The Unitary Fermi Gas: From Monte Carlo to Density Functionals

Abstract

In this chapter, we describe three related studies of the universal physics of two-component unitary Fermi gases with resonant short-ranged interactions. First we discuss an ab initio auxiliary field quantum Monte Carlo technique for calculating thermodynamic properties of the unitary gas from first principles. We then describe in detail a Density Functional Theory (DFT) fit to these thermodynamic properties: the Superfluid Local Density Approximation (SLDA) and its Asymmetric (ASLDA) generalization. We present several applications, including vortex structure, trapped systems, and a supersolid Larkin–Ovchinnikov (FFLO/LOFF) state. Finally, we discuss the time-dependent extension to the density functional (TDDFT) which can describe quantum dynamics in these systems, including non-adiabatic evolution, superfluid to normal transitions and other modes not accessible in traditional frameworks such as a Landau–Ginzburg, Gross–Pitaevskii, or quantum hydrodynamics.

Appendix

9.1.1 Formal Description of the DFT

Here we present a somewhat formal derivation of the variational property of the Kohn-Sham equations. Consider a general free-energy functional of the following form

$$ F = E(n_{A},n_{B},\cdots) + T\hbox{Tr}\left({\rho}\ln{\rho}\right) $$

(9.113)

where

$$ \begin{aligned} n_{A} &= \hbox{Tr}\left({\rho} {A}^{T}\right), \\ n_{B} &= \hbox{Tr}\left({\rho} {B}^{T}\right), \\ &\vdots \end{aligned} $$

are the various densities, anomalous densities, etc. expressed linearly in terms of the one-body density matrix ${\rho}.$ By varying the functional with respect to the density matrix ${\rho}$ subject to the appropriate constraints on density matrix form (discussed in ‘Fermions’), one obtains a solution of the form

$$ {\rho} = f_{\beta}({\mathbf{H}}[{\rho}]) $$

(9.114)

where $f_{\beta}(E)$ is the appropriate thermal distribution for the particles of interest, and ${\mathbf{H}}$ is a single-particle Hamiltonian that depends on ${\rho}$:

$$ {\mathbf{H}}[{\rho}] = \partial{E}{n_{A}}{\mathbf{A}} + \partial{E}{n_{B}}{\mathbf{B}} + \cdots. $$

(9.115)

The typical Kohn-Sham equations follow by diagonalizing the self-consistency condition (9.114) with a set of normalized Kohn-Sham eigenfunctions of definite energy:

$$ {\mathbf{H}}|{n}\rangle = E_{n}|{n}\rangle. $$

(9.116)

The density matrix is diagonal in this basis and expressed in terms of the appropriate distribution functions $f_{\beta}(E)$:

$$ {\rho} = \sum_{n} f_{\beta}(E_{n}) |{n}\rangle\langle{n}|. $$

(9.117)

All of the functionals considered in this chapter may be expressed in this form. Once the appropriate matrix structures A, B etc. are described, the form of the Kohn-Sham equations and potentials follows directly from these expressions.

9.1.1.1 Fermions

The only remaining complication is to impose the appropriate constraint on the density matrix ${\rho}.$ This ensures that the appropriate statistics of the particles is enforces. As we shall be interested in ‘Fermions’, the relevant constraint on the density matrix (dictated by the canonical commutation relationships) is

$$ {\rho} + {\mathbf{C}}{\rho}^{T}{\mathbf{C}} = 1 $$

(9.118)

where ${\mathbf{C}} = {\mathbf{C}}^{T}$ is the charge conjugation matrix:

$$ {\mathbf{C}}|{\psi}\rangle = |{\psi}\rangle^{\ast}. $$

(9.119)

This follows from the anti-commutation relationship for fermions and is discussed further in the ‘Single Particle Hamiltonian’. The constrained minimization of the functional $F({\rho})$ results in the standard Fermi distribution ⁷

$$ {\rho} = \frac{1} {1+e^{\beta \left({\mathbf{H}}[{\rho}] - {\mathbf{C}}{\mathbf{H}}^{T}[{\rho}]{\mathbf{C}}\right)}}, $$

