### Abstract

This review is written at the time of the twentieth anniversary of the discovery of high-temperature superconductors, which nearly coincides with the important discovery of the superfluid phases of ultracold trapped fermionic atoms. We show how these two subjects have much in common. Both have been addressed from the perspective of the BCS—Bose–Einstein condensation (BEC) crossover scenario, which is designed to treat short coherence length superfluids with transition temperatures which are “high” with respect to the Fermi energy. A generalized mean field treatment of BCS–BEC crossover at general temperatures, based on the BCS–Leggett ground state, has met with remarkable success in the fermionic atomic systems. Here we summarize this success in the context of four different cold atom experiments, all of which provide indications, direct or indirect, for the existence of a pseudogap. This scenario also provides a physical picture of the pseudogap phase in the underdoped cuprates which is a central focus of high research. We summarize successful applications of BCS–BEC crossover to key experiments in high systems, including the phase diagram, specific heat, and vortex core STM data, along with the Nernst effect, and exciting recent data on the superfluid density in very underdoped samples.

We gratefully acknowledge the help of our many close collaborators over the years: Jiri Maly, Boldizśar Jankó, Ioan Kosztin, Ying-Jer Kao, Andrew Iyengar, Shina Tan, and Yan He. We also thank our co-authors John Thomas, Andrey Turlapov, and Joe Kinast, as well as Thomas Lemberger, Brent Boyce, Joshua Milstein, Maria Luisa Chiofalo, and Murray Holland.

I. INTRODUCTION

A. Historical background

B. Fermionic pseudogaps and metastable pairs: two sides of the same coin

C. Introduction to high-superconductivity:pseudogap effects

D. Summary of cold atom experiments: crossover in the presence of Feshbach resonances

II. THEORETICAL FORMALISM FOR BCS–BEC CROSSOVER

A. Many-body Hamiltonian and two-body scattering theory

B. -matrix-based approaches to BCS–BEC crossover in the absence of Feshbach effects

C. Extending conventional crossover ground state to : -matrix scheme in the presence of closed-channel molecules

III. PHYSICAL IMPLICATIONS: ULTRACOLD ATOM SUPERFLUIDITY

A. calculations and trap effects

B. Thermodynamic experiments

C. Temperature dependent particle density profiles

D. rf pairing-gap spectroscopy

E. Collective breathing modes at

IV. PHYSICAL IMPLICATIONS: HIGH SUPERCONDUCTIVITY

A. Phase diagram and superconducting coherence

B. Electrodynamics in the superconducting phase

C. Bosonic power laws and pairbreaking effects

D. Anomalous normal state transport: Nernst coefficient

V. CONCLUSIONS

### Key Topics

- Mean field theory
- 52.0
- Superconductivity
- 45.0
- Band gap
- 44.0
- Bose Einstein condensates
- 44.0
- Superfluids
- 42.0

## Figures

Behavior of the chemical potential in the three regimes. is essentially pinned at the Fermi temperature in the BCS regime, whereas it becomes negative in the BEC regime. The PG (pseudogap) case corresponds to non-Fermi-liquid-based superconductivity in the intermediate regime.

Behavior of the chemical potential in the three regimes. is essentially pinned at the Fermi temperature in the BCS regime, whereas it becomes negative in the BEC regime. The PG (pseudogap) case corresponds to non-Fermi-liquid-based superconductivity in the intermediate regime.

Contrasting behavior of the excitation gap and order parameter versus temperature in the pseudogap regime. The height of the shaded region reflects the number of noncondensed pairs, at each temperature.

Contrasting behavior of the excitation gap and order parameter versus temperature in the pseudogap regime. The height of the shaded region reflects the number of noncondensed pairs, at each temperature.

Comparison of temperature dependence of excitation gaps in BCS and BEC limits. The gap vanishes at for the former while it is essentially -independent for the latter.

Comparison of temperature dependence of excitation gaps in BCS and BEC limits. The gap vanishes at for the former while it is essentially -independent for the latter.

The character of the excitations in the BCS–BEC crossover both above and below . The excitations are primarily fermionic Bogoliubov quasiparticles in the BCS limit and bosonic pairs (or “Feshbach bosons”) in the BEC limit. For atomic Fermi gases, the “virtual molecules ” in the PG case consist primarily of “Cooper” pairs of fermionic atoms.

The character of the excitations in the BCS–BEC crossover both above and below . The excitations are primarily fermionic Bogoliubov quasiparticles in the BCS limit and bosonic pairs (or “Feshbach bosons”) in the BEC limit. For atomic Fermi gases, the “virtual molecules ” in the PG case consist primarily of “Cooper” pairs of fermionic atoms.

Typical phase diagram of hole-doped high superconductors. The horizontal axis is hole doping concentration. There exists a pseudogap phase above in the underdoped regime. Here SC denotes superconductor, and is the temperature at which the pseudogap smoothly turns on.

Typical phase diagram of hole-doped high superconductors. The horizontal axis is hole doping concentration. There exists a pseudogap phase above in the underdoped regime. Here SC denotes superconductor, and is the temperature at which the pseudogap smoothly turns on.

