I summarize recent progress on obtaining rigorous upper bounds on superconducting transition temperature [Formula: see text] in two dimensions independent of pairing mechanism or interaction strength. These results are derived by finding a general upper bound for the superfluid stiffness for a multi-band system with arbitrary interactions, with the only assumption that the external vector potential couples to the kinetic energy and not to the interactions. This bound is then combined with the universal relation between the superfluid stiffness and the Berezinskii–Kosterlitz–Thouless [Formula: see text] in 2D. For parabolic dispersion, one obtains the simple result that [Formula: see text], which has been tested in recent experiments. More generally, the bounds are expressed in terms of the optical spectral weight and lead to stringent constraints for the [Formula: see text] of low-density, strongly correlated superconductors. Results for [Formula: see text] bounds for models of flat-band superconductors, where the kinetic energy vanishes and the vector potential must couple to interactions, are briefly summarized. Upper bounds on [Formula: see text] in 3D remains an open problem, and I describe how questions of universality underlie the challenges in 3D.
I summarize recent progress on obtaining rigorous upper bounds on superconducting transition temperature [Formula: see text] in two dimensions independent of pairing mechanism or interaction strength. These results are derived by finding a general upper bound for the superfluid stiffness for a multi-band system with arbitrary interactions, with the only assumption that the external vector potential couples to the kinetic energy and not to the interactions. This bound is then combined with the universal relation between the superfluid stiffness and the Berezinskii–Kosterlitz–Thouless [Formula: see text] in 2D. For parabolic dispersion, one obtains the simple result that [Formula: see text], which has been tested in recent experiments. More generally, the bounds are expressed in terms of the optical spectral weight and lead to stringent constraints for the [Formula: see text] of low-density, strongly correlated superconductors. Results for [Formula: see text] bounds for models of flat-band superconductors, where the kinetic energy vanishes and the vector potential must couple to interactions, are briefly summarized. Upper bounds on [Formula: see text] in 3D remains an open problem, and I describe how questions of universality underlie the challenges in 3D.
Understanding the material parameters that control the superconducting transition temperature T c is a problem of fundamental importance.In many novel superconductors phase fluctuations determine T c , rather than the collapse of the pairing amplitude.We derive rigorous upper bounds on the superfluid phase stiffness for multiband systems, valid in any dimension.This in turn leads to an upper bound on T c in two dimensions, which holds irrespective of pairing mechanism, interaction strength, or order-parameter symmetry.Our bound is particularly useful for the strongly correlated regime of low-density and narrowband systems, where mean-field theory fails.For a simple parabolic band in 2D with Fermi energy E F , we find that k B T c ≤ E F =8, an exact result that has direct implications for the 2D BCS-BEC crossover in ultracold Fermi gases.Applying our multiband bound to magic-angle twisted bilayer graphene, we find that band structure results constrain the maximum T c to be close to the experimentally observed value.Finally, we discuss the question of deriving rigorous upper bounds on T c in 3D.
We discuss standard and tighter upper bounds on the critical temperature Tc of two-dimensional superconductors and superfluids versus particle density n or filling factor ν, under the assumption that the transition from the normal to the superconducting (superfluid) phase is governed by the Berezinskii-Kosterlitz-Thouless (BKT) mechanism of vortex-antivortex binding and a direct relation between the superfluid density tensor and Tc exists. The standard critical temperature upper bound Tcup1 is obtained from the Glover-Ferrell-Tinkham sum rule for the optical conductivity, which constrains the superfluid density tensor components. However, we show that Tcup1 is only useful in the limit of low particle/carrier density, where it may be close to the critical temperature supremum Tcsup . For intermediate and high particle/carrier densities, Tcup1 is far beyond Tcsup for any given interaction strength. We demonstrate that it is imperative to consider at least the full effects of phase fluctuations of the order parameter for superconductivity (superfluidity) to establish tighter bounds over a wide range of densities. Using the renormalization group, we obtain the critical temperature supremum for phase fluctuations Tc and show that it is a much tighter upper bound to Tcsup than Tcup1 for all particle/carrier densities. We conclude by indicating that if the Tc is exceeded in experiments involving single band systems, then a non-BKT mechanism must be invoked.
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 Letter, we show how the 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 flatband 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. 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.
