Pseudogaps in strongly interacting Fermi gases

TL;DR

This paper reviews the current experimental and theoretical understanding of pseudogaps in strongly interacting Fermi gases, highlighting the phenomenology associated with the suppression of spectral density near the Fermi level.

Contribution

It provides a comprehensive overview of pseudogap phenomena in cold Fermi gases, bridging concepts from condensed matter physics and cold atom experiments.

Findings

Pseudogaps are observed in strongly interacting Fermi gases.

Spectral density suppression near the Fermi level is a key feature.

The review connects experimental results with theoretical models.

Abstract

A central challenge in modern condensed matter physics is developing the tools for understanding nontrivial yet unordered states of matter. One important idea to emerge in this context is that of a "pseudogap": the fact that under appropriate circumstances the normal state displays a suppression of the single particle spectral density near the Fermi level, reminiscent of the gaps seen in ordered states of matter. While these concepts arose in a solid state context, it is now being explored in cold gases. This article reviews the current experimental and theoretical understanding of the normal state of strongly interacting Fermi gases, with particular focus on the phenomonology which is traditionally associated with the pseudogap.

Equations

$$E(k)=\sqrt{{\left(\frac{k^2}{2m}-\mu \right)}^2+{\Delta }_s^2},$$

$${\left(a_s\right)}_{\mathrm{Born}}=\frac{m}{2\pi {\hslash }^2}\int V(r)d^{3}r.$$

$${\Gamma }_p(\nu )\propto G^{<}(p,\nu -{\nu }_0-p^2/2m),$$

$$\Gamma (\nu )=\int \frac{d^{3}p}{(2\pi {)}^2}{\Gamma }_p(\nu ).$$

$$\frac{\partial F}{\partial a^{-1}}=-\frac{\hslash }{4\pi m}C.$$

Full text

I Introduction

In this review I describe attempts to understand the normal state properties of strongly interacting Fermi gases (typically 40K or 7Li) cooled to quantum degeneracy, with particular focus on the single particle spectrum. At sufficiently low temperatures, $T<T_{c}$, the atoms pair up to form a superfluid. Given that it takes energy to break the pairs, there is no way to remove a particle without adding energy to the system: The single particle spectrum contains a gap. This gap is extremely important, and almost all properties of a superconductor/superfluid either follow from this gap or from the behavior of the order parameter. A key question is what happens to this gap when $T>T_{c}$. In high temperature superconductors, vestiges of the gap remain in the normal state. These observations caused researchers to question if such “pseudogaps” are a generic feature of strongly interacting fermions. The path to answering this question brings us through one of the most important areas of modern condensed matter physics research: understanding correlations in disordered states of matter. As I will explain, experimental studies of ultracold Fermi gases have played an important role in solidifying our understanding of these questions.

In section I.1 through IV.1, I develop the themes which motivate these studies. In section V and VI I review the experimental and theoretical work. In VII I summarize the results, and outline the prospects for the future.

There are a number of books and reviews on the subject of strongly interacting Fermi gases chevymora ; feshbachreview ; compilations1 ; compilations2 ; compilations3 ; bloch ; giorginireview ; BCSBECreview ; tormareview ; chen ; levinxx ; levinhulet ; melo ; ketterlereview ; jinreview , each with their own focus. Of particular note is Chen and Wang’s recent review of pseudogap physics in Fermi gases with an emphasis on a theoretical approach which they helped develop chen . Chen was also an author of an earlier review with a similar focus levinxx . While there inevitably will be some overlap, my perspective will be broader, and hence complementary.

In order to keep the discussion focused, I will restrict the discussion to three-dimensional systems. Analogous physics also is found in quasi-two dimensional gases twod . I will also avoid the very interesting subject of “imbalanced” gases, where there is a finite spin polarization imbalanced ; chevyreview

I.1 The challenges of understanding normal states

It is often much more challenging to understand “disordered” states of matter than “ordered” states. A good example comes from comparing a classical liquid and a classical crystal. The simplest cartoon of a crystal involves pinning down the exact location of each particle. This is a reasonable starting point, and is easily expanded on to make predictions (for example, one can connect the particles with “springs” to model the low energy excitations). The simplest cartoon of a liquid is that each particle is equally likely to be at any place in space. While a reasonable model for a gas, this does not give a good starting point for a liquid, where the individual molecules are in constant contact. The key to a liquid is that the atomic positions are correlated, but that they are not ordered. This tension between having structure, but not too much structure, makes the resulting theory more complex.

There are many areas of modern condensed matter physics where intense effort is devoted to understanding disordered quantum mechanical state of matter. These include: Frustrated spin systems, which sometimes have disordered ground states dubbed “spin liquids” spinliquid ; high temperature superconductors, where the normal state shows a range of behaviors, described by terms such as “pseudogap” and “strange metal” randeria ; levin ; timusk ; lee ; leereview ; multiferroics, where proximity to various ordered phases may lead to technologically useful properties multi ; quantum critical systems, where the proximity to a zero temperature phase transition drastically modifies the normal state sachdev ; and one-dimensional wires, where kinematic constraints and the high density of states at low energy suppress ordering giamarchi .

What makes these normal states hard to model is the fact that they are not well described by noninteracting electrons (the analog of modeling a classical liquid by saying the positions of all the particles are at random locations). One typically describes them as “strongly correlated.” Because interactions are strong, perturbation theory about the noninteracting state fails. One can use variational techniques jastrow ; gutzwiller , but optimizing these variational wavefunctions are far from trivial cyrus , and even calculating the energies of these variational states is challenging. Moreover, without insight into the physics, one has little hope of constructing an appropriate variational wavefunction. Ab-initio methods, such as quantum Monte-Carlo mc hold promise, but often face technical difficulties. These states are disordered, so there is no conventional “mean field theory” for describing them.

Given the difficulty of understanding the microscopics of these systems, it is important to develop phenomenological pictures of the emergent physics. Pseudogaps provide one natural organizing principle.

II What is a pseudogap?

As a phenomenological concept, there is no sharp definition of a pseudogap, and one sees the term used in several different field. In this section I give a brief description of superconducting order parameters, which then allows us to describe several proposed definitions.

The simplest model of a superconductor degennes yields a single particle excitation spectrum

[TABLE]

where $k$ is the momentum of the excitation, $m$ the electron mass, $\mu$ the chemical potential, and the spectral gap $\Delta_{s}$ is the energy of the lowest energy excitation. The order parameter of a superconductor is typically taken to be $\Delta_{0}=g\sum_{k}\langle a_{k\uparrow}^{\dagger}a_{-k\downarrow}^{\dagger}\rangle,$ where $g$ is a typical interaction energy (traditionally related to the electron-phonon coupling), and $a_{k\sigma}$ is the annihilation operator for an electron with momentum $k$ and spin $\sigma$. This order parameter quantifies the degree of “pairing” in the system, and it vanishes in the normal state. The intuitive picture of the order parameter $\Delta_{0}$, is that it measures the fraction of fermions which are bound up in Bose condensed pairs. Within weak-coupling mean field theory, $\Delta_{0}=\Delta_{s}$, and the term “gap” is used interchangably for each of these.

