Finite-temperature superfluid depletion of disordered Bose gases

Using inhomogeneous Bogoliubov theory and diagrammatic perturbation theory, this paper calculates finite-temperature disorder corrections to the normal fluid density of weakly interacting Bose gases in arbitrary dimensions, providing closed analytical expressions up to quadratic order in disorder strength.

The Gist

The Big Picture: A Perfect Dance Floor with a Few Hiccups

Imagine a massive, perfectly synchronized dance floor. Every dancer (an atom) moves in perfect unison, gliding across the floor without ever bumping into each other or getting tired. This is a Superfluid. It's a state of matter where a gas of atoms (Bose-Einstein Condensate) acts like a single, giant "super-atom" that flows with zero friction.

Now, imagine two things happen to this perfect dance:

1. The Music Gets Hot (Temperature): The dancers start getting a little jittery. They aren't perfectly synchronized anymore; some are doing their own little jig. This creates a "normal" part of the crowd that drags and creates friction.
2. The Floor Gets Bumpy (Disorder): The dance floor isn't perfectly smooth anymore. There are random bumps, pits, and obstacles (impurities) scattered around.

The Question: How much of the "perfect, frictionless flow" is lost when you combine the heat (jittery dancers) with the bumps (rough floor)?

The Problem: It's Hard to Do the Math

For a long time, physicists knew how to calculate the friction caused by heat in a perfectly smooth room (Landau's theory). They also knew how to calculate the friction caused by bumps in a frozen room (zero temperature).

But figuring out what happens when you have both heat and bumps together is incredibly difficult. It's like trying to predict how a crowd of jittery people will navigate a maze of random furniture. The math gets messy because the bumps change the way the dancers move, and the heat changes how they react to the bumps.

The Solution: A New Way to Count Steps

The author, Cord Müller, uses a clever mathematical toolkit called Inhomogeneous Bogoliubov Theory.

Think of the dance floor not as a flat, empty stage, but as a landscape that the dancers have already reshaped. If there's a bump, the dancers naturally spread out to avoid it, creating a "deformed" dance floor. Müller's approach calculates the flow based on this actual deformed floor, rather than pretending the floor is flat and the bumps are just random noise added on top.

He breaks the problem down into two types of "dancers" (excitations):

1. The Solo Dancers (Single-Bogolon)

These are individual dancers who get bumped by the obstacles.

• The Finding: The author confirms that these solo dancers create a specific amount of friction (normal fluid density) that depends on how bumpy the floor is.
• The Surprise: This specific type of friction does not change with temperature. Whether the room is freezing or warm, the solo dancers react to the bumps in the exact same way. This part of the friction was already known, but the author calculated it precisely for this specific setup.

2. The Dance Pairs (Pair-Bogolon)

This is the big discovery. These are pairs of dancers interacting with each other while navigating the bumps.

• The Finding: When you look at these pairs, temperature matters. The heat changes how the pairs react to the bumps.
• The Result: The author found a new formula that adds a "temperature-dependent correction" to the old rules.
  • Analogy: Imagine the solo dancers are like a car hitting a pothole; the damage is the same whether it's day or night. But the dance pairs are like a car driving over a pothole while the suspension is shaking due to a bumpy road. The shaking (heat) changes how the car handles the pothole.

The "Smooth" vs. "Rough" Floor

The paper also looks at two extremes of the "bumpiness":

1. The White Noise Limit (Super-Rough): Imagine the floor is covered in tiny, sharp gravel. The math gets very tricky here, and the friction can become huge.
2. The Thomas-Fermi Limit (Smooth Hills): Imagine the floor has gentle, rolling hills rather than sharp rocks.
   • The Result: In this smooth-hill scenario, the author found a very clean, simple formula. It shows that even with gentle hills, the heat makes the superfluid flow slightly worse than we previously thought, but the effect is predictable and calculable.

Why Does This Matter?

This paper is like a new instruction manual for engineers building ultra-precise quantum devices.