Superconductivity is a macroscopic manifestation of a quantum phenomenon where pairs of electrons delocalize and develop phase coherence over a long distance. A long-standing quest has been to address the underlying microscopic mechanisms that fundamentally limit the superconducting transition temperature, T c . A platform which serves as an ideal playground for realizing “high”-temperature superconductors are materials where the electrons’ kinetic energy is quenched and interactions provide the only energy scale in the problem. However, when the noninteracting bandwidth for a set of isolated bands is small compared to the interactions, the problem is inherently nonperturbative. In two spatial dimensions, T c is controlled by superconducting phase stiffness. Here, we present a theoretical framework for computing the electromagnetic response for generic model Hamiltonians, which controls the maximum possible superconducting phase stiffness and thereby T c , without resorting to any mean-field approximation. Our explicit computations demonstrate that the contribution to the phase stiffness arises from i) “integrating out” the remote bands that couple to the microscopic current operator and ii) the density–density interactions projected on to the isolated narrow bands. Our framework can be used to obtain an upper bound on the phase stiffness and relatedly T c for a range of physically inspired models involving both topological and nontopological narrow bands with density–density interactions. We discuss a number of salient aspects of this formalism by applying it to a specific model of interacting flat bands and compare the upper bound against the known T c from independent numerically exact computations.
We discuss standard and tighter upper bounds on the critical temperature Tc of two-dimensional (2D) superconductors and superfluids versus particle density n or filling factor ν for continuum and lattice systems from the Bardeen-Cooper-Schrieffer (BCS) to Bose regime. We discuss only one-band Hamiltonians, where the transition from the normal to superconducting (superfluid) phase is governed by Berezinskii-Kosterlitz-Thouless (BKT) mechanism of vortex-antivortex binding, such that a direct relation between the superfluid density tensor and Tc exists. We demonstrate that it is imperative to consider at least the full effect of phase fluctuations of order parameter for superconductivity (superfluidity) to establish tighter bounds. Using the renormalization group, we obtain phase-fluctuation critical temperature Tcθ, a much tighter upper bound to critical temperature supremum Tcsup than standard critical temperature upper bound Tcup1 from Ferrell-Glover-Tinkham sum rule. We go beyond textbook phase-fluctuation theories and show that chemical potential renormalization, order parameter equation, and Nelson-Kosterlitz relation need to be solved self-consistently and simultaneously with renormalization group flow equation to produce Tcθ over the entire BCS-Bose crossover. We note that an analytic theory including modulus fluctuations of the order parameter valid throughout the BCS-Bose evolution is still lacking, but the inclusion of modulus fluctuations can only produce a critical temperature lower than Tcθ and thus produce an even tighter bound to Tcsup. We conclude by indicating that if the measured critical temperature exceeds Tcθ in experiments involving 2D single-band systems, then a non-BKT mechanism must be invoked to describe the superconducting (superfluid) transition.
Abstract We discuss standard and tighter upper bounds on the critical temperature T c of two-dimensional superfluids and superconductors versus particle density n or filling factor ν for continuum and lattice systems from the Bardeen–Cooper–Schrieffer (BCS) to the Bose regime. We consider only one-band Hamiltonians, where the transition from the normal to the superfluid (superconducting) phase is governed by the Berezinskii–Kosterlitz–Thouless (BKT) mechanism of vortex-antivortex binding, such that a direct relation between the superfluid density tensor and T c exists. The standard critical temperature upper bound T c up 1 is obtained from the Ferrell-Glover-Tinkham sum rule for the optical conductivity, which constrains the superfluid density tensor components. We demonstrate that it is imperative to consider at least the full effect of phase fluctuations of the order parameter for superfluidity (superconductivity) and use the renormalization group to obtain the phase-fluctuation critical temperature T c θ , a much tighter bound to the critical temperature supremum than T c up 1 over a wide range of densities or filling factors. We also discuss a fundamental difference between superfluids and superconductors in regards to the vortex core energy dependence on density. Going beyond phase fluctuations, we note that theories including modulus fluctuations of the order parameter or particle-hole fluctuations valid throughout the BCS-Bose evolution are still lacking, but the inclusion of these fluctuations can only produce a critical temperature that is lower than T c θ and thus produce an even tighter bound to the critical temperature supremum. We conclude by indicating that if the measured critical temperature exceeds T c θ in experiments involving two-dimensional single-band systems, then a non-BKT mechanism must be invoked to describe the superfluid (superconducting) transition.