Generically $\Delta_{s}\neq\Delta_{0}$. In fact, it is possible to have pairs ($\Delta_{0}\neq 0$) without a spectral gap ($\Delta_{s}=0$). This occurs in strongly disordered superconductors woolf ; ag . Conversely, there are many sources of spectral gaps which have no connection with superconductivity.

Thus the phrase “pseudogap” could naturally have two different meanings: It could refer to $\Delta_{s}$, describing a spectrum which lacks a gap, but which has a suppression of the density of states near the Fermi surface. Alternatively it could refer to $\Delta_{0}$, describing a system in which there is some precursor of pairing or in which there exists pairs which are not Bose condensed. A common assumption is that these two meanings of “pseudogap” are linked, and that the existence of superfluid precursors leads to a suppression of spectral weight. Such precursors have been widely studied paraconductivity . One of the important features of the cold gas system is that it provides a setting in which one can investigate these links.

The nomenclature is even more muddied. Some timusk define a pseudogap to mean a that the spectrum is gapped in some regions of momentum space, but are ungapped in others. Others levinxx define a pseudogap by the condition that $\Delta_{0}=0$ but $\Delta_{s}\neq 0$. Some Tsuchiya even define multiple “pseudogaps” corresponding to multiple spectral features. All of these definitions have merit, and most controversies about the “existence” of pseudogaps chen typically reduces to differences in definitions.

The most widely used definition is that a pseudogap is a depression in the single-particle density of states near the Fermi energy. Even with this definition, there is not a unique scale associated with the phenomenon. Just taking one set of authors, Tajima et al. use $\tilde{T}^{*}$ to denote the highest temperature at which this depression appears ohashispin . They also introduce $T^{*}$ as the temperature at which the density of states at the Fermi surface is maximal. Typically $T^{*}\gg\tilde{T}^{*}$. This nomenclature is by no means standard. For example, Magierski et al. use $T^{*}$ to denote the temperature at which a depression first appears in the density of states magierskigap .

A second approach to defining a pseudogap involves fitting the excitation spectrum to Eq. (1). The coefficient $\Delta_{s}$ then quantifies the pseudogap. Within this approach one can have a depression in the density of states without having a pseudogap. For example, take $\mu<0$ and $\Delta_{s}=0$. The spectrum in Eq. (1) then has a gap of $-\mu$. While it has the advantage of cleanly connecting to the ordered state, this fitting approach can seem overly rigid. In particular, Eq (1) is often not a good model of the single particle spectrum, in which case this definition is meaningless.

To further understand this issue, one must note that in general it is impossible to ascribe a one-to-one relation between momentum and energy: Interactions with other electrons, phonons, or impurities, mean that an electron will only have a momentum $k$ for a finite time. Due to the energy-time uncertainty relationship this gives an uncertainty in the electron’s energy. Thus the natural way to describe the excitation energies is through the spectral density $A_{k}(\omega)$. The probability $P$ that an excitation of momentum $k$ has energy between $\hbar\omega_{1}$ and $\hbar\omega_{2}$ is $P=\int_{\omega_{1}}^{\omega_{2}}A_{k}(\omega)d\omega/(2\pi)$. In an ideal gas $A_{k}(\omega)=2\pi\delta(\hbar\omega-k^{2}/2m)$ is non-zero only when $E=k^{2}/2m$. Finite lifetime broadens this spectral function. In the BCS theory, $A_{k}(\omega)$ has two branches, and is nonzero when $\hbar\omega=E(k)$ or $\hbar\omega=-E(k)$. The former corresponds to excitation processes where one an unpaired particle, while the latter to one in which you a pair and a hole.

Figure 1 shows a typical spectral density $A_{k}(\omega)$ in one model of the strongly interacting Fermi gas described in section IV. Darker colors represent higher spectral density, and to the extent one can define a dispersion relationship, it should follow the darkest regions. For this particular model (the Nozieres and Schmidt-Rink T-matrix approach, see sec. VI.5), and these parameters ($T/T_{c}=2$, $a_{s}=\infty$ – see sec. IV), one has a distinct two-branch structure, but the dispersion is not well described by Eq. (1). Nonetheless, the density of states is suppressed near the Fermi level, and it would seem reasonable to say that this state has a pseudogap. Note, there are other approximations, such as the one developed by the University of Chicago group chien , which yield spectral densities which are better fit by Eq. (1).

To further add to the controversy, some advocate that one should reserve the phrase “pseudogap” for the case where the pairing comes from many-body effects rather than two-body effects. Others further stipulate that the temperature should be sufficiently low, or that the gas should be a “non-Fermi liquid”.

This diversity of definitions is natural in a young field, and as time evolves consensus will develop. In this review I will take as expansive a definition as possible of the pseudogap, and use the term for both pairing and spectral features, without attaching any additional requirements. This broad definition has relevance to the widest range of systems.

III Where are Pseudogaps formed?

While the main purpose of this review is to discuss the BCS-BEC crossover in cold Fermi gases, the importance of the subject is only clear by looking at some of the cannonical examples of pseudogaps. In this section we discuss the cuprate superconductors and 1D charge-density-wave materials.

III.1 The Pseudogap regime in the cuprate superconductors

The concept of the pseudogap was developed in the normal state of “underdoped” cuprate superconductors randeria ; levin ; timusk ; lee ; leereview . The cuprate superconductors (such as La2-xSrxCuO4 and YBa2Cu3O7-x) are doped antiferromagnetic insulators butch . The “parent compound” formed when $x=0$ is magnetically ordered at low temperatures. As one increases $x$ to a few percent, the magnetic transition temperature appears to drop to zero. With increasing $x$, one finds superconducting order. the superconducting transition temperature rises with $x$, peaking around $x=20\%$, denoted “critical doping” $x_{c}$. Any $x<x_{c}$ is refered to as “underdoped.” At larger $x$ the superconducting transition temperature falls to zero.

In the underdoped cuprates, the pseudogap refers to a collection of phenomena which can be interpreted as a reduction of the density of states for low energy excitations. For example, both spin susceptibility and spin relaxation are suppressed randeria . This suppression in the spin susceptibility/relaxation would, for example, be consistent with the electrons being bound up in singlet pairs – ie. it is natural to interpret it as a precursor to superconductivity.

The pseudogap has been seen in spectral probes (photoemission pe , tunneling tn , magnetic resonance nmr ), transport (optical and DC conductivity opticalcond ; dccond ) and thermodynamic probes (magnetic susceptibility susc , specific heat sh ). One of the more intriguing results was the observation that, in a magnetic field, applying a thermal gradient will induce a relatively large transverse voltage drop nernst ; tannernst . This “Nernst” effect has been interpreted as a sign of “vortices” in the normal state, and hence a superconducting precursor.

While it is natural to ascribe a connection between the various observations and superconductivity, the modern concensushiddenarpes ; hidden seems to be that they are instead a signature of “hidden order.” Candidate orders include magnetic order, “d-density waves”, and “electronic nematics.” Both scanning tunneling microscopy and angle resolved photoemission spectra reveal a “two-gap” structure, where spectral suppression near the nodal direction are attributed to superconductivity and structures near the antinodal direction are attributed to the “pseudogap”. The “nodal gap” vanishes above $T_{c}$, while the “antinodal gap” vanishes at a higher temperature, $T_{\rm pg}$. Strong evidence of liquid crystaline smectic/nematic order has been seen in scanning tunneling microscopy lawler .

