Finite temperature single-particle Green's function in the Lieb-Liniger model

The authors develop a Monte Carlo sampling algorithm to numerically evaluate the finite-temperature single-particle Green's function and spectral function for the repulsive Lieb-Liniger model across all interaction strengths and temperatures, including generalized Gibbs ensembles, with results showing excellent agreement against known limits.

The Gist

Imagine you are trying to predict the weather in a tiny, magical city called Lieb-Liniger. This city is one-dimensional (like a long, straight street) and is filled with invisible, bouncing balls (bosons) that push each other away when they get too close.

Physicists have known the rules of this city for decades. They know exactly how the balls move if they are frozen in time (zero temperature) or if they are just sitting still. But there is a huge problem: What happens when the city is hot, the balls are moving fast, and they are pushing each other hard?

Specifically, scientists wanted to know: If I poke one ball at a specific spot, how does that "poke" ripple through the entire city over time? This ripple is called the Green's function.

The Problem: The "Library of Infinite Books"

To calculate this ripple, you have to look at every possible way the city could be arranged. In quantum physics, these arrangements are called eigenstates.

Here is the catch: As the city gets bigger, the number of possible arrangements doesn't just grow; it explodes.

• If the city has 100 balls, the number of arrangements is roughly $10^{100}$.
• If you tried to write down every single arrangement in a book, you would need more books than there are atoms in the universe.

For a long time, scientists could only calculate the ripple for very cold cities (where the balls barely move) or very small cities. For a hot, big city, the math was impossible. It was like trying to count every grain of sand on a beach by picking them up one by one. You'd die of old age before you finished.

The Solution: The "Smart Sampling" Algorithm

The authors of this paper, Riccardo Senese and Fabian Essler, invented a new way to solve this. Instead of trying to read every book in the library, they built a smart robot (a Monte Carlo sampling algorithm).

Here is how the robot works, using a simple analogy:

The Analogy: The "Most Likely" Crowd
Imagine you want to know the average height of people in a massive stadium.

1. The Old Way: You try to measure every single person. Impossible.
2. The New Way (Their Algorithm): You send a robot into the stadium. The robot doesn't pick people randomly. It has a special sense that tells it: "Hey, most people are standing in the middle section. The VIPs are in the front. The empty seats are in the back."

The robot uses a Metropolis-Hastings algorithm (a fancy name for a "try-and-see" strategy). It picks a person, checks if they are important to the average, and decides whether to keep them in its sample or swap them for someone else.

• It quickly learns that while there are billions of people, only a specific "cloud" of people actually matters for the calculation.
• It ignores the empty seats (arrangements that contribute almost nothing).
• It focuses its energy on the "VIPs" (the specific quantum states that actually create the ripple).

What They Found

By using this "smart robot," the authors were able to simulate the Lieb-Liniger city for the first time in regimes that were previously impossible:

• High Temperatures: The city was hot and chaotic.
• Strong Interactions: The balls were pushing each other very hard.
• Generalized States: They even looked at "Generalized Gibbs Ensembles" (GGE), which are like cities that have been shaken up in a specific, non-standard way (like a city where the traffic lights are broken in a specific pattern).

The Results:
They found that even though the "library" of possibilities is infinite, the "important" books are actually much fewer than you'd think. They are clustered in a specific way.

• They confirmed that at infinite interaction strength (where the balls act like ghosts that can't pass through each other), their results matched known math perfectly.
• They mapped out the "spectral function," which is essentially a sound map of the city. It shows what frequencies of "noise" the city makes when you poke it. They saw that at low interactions, the city makes clear, sharp sounds (like a bell). At high interactions, the sound becomes a fuzzy, broad rumble (like a drum).

Why This Matters

This paper is a breakthrough because it bridges the gap between theory (what math says should happen) and reality (what happens in real experiments with cold atoms).

Before this, if you wanted to know how a hot, interacting quantum gas behaves, you were stuck guessing. Now, thanks to this "smart sampling" method, scientists can predict exactly how these systems will behave. This helps experimentalists in labs around the world design better experiments with cold atoms, potentially leading to new technologies in quantum computing and sensing.

