Damping of phonons in Bose gas at low temperatures

This paper computes the imaginary part of the phononic excitation spectrum for a homogeneous Bose gas at low temperatures and low momentum in the thermodynamic limit, utilizing two distinct perturbation theory approaches based on the standard Liouvillean to analyze resonances near zero and the 2-point correlation function.

The Gist

Imagine a giant, perfectly organized dance floor filled with millions of dancers. In the world of physics, these dancers are atoms in a Bose gas (a special state of matter where atoms act like a single giant wave). When the room is very cold, these dancers don't just move randomly; they move in perfect unison, forming a Bose-Einstein Condensate.

In this paper, the authors are studying what happens when one dancer tries to do a solo move (a "phonon" or a sound wave) while the rest of the crowd is still dancing in sync. Specifically, they want to know: How long does this solo move last before it gets messy and disappears?

Here is a breakdown of their findings using everyday analogies:

1. The Perfect Dance vs. The Real Dance

In a perfect, theoretical world (called the Bogoliubov approximation), the dancers are so perfectly coordinated that a solo move would travel forever without losing energy. It's like a perfectly smooth slide where a ball rolls forever.

But in the real world, the dancers bump into each other. They interact. This causes the solo move to slow down, wobble, and eventually fade away. This fading is called damping. The authors calculated exactly how fast this fading happens.

2. The Two Ways the Solo Move Can Die

The paper discovers that there are two distinct ways the solo move (the phonon) can get "damped" or lose its energy. Think of these as two different types of traffic jams on the dance floor.

A. The "Beliaev" Crash (The Split)

• The Analogy: Imagine a solo dancer (the phonon) is gliding across the floor. Suddenly, they trip and split into two smaller dancers who then run off in different directions.
• What happens: The original energy is conserved, but it's now shared between two new, smaller waves.
• When it happens: This happens even if the room is freezing cold (near absolute zero). It's an inherent flaw in the dance floor itself.
• The Result: The authors calculated that the faster the dancer moves (higher momentum), the more likely this split is to happen. Specifically, the "death rate" of the wave grows very quickly as the speed increases (proportional to the speed to the 5th power!).

B. The "Landau" Collision (The Crowd Interaction)

• The Analogy: Now imagine the room is a bit warmer. The dancers aren't perfectly still; they are jiggling and shuffling around (thermal energy). The solo dancer tries to glide, but they keep bumping into these jiggling dancers.
• What happens: The solo dancer transfers some of their energy to the jiggling crowd, slowing themselves down.
• When it happens: This only happens if the room is warm (positive temperature). If the room is absolute zero, there is no jiggling crowd, so this type of damping disappears.
• The Result: At very low temperatures, this effect is weak. But as the temperature rises, it becomes a major factor.

3. The Temperature Tug-of-War

The authors found a fascinating battle between these two effects depending on how cold the room is:

• Super Cold (Near Absolute Zero): The "Beliaev" split is the boss. The solo dancer mostly dies because they split in two. The "Landau" crowd collision is negligible because the crowd is frozen.
• Slightly Warmer: As the temperature rises, the "Landau" effect starts to grow. The authors found that in a specific "Goldilocks" zone (very cold, but not too cold), the Landau effect (bumping into the crowd) actually becomes the dominant reason the wave dies, even though it's still quite cold.

4. The Math Behind the Magic

To figure this out, the authors used two different "lenses" to look at the problem:

1. The Operator Lens: They looked at the mathematical rules governing the dancers' movements (using something called "Liouvillians," which is just a fancy way of tracking how the system evolves over time).
2. The Correlation Lens: They looked at how the dancers' movements are linked to each other (like watching a video of the dance floor and measuring how one dancer's move predicts another's).

Surprisingly, both lenses gave them the exact same answer, confirming their calculations are solid.

5. Why Does This Matter?

You might ask, "Who cares about a theoretical dance floor?"

• Real-World Connection: This isn't just theory. This math helps explain what happens in liquid Helium (the stuff used to cool superconductors) and in ultra-cold atomic gases created in labs.
• The "Roton" Mystery: In liquid Helium, the waves don't just move in a simple line; they have a weird shape with bumps and dips (called rotons and maxons). The authors showed that their math works even if the dance floor has these weird bumps, which is a big deal because it connects their simple model to the complex reality of superfluids.