(9.120)

which is the fermionic form of the self-consistency condition (9.114) for the density matrix ${\rho}.$ In practise, one does not iterate the entire density matrix. Instead, one stores only the densities $n_{A}, n_{B},$ etc. Through (9.115), these define the Kohn-Sham Hamiltonian H, which is then diagonalized to form the new density matrix and finally the new densities. If, for example, symmetries allow the Hamiltonian H to be block diagonalized, then one can construct and accumulate the densities in parallel over each block. Finally, the densities represent far fewer parameters than the full density matrix. Thus, more sophisticated root-finding techniques such as Broyden’s method [201] may be efficiently employed: Applying these techniques to the full density matrix would be significantly more expensive.

9.1.2 Single Particle Hamiltonian

It is convenient to express these concepts in the language of second quantization. The Hamiltonian will appear as a quadratic operator of the form

$$ {H}_{s} = \frac{1}{2}{\Uppsi}^{\dagger}{{\fancyscript{H}}}_{s}{\Uppsi} $$

(9.121)

where ${\Uppsi}$ has several components and ${{\fancyscript{H}}}_{s}$ is a matrix. The factor of $1/2$ accounts for the double counting to be discussed below. For a two component system, the most general ${\Uppsi}$ that allows for all possible pairings has four components:

$$ {\Uppsi} = \left(\begin{array}{l} {a}\\ {b}\\ {a}^{\dagger}\\ {b}^{\dagger}.\end{array}\right). $$

(9.122)

In terms of components of the wavefunction, we will write ${\fancyscript{H}}_{s} \psi = E\psi$ where:

$$ \psi = \left(\begin{array}{l} u_{a}\\ u_{b}\\ {\it v}_{a}\\ {\it v}_{b} \end{array} \right) . $$

(9.123)

The naming of these components is conventional (see for example [117]) and the functions u and v are typically called “Coherence Factors”. Note that the convention is that ${\it v}_{a,b}^{\ast}({\mathbf{r}}, t)$ are the wavefunctions of the particles. In this formulation the Hamiltonian has the form presented in (9.100):

$$ {\fancyscript{H}}_{s} = \left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} h_a+U_a & 0 & 0 & \Updelta \\ 0 & h_b+U_b & -\Updelta & 0 \\ 0 & -\Updelta^{\ast}& -h_a^{\ast}-U_a & 0 \\ \Updelta^{\ast}& 0 & 0& -h_b^{\ast}-U_b \end{array}\right) $$

(9.124)

9.1.2.1 Four-Component Formalism

We shall start with this full four-component formalism but soon utilized a reduction: If the superfluid pairing $\Updelta \sim \langle{{a}{b}}\rangle$ channel is attractive, then often the “Fock” channel is repulsive so we can take $\langle{{a}^{\dagger}{b}}\rangle = 0.$ In combination with the double-counting discussed below, this will allow us to fully express the system in terms of two components.

The four-component formalism double counts the degrees of freedom: ${\Uppsi}$ contains both a and ${a}^{\dagger}.$ This degeneracy is described in terms of the charge conjugation matrix ${\fancyscript{C}}{:}$

$$ {\Uppsi}^{\dagger} = {\fancyscript{C}}{\Uppsi} \qquad \hbox{where} \qquad {\fancyscript{C}} = \left(\begin{array}{ll} {\bf 0} & {\bf 1}\\ {\bf 1} & {\bf 0}\end{array}\right) . $$

(9.125)

The operator ${\Uppsi}$ will satisfy the single-particle Shrödinger equations

$$ {H}_{s}|{\Uppsi}\rangle = E|{\Uppsi}\rangle $$

(9.126)

where the Hamiltonian ${H}_{s}={\Uppsi}^{\dagger}{\fancyscript{H}}_{s}{\Uppsi}$ can be chosen to satisfy (the sign implements Fermi statistics)

$$ {\fancyscript{C}}{\fancyscript{H}}_{s}^{T}{\fancyscript{C}} = -{\fancyscript{H}}_{s}. $$

