Nature abhors a vacuum: A simple rigorous example of thermalization in an isolated macroscopic quantum system

This paper provides a mathematically rigorous proof that a low-density free fermion chain thermalizes from a specific nonequilibrium initial state by demonstrating that the absence of energy degeneracy and specific particle distribution properties ensure a large effective dimension, leading the system to reach equilibrium values with high probability.

The Gist

The Big Question: Why Does a Messy Room Eventually Get Clean?

Imagine you have a very long hallway (a quantum system) with a bunch of people (particles) standing on the left side, and the right side is completely empty. This is a "messy" or nonequilibrium state.

In the real world, if you open a door between a crowded room and an empty one, the people will eventually spread out until the crowd is evenly distributed. This is called thermalization (or reaching equilibrium).

But here is the puzzle: In the quantum world, the rules of physics (specifically, unitary time evolution) are like a perfect, reversible movie. If you played the movie backward, the people would magically run back to the left side. So, how can a quantum system "forget" its messy start and settle down into a calm, spread-out state?

For decades, physicists have believed this happens, but proving it mathematically without making up "magic rules" has been incredibly hard.

The Paper's Achievement: A Rigorous Proof

This paper says: "We found a specific, simple quantum system where we can prove, 100% mathematically, that thermalization happens."

They didn't just guess; they built a mathematical fortress to prove it. Here is how they did it, using some simple metaphors.

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1. The Setup: The "Half-Filled" Hallway

Imagine a long chain of lockers (the lattice).

• The System: A row of $L$ lockers.
• The Particles: $N$ fermions (a type of quantum particle that hates sharing lockers; only one can fit in each).
• The Initial State: All $N$ particles are crammed into the left half of the hallway. The right half is a vacuum (empty).
• The Twist: The particles are "free fermions," meaning they don't bump into each other or talk to each other. They just hop around. Usually, physicists think these "lazy" particles never thermalize because they don't interact.

2. The Secret Weapon: The "Effective Dimension"

To prove the particles spread out, the authors used a concept called Effective Dimension.

The Analogy: The Dice Roll
Imagine you have a giant bag of dice.

• Low Effective Dimension: If you only have 2 dice, your future is very predictable. You can only roll a few combinations. The system is "rigid."
• High Effective Dimension: If you have a million dice, the number of possible combinations is astronomical. The system is "chaotic" and flexible.

The authors proved that if you start with a random arrangement of particles in the left half, the system has a massive Effective Dimension. It's like having a million dice. Because there are so many ways the particles can arrange themselves, the system gets "lost" in the possibilities and naturally drifts toward the most likely state: an even spread.

3. The Two Magic Rules (Assumptions)

To make their proof work, they had to verify two specific rules for their system:

Rule A: No "Twin" Energies (Non-degeneracy)
In quantum mechanics, different states can sometimes have the exact same energy (like two different songs having the same pitch). This is called degeneracy.

• The Problem: If energies are twins, the system can get stuck in a loop.
• The Fix: The authors used a clever trick involving prime numbers and a "phase shift" (a tiny mathematical nudge) to ensure that every single energy state in their system is unique. No twins allowed. They used number theory (math about prime numbers) to prove this rigorously.

Rule B: The "Vacuum" is Rare
They needed to prove that in any possible energy state of the system, it is extremely unlikely to find all the particles still stuck in the left half.

• The Analogy: Imagine shuffling a deck of cards. The chance that all the Aces end up in the top half of the deck is tiny.
• The Proof: They showed that for their specific chain, the probability of finding all particles in the left half is so small ($2^{-N}$) that it's effectively zero. This means the system must spread out.

4. The Result: Nature Hates a Vacuum

Once these two rules were proven, the math took over. They showed that:

1. You start with all particles on the left.
2. You let time pass.
3. If you look at the system at a "typical" time (not a weird, specific moment), and you count how many particles are in the left half...
4. The result will be almost exactly half.

The system has thermalized. It has forgotten it started on the left and settled into the equilibrium state (half left, half right).

Why is this a Big Deal?