• Real-world application: Scientists are currently building quantum computers and sensors using these super-cold gases. These devices are often placed in environments that aren't perfectly perfect (they have noise and disorder).
• The Takeaway: If you want to build a device that relies on frictionless flow (superfluidity), you can't just ignore the temperature and the imperfections of your setup. This paper gives you the exact math to predict how much your "perfect flow" will degrade when both factors are present.

Summary in a Nutshell

• Old View: Heat causes friction; Bumps cause friction. We knew how to calculate them separately.
• New View: When you mix heat and bumps, they interact in a complex way.
• The Discovery: The author developed a new method to calculate this mix. He found that while some friction is constant, a specific type of friction caused by pairs of atoms gets worse as the temperature rises, even in a disordered environment.
• The Analogy: It's the difference between a single person stumbling on a rock (always the same) versus a whole group of people trying to dance in a circle on a bumpy floor while the music speeds up (the chaos changes how they stumble).

This work provides the "closed analytical expressions" (the final, neat math formulas) that allow scientists to predict exactly how much "super" is lost in their experiments, ensuring their quantum technologies work as intended.

Technical Summary

1. Problem Statement

The paper addresses the fundamental question of how spatial inhomogeneity (disorder) and finite temperature jointly affect the superfluid fraction of interacting Bose-condensed gases.

• Context: In a homogeneous system at zero temperature, a Bose-Einstein condensate (BEC) is entirely superfluid. Finite temperatures introduce a normal fluid component ($\rho_n$) due to thermal excitations, described by Landau's two-fluid theory.
• The Gap: While Landau's formula for the normal density is well-established for homogeneous systems, it relies on translation invariance. It is unclear how this relation is modified when the system is subjected to static external potentials (disorder) that break translational symmetry.
• Specific Challenge: Previous theories often struggled to quantitatively describe the combined effects of disorder and temperature, particularly regarding the role of spatial correlations in the disorder potential and the distinction between single-particle and pair-excitation contributions to the normal density.

2. Methodology

The author employs a microscopic approach based on inhomogeneous Bogoliubov theory and diagrammatic perturbation theory.

• System Model: A dilute gas of massive bosons with short-range repulsive interactions ($s$-wave, $g>0$) in a $d$-dimensional volume, subject to a stationary, spatially ergodic random potential $V(\mathbf{r})$ with zero mean and finite variance.
• Theoretical Framework:
  • The boson field is split into a macroscopic mean-field condensate $\Phi(\mathbf{r})$ (solving the Gross-Pitaevskii equation) and fluctuations $\delta\hat{\psi}$.
  • The Hamiltonian is expanded around the deformed condensate (the inhomogeneous Bogoliubov vacuum) to yield an effective quadratic Hamiltonian for non-interacting Bogoliubov excitations ("bogolons").
  • Crucially, the effective potential in this Hamiltonian encodes the nonlinear response of the condensate to the disorder.
• Calculation of Normal Density ($\rho_n$):
  • The normal density is defined via the transverse current-current correlation function in the limit of zero momentum and zero frequency (Matsubara formalism).
  • The current operator is decomposed into single-bogolon (linear in fluctuations) and pair-bogolon (quadratic in fluctuations) contributions.
  • Wick's Theorem is applied to evaluate the thermal averages, separating the total normal density into two distinct parts: $\rho_n = \rho_n^{[1]} + \rho_n^{[2]}$.
• Perturbation Order: The analysis is performed to second order ($O(V^2)$) in the strength of the external potential, providing closed analytical expressions.

