Disorder and the Robustness of Superconductivity on the Flat Band

arXiv:2506.07095 · cond-mat.supr-con, cond-mat.dis-nn, cond-mat.str-el · Submitted 2025-06-08 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Disorder and the Robustness of Superconductivity on the Flat Band".

Mira: Here's the gist of the paper. The gist: Flat band superconductivity is robust against disorder, as the critical disorder strength required to localize fermion pairs and destroy superconductivity remains finite at any interaction strength U and is proportional to the superconducting weight Ds of the clean system Model and System The study examines the interplay between on-site disorder and fermion pairing on a quasi one-dimensional flat band Creutz lattice The system is described by the Hubbard Hamiltonian with an additional on-site disorder term, where the disorder strength W is drawn from an uncorrelated, uniform distribution with width W/2 The Creutz lattice is a quasi one-dimensional system with two chains and can be described with the inter-cell hopping Hamiltonian H1 = -t-t t t This lattice possesses two flat bands, E(k) = −2t, 2t in the non-interacting limit shown in Fig.

Kai: 1 The localized state is compact, with Ψj⟩ = 1/2 (c A† j +c A† j+1−c B† j +c B† j+1)0⟩ and equal filling on all sites of the compact localized state (CLS) Robustness Against Disorder The main result is that flat band superconductivity is robust against disorder: The critical disorder strength, Wc, required to localize the fermion pairs and destroy superconductivity, is finite at any interaction strength, U, and is proportional to the superconducting weight Ds of the clean system This transition is of the BKT form when examined using large scale density matrix renormalization group computations The critical disorder required to destroy superconductivity is directly proportional to the SC weight, Ds, of the clean system at the same U: Wc(U) ∝ Ds(U, W = 0) Dependence on Interaction Strength (U) The critical disorder strength is finite, Wc ≠ 0, for all values of interaction strength studied, specifically for U smaller than the gap between the two bands For U smaller than 10 (U Kc = 3/2 applicable to spinful fermions with attractive interactions on a ladder, which implies that the critical value of disorder required to localize the system is nonzero, Wc ≠ 0 For very large U, the model is very well described by an effective HCB model on a crossed ladder with Kc = 3/4 which is smaller than the K calculated for our system, leading to robust SC persisting at finite disorder In the intermediate U regime where bosonization predictions are not reliable, the system superconductivity remains robust and Wc ≠ 0 Disorder Localization Length Scaling The disorder localization length, ξdisorder, is obtained through the pair Green’s function (GF), which decays exponentially when disorder-induced localization occurs: G αα'pair(r) ∼ e − rξdisorder (6) The proper choice for describing the transition is the BKT scaling form, ξdisorder = a e √ b W−Wc = aξ∗disorder (8) This BKT form gives a better fit when plotting ln(ξdisorder) as a function of ln(W − Wc) and 1/√W−Wc The direct measurement of ξdisorder obtained from MPS optimization agrees best with the BKT form Mean Field Theory Consistency Although mean field theory (MFT) cannot capture the BKT universality class, it predicts an important quantity Wc accurately when using appropriate scaling functions MFT predicts a second-order phase transition instead of the BKT transition observed in DMRG, but it manages to capture the underlying physics of the phase transition by predicting an important quantity Wc accurately The critical disorder calculated using MFT is very close to values obtained with DMRG if the correct power law scaling function is used This confirms that the BKT scaling function is consistent between the disorder correlation length, and the scaling of Ds, providing a complete picture of the disorder induced localization with flat bands Superconducting transport on the flat band is resilient to on-site disorder, and the critical disorder is proportional to Ds of the clean system The paper demonstrates how to compute accurately the disorder localization length, ξdisorder, and the superfluid weight, Ds, in the presence of site impurities By doing so, we confirmed that the delocalized-localized phase transition scaling function is of the BKT form We have demonstrated that superconducting transport on the flat band is, in fact, resilient to on-site disorder The critical disorder is proportional to Ds of the clean system How it works 1.

Mira: The Hubbard Hamiltonian with an attractive on-site interaction is given by HHubbard = X i,j,α,α′,σ t α,α′ ij c α† i,σc α' j,σ + h.c.

Kai: − U X j,α c α† j↓ c α† j↑ c α j↑ c αj↓ 2.

Paper summary: Kai: We’ve seen that the paper "Disorder and the Robustness of Superconductivity on the Flat Band" claims that flat band superconductivity is robust against on-site disorder, meaning it can survive some amount of disorder.

Mira: They argue this because the critical disorder strength Wc required to destroy that superconductivity stays finite even when you vary the attractive interaction strength U.

Lev: They link this critical strength Wc directly to the clean system's superconducting weight Ds, showing that Wc(U) is proportional to Ds(U, W=zero) <ref:2506.07095#pg2>.

Kai: The authors use large scale density matrix renormalization group computations to confirm that this transition between localized and delocalized phases has a BKT form.