1. No Magic Assumptions: Most previous proofs said, "If we assume the system is chaotic enough, then thermalization happens." This paper said, "We proved the system is chaotic enough using hard math."
2. It Works for "Lazy" Particles: They proved this even for free fermions (particles that don't interact). Usually, you need particles to crash into each other to mix. Here, the sheer complexity of the quantum possibilities did the mixing.
3. The Catch: The proof works best when the hallway is very empty (low density). If the hallway is packed tight, the math gets messy. But for a sparse system, the proof is rock solid.

The Bottom Line

The authors took a simple, one-dimensional quantum chain, filled half of it with particles, and proved mathematically that nature will inevitably force those particles to spread out evenly. They didn't rely on guesses; they used the properties of prime numbers and probability to show that thermalization is a mathematical certainty in this specific setup.

In short: Even in a world of perfect, reversible quantum laws, if you start with a vacuum on one side, nature will eventually fill it, simply because there are too many other ways for the particles to be arranged.

Technical Summary

1. Problem Statement

The fundamental question addressed is whether the unitary time evolution of an isolated macroscopic quantum system can rigorously explain the phenomenon of thermalization (approach to thermal equilibrium).

• The Challenge: While the Eigenstate Thermalization Hypothesis (ETH) and large effective dimensions are widely believed to cause thermalization in complex systems, rigorous mathematical proofs for concrete models with short-range interactions have been elusive. Most existing proofs rely on unproven assumptions (like ETH) or apply only to integrable systems (which do not thermalize) or systems with many-body localization (which do not thermalize).
• The Specific Goal: The authors aim to provide a mathematically rigorous, non-perturbative proof of thermalization in a concrete model without relying on unproven assumptions. They focus on a restricted sense of thermalization: the relaxation of a specific macroscopic observable (particle number in a region) to its equilibrium value.

2. Methodology and Framework

The paper is structured in two parts: a general theoretical framework and a concrete application to a specific model.

A. General Theoretical Framework (Section 2)

The authors establish a theorem for thermalization in low-density lattice gases based on two key assumptions:

1. Assumption 2.1 (Non-degeneracy): The energy eigenvalues of the Hamiltonian are non-degenerate.
2. Assumption 2.2 (Particle Distribution): For any energy eigenstate $|\Psi_j\rangle$, the probability of finding all $N$ particles confined in a specific half-lattice $\Lambda_1$ is exponentially small:
   $$\langle \Psi_j | \hat{P}_1 | \Psi_j \rangle \le 2^{-N}$$
   where $\hat{P}_1$ projects onto the subspace where all particles are in $\Lambda_1$.

Key Theoretical Insight (Effective Dimension):
The authors utilize the concept of Effective Dimension ($D_{\text{eff}}$), defined as:
$$D_{\text{eff}} = \left( \sum_{j} |\langle \Phi(0) | \Psi_j \rangle|^4 \right)^{-1}$$
They prove (Theorem 2.3) that if Assumption 2.2 holds and the particle density $\rho$ is low ($\rho \le 1/5$), a randomly chosen initial state from the subspace where all particles are in $\Lambda_1$ has an effective dimension $D_{\text{eff}}$ that is exponentially close to the total Hilbert space dimension $D_{\text{tot}}$.

• Specifically, $D_{\text{tot}} / D_{\text{eff}} \le e^{\rho N}$ with high probability.
• A large effective dimension implies that the system explores the Hilbert space sufficiently to mimic equilibrium.

Thermalization Result (Theorem 2.4):
Combining non-degeneracy (Assumption 2.1) and the large effective dimension, they prove that for a macroscopic region $\Gamma$, the measured particle number $N_\Gamma$ in the time-evolved state $|\Phi(t)\rangle$ will, with probability close to 1, be close to the equilibrium value $\gamma N$ (where $\gamma$ is the fraction of the lattice size).

B. Concrete Model Application (Section 3)

To make the theory rigorous, the authors apply it to a 1D free fermion chain with $N$ particles on $L$ sites.