(9.127)

In this form, the charge conjugation symmetry ensures that the eigenstates will appear in $\pm E$ pairs. ⁸ Keeping only one set of pairs will ensure that we do not double count. Using this symmetry, we can formally diagonalize the Hamiltonian by a unitary transformation ${\fancyscript{U}}$ such that:

$$ {\fancyscript{U}}^{\dagger}{\fancyscript{H}}_{s}{\fancyscript{U}} = \frac{1}{2} \left(\begin{array}{ll}{\mathbf{E}} & {\bf 0}\\ {\bf 0} & -{\mathbf{E}}\end{array}\right) $$

(9.128)

where ${\mathbf{E}} = \hbox{diag}(E_{i})$ is diagonal. The columns of the matrix ${\fancyscript{U}}$ are the (ortho) normalized wave-functions and describe the “coherence” factors. To determine the correct expressions for the densities in terms of the wavefunctions we form them in the diagonal basis and then transform back to the original basis using ${\fancyscript{U}}.$

Despite this formal degeneracy of eigenstates, we are not aware of a general technique to block-diagonalize the original Hamiltonian in the presence of non-zero terms of the form $\langle{{a}^{\dagger}{b}}\rangle,$ though perhaps the symmetry might be incorporated into the eigensolver.

9.1.2.2 Two-Component Formalism

If $\langle{{a}^{\dagger}{b}}\rangle = 0,$ however, then the Hamiltonian is naturally block diagonal:

$$ {\fancyscript{H}}_{s} = \frac{1}{2} \left( \begin{array}{ll} {\mathbf{H}}_{s} & {\mathbf{0}}\\ {\mathbf{0}} & -{\mathbf{H}}_{s}^{T}\end{array}\right) , \qquad {H}_{s} = {\psi}^{\dagger}{\mathbf{H}}_{s}{\psi} + \hbox{const}, $$

(9.129)

and one may consider only a single block in terms of the reduced set of operators

$$ {\psi} = \left(\begin{array}{l} {a}\\ {b}^{\dagger} \end{array}\right) . $$

(9.130)

This directly avoids any double counting issues. This system may be diagonalized:

$$ {\mathbf{H}}_{s}{\mathbf{U}} = {\mathbf{U}}{\mathbf{E}}. $$

(9.131)

The matrix ${\mathbf{U}}$ defines the single “quasi”-particle operators ${\phi}$ as linear combination of the physical particle operators contained in ${\psi}$:

$$ {\phi} = {\mathbf{U}}^{\dagger} {\psi}. $$

(9.132)

The Hamiltonian is diagonal in this basis

$$ {H}_{s} = {\phi}^{\dagger}\cdot{\mathbf{E}}\cdot{\phi} $$

(9.133)

and hence expectation values may be directly expressed

$$ \langle{{\phi}{\phi}^{\dagger}}\rangle = \theta_{\beta}({\mathbf{E}}) = \left(\begin{array}{llll} \theta_{\beta}(E_{0})\\ & \theta_{\beta}(E_{1})\\ && \ddots\\ &&& \theta_{\beta}(E_{n})\end{array}\right) $$

where $1-\theta_{\beta}(E) = f_{\beta}(E)$ is the appropriate distribution function: For fermions we have

$$ \theta_{\beta}(E) = \frac{1} {1+e^{-\beta E}}. $$

(9.134)

At $T=0$ this reduces to $\theta_{0}(E) = \theta(E)$ and is equivalent to the zero-temperature property that negative energy states are filled while positive energy states are empty. This may be simply transformed back into the original densities (on the diagonal) and anomalous densities (off-diagonal):