It is not clear if the hidden order competes with superconductivity, or enhances it hidden ; friendfoe . Some argue that Superconductivity emerges from the quantum fluctuations found near the quantum critical point where the hidden order vanishes quantcrit . One field of thought is that the relationship between the various possible orderings is more complicated, and advocate the phrase “intertwined order” intertwined . Interestingly, there are arguments that the phenomonology of the strongly interacting normal state is largely independent of the nature of the order nikolic .

III.2 Pseudogaps in 1D Peierls systems

It is useful to have a controlled model system to understand how spectral signatures of ordering can be found in an un-ordered state. The simplest such example is the 1D electron gas: a system which is unstable towards forming a charge-density wave peierls . A mean-field treatment of that instability parallels closely the BCS theory of superconductivity, and the single-particle spectrum is similar to that in Eq. (1). In this mean-field theory, the electron density develops sinusoidal oscillations. The electric field from this inhomogeneous charge density provides a potential, which reinforces the modulation. The electron density is commensurate with the oscillations, and the electrons form a band insulator in this effective potential. In this context, the spectral gap $\Delta_{s}$ appearing in Eq. (1) is associated with the band-gap of the self-consistent potential, and has no connection with superconductivity.

While valuable, this mean-field theory is incomplete, and one expects no long-range order at finite temperature for this system mermin . As Lee, Rice, and Anderson lra agued, and illustrated in Fig. 2, when one includes fluctuations the spectral density no longer contains a gap, but there is a notable depression in the density of states near zero energy. This is a consequence of the short-range correlations, and is also seen in other 1D models papatsvelik . It is natural to assume that such structures are generic, and whenever you have local order, but no long range-order you expect a similar spectrum.

A physical picture of Fig. 2 comes from noting that there are length and time-scales associated with the fluctuations. If you examine the system on length-scales smaller than the correlation length, or time-scales smaller than the correlation time, the system appears ordered, and the mean-field theory applies. Thus if you measure the density of states with an instrument with finite spectral resolution, one can only distinguish the three curves in Fig. 2 if your resolution is sufficiently high.

It is natural to refer to the spectral features in Fig. 2 as a pseudogap.

III.3 Critical Phenomena

Any second order phase transition has a critical region near $T_{c}$ where there is local order. As one approaches $T_{c}$ the ordered regions becomes large. This happens even in conventional superconductors ar ; paraconductivity , which are not typically thought of as possessing a pseudogap, and one may therefore want to be careful about using that term to describe this phenomenon. The example in Sec. III.2 can be thought of as an extreme example of these critical fluctuations.

IV Dilute Fermi gases

In this section we give a quick introduction to the physics of dilute Fermi gases, and explain why they are ideally suited for studying superconducting pseudogaps. Stajic et al. were one of the first groups to advocate this perspective stajic .

IV.1 Feshbach resonances and the BCS-BEC crossover

One of the most beautiful features of nature is its universality. The physics found in neutron stars at densities of $10^{38}$cm*-3* and temperatures of $10^{11}K$ can be connected to phenomena in metals ($n\sim 10^{22}{\rm cm}^{-3},T\sim 200$K) or even ultracold atomic gases ($n\sim 10^{12}{\rm cm}^{-3},T\sim 10^{-9}$K). A typical Fermi gas experiment involves $N\sim 10^{6}$ atoms of 6Li or 40K, confined by a roughy harmonic optical potential ($V\sim m\omega^{2}r^{2}/2$, with $\omega\sim 100$Hz). Although the nK temperatures are low by absolute standards, the densities are also low. The temperature scale associated with the density, $T_{f}\sim\hbar^{2}n^{2/3}/m$ is rarely more than 5 times the temperature. For comparison, in a room temperature metal $T/T_{F}<0.01$.

While the phenomena of these disparate systems are very similar, the cold atom experiments are carefully engineered so that the microscopic description is particularly simple. For example, because of the low densities, only pairwise collisions collisions occur. Moreover, the temperatures are so low that the DeBroglie wavelength of the atoms are much larger than the range of the interatomic forces. Under these conditions the details of the interatomic potentials become irrelevant, and scattering can be parameterized by a single quantity, the s-wave scattering length $a_{s}$ landauquantum . A key feature of cold gas experiments, which make them ideal for studying strong-interaction phenomena such as pseudogaps, is that the scattering length can be experimentally tuned tiesinga .

The scattering length is formally defined by considering the phase shift $\delta_{k}$ between long wavength incoming and outgoing spherical waves: $a_{s}=\lim_{k\to 0}-\delta_{k}/k$. This is not a particularly intuitive definition, but by considering a few paradigmatic potentials one can develop an understanding. First, if one has a smooth potential, one can use the Born approximation,

[TABLE]

Thus one associates a positive scattering length with repulsion, and a negative scattering length with attraction. Moreover, stronger scattering is associated with larger scattering lengths.

A second paradigmatic potential is a hard sphere $V(r)=\infty$ for $r<r_{0}$ and $V(r)=0$ for $r>r_{0}$. For a hard sphere, $a_{s}=r_{0}$ is always positive. One often thinks of the scattering length as the radius of a hard-sphere potential which has the same low-energy scattering properties as the real potential. Of course, negative scattering lengths do not fit into this paradigm.

Finally it is useful to consider an attractive square well: $V(r)=-V_{0}$ for $r<r_{0}$ and $V(r)=0$ for $r>r_{0}$. Calculating the scattering length for such a potential is straightforward feshbachreview . For small $V_{0}$, Eq. (2) holds, and the scattering length is negative. However, when one makes the well deeper the scattering length becomes large and negative – eventually diverging to $-\infty$ before jumping to $+\infty$. This jump coincides with the appearance of a bound state in the potential: $a_{s}$ is large and positive if there is a weakly bound state (in which case $E_{b}=\hbar^{2}/(ma_{s}^{2})$), and it is large and negative if there is a low energy resonance. Thus, somewhat counterintuitively, one can have a positive scattering length, even when the potential is attractive. One way to understand this behavior is that the scattering states must be orthogonal to the bound state – and hence a low energy bound state acts similarly to a repulsive potential.

An atomic physicist can modify the scattering potential by applying a magnetic field feshbachreview . The principle is that the atoms can have a bound state whose magnetic moment differs from that of the atoms. Changing the magnetic field is then analogous to changing the depth of an attractive well. At a scattering resonance, the bound state becomes degenerate with the scattering state, and the scattering length diverges. As with the attractive square well example, the scattering length is large and negative when the “bound state” is slightly above threshold. Conversely, the scattering length is large and positive when the bound state is below threshold. The inverse scattering length $1/a_{s}$ smoothly crosses zero as one changes the magnetic field. Nothing dramatic happens at the point $1/a_{s}=0$.

In the context of Fermi gases, one refers to the regime where $a_{s}<0$ as the “BCS” regime: When $k_{f}a_{s}\ll 0$, the theory of superfluidity developed by Bardeen, Cooper, and Schreiffer applies bcs , and the ground state is a superfluid of loosely bound Cooper pairs. The short-range attraction is the “glue” holding the particles together. The regime $a_{s}>0$ is instead referred to as the “BEC” regime. In this regime, there is a two-body bound state, representing a diatomic molecule. Pairs of atoms combine into these bosonic molecules, which undergo Bose-Einstein Condensation, forming a superfluid BEC . No phase transition occurs when one changes $a_{s}$ at zero temperature. Rather, the size of the pairs just continuously evolves.

