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Physical review. B./Physical review. B

Superfluid stiffness within Eliashberg theory: The role of vertex corrections

Zachary M. Raines, Shang-Shun Zhang, Andrey V. Chubukov

Published Apr 5, 2024• 5 citations
Publisher

Abstract

The superfluid stiffness ${D}_{s}$($T$) is one of the key characteristics of a superconductor, determining the magnetic penetration length and, in two dimensions, the critical temperature. For Galilean invariant systems, it is known that the zero-temperature stiffness takes a universal value ${D}_{s}^{G\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}l}$ = $\frac{n}{m}$ ($T$=0), while for non-Galilean invariant systems the superfluid stiffness may be suppressed. Here, the authors demonstrate when and how vertex corrections may cancel suppression of the superfluid stiffness at strong coupling.

DOI
10.1103/physrevb.109.144505

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Superfluid stiffness within Eliashberg theory: the role of vertex corrections

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In this work we consider the superfluid stiffness of a generically non-Galilean invariant interacting system and investigate under what conditions the stiffness may nonetheless approach the Galilean-invariant value $n/m$. Within Eliashberg theory we find that the renormalized stiffness is approximately given by $n/m$ in the case when the $l=0$ and $l=1$ components of the effective Fermi-surface projected interaction are approximately equal over a range of frequencies. This holds, in particular, when the interaction is peaked at zero momentum transfer. We examine this result through three complementary lenses: the $δ(ω)$ term in the conductivity, the phase dependence of the Luttinger-Ward free energy, and the coupling of the amplitude and phase sectors in the Hubbard-Stratonovich collective mode action. From these considerations we show that the value of the stiffness is determined by the strength of renormalization of the current vertex and that the latter can be interpreted as the shift of the self-consistent solution due to flow of the condensate, or alternatively as coupling of the phase mode to $l=1$ fluctuations of the order parameter. We highlight that even though the superfluid stiffness in some non-Galilean systems approaches the Galilean value, this is not enforced by symmetry, and in general the stiffness may be strongly suppressed from its BCS value. As a corollary we obtain the generic form of the phase action within Eliashberg theory and charge and spin Ward identities for a superconductor with frequency dependent gap function.

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Migdal-Eliashberg theory of boson-mediated superconductivity contains a $\sqrt{\ensuremath{\lambda}}$ divergence in the critical temperature ${T}_{c}$ for strong electron-boson coupling $\ensuremath{\lambda}$. In the conventional Migdal-Eliashberg theory, the strong-coupling regime can be accessed only in the limit that ${\ensuremath{\lambda}}_{E}=\ensuremath{\lambda}\phantom{\rule{0.16em}{0ex}}{\ensuremath{\omega}}_{D}/{\ensuremath{\varepsilon}}_{F}\ensuremath{\ll}1$, where ${\ensuremath{\omega}}_{D}$ is the Debye frequency and ${\ensuremath{\varepsilon}}_{F}$ is the Fermi energy. Here we go beyond this restriction in the context of the two-dimensional Yukawa-SYK (Y-SYK) model, which is solvable for arbitrary values of ${\ensuremath{\lambda}}_{E}$. We find that ${T}_{c}\ensuremath{\approx}0.18\phantom{\rule{0.16em}{0ex}}{\ensuremath{\omega}}_{D}\sqrt{\ensuremath{\lambda}}$ for large $\ensuremath{\lambda}$, provided ${\ensuremath{\lambda}}_{E}$ remains small, and crosses over to a universal value of ${T}_{c}\ensuremath{\approx}0.04\phantom{\rule{0.16em}{0ex}}{\ensuremath{\varepsilon}}_{F}$ for large ${\ensuremath{\lambda}}_{E}$. The saturation of ${T}_{c}$ is due to a self-consistent account of the boson dynamics for large ${\ensuremath{\lambda}}_{E}$ and remains valid provided the vertex corrections are negligible. Depending on the value of $\ensuremath{\lambda}$, this self-consistent approach leads to pairing that describes multiple classes of quantum critical electronic systems. These results demonstrate how the $\sqrt{\ensuremath{\lambda}}$ growth of ${T}_{c}$ in Migdal-Eliashberg theory saturates to a universal value independent of $\ensuremath{\lambda}$ and ${\ensuremath{\omega}}_{D}$, providing an upper bound on the critical temperature at strong electron-boson coupling.