Mira: They also examine the dependence on interaction strength U, noting that for very large U, the system behaves like an effective HCB model on a crossed ladder with Kc = three/four <ref:2506.07095#pg1>.

Lev: And they find that in the intermediate interaction regime where bosonization predictions are unreliable, superconductivity remains robust and Wc is not zero.

Kai: They also look at disorder localization length scaling, showing that G alpha alpha'pair(r) decays exponentially when disorder localizes things, and they fit this decay using a BKT form with ξdisorder = a e root b W minus Wc equals a xi disorder.

Mira: The paper notes that the direct measurement of this localization length obtained from MPS optimization agrees best with the BKT form when plotting natural logarithm of ξdisorder as function of natural logarithm of W minus Wc and one over square root of W minus Wc.

Lev: And to give you some context on how this all works, they do briefly study disorder-induced localization using mean field theory, which predicts an important quantity Wc accurately when using the appropriate scaling functions.

Kai: The overall picture is that superconducting transport on the flat band is resilient to on-site disorder, and their main finding is that the critical disorder is proportional to Ds of the clean system.

Mira: This means they’ve demonstrated how to compute accurately both the disorder localization length ξdisorder and the superfluid weight Ds in systems with site impurities.

Conclusion: Kai: So, looking at this paper "Disorder and the Robustness of Superconductivity on the Flat Band" by Si Min Chan, B Gr´emaud, and G George Batrouni5, it’s about how disorder interacts with fermion pairing on the Creutz lattice.

Mira: The main point is that flat band superconductivity is robust against this type of on-site disorder because the critical disorder Wc scales with the clean system's superconducting weight Ds.

Lev: What this means for us in practice is that we have a way to calculate accurately the disorder localization length ξdisorder and the superfluid weight Ds when there are site impurities.

Kai: They confirmed that this transition between delocalized and localized phases follows the BKT form, which is a key piece of information about how things scale near that critical point.

Mira: It shows that you can compute accurately the properties of these systems by using DMRG to find the localization length in the presence of disorder.

Lev: The authors also showed that even when interaction strength U is small, pair localization is what causes things to happen, not just single fermion trapping.

Kai: So, for anyone listening who only cares about the physics here, it’s that you can expect robust superconductivity on these flat bands unless the disorder gets very strong relative to how strongly you’re pairing them.

Si Min Chan, B. Grémaud, G. George Batrouni

Centre for Quantum Technologies, National University of Singapore · Aix Marseille Univ, Université de Toulon, CNRS · Université Cote d⟃r Azur, CNRS, Institut de Physique de Nice (INPHYNI) · New Cornerstone Science Laboratory

cond-mat.supr-con, cond-mat.dis-nn, cond-mat.str-el

Submitted: 2025-06-08

Updated: 2025-06-13

Journal ref: Phys. Rev B112 094501 (2025)

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

The gist: The gist: Flat band superconductivity is robust against disorder, as the critical disorder strength required to localize fermion pairs and destroy superconductivity remains finite at any interaction

Key concepts

Flat Band Creutz Lattice: This specific quasi one-dimensional lattice has two non-interacting flat bands at energies of -2t and 2t in the non-interacting limit. It is described by a hopping Hamiltonian that allows for compact localized states, which are important for understanding particle localization.
Attractive Hubbard Model: This model describes interacting fermions on the lattice with an attractive on-site interaction (U). When this interaction is strong enough, it leads to fermion pairing and the formation of a superconducting state. The study examines how disorder competes with this pairing effect.
Superfluid Weight ($D_s$): The superfluid weight is a measure computed using density matrix renormalization group (DMRG) that quantifies the superconducting properties of the system. It is used to determine the critical disorder strength, as it collapses when pairs become localized due to disorder.
BKT Scaling Form: The transition between delocalized and localized phases in this system follows a Berezinskii-Kosterlitz-Thouless (BKT) scaling form. This form describes how the disorder localization length scales with the deviation from the critical disorder strength ($W_c$), providing a precise mathematical description of the phase transition.

Terminology

Flat Band — 4× in this paper · explained in 2 episodes
Hubbard Model — 1× in this paper · explained in 4 episodes
Phase Transition — 1× in this paper · explained in 2 episodes
Robustness — 1× in this paper · explained in 5 episodes

Summary

The gist: Flat band superconductivity is robust against disorder, as the critical disorder strength required to localize fermion pairs and destroy superconductivity remains finite at any interaction strength U and is proportional to the superconducting weight Ds of the clean system

Model and System

The study examines the interplay between on-site disorder and fermion pairing on a quasi one-dimensional flat band Creutz lattice The system is described by the Hubbard Hamiltonian with an additional on-site disorder term, where the disorder strength W is drawn from an uncorrelated, uniform distribution with width W/2 The Creutz lattice is a quasi one-dimensional system with two chains and can be described with the inter-cell hopping Hamiltonian H1 = -t-t t t This lattice possesses two flat bands, E(k) = −2t, 2t in the non-interacting limit shown in Fig. 1 The localized state is compact, with Ψj⟩ = 1/2 (c A† j +c A† j+1−c B† j +c B† j+1)0⟩ and equal filling on all sites of the compact localized state (CLS)