Temperature dependence of the excitation gap at the antinodal point in (BSCCO) for three different doping concentrations from near-optimal (discs) to heavy underdoping (inverted triangles), as measured by angle-resolved photoemission spectroscopy (from Ref. 3).

Temperature dependence of the excitation gap at the antinodal point in (BSCCO) for three different doping concentrations from near-optimal (discs) to heavy underdoping (inverted triangles), as measured by angle-resolved photoemission spectroscopy (from Ref. 3).

Temperature dependence of fermionic excitation gaps and superfluid density for various doping concentrations (from Ref. 48). When , there is little correlation between and ; this figure suggests that something other than fermionic quasiparticles (e.g., bosonic excitations) may be responsible for the disappearance of superconductivity with increasing . Figure shows a quasi-universal behavior for the sloped at different doping concentrations, despite the highly non-universal behavior for .

Temperature dependence of fermionic excitation gaps and superfluid density for various doping concentrations (from Ref. 48). When , there is little correlation between and ; this figure suggests that something other than fermionic quasiparticles (e.g., bosonic excitations) may be responsible for the disappearance of superconductivity with increasing . Figure shows a quasi-universal behavior for the sloped at different doping concentrations, despite the highly non-universal behavior for .

Spatial density profiles of a molecular cloud of trapped atoms in the BEC regime in the transverse directions after of free expansion (from Ref. 61), showing a thermal molecular cloud above (left) and a molecular condensate (right) below . Part shows the surface plots and part shows cross sections through the images (dots) with bimodal fits (lines).

Spatial density profiles of a molecular cloud of trapped atoms in the BEC regime in the transverse directions after of free expansion (from Ref. 61), showing a thermal molecular cloud above (left) and a molecular condensate (right) below . Part shows the surface plots and part shows cross sections through the images (dots) with bimodal fits (lines).

Characteristic behavior of the scattering length for in the three regimes.

Characteristic behavior of the scattering length for in the three regimes.

Typical behavior of as a function of in a homogeneous system. follows the BCS predictions and approaches the BEC asymptote in the BEC limit. In the intermediate regime, it reaches a maximum around and a minimum around where .

Typical behavior of as a function of in a homogeneous system. follows the BCS predictions and approaches the BEC asymptote in the BEC limit. In the intermediate regime, it reaches a maximum around and a minimum around where .

Typical behavior of of a Fermi gas in a trap as a function of . It follows BCS prediction in the weak coupling limit, , and approaches the BEC asymptote in the limit . In contrast to the homogeneous case in Fig. 10, the BEC asymptote is much higher due to a compressed profile for trapped bosons.

Typical behavior of of a Fermi gas in a trap as a function of . It follows BCS prediction in the weak coupling limit, , and approaches the BEC asymptote in the limit . In contrast to the homogeneous case in Fig. 10, the BEC asymptote is much higher due to a compressed profile for trapped bosons.

Typical spatial profile of density and fermionic excitation gap of a Fermi gas in a trap. The curves are computed at unitarity, where . Here is the Thomas–Fermi radius.

Typical spatial profile of density and fermionic excitation gap of a Fermi gas in a trap. The curves are computed at unitarity, where . Here is the Thomas–Fermi radius.

Energy vs. physical temperature . The upper curve and data points correspond to the BCS or essentially free Fermi gas case, and the lower curve and data correspond to unitarity. The latter provide indications for a phase transition. The inset shows how temperature must be recalibrated below . From Ref. 77.

Energy vs. physical temperature . The upper curve and data points correspond to the BCS or essentially free Fermi gas case, and the lower curve and data correspond to unitarity. The latter provide indications for a phase transition. The inset shows how temperature must be recalibrated below . From Ref. 77.

Low-temperature comparison of theory (curves) and experiments (symbols) in terms of per atom as a function of , for both unitary and noninteracting gases in a Gaussian trap. From Ref. 77.

Low-temperature comparison of theory (curves) and experiments (symbols) in terms of per atom as a function of , for both unitary and noninteracting gases in a Gaussian trap. From Ref. 77.

Same as Fig. 14 but for a much larger range of temperature. The quantitative agreement between theory and experiment is very good. The fact that the two experimental (and the two theoretical) curves do not merge until higher is consistent with the presence of a pseudogap.

Same as Fig. 14 but for a much larger range of temperature. The quantitative agreement between theory and experiment is very good. The fact that the two experimental (and the two theoretical) curves do not merge until higher is consistent with the presence of a pseudogap.

Temperature dependence of experimental one-dimensional spatial profiles (circles) and TF fit (line) from Ref. 99, TF fits (line) to theory both at (circles) and overlay of experimental (circles) and theoretical (line) profiles, as well as relative rms deviations associated with these fits to theory at unitarity. The circles in are shown as the line in . The profiles have been normalized so that , and we set in order to overlay the two curves. reaches a maximum around .