In a nutshell: They built a digital "spotlight" that ignores the dark, empty corners of the quantum universe and shines a light only on the parts that actually matter, allowing us to finally see the dynamics of hot, interacting quantum matter.

Technical Summary

Here is a detailed technical summary of the paper "Finite temperature single-particle Green's function in the Lieb-Liniger model" by Riccardo Senese and Fabian H.L. Essler.

1. Problem Statement

The Lieb-Liniger (LL) model describes a one-dimensional gas of bosons with repulsive delta-function interactions. While the model is exactly solvable via the Bethe Ansatz, calculating dynamical correlation functions (specifically the single-particle Green's function) at finite temperatures and general interaction strengths remains a major open problem.

• The Bottleneck: Dynamical correlators are expressed via the Lehmann representation, which involves a sum over all "intermediate" eigenstates $|\mu\rangle$. For a system of size $L$, the number of relevant eigenstates scales exponentially ($e^{L}$).
• Limitations of Previous Methods:
  • Zero/Low Temperature: Algorithms like ABACUS (Algebraic Bethe Ansatz for Correlation Functions) work well at $T=0$ or very low $T$ because only a few low-energy excitations contribute significantly.
  • Fredholm Determinants: While exact analytical expressions exist for the impenetrable limit ($c \to \infty$) using Fredholm determinants, these have not been successfully extended to finite interactions or generalized Gibbs ensembles (GGE) for dynamical quantities.
  • Eigenstate Thermalization: Unlike generic chaotic systems where matrix elements are exponentially small and sparse, integrable systems have a complex structure where a sub-entropic but still exponentially large fraction of states contributes to the sum.

The authors aim to compute the single-particle Green's function $C(x,t) = \langle \lambda | \phi^\dagger(x,t) \phi(0,0) | \lambda \rangle$ across the full range of temperatures ($T$), interaction strengths ($c$), and system sizes, including Generalized Gibbs Ensembles (GGE).

2. Methodology: Monte Carlo Sampling of Lehmann Sums

The core innovation is a Markov Chain Monte Carlo (MCMC) algorithm designed to stochastically sample the relevant eigenstates $|\mu\rangle$ that contribute to the Lehmann sum, rather than summing over all states explicitly.

A. Reformulation as a Statistical Problem

The authors rewrite the Lehmann representation:
$$C(x, t) = \sum_{\mu} g_{\lambda,\mu}(x, t) | \langle \mu | \phi(0,0) | \lambda \rangle |^2$$
They define an "energy" term $M_{\lambda,\mu} = -\ln | \langle \mu | \phi(0,0) | \lambda \rangle |^2$. The sum is then interpreted as an expectation value over a normalized Gibbs distribution:
$$P_\lambda(\mu) = \frac{e^{-M_{\lambda,\mu}}}{Z_\lambda}$$
where $Z_\lambda = \langle \lambda | \phi^\dagger \phi | \lambda \rangle = n$ (particle density).

B. The Sampling Algorithm

1. State Representation: Eigenstates $|\mu\rangle$ (with $N-1$ particles) are uniquely defined by a set of $N-1$ distinct integers (quantum numbers) $\{J_j\}$ via the Bethe equations.
2. Metropolis-Hastings Updates: The algorithm performs single-integer updates on the set $\{J_j\}$.
   • Select a random quantum number $J_s$.
   • Propose a move to a nearby unoccupied integer $J^*$.
   • Calculate the change in $M_{\lambda,\mu}$ using exact form factor formulas (derived from the Algebraic Bethe Ansatz).
   • Accept or reject the move based on the Metropolis probability $\min(1, e^{-(M_{new} - M_{old})})$.
3. Efficiency: The authors find that the number of MCMC steps required to converge is drastically lower than the total number of contributing states. They identify that the relevant states have a "cost" $M_{\lambda,\mu}$ that scales linearly with system size $L$ ($M_{\lambda,\mu} \approx L \cdot m[\rho]$), where $m[\rho]$ is strictly less than the entropy density $s[\rho]$. This implies that while the number of relevant states is exponential ($e^{L m[\rho]}$), it is a vanishingly small fraction of the total Hilbert space ($e^{L s[\rho]}$).