• Hamiltonian: A standard nearest-neighbor hopping model with an artificial phase factor $\theta$ to break reflection symmetry:
  $$\hat{H} = \sum_{x=1}^L \left( e^{i\theta} \hat{c}^\dagger_x \hat{c}_{x+1} + e^{-i\theta} \hat{c}^\dagger_{x+1} \hat{c}_x \right)$$
• Initial State: A pure state drawn randomly from the subspace where all particles are confined to the left half of the chain ($\Lambda_1$). This represents a "vacuum" on the right and an infinite-temperature gas on the left.

Rigorous Justification of Assumptions:

1. Non-degeneracy (Assumption 2.1): The authors prove that for an odd prime $L$ and a suitably chosen $\theta$ (e.g., $\theta = (4N+2L)^{-(L-1)/2}$), the many-body energy eigenvalues are strictly non-degenerate. This relies on number-theoretic results (irreducibility of cyclotomic polynomials and bounds on algebraic integers) to show that no two distinct sets of momenta yield the same energy sum.
2. Particle Distribution (Assumption 2.2): They prove that for free fermions, the probability of all $N$ particles being in the left half in any energy eigenstate is bounded by $2^{-N}$. This is derived by decomposing the creation operators into left and right parts and estimating the operator norms.

3. Key Results

1. Rigorous Proof of Thermalization: The paper provides the first rigorous proof that a concrete, short-range interacting (in this case, non-interacting but complex) quantum system thermalizes without assuming ETH or other unproven hypotheses.
2. Theorem 1.1 (Main Result): For a free fermion chain with low density $\rho \le 1/5$ and large $N$, if the initial state is random within the "half-filled" subspace, then at sufficiently large typical times, the particle number in the left half, $N_{\text{left}}$, satisfies:
   $$\left| \frac{N_{\text{left}}}{N} - \frac{1}{2} \right| \le \varepsilon_0(\rho)$$
   with probability $> 1 - e^{-(\rho/4)N}$. Here $\varepsilon_0(\rho) = \sqrt{1.5\rho}$.
3. Irreversibility: The system evolves from a state where $N_{\text{left}}/N = 1$ to a state where it is $\approx 1/2$, demonstrating irreversible behavior driven solely by unitary dynamics.
4. Extension to Deterministic States (Appendix C): The authors show that while random initial states are required for free fermions to achieve maximal effective dimension, specific deterministic initial configurations (based on Golomb rulers) also possess large effective dimensions and thermalize, provided the density is extremely low ($\rho \sim N^{-1}$).

4. Significance and Limitations

Significance:

• Mathematical Rigor: It bridges the gap between physical intuition (ETH) and mathematical proof for a concrete model.
• Role of Initial States: It highlights that thermalization depends critically on the "complexity" of the initial state (large effective dimension). Even in integrable systems (free fermions), thermalization can occur if the initial state is sufficiently complex (random or Golomb-ruler based), whereas simple states (like periodic configurations) do not thermalize.
• Generalizability: The framework suggests that thermalization is a generic feature of systems satisfying non-degeneracy and uniform particle distribution in eigenstates, potentially applying to non-integrable models as well.

Limitations:

• Low Density Requirement: The precision of the thermalization ($\varepsilon_0$) depends on the density $\rho$. High precision requires very low density ($\rho \ll 1$). This is a consequence of the strategy to use only mild, verifiable assumptions.
• Restricted Observables: The proof currently guarantees thermalization only for the particle number in a macroscopic region. It does not yet rigorously prove thermalization for observables involving particle-particle correlations or more complex single-particle properties.
• Free Fermions: While the proof is rigorous for free fermions, the authors acknowledge that free fermions are integrable and do not exhibit "full-fledged" thermalization from arbitrary initial states. Their result is specific to the chosen class of complex initial states.

5. Conclusion

The paper successfully demonstrates that "Nature abhors a vacuum" in a quantum mechanical sense: a system prepared with all particles in one half of a chain will, under unitary evolution, spread out to fill the chain uniformly, provided the system is large, the density is low, and the initial state is sufficiently complex. This is achieved by proving that such initial states have an effective dimension nearly equal to the total Hilbert space dimension, forcing the system to explore the equilibrium subspace.