Charged moments and symmetry-resolved entanglement from Ballistic Fluctuation Theory

This paper extends Ballistic Fluctuation Theory to composite branch-point twist fields to derive analytic expressions for charged Rényi entropies in free fermion systems at equilibrium and following quantum quenches, confirming consistency with the quasiparticle picture conjecture.

The Gist

The Big Picture: Untangling a Messy Room

Imagine you have a massive, incredibly complex room (a quantum system) filled with billions of tiny, invisible marbles (particles). These marbles are constantly moving, bouncing off each other, and getting "entangled."

Entanglement is like a magical invisible thread connecting two marbles. If you look at one, you instantly know something about the other, no matter how far apart they are. Physicists love to measure how "tangled" a specific corner of the room (a subsystem) is with the rest of the room. This measurement is called Entanglement Entropy.

But here's the twist: The marbles in this room aren't just random; they have a rule. They must obey a symmetry. Think of it like a strict bouncer at a club: the total number of "red" marbles minus "blue" marbles must always stay the same. This is a U(1) symmetry (like electric charge).

The Problem: The "Blind" Measurement

Usually, when physicists measure entanglement, they look at the whole mess of marbles in the corner without caring about their colors. They ask, "How tangled is this corner?"

But this paper asks a deeper question: "How is the entanglement distributed among the different colors?"

• How much entanglement exists between the red marbles in the corner and the rest of the room?
• How much is there for the blue marbles?

This is called Symmetry-Resolved Entanglement. To answer this, the authors needed a new way to count the threads, not just the marbles.

The Tool: The "Charged Moment" (The Magical Flashlight)

To solve this, the authors use a mathematical tool called a Charged Moment.

Imagine you have a Flashlight (the measurement tool).

1. Standard Flashlight: Shines on the whole corner, giving you a total brightness (standard entanglement).
2. Charged Flashlight: This special flashlight has a filter. It doesn't just shine; it "charges" the marbles as it scans them. It asks, "If I count the marbles with a specific 'charge' (color), how tangled are they?"

By turning the dial on this flashlight (changing a variable called $\alpha$), the authors can isolate the entanglement of specific groups of marbles.

The Theory: Ballistic Fluctuation Theory (BFT)

How do they calculate this without simulating billions of marbles? They use a theory called Ballistic Fluctuation Theory (BFT).

The Analogy: The Highway and the Traffic Jam
Imagine the marbles are cars on a highway.

• Ballistic: The cars are driving at full speed, never stopping or crashing (this is a "free fermion" system, meaning the cars don't interact much).
• Fluctuation: Sometimes, you get a random surge of traffic (a fluctuation).
• BFT: This theory is like a super-advanced traffic model. It predicts how likely it is to see a massive traffic jam or a clear road, based on the speed and density of the cars.

The authors realized that the "Charged Flashlight" is mathematically identical to counting how many cars pass a specific point on the highway over a certain time. Instead of counting cars, they are counting "twists" in the quantum threads.

The Two Scenarios: Calm vs. Chaos

The paper looks at two different situations:

1. Equilibrium (The Calm Lake)

The system has been sitting still for a long time. The marbles have settled into a comfortable pattern.

• The Result: The authors found a simple formula. The amount of entanglement in a specific "color" group depends on the temperature and the density of the marbles. It's like predicting the water level in a calm lake; it's stable and predictable.

2. Out-of-Equilibrium (The Earthquake)

Suddenly, the room shakes! This is a Quantum Quench. The marbles are thrown into chaos.

• The Setup: Imagine the marbles were arranged in a perfect checkerboard pattern (like a chessboard). Suddenly, the rules change, and they start flying apart in pairs.
• The Process: These pairs fly out in opposite directions like rockets.
• The Discovery: The authors tracked how the "Charged Flashlight" sees these flying pairs.
  • Short Time: The rockets haven't reached the corner yet. The entanglement is low.
  • Long Time: The rockets have filled the corner. The entanglement grows linearly (like a straight line going up) until it hits a limit.
  • The Surprise: They found that the "odd" fluctuations (weird, asymmetrical wiggles in the data) cancel out perfectly. It's as if the rockets always come in perfect pairs, so the "left-right" balance is always maintained.

The "Height Field" and "Twist Fields" (The Secret Sauce)

You might wonder, "How did they turn a complex quantum problem into a traffic model?"

They used a trick called the Height Field Formulation.