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The superfluid stiffness Ds{D}_{s}Ds​(TTT) is one of the key characteristics of a superconductor, determining the magnetic penetration length and, in two dimensions, the critical temperature. For Galilean invariant systems, it is known that the zero-temperature stiffness takes a universal value {D}_{s}^{G\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}l} = nm\frac{n}{m}mn​ (TTT=0), while for non-Galilean invariant systems the superfluid stiffness may be suppressed. Here, the authors demonstrate when and how vertex corrections may cancel suppression of the superfluid stiffness at strong coupling.
In this work we consider the superfluid stiffness of a generically non-Galilean invariant interacting system and investigate under what conditions the stiffness may nonetheless approach the Galilean-invariant value n/mn/mn/m. Within Eliashberg theory we find that the renormalized stiffness is approximately given by n/mn/mn/m in the case when the l=0l=0l=0 and l=1l=1l=1 components of the effective Fermi-surface projected interaction are approximately equal over a range of frequencies. This holds, in particular, when the interaction is peaked at zero momentum transfer. We examine this result through three complementary lenses: the δ(ω)δ(ω)δ(ω) term in the conductivity, the phase dependence of the Luttinger-Ward free energy, and the coupling of the amplitude and phase sectors in the Hubbard-Stratonovich collective mode action. From these considerations we show that the value of the stiffness is determined by the strength of renormalization of the current vertex and that the latter can be interpreted as the shift of the self-consistent solution due to flow of the condensate, or alternatively as coupling of the phase mode to l=1l=1l=1 fluctuations of the order parameter. We highlight that even though the superfluid stiffness in some non-Galilean systems approaches the Galilean value, this is not enforced by symmetry, and in general the stiffness may be strongly suppressed from its BCS value. As a corollary we obtain the generic form of the phase action within Eliashberg theory and charge and spin Ward identities for a superconductor with frequency dependent gap function.
Migdal-Eliashberg theory of boson-mediated superconductivity contains a $\sqrt{\ensuremath{\lambda}}$ divergence in the critical temperature Tc{T}_{c}Tc​ for strong electron-boson coupling $\ensuremath{\lambda}$. In the conventional Migdal-Eliashberg theory, the strong-coupling regime can be accessed only in the limit that ${\ensuremath{\lambda}}_{E}=\ensuremath{\lambda}\phantom{\rule{0.16em}{0ex}}{\ensuremath{\omega}}_{D}/{\ensuremath{\varepsilon}}_{F}\ensuremath{\ll}1$, where ${\ensuremath{\omega}}_{D}$ is the Debye frequency and ${\ensuremath{\varepsilon}}_{F}$ is the Fermi energy. Here we go beyond this restriction in the context of the two-dimensional Yukawa-SYK (Y-SYK) model, which is solvable for arbitrary values of ${\ensuremath{\lambda}}_{E}$. We find that ${T}_{c}\ensuremath{\approx}0.18\phantom{\rule{0.16em}{0ex}}{\ensuremath{\omega}}_{D}\sqrt{\ensuremath{\lambda}}$ for large $\ensuremath{\lambda}$, provided ${\ensuremath{\lambda}}_{E}$ remains small, and crosses over to a universal value of ${T}_{c}\ensuremath{\approx}0.04\phantom{\rule{0.16em}{0ex}}{\ensuremath{\varepsilon}}_{F}$ for large ${\ensuremath{\lambda}}_{E}$. The saturation of Tc{T}_{c}Tc​ is due to a self-consistent account of the boson dynamics for large ${\ensuremath{\lambda}}_{E}$ and remains valid provided the vertex corrections are negligible. Depending on the value of $\ensuremath{\lambda}$, this self-consistent approach leads to pairing that describes multiple classes of quantum critical electronic systems. These results demonstrate how the $\sqrt{\ensuremath{\lambda}}$ growth of Tc{T}_{c}Tc​ in Migdal-Eliashberg theory saturates to a universal value independent of $\ensuremath{\lambda}$ and ${\ensuremath{\omega}}_{D}$, providing an upper bound on the critical temperature at strong electron-boson coupling.