3. Key Contributions

The paper makes several distinct theoretical advancements:

1. Inhomogeneous Bogoliubov Formalism: It utilizes a formalism that describes excitations relative to the deformed condensate rather than a fictitious homogeneous one, allowing for the rigorous treatment of arbitrary spatial correlations in the disorder potential.
2. Separation of Contributions: It explicitly distinguishes between:
   • Single-bogolon contribution: Arising from the deformation of the condensate density.
   • Pair-bogolon contribution: Arising from the scattering of excitation pairs, which includes both self-energy and vertex corrections.
3. Vertex Corrections: The author identifies and computes vertex corrections (Type II in Fig. 3) for the pair-bogolon current response. These were previously uncalculated in this context and are essential for obtaining the correct temperature-dependent disorder corrections to Landau's formula.
4. Analytical Results for Correlated Disorder: The work provides closed-form analytical expressions for the normal fraction that account for the correlation length ($\sigma$) of the disorder relative to the condensate healing length ($\xi$).

4. Key Results

A. Single-Bogolon Contribution ($\rho_n^{[1]}$)

• Nature: This term arises from the static deformation of the condensate density by the disorder.
• Temperature Dependence: It is temperature-independent.
• Result: It reproduces known zero-temperature results where disorder depletes the superfluid fraction. The depletion depends on the disorder power spectrum and the ratio $\zeta = \sigma/\xi$.
  • In the Thomas-Fermi limit (smooth potentials, $\zeta \gg 1$), the normal fraction approaches a universal limit: $f_n^{[1]} \approx v^2/d$, where $v^2 = V^2/\mu^2$.
  • In the white-noise limit ($\zeta \to 0$), the depletion depends on non-universal constants related to the potential correlator.
• Implication: This term does not modify Landau's finite-temperature relation; it simply adds a temperature-independent offset.

B. Pair-Bogolon Contribution ($\rho_n^{[2]}$)

• Nature: This term arises from the thermal population of Bogoliubov excitations and their scattering.
• Temperature Dependence: This is the source of finite-temperature disorder corrections to Landau's formula.
• Thomas-Fermi Regime Results: For smooth potentials ($\zeta \to \infty$), the paper derives a corrected normal fraction:
  $$f_n^{TF}(T) \approx f_{n0}(T) \left[ 1 + \frac{\Gamma(d+4)}{2\Gamma(d+2)} \frac{V^2}{\mu^2} \right]$$
  where $f_{n0}(T)$ is the clean-system normal fraction.
• Mechanism: The correction involves a competition between:
  1. Self-energy terms (negative contribution, related to dispersion shifts).
  2. Vertex corrections (positive contribution).
     The net result is a positive enhancement of the normal fraction (further depletion of superfluidity) that scales with the disorder strength $V^2$.
• Dimensionality: The effect is more pronounced in lower dimensions ($d=1, 2$) and for longer correlation lengths.

C. Temperature-Independent Pair Correction

• The paper also calculates a temperature-independent pair-bogolon term (Appendix B).
• Significance: This term is found to be negligible (orders of magnitude smaller than the single-bogolon term) due to the small gas parameter ($1/n\xi^d$), justifying its omission in leading-order analyses.

5. Significance and Outlook

• Quantitative Precision: The paper provides the first systematic, closed analytical derivation of finite-temperature disorder corrections to the superfluid fraction, resolving ambiguities in how spatial inhomogeneity modifies Landau's two-fluid theory.
• Experimental Relevance: The results are directly applicable to ultracold atomic gases in optical lattices and random potentials, where disorder correlation lengths can be tuned. The distinction between smooth (Thomas-Fermi) and rough (white-noise) potentials is critical for interpreting experimental data.
• Theoretical Foundation: By successfully incorporating vertex corrections and spatial correlations within a quadratic Bogoliubov framework, the work bridges the gap between simple mean-field theories and complex many-body treatments.
• Limitations: As a perturbative approach, the results are valid for weak disorder. The paper notes that extrapolating to strong disorder (to predict the Bose glass transition) requires non-perturbative methods (e.g., maximally crossed diagrams for Anderson localization), which remains an open area for future research.

In summary, Müller demonstrates that while disorder induces a temperature-independent superfluid depletion at zero temperature, it also introduces a specific, calculable temperature-dependent correction to the normal fluid density, driven by the interplay of thermal excitations and disorder-induced vertex corrections.