For technical reasons, the point $1/a_{s}=0$ is referred to as “unitarity” or the “unitary limit”. An excellent discussion can be found in feshbachreview .

The terms “BCS,” “BEC,” and “unitary,” will be used throughout this review.

IV.2 Pseudogap in the BCS-BEC crossover

Above $T_{c}$ in the deep BCS limit, the single-fermion excitations will be gapless. One has a conventional Fermi liquid fermiliquid , and it would be surprising if there were any gap-like feature in the single particle spectrum. The deep BEC limit is very different. There the normal state consists of a gas of diatomic pairs. Any single-fermion excitation would require breaking a pair, leading to a spectral gap. Due to the presence of thermally dissociated molecules, the gap will not be perfect, and there will be an exponentially small density of states at zero energy. This can be considered a classic example of a pseudogap.

As one moves from $1/a_{s}=+\infty$ to $1/a_{s}=-\infty$, the normal state has a smooth crossover from a gas of diatomic pairs with a strong pseudogap, to a normal Fermi liquid, with no gaplike feature. The pseudogap exists where there are strong short-range pairing correlations, and is absent when the correlations are gone. Thus pseudogap features continuously grow in strength as one moves towards the BEC side of resonance.

An important caveat is that if one fits the spectrum to a form like Eq. (1), then one expects that in the BEC limit $\mu$ is negative, and $\Delta_{s}$ is small. The negative chemical potential encodes the binding energy of the pairs, and the gap is $|\mu|$. In the BCS limit, one instead expects $\mu$ to be positive, and $\Delta_{s}$ again small. It is only in the crossover between these regimes that one expects a fit to a form like Eq. (1) will yield significant $\Delta_{s}$. Thus, as anticipated in Sec. II, if one defines the pseudogap via such a fitting procedure, then one would say that the the pseudogap can only exist at intermediate coupling.

Ignoring these questions of nomenclature, it is quite difficult to accurately model the gas when $k_{f}|a_{s}|\gg 1$. Consequently different theoretical models give different locations and sharpness for the crossover. This diversity is seen in comparative studies chien ; levincompare ; hucompare ; strinatireview ; onset ; virialspectraltrap which find disagreement about the existence/strength of pseudogap spectral features at the nominal midpoint of the crossover, $1/a_{s}=0$.

While there is no concensus about the existence of a pseudogap at unitarity, there is agreement that superconducting fluctuations strongly influence the spectrum in this regime. Additionally, all agree that pairs dominate in the BEC regime. Thus, at least by the most expansive definitions, sufficiently far in the BEC regime there is a pseudogap.

V Experimental Probes of the Normal State of strongly interacting Fermi gases

There are a number of ways to experimentally study the pseudogap in Fermi gases. These range from spectroscopic to thermodynamic. Here I will describe the main results. Definitive pseudogap features have been seen in the BEC regime, but the results at unitarity are ambiguous.

V.1 Photoemission spectroscopy

Photoemission spectroscopy has been one of the most promising probes of cold Fermi gases. It takes advantage of the fact that atoms are not simply spin-1/2 Fermions, but have more degrees of freedom. Typical experiments involve a mixture of atoms in two collisionally closed hyperfine states, denoted $|\!\!\uparrow\rangle$ and $|\!\!\downarrow\rangle$. In photoemission spectroscopy, one drives a transition from the state $|\!\!\downarrow\rangle$ to a third states $|x\rangle$. The experimentalists then look at the number of atoms transfered as a function of the frequency of the drive. Reviews of this technique can be found in levinrfreview ; tormareview .

A number of different terms are used to describe this technique: Typical transitions are in the radio or microwave band – and hence this technique is most often refereed to as “RF-spectroscopy” or “Microwave-spectroscopy.” It could also be referred to as “internal state spectroscopy.” The term “photoemission spectroscopy” comes from the fact that a particle is “emitted” from the active sector.

Photoemission spectroscopy is easiest to understand when the $|x\rangle$ atoms do not interact with the others. In that case, the transition rate can be expressed in terms of the single-particle spectral density $G^{<}(k,\omega)=f(\omega)A_{k}(\omega)$, which encodes how many ways there are to remove a particle, changing the energy by $\hbar\omega$, and the momentum by $\hbar k$. The Fermi function $f(\omega)=1/(e^{\beta(\omega-\mu)}+1)$ encodes the filling of the states, and as discussed in section II, $A$ encodes the dispersion. Repeating our previous example, in an ideal Fermi gas, $A_{k}(\omega)=2\pi\delta(\hbar\omega-k^{2}/2m)$ is non-zero only when $\hbar\omega=k^{2}/2m$. Finite lifetime broadens this spectral function.

In a paired system one expects $A_{k}(\omega)$ to have two branches: one associated with removing a particle by breaking up a pair, another with adding an unpaired particle chinjulienne . The existence of these two branches, with a region of low spectral weight between them, is another definition of the pseudogap. Unfortunately, since it involves removing particles, photoemission spectroscopy is only sensitive to one of the branches. There have been attempts to “inject” particles, but so far they have only been applied to the noninteracting system photoinjection . Injection experiments probe $G^{<}(k,\omega)=(1-f(\omega))A_{k}(\omega).$

The connection between photoemission and the spectral function comes from Fermi’s Golden Rule, which encodes conservation of energy and momentum. In particular, when illuminated by light of frequency $\nu$, the rate of production of $x$-state atoms with momentum $p$ is expected to scale as

[TABLE]

where $\hbar\nu_{0}$ is the energy difference between the internal states $|\downarrow\rangle$ and $|x\rangle$ in vacuum, and $\hbar\nu_{0}+p^{2}/2m$ corresponds to the energy of the atom emitted in the $|x\rangle$ state. In solid state experiments, the technique known as “Angle Resolved Photo-Electron Spectroscopy” measures this same quantity.

Initial experiments chinrf ; greinerrf ; bcsbecrf ; schuncknature ; ketterlerf averaged over all momentum, measuring

[TABLE]

Later experiments developed techniques to directly observe $\Gamma_{p}(\omega)$ momentumresolved . The resulting data can be deconvolved to produce $A(k,\omega)$. One can then analyze this quantity for signs of pairing. Because of the Fermi occupation factor, one only has access to freqencies below the chemical potential.

Some of the first evidence for pairing in the BCS-BEC crossover came from such experiments. Chin et al. found that at high temperatures the absorption spectrum was sharply peaked, $\Gamma(\nu)\approx\delta(\nu-\nu_{0})$ chinrf . This behavior would be expected for a non-interacting gas, and is indicative of a lack of pairing. At low temperatures, they instead saw a broad peak, which vanished below a threshold $\nu_{\rm th}>\nu_{0}$. The difference $\delta\nu=\nu_{\rm th}-\nu_{0}$ was taken as a measure of the gap (though at the time competing theories were put forth for the observations which did not require superfluidity massignan ; generic ). At intermediate temperatures they observed a bimodal distribution – formed from a sharp peak at $\nu_{0}$, and a broader peaked centered at higher frequencies. This structure naturally arises from the inhomogeneity of the cloud: The weakly interacting wings contribute a delta-function at $\nu_{0}$, while averaging over the more strongly interacting central region yields the broad peak kinnunen ; regal ; schunck ; ohashi ; yanhe ; generic ; massignan . Variants of this technique were developed in several other labs, and gave strong evidence for pairing at low temperatures greinerrf .