This thesis is organized in two independent parts, in which I study two different aspects of high Tc superconductivity. The first part begins with an introduction aimed to briefly introduce some relevant experimental and theoretical works performed in recent years, that have helped us to think about cuprates the way we do now. Afterwards, I introduce Landau Fermi liquid theory in a standard text book way. The question of validity of Fermi liquid theory in 2-d is then raised and investigated by searching for singularities in Landau's /-function. I show that the interaction function between two quasiparticles whose momenta approach each other near a curved point of the Fermi surface, contains a 1-d singularity not strong enough to change the Fermi liquid behavior. On the other hand, inflection points provide 2-d singularities that have to be taken seriously in Fermi liquid considerations. I then introduce nearly antiferromagnetic Fermi liquid theory (NAFL), which is a phenomenological theory proposed to describe high Tc systems. I mainly focus on the self-consistency of the theory in calculations. I criticize the theory on the basis of overlooking the vertex corrections in the strong coupling calculations of the transition temperature Tc. I calculate the first vertex correction for an optimally doped system and show that it is of the same order of magnitude as the bare vertex. Migdal's theorem is therefore not valid and Eliashberg formalism is not applicable to this situation. The same conclusion is obtained even after inclusion of the quasiparticle residue Z to the calculation. The sign of the vertex correction is then considered. I show that the positive sign of the vertex correction for the optimally doped system requires a phase transition of some sort as the doping is decreased. Part II of the thesis is devoted to the vortex lattice properties of high Tc superconductors. I establish a method to study vortex lattice properties of d-wave superconductors based on a generalization of the London model. The method has the advantage of simplicity as well as having very few free parameters (one at most) compared to other methods. The generalized London free energy is obtained from an s-d mixing Ginsburg-Landau free energy and also from the microscopic theory of Gorkov. The generalized London equation is found to be analytic at high temperatures. At very low temperatures however nonanalyticities arise as a result of the nodes on the superconducting gap. I then present the results of our calculations of some measurable quantities, such as the vortex lattice geometry and the effective penetration depth (as denned in μSR experiments). Comparison between our results and different experimental data is then performed. Especially our prediction for the magnetic field dependence of the effective penetration depth at T = 0, which is recently observed in //SR experiments with excellent agreement, is discussed.
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.
The Landau theory of superfluidity associates the low-temperature flow of the normal component with the phonon wind. This picture does not apply to superfluids in which Galilean invariance is broken either by disorder, porous media, or lattice potential, and the phonon wind is no longer solely responsible for depletion of the superfluid component. Based on Popov's hydrodynamic action with anharmonic terms, we present a general theory for low-temperature (T) dependence of the superfluid stiffness, which reproduces the Landau result as a special case when several parameters of the hydrodynamic action are fixed by Galilean invariance. We validate our theory with numerical simulations of interacting lattice bosons. In a broader context, our approach reveals universal low-temperature thermodynamics of superfluids with an intrinsic connection between finite-T and finite-size (L) effects implying universal scaling, T^{d+1} and 1/L^{d+1}, respectively, for a large class of thermodynamic quantities. We discuss the experimental detection of this law, and compare our prediction to the existing literature.
We analyze the corrections to the superfluid density due to phase fluctuations within both a continuum and a lattice model for s- and d-wave superconductors. We expand the phase-only action beyond the Gaussian level and compare our results with the quantum XY\mathrm{XY}XY model both in the quantum and in the classical regime. We find dynamic anharmonic vertices, absent in the quantum XY\mathrm{XY}XY model, which are responsible for the vanishing of the correction to the superfluid density at zero temperature in a continuum (Galilean invariant) model. Moreover, the phase-fluctuation effects are reduced with respect to the XY\mathrm{XY}XY model by a factor at least of order ${1/(k}_{F}{\ensuremath{\xi}}_{0}{)}^{2}.$