Summary

In short, this paper is a detailed recipe for calculating how fast a sound wave in a super-cold gas will die out. They found that the wave can die in two ways: by splitting in two (which happens even in the deep freeze) or by bumping into the warm crowd (which only happens when it's slightly warmer). They proved that depending on the temperature, one of these two "death mechanisms" takes over the other.

It's a bit like predicting whether a snowball will melt because it's too hot (Landau) or because it crumbles under its own weight (Beliaev). The authors figured out exactly when and why each happens.

Technical Summary

1. Problem Statement

The paper investigates the damping (decay) of phonons (elementary quasiparticle excitations) in a homogeneous Bose gas at low temperatures and low momenta. While the Bogoliubov approximation successfully predicts the real part of the phonon dispersion relation (linear at low momenta), it treats the gas as a collection of non-interacting quasiparticles, ignoring the broadening of these excitations due to interactions.

The authors aim to rigorously compute the imaginary part of the phonon dispersion relation (the decay rate) in the thermodynamic limit ($L \to \infty$) using perturbation theory. Specifically, they analyze the lowest non-trivial order of the interaction strength, which corresponds to the Fermi Golden Rule for three-body interactions. The study addresses two distinct damping mechanisms:

1. Beliaev Damping ($\gamma_B$): A process where a single quasiparticle decays into two quasiparticles. This persists even at zero temperature.
2. Landau Damping ($\gamma_L$): A process involving the interaction between a quasiparticle and thermal excitations (specifically, a left quasiparticle decaying into a left and a right quasiparticle). This vanishes at zero temperature.

2. Methodology

The authors employ two distinct but consistent mathematical formalisms to derive the damping rates, both based on the Liouvillean formalism (doubling the Hilbert space to handle finite temperatures via the KMS condition).

A. Theoretical Framework

• Hamiltonian: They start with a Bose gas in a large periodic box with a two-body potential $v(x)$ having a positive Fourier transform. They use the c-number substitution (replacing the zero-momentum mode operator with $\sqrt{\nu}$, where $\nu$ is related to the chemical potential) to define the Hamiltonian $H_\nu$.
• Liouvillean Approach: Instead of the Hamiltonian, they work with the Liouvillean $L = [H, \cdot]$. This allows for a unified treatment of zero and positive temperatures. The system is described in the Araki-Woods representation, where the thermal state is the vacuum, and excitations are "left" (physical) and "right" (hole) quasiparticles.
• Perturbation Theory: The interaction is expanded in a coupling constant $\kappa$ (replacing $v \to \kappa v$). The damping rate is calculated at order $\kappa^1$.

B. Two Approaches

1. Standard Representation Approach (Operator Algebras):

   • Based on modular theory of $W^*$-algebras.
   • Analyzes resonances near the zero of the Liouvillean spectrum.
   • Computes the shift of the eigenvalue for one-quasiparticle states embedded in the continuous spectrum.
   • Derives the damping rate as the imaginary part of the energy shift.

2. Green Function Approach (Correlation Functions):

   • Analyzes time-ordered two-point correlation functions in energy-momentum space.
   • Uses the Lehmann representation to relate correlation functions to the resolvent of the Liouvillean.
   • The broadening of the spectral "ridge" (singularity) corresponds to the imaginary part of the dispersion relation.

Both approaches yield identical formulas for the damping rates, validating the consistency of the methods.

3. Key Contributions

• Rigorous Derivation of Damping Rates: The paper provides a mathematically rigorous derivation of the Beliaev and Landau damping rates for a general class of potentials with positive Fourier transforms, moving beyond heuristic physics arguments.
• Unified Treatment of Temperatures: The authors derive formulas valid for both zero and positive temperatures, explicitly showing how the Landau damping term vanishes as $T \to 0$.
• Asymptotic Analysis: They perform a detailed asymptotic analysis of the damping rates in various regimes of momentum ($k$) and temperature ($\beta$), specifically focusing on the ratios $\beta \sqrt{\nu} |k|$ and $1/(\beta \nu)$.
• Resolution of Real Part Divergence: The paper notes that while their method successfully computes the imaginary part (damping), the real part of the energy shift (mass renormalization) computed via the c-number substitution suffers from infrared divergences. They attribute this to the crudeness of the c-number substitution for the four-body term ($H_4$) and suggest that a more precise treatment (canonical vs. grand-canonical) is needed for the real part, a topic reserved for future work.