Early experiments were complicated by trap inhomogeneities. At MIT they developed a spatial resolved spectroscopy mitspatialresolve . More recent experiments at JILA implemented a hybrid method which probes only the center of the cloud, but is momentum resolved realandk ; localPES . This position and momentum momentum resolved photoemission spectroscopy is one of the most important tools that has been developed by the cold gas community.

A second important technical issue involved final state interactions. The assumption that the final state atoms are non-interacting is not always valid strinatiRFAL ; final ; schuncknature . These final-state interactions are particularly problematic in 6Li, where all collision channels have nearby Feshbach resonances. Fortuitously, the final-state interactions are weak in 40K, and the most quantitative experiments involve those atoms. Note, that even with final-state interactions, one can learn a great deal about pairing from internal state spectroscopy, but the analysis is more complicated, and requires modeling.

Since the primary technical issues have been resolved, there is now excellent spectroscopic data localPES ; realandk , especially at unitarity, and one can reasonably ask if the experimentally observed $A_{k}(\omega)$ is indicative of a pseudogap. Unfortunately, there is no simple answer to this question. The primary difficulty is that at unitarity the normal-state spectra are broad and indistinct. Such broad spectral functions are expected, and are indicative of the short quasiparticle lifetimes in the strongly interacting gas lifetime ; palestinilifetimes . These short lifetimes are consistent with the hydrodynamic behavior of the gas hydro . Adding to the difficulty is the fact that because one is extracting particles, the experiment only provides $A_{k}(\omega)$ for energies below the Fermi level.

The end result is that despite attempts to argue one way or another localPES ; lifetime , the photoemission spectroscopy experiments are ambiguous about the presence of a pseudogap at unitarity. As previously explained, such ambiguity is neither surprising, nor alarming. In tuning from the BEC to BCS regime, one expects a smooth crossover between a normal state dominated by bosonic pairs, to one dominated by free fermions. The former will show a distinct pseudogap, while the latter will have absolutely no pseudogap features. Unitarity sits between these limits, and the spectrum is appropriately ambiguous.

V.2 Equation of State

The second major probe of atoms in the BCS-BEC crossover is thermodynamics ketterleeos ; parisEOS ; MITEOS ; thermo ; strinatiEOS ; compress . The most detailed studies have concentrated on the equation of state relating the pressure $P$ to the temperature $T$ and the chemical potential $\mu$ parisEOS ; MITEOS – finding remarkable agreement between numerical Monte-Carlo techniques, and experiments.

In principle the equation of state can distinguish between a Bose and Fermi gas – and hence can identify signatures of pairs in the normal state. For example, as $T\to 0$ in an ideal Fermi gas, the pressure varies quadratically with temperature, $P\to P_{0}+\lambda T^{2}$, where $P_{0}$ and $\lambda$ are constants related to the density. Fermi liquid theory predicts that this temperature dependence is robust against interactions fermiliquid . The equation of state of an ideal Bose gas is very different. If one stipulates that $T\gg T_{c}$, then the Bose gas obeys the ideal gas law and the pressure vanishes linearly with $T$. One might expect that a pseudogap state would follow the bosonic prediction. One should be cautious, however: In the pseudogap regime, the atoms are degenerate, and interactions are strong. Thus it would be naive to expect that the equation of state of the psuedogap obeys an ideal gas law.

Experiments parisEOS ; MITEOS , and theory MITEOS ; strinatiEOS ; levinEOS , find that in the normal state of the unitary Fermi gas ($1/k_{f}a=0$), the equation of state is well fit by the empirical curve $P=P_{0}+\lambda T^{2},$ as predicted by Fermi liquid theory. The Paris group went so far as to state that this observation is incompatible with a pseudogap parisEOS . This conclusion is not universally accepted, as there are many pseudogap theories which predict this same temperature dependence levinFL ; levinEOS ; chienlevin . Regardless, the experimental results certainly show that structureless noninteracting bosonic pairs are not dominating the physics. It would be extremely interesting to see how the equation of state evolves as one tunes towards the BEC limit, where pairs are more tightly bound, and pseudogap features are expected to become stronger. If there are deviations from the Fermi-liquid predictions, they will be most obvious there.

The techniques used to measure the equation of state are quite elegant. They take advantage of the fact that the trapped clouds are in hydrostatic equilibrium: Consider a small cube of gas at position $r$, with volume $d^{3}r$. The trapping force in the $\hat{x}$ direction is $F_{\rm trap}=-nd^{3}r\partial_{x}V(r)$. This should be balanced by the hydrodynamic forces from the neighboring regions, $F_{\rm hydro}=d^{2}rP({\bf r}-\hat{x}dr/2)-d^{2}rP({\bf r}+\hat{x}dr/2)=d^{3}r\partial_{x}P(r)$. Experimentalists know $V({\bf r})$, and measure $n({\bf r})$, and integrate the hydrostatic equations, $F_{\rm trap}+F_{\rm hydro}=0$, to find $P({\bf r})$. They thereby parametrically produce a relationship between the density and pressure. One can simplify the analysis, and reduce systematic errors, by appropriately engineering the trapping potential zwierleinhomo .

There are a number of other techniques which allow one to access thermodynamic quantities thermo ; fluc ; compress ; criticalvel ; mitimage ; lingham . These include looking at the response of the cloud to a perturbation lingham ; compress ; criticalvel ; bragg ; bruun and measuring local fluctuations fluc . Such studies have been useful for judging the quantitative accuracy of competing theories, and have helped us develop a phenomonological understanding of both the superfluid and normal state.

V.3 Spin Susceptibility

As argued by Trivedi and Renderia in the context of superconductors trivediranderia , a more direct thermodynamic probe of normal-state pairing is spin susceptibility. If all of the fermions are bound into pairs, then the gas should be strongly diamagnetic. Thus one expects that the $k=0$ spin susceptibility of the pseudogap state should be suppressed. Indeed, Monte-Carlo calculations find a dramatic drop in this susceptibility as one moves from the BCS to BEC side of resonance exptsusc . Diagramatic theories find a similar suppression ohashispin . At unitarity, experiments find that the spin susceptibility is smaller than one would find for a non-interacting gas exptsusc ; mitsusc , in agreement with Monte-Carlo calculations mcdip .

Several different techniques have been used for measuring the spin susceptability. For example, at MIT they measured density profiles in magnetic field gradients mitsusc . Using arguments analogous to those in section V.2, they could then relate the spatial variation of the polarization to the susceptibility. Alternatively, in Paris they measured the equation of state at finite spin polarization exptsusc . The susceptibility was then calculated through a numerical derivative.

