Bootstrapping Flat-band Superconductors: Rigorous Lower Bounds on Superfluid Stiffness

Abstract

The superfluid stiffness fundamentally constrains the transition temperature of superconductors, especially in the strongly coupled regime. However, accurately determining this inherently quantum many-body property in microscopic models remains a significant challenge. In this work, we show how the \textit{quantum many-body bootstrap} framework, specifically the reduced density matrix (RDM) bootstrap, can be leveraged to obtain rigorous lower bounds on the superfluid stiffness in frustration-free interacting models with superconducting ground state. We numerically apply the method to a special class of frustration free models, which are known as quantum geometric nesting models, for flat-band superconductivity, where we uncover a general relation between the stiffness and the pair mass. Going beyond the familiar Hubbard case within this class, we find how additional interactions, notably simple magnetic couplings, can enhance the superfluid stiffness. Furthermore, we find that the RDM bootstrap unexpectedly reveals that the trion-type correlations are essential for bounding the stiffness, offering new insights on the structure of these models. A straightforward generalization of the method can lead to bounds on susceptibilities complementary to variational approaches. Our findings underscore the immense potential of the quantum many-body bootstrap as a powerful tool to derive rigorous bounds on physical quantities beyond energy.

Figures (4)

• Figure 1: (a) Schematic illustration of classes of models with exact superconducting ground states. Our lower-bounding method applies to the most general class: frustration-free (FF) models. Within the FF set, quantum geometric nesting (QGN) models allow a variational upper bound based on pair mass (Eq.\ref{['lower_bound_formula']}), which we prove in the SM and find to be saturated by the bootstrap lower bound, suggesting its exactness. A subset of QGN models with Hubbard interaction and uniform pairing condition (UPC) previously have MF result for stiffness related to the minimal metric TormaNatPhy2015PhysRevB.94.245149PhysRevB.106.014518herzog2022many. (b) Schematic plot of feasible regions at different levels in the 2RDM bootstrap hierarchy. The red dot labels the frustration-free point where all feasible boundaries of ($2,p$) constraints for $p=2,\cdots, N$ meet. The two arrows represent the optimization direction when choosing different Hamiltonians (with different external parameter $A$). The upright corner shows schematically the energy $E(A)$ after imposing $(2,p)$ constraints at different levels.
• Figure 2: (a) Filling dependence of the superfluid stiffness in an attractive Hubbard model projected to a topological FB (Model I PhysRevB.102.201112), with system size $7\times 7$ (squares). The square with a black outline is an exact result in the two-particle sector. The dashed curve is $m_{\text{pair}}^{-1}\nu(1-\nu)$ with the pair mass $m_{\text{pair}}^{-1}$ fixed by the 2-particle ED results for $7\times 7$. The determinant QMC calculation Note_on_DQMC_data is also shown using a hexagram with an error bar. The inset shows the system scaling of the $D_\text{s}/(\nu(1-\nu))$ obtained from bootstrapping different system sizes Note_on_many_body_pair_mass, indicating a finite size correction of order $1/N_k$ to the many-body pair mass, where the red dot represents the extrapolated thermodynamic value. (b) The comparison between exact diagonalization (ED) and bootstrap for the same system size ($5\times 5$) and same fillings, showing the exact agreement between each other. The inset shows in semi-log scale the total run time as a function of the particle number $N$. (c) Quantum geometry dependence of $D_\text{s}$ in an attractive Hubbard model projected to a topologically trivial FB (Model II PhysRevLett.130.226001), where $\xi$ is a tuning parameter for quantum geometry with the minimal quantum metric $g = \xi^2/4$.
• Figure 3: Tuning the additional nearest-neighbor $S^z$-$S^z$ interaction, $J$, in a QGN model going beyond the Hubbard limit (Model I$'$, see the main text). The stiffness is shown as a function of $J$ with $|U|=1$ fixed. The plot range is divided into four regions: $F^{+(-)},S^{+(-)}$ representing (not) frustration-free and (not) sign-problem-free (e.g., $F^+,S^-$ means frustration free but sign problematic). Two different filling fractions are shown: $N=20$ (blue) and $N=10$ (red) for a system size of $5\times 5$. The inset shows the exact result of single-particle excitation energy which is independent of filling.
• Figure 4: The 2RDM in ${\bm{Q}}=0,S^z=0$ sector at the frustration free point obtained from bootstrap with T2 (left panel) and its deviation from the exact AGP 2RDM \ref{['AGP_2RDM']} (right panel), indicating a perfect agreement.

Theorems & Definitions (7)

• Definition 1: $(2,p)$ constraints
• Remark 1
• Definition 2: $p$-body exactness/frustration-freeness
• Remark 2
• Theorem 1
• proof
• Remark 3