensuremath{\rangle},\ensuremath⟨g\ensuremath⟩, exhibits two different types of behavior. Despite the fact that the pairings differs drastically, Ds{D}_{s}Ds​ scales linearly with the QM only when its \ensuremath⟨g\ensuremath⟩\ensuremath{\langle}g\ensuremath{\rangle}\ensuremath⟨g\ensuremath⟩ is large enough (small gap limit). On the other hand, when the QM is of small amplitude an unusual power law is found, more precisely Ds\ensuremath∝\ensuremath⟨g\ensuremath⟩\ensuremathν{D}_{s}\ensuremath{\propto}{\ensuremath{\langle}g\ensuremath{\rangle}}^{\ensuremath{\nu}}Ds​\ensuremath∝\ensuremath⟨g\ensuremath⟩\ensuremathν, where \ensuremathν\ensuremath→2\ensuremath{\nu}\ensuremath{\rightarrow}2\ensuremathν\ensuremath→2 in the limit of large single particle gap. In addition to the numerical calculations, we have provided several analytical results which shed light on the physics in both the weak and strong coupling regime. Finally, we have addressed the impact of the thermal fluctuations on the superfluid weight.
We construct quasi-one-dimensional topological and nontopological three-band lattices with a tunable band gap and winding number of the flat band. Using full multiband mean field (MF) and exact density matrix renormalization group (DMRG) calculations, we show explicitly how the band gap affects pairing and superconductivity (SC) in a Hubbard model with attractive interactions. We obtain excellent agreement between MF and DMRG calculations. The SC weight Ds{D}_{s}Ds​ on the gapped topological, $W\ensuremath{\ne}0$, flat band increases linearly with interaction strength UUU for low values and with a slope that depends on the details of the compact localized state at U=0U=0U=0. As $U\ensuremath{\rightarrow}0$ for the gapped nontopological flat band (W=0W=0W=0), Ds{D}_{s}Ds​ decays with a power law faster than quadratically but slower than exponentially. In the gapless case (flat band touching the band above it), we find at low UUU (for both W=0W=0W=0 and $W\ensuremath{\ne}0$) that ${D}_{s}\ensuremath{\propto}{U}^{\ensuremath{\varphi}}$, with $\ensuremath{\varphi}<1$. In other words, Ds{D}_{s}Ds​ increases faster than linearly for low UUU, thus favoring SC at weak interaction more than the gapped case. Our results reestablish that the BCS mean field and quantum metric alone are insufficient to characterize SC at weak coupling.
The possibility of an unconventional form of high temperature superconductivity in flat band (FB) material does not cease to challenge our understanding of the physics in correlated systems. Here, we calculate the normal and anomalous one-particle correlation functions in various one and two dimensional FB systems and systematically extract the characteristic lengthscales. When the Fermi energy is located in the FB, it is found that the coherence length ( \xi ξ ) is of the order of the lattice spacing and weakly sensitive to the strength of the electron-electron interaction. Recently, it has been argued that in FB compounds \xi ξ could be decomposed into a conventional part of BCS type ( \xi_{BCS} ξ B C S ) and a geometric contribution which characterises the FB eigenstates, the quantum metric ( \langle g \rangle ⟨ g ⟩ ). However, by calculating the coherence length in two possible ways, our calculations show that \xi ≠ \sqrt{\langle g \rangle.} ξ ≠ ⟨ g ⟩ . This may suggest that the link between QM and coherence length is more complex, and leaves us with the open question: what is the appropriate definition of the coherence length in flat-band systems?