4. Main Results

The total damping rate is given by $\gamma = \gamma_B + \gamma_L$. The authors derive explicit integral formulas (Equations 1.7 and 1.8) and then analyze their asymptotic behavior.

A. Beliaev Damping ($\gamma_B$)

• Mechanism: Decay of one phonon into two phonons ($k \to p + (k-p)$).
• Zero Temperature Limit: Reproduces the classic Beliaev result where $\gamma_B \propto |k|^5$.
• Low Temperature/High Momentum Regime ($\beta \sqrt{\nu} |k| \to 0$):
  $$\gamma_B \sim \frac{3 \hat{v}(0) \nu^{3/2}}{640\pi} \frac{|k|^5}{\nu^{5/2}}$$
  This confirms the standard $|k|^5$ dependence.
• Low Temperature/Low Momentum Regime ($1/(\beta \sqrt{\nu} |k|) \to 0$):
  $$\gamma_B \sim \frac{3 \hat{v}(0) \nu^{3/2}}{128\pi} \frac{|k|^4}{\nu^2} \frac{1}{\beta \nu}$$
  This represents a novel temperature-dependent correction where the damping scales as $|k|^4 T$.

B. Landau Damping ($\gamma_L$)

• Mechanism: Interaction with thermal phonons (involving "right" quasiparticles).
• Low Temperature/Low Momentum Regime ($\beta \sqrt{\nu} |k| \to 0$):
  $$\gamma_L \sim \frac{3\pi^3 \hat{v}(0) \nu^{3/2}}{40} \frac{|k|}{\sqrt{\nu}} \frac{1}{(\beta \nu)^4}$$
  This scales as $|k| T^4$.
• Low Temperature/High Momentum Regime ($1/(\beta \sqrt{\nu} |k|) \to 0$):
  $$\gamma_L \sim \frac{9\zeta(3) \hat{v}(0) \nu^{3/2}}{16\pi} \frac{1}{(\beta \nu)^3} \frac{|k|^2}{\nu}$$
  This scales as $|k|^2 T^3$.

C. Comparison of Regimes

• Dominance:
  • In the limit $\beta \sqrt{\nu} |k| \to 0$ (very low temperature relative to momentum), Landau damping dominates over Beliaev damping ($\gamma_L / \gamma_B \sim (\beta \sqrt{\nu} |k|)^{-3}$).
  • In the limit $1/(\beta \sqrt{\nu} |k|) \to 0$ (higher temperature or very low momentum), Beliaev damping dominates ($\gamma_B / \gamma_L \sim (\beta \sqrt{\nu} |k|)^3$).
• High Temperature: The paper briefly analyzes the high-temperature limit ($\beta \nu \to 0$), finding that the Landau damping rate diverges, indicating the breakdown of the low-density/weak-coupling approximation in this regime.

5. Significance

• Validation of Experiments: The results align with experimental observations in superfluid $^4$He and alkali Bose-Einstein condensates, where phonon linewidths are observed to be sharp but non-zero.
• Mathematical Rigor: The paper bridges the gap between physical intuition (Fermi Golden Rule) and rigorous mathematical physics, providing precise error estimates and conditions on the interaction potential (e.g., smoothness, positivity of Fourier transform).
• Clarification of Mechanisms: It clearly delineates the regimes where Beliaev (zero-temperature quantum decay) and Landau (thermal scattering) damping are dominant, resolving ambiguities in previous literature regarding their relative importance at low temperatures.
• Methodological Advancement: The successful application of operator algebraic methods (standard representation) to derive physical decay rates in a translation-invariant system offers a robust framework for future studies of interacting quantum many-body systems.

In summary, this paper provides a comprehensive and rigorous calculation of phonon damping in Bose gases, confirming known zero-temperature results while deriving new, precise temperature-dependent corrections for both Beliaev and Landau damping mechanisms.