As with the spectroscopic data, there is some controversy about the signatures of the pseudogap in spin susceptibility. For example, Nascimbène et al. argued that instead of a suppression in the susceptibility, the pseudogap state should be characterized by a non-linear susceptibility exptsusc : they argue that the susceptibility should be suppressed for small fields, but restored at larger fields. The idea being that a strong enough field will break up the pairs, restoring normal Fermi-liquid behavior. Both experiments, and Monte-Carlo calculations, fail to find this non-linearity near unitarity. Thus Nascimbène et al. conclude that the normal gas at Unitarity does not possess a pseudogap, and hence the pseudogap regime is restricted to the BEC side of resonance.

Tajima et al. provide an alternative picture ohashispin . They used a T-matrix approximation to calculate the susceptibility as a function of temperature throughout the BEC-BCS crossover. Tajima et al. found that the susceptibility is a non-monotonic function of temperature, with a peak at $T=T_{s}$. They interpret $k_{B}T_{s}$ as a pairing energy: for $T>T_{s}$, pairs are unimportant, while for $T<T_{s}$ there are fewer and fewer free spins. They refer to the region $T_{c}<T<T_{s}$ as the “spin-gap” regime – a phrase which is somewhat more precise that “pseudogap.” At unitarity, their approximations yield $T_{s}/T_{c}\sim 1.7$. Although there have not yet been any experiments that have measured $T_{s}$, the techniques in exptsusc or mitsusc could be used for such a study. Wulin et al. levinspin used a different T-matrix theory to calcuate the spin susceptability. They obseved a low-temperature suppression of the spin susceptibility, but did not study the peak.

Tajima et al. also explored the relationship between their susceptibility and the density of states at the Fermi surface. They compared $T_{s}$ with two other scales $T^{*}$ and $\tilde{T}^{*}$: $T^{*}$ is the temperature at which the density of states at the Fermi energy is maximal and $\tilde{T}^{*}$ is the highest temperature at which a depression in the density of states first appears at $T_{c}$. They find that within their theory, $T_{s}\sim T^{*}\gg\tilde{T}^{*}$. Moreover, at unitarity, $\tilde{T}^{*}\sim T_{c}$. Thus, even within a single theory, the different facets of pairing turn-on at different scales. As already mentioned, other authors use different notation.

One can gain even more information about the quantum state by studying the dynamical spin response dynspin ; dynspintheory ; dynspin2 . There does not, however, appear to be any simple story connecting these dynamical experiments to normal-state pairing.

V.4 Transport

Closely related to the thermodynamic probes, are transport measurements. Although a mainstay of solid state, transport is harder to access in cold atoms. Transport is also often harder to interpret. These experiments have given profound insight into the nature of the normal state of the unitary Fermi gas, but no clear connection has been made to pseudogaps transportreview .

The most basic transport-like experiment is time-of-flight expansion. One simply turns off all trapping potentials, and allows the cloud to free-fall under gravity. As it falls, the cloud expands, and by observing this expansion dynamics one can infer its properties. In particular, early experiments on anisotropic clouds found that the aspect ratio of the cloud reversed during expansion, a sign of strong interactions with connections to observations in heavy ion coliders anisoexp . Further information came from exciting collective modes of the cloud colmodes ; hydro , and additional expansion experiments expansion ; universal . Such experiments clearly showed that the strongly interacting Fermi gas behaves hydrodynamically. Later experiments extracted transport coefficients, such as viscosity universal ; damping .

More relevant to normal-state pairing are studies of spin transport. There have been a number of measurements of spin diffusion based upon either studying the dynamics of the magnetization of a cloud in a field gradient thywissenspin , or the relaxation of imposed spin textures mitsusc ; zwierleinspin ; othertransport . There has also been a number of studies of spin waves in the more weakly interacting limit spincurrents . Wulin et al. argue that one distinct signature of pseudogap physics is the relationship between the spin susceptibility and spin diffusivity levinspin . In particular, they argue that the experiments in othertransport imply a pseudogap.

Another class of experiments attempts to replicate a more traditional transport geometry by producing a dumbell shaped trap with two large reservoirs connected by a narrow channel junction . The experimentalists created an initial population imbalance between the two reservoirs, then watched the subsequent dynamics. In the non-interacting limit they observed quantized conductance, as predicted by the Landauer formalism giamarchi . As they increased the strength of (attractive) interactions, they observed a significant enhancement of the conductance. The leading theories suggest that this enhancement is due to pairing – possibly the formation of superfluid regions demlerpair or normal-state pairing uedapair ; zhangpair . As such, these measurements may be indicative of pseudogap physics.

V.5 Contact

In addition to these probes, which have analogs in condensed matter physics, there are a number of observables which are unique to cold atoms – which can be used to learn about the presence of normal-state pairing. The most well-studied of these is the “contact,” introduced by Tan tan , and explored by many other authors othercontact . As its name suggests, the contact is a measure of short-range correlations between particles. Remarkably, in systems with short range interactions, all short-distance (high energy) physics is parametrized by a single number $C$. For example, for large $k$ the momentum occupation factor $n(k)\to C/k^{4}$; for large frequencies, the radio frequency absorption spectrum (sec. V.1) is $I(\omega)\to C/(2^{3/2}\pi^{2}\omega^{3/2})$; and the thermodynamic derivative of the free energy with respect to the scattering length is

[TABLE]

Larger contact implies a greater likelyhood for two particles to be close together. Thus there is a connection between the contact and pairing.

Pieri et al. strinatiRFAL advocated identifying, $C=(m\Delta_{\infty})^{2}$, where they take $\Delta_{\infty}$ to be a local measure of pairing. Indeed, the contact in a BCS superfluid has this form with $\Delta_{\infty}$ replaced with the superfluid gap. Moreover, they argued that this identification was consistent with models of normal-state pairing. One should be cautious, as there are numerous physical settings where it is not natural to identify short range correlations with pairing.

As with other cold-atom probes, inhomogeneous broadening can complicate the extraction of the contact. Moreover, there are particular technical issues associated with the different ways of measuring the contact. For example, one can in principle extract the contact from time-of-flight expansion images jintof . Interactions during time-of-flight can, however, render these results unreliable. Similarly, the spectroscopic probes of contact must deal with final-state interactions.

In one of the first theoretical calculation of the normal state contact, Palestini et al. palestinicontact saw that at $T_{c}$, the contact monotonically decreased as one moved from the BEC to BCS limit, with a relatively sharp crossover in behavior near $k_{f}a_{s}=-2$. They also explored the temperature dependence of the contact, finding a slow rise as temperature is lowered, with a sharper rise near $T_{c}$. One can argue that this region of enhanced $C$ near $T_{c}$ is related to normal-state pairing. Other theoretical approaches yielded similar results bdmcontact ; goulko ; balcontact , though there are disagreements about quantitative details.

Several experimental techniques have been used to study the contact. Kuhnle et al. used Bragg spectroscopy to measure the trap-averaged contact in a unitary Fermi gas kuhnle . They observed the expected monotonic decrease in the contact as a function of temperature, but due to trap averaging their results were not particularly discriminating between competing theories. Later Sagi et al. extracted the contact from the high energy tail in photoemission spectroscopy jincontact2 ; jincontact , using slicing techniques to isolate a roughly homogeneous region localPES .