In this thesis, superconductivity of fermions in two-dimensional Hubbard lattice models is theoretically studied. Each of the studied lattice structures features intriguing properties that provide valuable insight to understand superconductivity in general. While real condensed matter systems are too complicated to tackle with theoretical tools, the Hubbard model is one of the simplest quantum many-body models to possess a wide spectrum of physically relevant phenomena, from superconductivity to magnetism and topological order. Hubbard models are also interesting as they can be experimentally realized and accurately controlled by utilizing ultracold gases. The first part of the thesis explains the required theoretical background of superconductivity in the level of mean-field and linear response theories. Specifically, we primarily focus on the superfluid weight which implies the existence of the Meissner effect and dissipationless current, defining properties of superconductors. We further discuss how the Berezinskii-Kosterlitz-Thouless (BKT) transition, related to the loss of superconductivity due to the thermal fluctuations of the superconducting order parameter in high enough temperatures, can be accessed via the superfluid weight. In the second part, the main results are discussed. We show how the superconductivity of lattice models featuring dispersionless Bloch bands, so-called flat bands, can be large and originating from the quantum geometric properties of the Bloch states of the flat bands. This is in contrast to conventional dispersive bands where superconductivity is determined by the effective mass of the particles, i.e. by the dispersion relation of the Bloch bands alone and not by the Bloch states. This is well demonstrated in the Lieb lattice geometry, studied in Publication I, which features both a strictly flat and two dispersive Bloch bands, allowing direct comparison of superconductivity in dispersive and flat bands. We show how the superconductivity of the flat band is more robust than that of the dispersive bands and how the superfluid weight behaves non-monotonically as a function of the interaction strength. This non-monotonic behavior should be experimentally accessible. Flat band superconductivity is also the topic of Publication III, where the experimentally realized twisted bilayer graphene (TBG) is studied with local and non-local attractive interactions. We predict qualitative differences between local and non-local interaction schemes which could be distinguished experimentally and reveal the quantum geometric origin of superconductivity in TBG systems in case of both the assumed interaction schemes. Exotic Fulde-Ferrell (FF) superfluid phases, characterized by the non-zero momentum of Cooper pairs, were explored in Publication II. We show how the FF phases can be stabilized against thermal fluctuations in the presence of spin-orbit coupling (SOC) and that topologically non-trivial FF states, created with SOC and Zeeman fields, can be found at finite temperatures.; Tassa…
We use unbiased numerical methods to study the onset of pair superfluidity in a system that displays flat bands in the noninteracting regime. This is achieved by using a known example of flat band systems, namely, the Creutz lattice, where we investigate the role of local attractive interactions in the U<0U<0U<0 Hubbard model. Going beyond the standard approach used in these systems where weak interactions are considered, we map the superfluid behavior for a wide range of interaction strengths and exhibit a crossover between BCS and tightly bound bosonic fermion pairs. We further contrast these results with a standard two-leg fermionic ladder, showing that the pair correlations, although displaying algebraic decay in both cases, are longer ranged in the Creutz lattice, signifying the robustness of pairing in this system.
We study the interplay between on-site disorder and fermion pairing on the quasi-one-dimensional flat-band Creutz lattice. Both disorder and flat bands localize particles, but an attractive interaction results in pair formation and delocalization giving rise to superconductivity. In this work, we examine the attractive Hubbard model on the Creutz lattice to study the competition between these two effects and elucidate the properties of the superconducting phase and the localization quantum phase transition as the disorder strength is increased. Our main result is that flat-band superconductivity is robust against disorder: The critical disorder strength Wc{W}_{c}Wc​ required to localize the fermion pairs and destroy superconductivity is finite at any interaction strength UUU and is proportional to the superconducting weight Ds{D}_{s}Ds​ of the clean system. Using large-scale density matrix renormalization group computations, we show that this transition is of the Berezinskii-Kosterlitz-Thouless form. In addition, even at very small interaction strength, the localization is not due to single-fermion localization but to pair localization. For completeness, we briefly study this disorder-induced localization with mean field theory and show that Wc{W}_{c}Wc​ can be accurately determined by using an appropriate scaling function.
Flat-band systems such as magic-angle twisted bilayer graphene host strong-correlation superconductivity at vanishingly weak coupling, yet how quantum geometry and pairing fluctuations conspire to drive this phenomenon remains an open question. We investigate finite-temperature superfluidity in a quasi-two-dimensional Lieb lattice using a pairing fluctuation theory with a band-uniform attractive interaction g<0g<0g<0 that isolates the intrinsic quantum geometric contributions. Quantum geometry significantly amplifies superfluidity; the geometric pair hopping integral surpasses its conventional counterpart, and the geometric superfluid density becomes the dominant in-plane transport component. When the Fermi level enters the flat band, the BCS paradigm breaks down entirely; the pairing gap and TcT_\text{c}Tc​ shift from exponential to anomalous power-law scaling Δ,Tc∝∣g∣νΔ, T_\text{c} \propto |g|^νΔ,Tc​∝∣g∣ν (ν>1ν>1ν>1), and the superfluid density inherits an unconventional power-law temperature dependence at low temperatures. In the 2D limit (tz=0t_z=0tz​=0), the pseudogap at TcT_\text{c}Tc​ nearly saturates the zero-temperature gap even at ∣g∣/t=0.001|g|/t=0.001∣g∣/t=0.001, placing the system in a strong-pseudogap regime that would otherwise require unitary or BEC-scale interactions. These findings establish a microscopic mechanism for flat-band enhanced superfluidity and offer testable predictions for ultracold atom experiments.