Temperature dependence of experimental one-dimensional spatial profiles (circles) and TF fit (line) from Ref. 99, TF fits (line) to theory both at (circles) and overlay of experimental (circles) and theoretical (line) profiles, as well as relative rms deviations associated with these fits to theory at unitarity. The circles in are shown as the line in . The profiles have been normalized so that , and we set in order to overlay the two curves. reaches a maximum around .

Decomposition of density profiles at various temperatures at unitarity. Here (light gray) refers to the condensate, (dark gray) to the noncondensed pairs, and (black) to the excited fermionic states. , and is the Thomas–Fermi radius.

Decomposition of density profiles at various temperatures at unitarity. Here (light gray) refers to the condensate, (dark gray) to the noncondensed pairs, and (black) to the excited fermionic states. , and is the Thomas–Fermi radius.

Experimental rf spectra at unitarity. The temperatures labeled in the figure were computed theoretically at unitarity based on adiabatic sweeps from BEC. The two top curves thus correspond to the normal phase, thereby indicating pseudogap effects. Here , or . From Ref. 72.

Experimental rf spectra at unitarity. The temperatures labeled in the figure were computed theoretically at unitarity based on adiabatic sweeps from BEC. The two top curves thus correspond to the normal phase, thereby indicating pseudogap effects. Here , or . From Ref. 72.

Comparison of calculated rf spectra (solid curve, ) with experiment (symbols) in a harmonic trap calculated at for the two lower temperatures. The temperatures were chosen based on Ref. 72. The particle number was reduced by a factor of 2, as found to be necessary in addressing another class of experiments.^{88} The dashed lines are a guide to the eye. From Ref. 95.

Comparison of calculated rf spectra (solid curve, ) with experiment (symbols) in a harmonic trap calculated at for the two lower temperatures. The temperatures were chosen based on Ref. 72. The particle number was reduced by a factor of 2, as found to be necessary in addressing another class of experiments.^{88} The dashed lines are a guide to the eye. From Ref. 95.

Breathing mode frequencies as a function of , from Tosi *et al.* ^{10} The main figure and inset plot the transverse and axial frequencies, respectively. The solid curves are calculations^{101} based on BCS–BEC crossover theory at , and the symbols plot the experimental data from Kinast *et al.* ^{71} Here is total atom number, and the harmonic oscillator length.

Breathing mode frequencies as a function of , from Tosi *et al.* ^{10} The main figure and inset plot the transverse and axial frequencies, respectively. The solid curves are calculations^{101} based on BCS–BEC crossover theory at , and the symbols plot the experimental data from Kinast *et al.* ^{71} Here is total atom number, and the harmonic oscillator length.

Typical phase diagram for a quasi-two dimensional -wave superconductor on a tight-binding lattice at high filling per unit cell; here the horizontal axis corresponds to , where is the in-plane hopping matrix element.

Typical phase diagram for a quasi-two dimensional -wave superconductor on a tight-binding lattice at high filling per unit cell; here the horizontal axis corresponds to , where is the in-plane hopping matrix element.

Extrapolated normal state (PG) and superconducting state (SC) contributions to SIN tunneling and thermodynamics (left), as well as comparison with experiments (right) on tunneling for BSCCO^{33} and on specific heat for .^{113} The theoretical SIN curve is calculated for , while the experimental curves are measured outside (dashed line) and inside (solid line) a vortex core.

Extrapolated normal state (PG) and superconducting state (SC) contributions to SIN tunneling and thermodynamics (left), as well as comparison with experiments (right) on tunneling for BSCCO^{33} and on specific heat for .^{113} The theoretical SIN curve is calculated for , while the experimental curves are measured outside (dashed line) and inside (solid line) a vortex core.

Comparison between calculated lower critical field, , as a function of (upper left panel), and experimental data (lower left) from Ref. 114, with variable doping concentration . The right column shows normalized plots, versus , for theory and experiment, respectively, revealing a quasi-universal behavior with respect to doping, with the exception of the ortho-II phase. Both theoretical plots share the same legends. The quantitative agreement between theory and experiment is quite good.

Comparison between calculated lower critical field, , as a function of (upper left panel), and experimental data (lower left) from Ref. 114, with variable doping concentration . The right column shows normalized plots, versus , for theory and experiment, respectively, revealing a quasi-universal behavior with respect to doping, with the exception of the ortho-II phase. Both theoretical plots share the same legends. The quantitative agreement between theory and experiment is quite good.

Comparison of theoretically calculated low- slope (main figure) for various doping concentrations (corresponding to different ) in the underdoped regime with experimental data (inset) from Ref. 114. The theoretical slopes are estimated using the low-temperature data points accessed experimentally. The quantitative agreement is very good.

Comparison of theoretically calculated low- slope (main figure) for various doping concentrations (corresponding to different ) in the underdoped regime with experimental data (inset) from Ref. 114. The theoretical slopes are estimated using the low-temperature data points accessed experimentally. The quantitative agreement is very good.

Calculated transverse thermoelectric response, which appears in the Nernst coefficient, as a function of temperature for the underdoped cuprates.

Calculated transverse thermoelectric response, which appears in the Nernst coefficient, as a function of temperature for the underdoped cuprates.

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