$$ \begin{aligned} {\mathbf{F}}_{+} = \langle{{\psi}{\psi}^{\dagger}}\rangle =& \left(\begin{array}{ll} \langle{{a}{a}^{\dagger}}\rangle & \langle{{a}{b}}\rangle\\ \langle{{b}^{\dagger}{a}^{\dagger}}\rangle & \langle{{b}^{\dagger}{b}}\rangle\end{array}\right) = {\mathbf{U}}\theta_{\beta}({\mathbf{E}}){\mathbf{U}}^{\dagger},\\ {\mathbf{F}}_{-}^{T} = \langle{{\psi}^{\ast}{\psi}^{T}}\rangle =& \left(\begin{array}{ll} \langle{{a}^{\dagger}{a}}\rangle & \langle{{a}^{\dagger}{b}^{\dagger}}\rangle\\ \langle{{b}{a}}\rangle & \langle{{b}{b}^{\dagger}}\rangle \end{array}\right) = {\mathbf{U}}^{\ast}\theta_{\beta}(-{\mathbf{E}}){\mathbf{U}}^{T}. \end{aligned} $$

Fermi statistics demands ${\mathbf{F}}_{-} + {\mathbf{F}}_{+} = {\mathbf{1}}$ but we may have to relax this requirement somewhat in order to regulate the theory in terms of an energy cutoff $\theta_{c}(E).$ The columns of ${\mathbf{U}}_{n}$ of U correspond to the single-particle “wavefunctions” for the state with energy $E_{n}.$ We partition these into two components sometimes referred to as “coherence factors”

$$ {\mathbf{U}}_{n} = \left(\begin{array}{l} {\mathbf{u}}_{n}\\ {\mathbf{v}}_{n}\end{array}\right) . $$

(9.135)

The unitarity of U imposes the conditions that

$$ {\mathbf{u}}_{m}^{\dagger}{\mathbf{u}}_{n} + {\mathbf{v}}_{m}^{\dagger}{\mathbf{v}}_{n} =\delta_{mn} $$

(9.136a)

$$ \sum_{n}{\mathbf{u}}_{n}{\mathbf{u}}_{n}^{\dagger} = \sum_{n}{\mathbf{v}}_{n}{\mathbf{v}}_{n}^{\dagger} = 1, $$

(9.136b)

$$ \sum_{n}{\mathbf{u}}_{n}{\mathbf{v}}_{n}^{\dagger} = \sum_{n}{\mathbf{v}}_{n}{\mathbf{u}}_{n}^{\dagger} = 0. $$

(9.136c)

From this we may read off the expressions for the densities

$$ {\mathbf{n}}_{a} = \langle{{a}^{\dagger}{a}}\rangle = \sum_{n} {\mathbf{u}}_{n}^{\ast}{\mathbf{u}}_{n}^{T} \theta_{\beta}(-E_{n}), $$

(9.137a)

$$ {\mathbf{n}}_{b} = \langle{{b}^{\dagger}{b}}\rangle = \sum_{n} {\mathbf{v}}_{n}{\mathbf{v}}_{n}^{\dagger} \theta_{\beta}(E_{n}), $$

(9.137b)

$$ {\nu} = \langle{{a}{b}}\rangle = \sum_{n} {\mathbf{u}}_{n}{\mathbf{v}}_{n}^{\dagger} \theta_{\beta}(E_{n}) $$

(9.137c)

$$ =-\sum_{n} {\mathbf{u}}_{n}{\mathbf{v}}_{n}^{\dagger} \theta_{\beta}(-E_{n}) $$

(9.137d)

$$ = \sum_{n} {\mathbf{u}}_{n}{\mathbf{v}}_{n}^{\dagger} \frac{\theta_{\beta}(E_{n})-\theta_{\beta}(-E_{n})} {2}. $$

(9.137e)

The last form for ${\nu}$ must be used if the regulator is implemented such that $\theta_{c}(E) + \theta_{c}(-E)\neq 1,$ in particular, if $\theta_{c}(E)=0$ for $\lvert{E}>E_{c}\rvert.$ Note that these expressions are basis independent, e.g. in position space:

$$ n_{a}({\mathbf{r}},{\mathbf{r}}^{\prime}) = \sum_{n} u_{n}({\mathbf{r}})^{\ast}u_{n}({\mathbf{r}}^{\prime})^{T} \theta_{\beta}(-E_{n}). $$

(9.138)

The energy $E_{n}$ here is the energy determined by solving these equations and will contain both positive and negative energies.