C. Data Reconstruction

• Real Space: The algorithm directly samples $C(x,t)$.
• Frequency Space: To obtain the spectral function $\tilde{C}(k, \omega)$, the authors sample the integrated correlator $\tilde{C}_{int}(k, \omega) = \int_{-\infty}^\omega \tilde{C}(k, y) dy$ (a sum of Heaviside steps) and then perform a numerical derivative (using spline interpolation and low-pass filtering) to recover the smooth spectral density.

3. Key Contributions

1. First Finite-Temperature Dynamics for General Interactions: The paper provides the first numerical evaluation of the single-particle Green's function for the LL model at arbitrary finite temperatures and interaction strengths ($0 < c < \infty$), a regime previously inaccessible to exact diagonalization or low-$T$ expansions.
2. Generalized Gibbs Ensembles (GGE): The method is successfully applied to GGEs, allowing the study of non-thermal steady states (e.g., after a quantum quench) by sampling eigenstates corresponding to specific sets of conserved charges.
3. Validation of "Sub-Entropic" Sampling: The work numerically confirms the theoretical prediction (from Ref. [48]) that only a specific subset of eigenstates (those with "anomalous" rapidity distributions close to the reference state) dominates the dynamical correlators, justifying the stochastic approach.
4. Benchmarking: The method is rigorously tested against:
   • Exact Fredholm determinant results for the impenetrable limit ($c \to \infty$).
   • Known static correlation lengths for finite $c$.
   • Zero-temperature results from previous literature.

4. Key Results

• Spectral Functions: The authors present density plots of the spectral function $\tilde{C}(k, \omega)$ for various $\gamma$ (interaction) and $\tau$ (temperature).
  • Weak Coupling ($\gamma \to 0$): The spectrum shows sharp, well-defined quasiparticle peaks following the free dispersion $\omega = -k^2 + h$.
  • Strong Coupling ($\gamma \to \infty$): The spectral weight broadens significantly. At low temperatures, a "Type II" dispersion (associated with particle-hole excitations) is visible. At higher temperatures, the spectrum becomes a broad continuum.
  • Intermediate Coupling: The transition from quasiparticle-like behavior to a continuum is captured smoothly.
• Finite Size Effects: The method demonstrates that system sizes of $L \approx 200$ (with $N \approx 200$) are sufficient to approximate the thermodynamic limit for the spectral function, as finite-size effects are negligible in the relevant frequency/momentum range.
• GGE Dynamics: Results for GGEs show that breaking parity (via non-zero $\beta_3$) leads to distinct asymmetries in the spectral function compared to thermal states, highlighting the method's ability to capture non-equilibrium physics.

5. Significance and Future Outlook

• Overcoming the "Curse of Dimensionality": This work demonstrates that stochastic sampling of the Lehmann representation is a viable and powerful alternative to exact summation for integrable models at finite temperature. It opens the door to studying dynamics in regimes where the Hilbert space is too large for deterministic methods.
• Operator Dependence: The authors discuss a crucial distinction between the Bose field $\phi(x)$ and the density operator $\rho(x) = \phi^\dagger(x)\phi(x)$. While density correlators can be reconstructed up to $N \approx 50-100$ using explicit sums (due to sparser matrix elements), the Bose field requires sampling even at moderate sizes because its matrix elements are less sparse and decay more slowly.
• Applicability: The framework is not limited to the LL model. It can be extended to other integrable models, quantum quenches, and potentially to the study of off-diagonal matrix elements in other integrable systems.

In summary, Senese and Essler have developed a robust numerical tool that bridges the gap between exact integrability and finite-temperature dynamics, providing the first comprehensive view of the single-particle spectral function of the Lieb-Liniger gas across the entire phase diagram.