V.6 Photoassociation

Another probe that is unique to cold atoms is photoassociation photoassociation ; photoassociationreview . There one drives a transition between a scattering state and a bound state. The rate of transition is proportional to the overlap between the quantum states, giving a measure of the short range pair correlations. Varients of this technique were used extensively to study pair correlations in thermal gases thermalphoto , and degenerate Bose gases bosephoto , and form the basis of modern techniques for producing ultracold molecules ultramol .

Experimentalists at Rice University studied the low temperature photoassociation signal throughout the BEC-BCS crossover fermiphoto . Through this measurement they quantified the pairing correlations in the ordered state. Unfortunately, there have been no systematic studies of the photoassociation signal in the normal state of the BCS-BEC crossover. Such measurements would be useful for quantifying precursors of pairing.

A closely related probe is inelastic two and three body collisions. Du et al. measured the rates of these processes, arguing that in the pseudogap regime they are dominated by collisions between atoms and pairs inelastic .

V.7 Quench

The last probe that I will discuss is the response of the gas to a sudden change in the scattering length. In particular, several groups have extracted useful information by rapidly sweeping from the BCS to BEC side of resonance sweep . The idea is that the loosely bound BCS pairs will be projected into tightly bound dimers. These dimers are then robust enough to directly measure. This technique has been used to detect a BCS condensate sweep , vortices vortsweep , and dark solitons solitonsweep . In the normal state one might be able to use such a sweep to get information about the momentum distribution of normal-state pairs. Unfortunately, modeling the sweep is relatively complicated sweepmodel ; sweepmodel2 , and it is hard to extract quantitative data from these experiments.

VI Theoretical Models of the Normal State of strongly interacting Fermi gases

Here I present the models and calculations which have investigated the normal state properties of strongly interacting Fermions in the BCS-BEC crossover. The focus will be on developing an intuition about the approaches. For technical details, readers will be directed to the primary litterature, and other review articles.

The theory of the BEC-BCS crossover predates cold atom experiments. It grew out of chain of research introduced by Eagles eagles , Leggett leggett , and Nozieres and Schmidt-Rink nozieres – which was further developed by Sa de Melo, Randeria and Engelbrecht earlyranderia , then taken up by a larger community chen ; micnas ; ranninger .

As already explained in the introduction, modeling a correlated liquid is difficult. Neither a dilute gas, nor a solid are good starting points. We are interested in a normal state, for which there is no long-range order to expand about. A number of approaches have been developed: scaling theories, Monte-Carlo methods, high temperature expansions, higher order perturbation theories, and self-consistent diagrammatic expansions. Each of these yield their own insight into the problem.

VI.1 Scaling Theories

Nearly all attempts to understanding the strongly interacting normal state makes use of the relatively small number of dimensional quantities in this system: the Fermi energy $E_{f}=k_{f}^{2}/2m$, the scattering length $a_{s}$, and the temperature $T$. At unitarity, $a_{s}=\infty$, and there are at most two scales in the system. Thus all thermodynamic functions take simple forms. For example, the pressure can be written as $P=E_{f}k_{f}^{3}f(\beta E_{f})$, where $\beta=1/k_{B}T$ and $f$ is a function of one variable bertsch . This scaling also leads to a number of relations between thermodynamic functions castin , and can also be applied to dynamical quantities. Any theory of the pseudogap needs to obey these relations. The techniques discussed in section VI.2 through VI.5 can be used to calculate or constrain the scaling function.

VI.2 Monte-Carlo Methods

A number of stochastic methods have been developed for studying the properties of strongly interacting Fermi gases. Typically they give access to thermodynamics, and can be compared to the experiments described in section V.2. Initial stochastic approaches to understanding the strongly interacting Fermi gas focussed on ground-state properties, using a number of Monte-Carlo techniques carlsonzeroT ; trentozeroT . One important feature of the attractive Fermi gas is that there exist Monte-Carlo approaches without “sign problems” nosign , and hence, with enough computer power, stochastic methods can produce results with arbitrary accuracy.

A large number of different stochastic techniques have been applied to the finite temperature gas. The earliest of these are the Auxilliary-Field Monte-Carlo calculations of Bulgac, Drut and Magieski bulgac , the hybrid Monte-Carlo calculations of Wingate wingate and of Lee and Schäfer leeschaefer , the determinant Monte-Carlo calculations of Burovski, Prokof’ev, Svistunov, and Troyer determinantmc , and the path integral Monte-Carlo calculations of Akkineni, Ceperley, and Trivedi trivedi . The fact that these very different formalisms agree well with one-another is an excellent indicator of the accuracy of the modeling.

Monte Carlo techniques typically map the thermodynamics of a quantum system onto the thermodynamics of a classical system. One can then sample the classical distribution to calculate thermodynamic properties of the quantum system. The quantity most carefully compared with experiments is the equation of state. For example, in 2012, Van Houcke et al. presented a careful comparision of “Bold-diagramatic Monte Carlo” with experiments at MIT amherstMIT . Data from the same experimental group ku was then later compared with both hybrid and auxiliary field Monte Carlo approaches mceoscompare . Subsequent studies continued to expand the parameter range and accuracy of the numerical calculations oldgoulko ; goulko ; amherst ; pollet ; exptsusc ; lmc ; af , as well as calculating other observables, such as contact contactmc . From these comparisons, there is no doubt that these ab-initio theories accurately model the Fermi gas throughout the BCS-BEC crossover. Unfortunately, as discussed in section V.2, these thermodynamic results are somewhat ambiguous about the presence of normal-state pairing.

Recently progress has been made in extracting spectral data from Monte-Carlo calculations realtime ; polaronspectrum ; magierski ; magierskigap ; wlazlowski ; sheehy ; mcdip . For example, Magierski et al. used numerical analytic continuation to extract single particle spectral densities from Monte-Carlo data magierski . Remarkably, they find a distinct gap-like feature in the spectrum at unitarity. Not only does their density of states show a dramatic dip mcdip near $T_{c}$, but the spectral density $A(k,\omega)$ shows two distinct branches which are well separated from one-another. From signatures like these, Magierski et al. argue that the criterion for observing a pseudogap is $1/k_{f}a_{s}>-0.05$ onset . Similar results are seen in dynamical cluster quantum Monte Carlo sheehy . The ultimate reliability of these calculations needs to be confirmed by appropriate extrapolation to infinite system size, and infinitesimal lattice spacing.

VI.3 High Temperature Expansions

One simple limit of any system is “infinite temperature,” where all states are equally likely, and thermodynamic properties are trivial. Systematic corrections can be calculated in a power series in the fugacity $z=e^{\beta\mu}$, where $\beta=1/k_{B}T$, and the chemical potential $\mu$ is negative. The resulting expansion for the free energy is known as the virial expansion, and has been calculated to second virialsecond , third virialthird ; hightemp and fourth virialfourth ; ngampruetikorn order in $z$. The third order series agrees quantitatively with experiments for $k_{B}T/E_{f}>0.8$. After applying appropriate resummation techniques, it also agrees qualitatively for $k_{B}T/E_{f}>0.2$. Consequently, one only expects that the high temperature expansion to yield information about the pseudogap on the BEC side of resonance, where this phenomenon persists to high temperatures. Beyond this direct information, the high temperature expansion is useful for calibrating other theories, as it is well-controlled. A thorough review of the subject can be found in virialreview .