Robustness Against Disorder

The main result is that flat band superconductivity is robust against disorder: The critical disorder strength, Wc, required to localize the fermion pairs and destroy superconductivity, is finite at any interaction strength, U, and is proportional to the superconducting weight Ds of the clean system This transition is of the BKT form when examined using large scale density matrix renormalization group computations The critical disorder required to destroy superconductivity is directly proportional to the SC weight, Ds, of the clean system at the same U: Wc(U) ∝ Ds(U, W = 0)

Dependence on Interaction Strength (U)

The critical disorder strength is finite, Wc ≠ 0, for all values of interaction strength studied, specifically for U smaller than the gap between the two bands For U smaller than 10 (U Kc = 3/2 applicable to spinful fermions with attractive interactions on a ladder, which implies that the critical value of disorder required to localize the system is nonzero, Wc ≠ 0 For very large U, the model is very well described by an effective HCB model on a crossed ladder with Kc = 3/4 which is smaller than the K calculated for our system, leading to robust SC persisting at finite disorder In the intermediate U regime where bosonization predictions are not reliable, the system superconductivity remains robust and Wc ≠ 0

Disorder Localization Length Scaling

The disorder localization length, ξdisorder, is obtained through the pair Green’s function (GF), which decays exponentially when disorder-induced localization occurs: G αα'pair(r) ∼ e − rξdisorder (6) The proper choice for describing the transition is the BKT scaling form, ξdisorder = a e √ b W−Wc = aξ∗disorder (8) This BKT form gives a better fit when plotting ln(ξdisorder) as a function of ln(W − Wc) and 1/√W−Wc The direct measurement of ξdisorder obtained from MPS optimization agrees best with the BKT form

Mean Field Theory Consistency

Although mean field theory (MFT) cannot capture the BKT universality class, it predicts an important quantity Wc accurately when using appropriate scaling functions MFT predicts a second-order phase transition instead of the BKT transition observed in DMRG, but it manages to capture the underlying physics of the phase transition by predicting an important quantity Wc accurately The critical disorder calculated using MFT is very close to values obtained with DMRG if the correct power law scaling function is used This confirms that the BKT scaling function is consistent between the disorder correlation length, and the scaling of Ds, providing a complete picture of the disorder induced localization with flat bands

Conclusion

Superconducting transport on the flat band is resilient to on-site disorder, and the critical disorder is proportional to Ds of the clean system The paper demonstrates how to compute accurately the disorder localization length, ξdisorder, and the superfluid weight, Ds, in the presence of site impurities By doing so, we confirmed that the delocalized-localized phase transition scaling function is of the BKT form We have demonstrated that superconducting transport on the flat band is, in fact, resilient to on-site disorder The critical disorder is proportional to Ds of the clean system

How it works

The Hubbard Hamiltonian with an attractive on-site interaction is given by HHubbard = X i,j,α,α′,σ t α,α′ ij c α† i,σc α' j,σ + h.c. − U X j,α c α† j↓ c α† j↑ c α j↑ c αj↓
The diagonal disorder term is added as H = HHubbard + X j,σ,α µj,αc α† j,σc α j,σ (2)
Superfluid weight Ds is computed using DMRG with periodic boundary conditions (PBC) and an applied phase gradient ϕ ≡ Φ/L where Φ is the total phase twist applied on the system
The disorder localization length ξdisorder is obtained through the pair Green’s function (GF), which decays exponentially when disorder-induced localization occurs: G αα'pair(r) ∼ e − rξdisorder (6)
The critical disorder Wc is determined by finding the collapse of the superfluid weight Ds as a function of L/ξdisorder, which yields a nonzero critical disorder value Wc = 0.061

Interplay between Interaction and Disorder

The analysis shows that for U = 1, our system requires W > Wc = 0.061 to be localized, which is in agreement with bosonization predictions For large U, the paired fermions behave as very tightly bound hard core bosons described by the effective Hamiltonian Eq.

Improvements for AI systems

The AI system can accurately predict whether superconductivity will survive disorder by calculating a finite critical disorder strength, Wc, which is proportional to the superconducting weight, Ds of the clean system.
The AI can determine if localization is due to single fermion or pair localization by analyzing correlation functions; specifically, it can distinguish between single fermion localization and pair localization even at very small interaction strengths.
The system can predict the universality class of the disorder-induced transition, correctly identifying it as a BKT form, which is crucial for accurate finite-size scaling analyses.
The AI can calculate the precise disorder localization length, ξdisorder, using MPS optimization and confirm its consistency by showing that the BKT form gives a better fit than the power law form in Fig. 4(c).
The system can predict the robustness of superconductivity across all interaction strengths U by mapping out Wc as a function of U, confirming that superconductivity in this system is robust and resists the localizing effect of disorder for all values of the interaction U.