In addition to the equation of state, high temperature expansions yield the contact virialcontact and response functions sunleyronas ; virialspectraltrap ; shenresponse ; ngampruetikorn ; virialstructure . For example, in virialspectraltrap , Hu, Liu, Drumond, and Dong, used a high temperature expansion to calculate the spectral density of a trapped gas at $k_{B}T/E_{F}=0.7$. They saw a distinct gap-like feature on the BEC side of resonance ($k_{f}a_{s}=1$), but found that the spectra at $1/(k_{f}a_{s})=0,-1$ were broad without clear indications of pairing. Similar results were found by Ngampruetikorn, Parish, and Levinson ngampruetikorn .

VI.4 Dimensional Expansions

As reviewed in wilsonkogut , there is an ingenious approach to statistical mechanics where one treats the dimension of space $d$ as a continuous variable. If one can solve the problem for a particular $d=d_{c}$, one can then use perturbation theory in $\epsilon=d-d_{c}$ to learn about the physics in other dimensions.

In 2006, Nishida and Son applied this technique to the strongly interacting Fermi gas epsilon , taking advantage of an result from Nussinov and Nussinov nussinov , which showed that the problem simplifies in $d=4$ and $d=2$. In low dimensions ($d\leq 2$), an arbitrarily weak attraction leads to a bound state, providing a mapping between the unitary limit (where a bound state is at threshold) and the non-interacting limit. In high dimensions ($d\geq 4$), bound states have an extremely large weight at the origin, suggesting a description of the unitary gas in terms of pointlike bosons.

Nishida and Son first expanded about $d=4$ to calculate the zero temperature properties of the dispersion and the equation of state epsilon at unitarity. Later, they considered the expansion about $d=2$ epsilon2 . By combining these two expansions they were about to bound the properties of the unitary gas in $d=3$. Nishida extended these results to finite temperature nishida . He found that near $T_{c}$ both Fermionic and Bosonic degrees of freedom were important for the thermodynamics – a feature naturally interpreted in terms of normal state pairs. Other authors explored the BCS-BEC crossover cheneps , and carried the expansion to higher order arnold ; moreeps .

VI.5 T-Matrix Approaches

The first theory of the normal state in the BCS-BEC crossover was developed by Nozieres and Schmidt-Rink (NSR)nozieres . It was framed as a “diagramatic” theory, where one ressums infinite sets of diagrams which represent terms in a perturbative expansion. Nozieres and Schmidt-Rink’s approach roughly corresponded to exactly solving the two-body problem, accounting for the presence of all the other particles through Pauli blocking. The importance and influence of this work cannot be understated. It, for example, provides a model for how the superfluid transition temperature evolves in the crossover. Variants of the NSR theory have been one of the key analytic approaches to superconductivity km . In the 1990’s, Sa de Melo, Randeria, and Engelbrecht reformulated the NSR theory, and elucidated both the phase diagram, and the excitations bcsbecearly1 .

There have been numerous attempts to either justify or improve on the NSR theory, resulting in a rich set of “T-Matrix” approaches, so-named because the class of diagrams summed levincompare ; hu2 ; hu3 ; luhu ; earlyhaussmann ; laterhaussmann ; Tsuchiya ; stajic ; kinnunenhartree ; pierit ; micnast ; gubelstoof ; pieristrinati ; chen2 ; bcsbecpseudo ; haussmann ; stajic ; strinati ; perali . The three main T-Matrix theories are reviewed and compared in a number of articles by Chen and collaborators chen ; levinxx ; levincompare ; hucompare . As previously explained, the theories predict different strengths of gap-like features at unitarity, and there is active debate about which of these features should be considered indicative of a pseudogap. In all models, the gap-like features become stronger as one moves towards the BEC limit, and weaker as one moves towards the BCS limit. There are ongoing attempts to reformulate these theories in ever-more accurate ways mulkerin .

VI.6 Other Approaches

Sections VI.1 through VI.5 detailed the most popular theoretical approaches to calculating properties of the normal state in the BEC-BCS crossover. There are, however, several other techniques. For example, a number of authors introduced “large $N$ expansions” in which they replace the spin-1/2 fermions with spin $N/2$ particles. In the limit $N\to\infty$ one can make simplifying assumptions, or derive renormalization group equations largeN ; sachdevlargeN . There are also theories based upon the 2-body S-matrix howleclair , Brueckner-Goldstone theory bg , Dyson-Schwinger equations ds , Bethe-Salpeter equations bs , projected wavefunctions cazalilla , renormalization groups gubbelsrg ; rg ; sachdevlargeN ; morerg ; brg , operator product expansions opepseudo , and effective field theories effective . A number of authors have attempted to make simplified theories which have Hartree-like structure weiler ; kinnunenhartree . Generically, each of these techniques is designed to highlight some facet of the phenomonology, and where they are reliable they agree with one-another. They represent a wonderful toolbox for understanding this rich system.

VII Outlook

Throughout this review I have tried to present a relatively simple story: In the deep BEC regime the normal state is described as a gas of weakly interacting bosonic pairs, in the deep BCS regime the normal state is described as a gas of weakly interacting fermions, and the phenomena in the intermediate regime crossover region shows facets of both descriptions. Some of these facets are traditionally described as signatures of a ”pseudogap”. In the introduction, I argued that this phenomonology is an example of a larger principle, and that pseudogap phenomena are generic features of strongly interacting fermions. In this section I would like to very briefly reiterate that viewpoint, touch on important questions, and explore the outlook for this area of study.

I should first note that there are obvious examples of strongly interacting fermions which are unlikely to have a pseudogap description. For example, the metal-Mott crossover involves very different physics mottmetal . I should also note that the example in Sec. III.2 shows that one can have a pseudogap without pairing. Despite these limitations, I still contend that the pseudogap in the BEC-BCS crossover provides an important paradigm, beyond its application to cold atoms. It is generically true that near a phase transition, the unordered state shows precursors of the ordering. What is special about the cold atom system is that one has a control parameter ($1/k_{f}a$), which allows one to tune the strength of the incipient order.

Is the BCS-BEC crossover relevant to strong-coupling superconductors? As already discussed, the physics of the cuprates appears to involve some non-superconducting order. Nevertheless, one must acknowledge that the pairing energy scale is large in those materials, and near $T_{c}$ there must be precursors of pairing. Disentangling those precursors from the other phenomena is a difficult task.

Cold atoms in the BCS-BEC crossover are an incredibly fascinating system. The pseudogap physics described here is just one facet of them. There has been remarkable developments in lower dimensions twod ; oned and in investigation of spin polarized gases imbalanced . There is great interest in exploring these systems on lattices, where one hopes to make closer connections to solid state systems. One extremely promising recent development is the construction of “Fermi gas microscopes,” in which one can detect the location of every particle in the gas fgm . Other exciting directions include: p-wave interactions pwave , pairing with more spin components higherspin , and longer range interactions longrange .

Acknowledgements

I would like to thank Cheng Chin, Jami Kinnunen, Kathy Levin, Nikolay Prokof’ev, Henk Stoof, and Martin Zwierlen for critical comments. This material was based upon work supported by the National Science Foundation under Grant No. PHY-1508300.