• Imagine the marbles are standing on a hilly landscape. The "height" of the hill at any point represents the number of marbles there.
• The "Twist Field" is like a magical spiral staircase that connects different copies of the room.
• The authors realized that counting the marbles on the staircase is the same as measuring the "height" of the hill. This allowed them to use the traffic model (BFT) to solve the quantum problem.

Why Does This Matter?

1. It's a New Lens: It allows us to see how entanglement is organized, not just how much there is. It's like looking at a painting and realizing the red paint is all on the left and the blue is on the right, rather than just seeing a gray blur.
2. It Works for Chaos: It proves that even when a quantum system is thrown into chaos (a quench), there are still simple, predictable rules governing how information spreads, as long as the particles are "free" (not crashing into each other).
3. It Connects Worlds: It bridges the gap between abstract quantum math (twist fields) and practical physics (traffic flow/fluctuations).

The Bottom Line

This paper is like a master key. It unlocks a way to measure the hidden, color-coded structure of quantum entanglement. By treating quantum particles like cars on a highway and using a "charged" flashlight, the authors showed us that even in the chaotic aftermath of a quantum explosion, the universe keeps a perfect, balanced ledger of how its parts are connected.

Technical Summary

1. Problem Statement

The paper addresses the calculation of symmetry-resolved entanglement entropies in quantum many-body systems possessing a global internal symmetry (specifically $U(1)$). While standard entanglement entropy quantifies total entanglement, symmetry-resolved entanglement decomposes this into sectors defined by the conserved charge (e.g., particle number).

The core challenge lies in computing the charged moments of the reduced density matrix (RDM), defined as:
$$Z_m(\alpha) = \text{tr}(\rho_A^m e^{i\alpha Q_A})$$
where $\rho_A$ is the RDM of a subsystem $A$, $Q_A$ is the charge restricted to $A$, and $\alpha$ is a flux parameter. These moments are the generating functions for the symmetry-resolved Rényi entropies.

Existing methods often rely on exact diagonalization (limited to small systems) or specific free-fermion techniques. The authors aim to derive analytic expressions for these quantities in both equilibrium and non-equilibrium settings using Ballistic Fluctuation Theory (BFT), a hydrodynamic framework describing large-scale fluctuations of conserved charges and currents.

2. Methodology

The authors employ a multi-step theoretical framework combining replica theory, twist fields, and hydrodynamic fluctuation theory:

• Twist Field Formulation: The charged moments are mapped to the correlation functions of composite branch-point twist fields, $T_{m,\alpha}(x)$. These fields are constructed by inserting a $U(1)$ flux $\alpha$ into the standard branch-point twist fields used for Rényi entropies.
• Height-Field Formulation: The authors utilize the "height-field" representation of twist fields. In this formulation, twist fields are expressed as exponentials of integrated charge densities (height fields). This links the correlation functions of twist fields directly to the Full Counting Statistics (FCS) of conserved charges.
• Ballistic Fluctuation Theory (BFT): BFT provides a large-deviation description of charge and current fluctuations in homogeneous, stationary states.
  • Equilibrium: The FCS is governed by the Scaled Cumulant Generating Function (SCGF) derived from the free energy density of Generalized Gibbs Ensembles (GGE).
  • Non-Equilibrium: For quantum quenches, the authors map the problem to the FCS of time-integrated currents. They utilize a contour deformation technique (originally introduced in Ref. [1]) to handle long-range correlations in the initial state, allowing the application of BFT along specific space-time paths.
• Replica Diagonalization: To handle the composite twist fields in the replica theory ($m$ copies), the authors perform a discrete Fourier transform on the replica index. This diagonalizes the twist field action, reducing the problem to a sum of independent $U(1)$ twist fields in the Fourier basis.

3. Key Contributions

1. Extension of BFT to Charged Moments: The paper extends the application of BFT from standard Rényi entropies (branch-point twist fields) to charged moments (composite twist fields with flux insertion).
2. Analytic Derivation for Free Fermions: The authors derive fully analytic, closed-form expressions for charged moments in free fermion systems for:
   • Equilibrium: Generalized Gibbs Ensembles (GGE).
   • Non-Equilibrium: Quantum quenches from a specific class of integrable, pair-producing initial states (generalizing Néel and Dimer states) that preserve the $U(1)$ symmetry.
3. Unification of Regimes: The work provides a unified hydrodynamic description that connects the asymptotic behavior of entanglement growth to the statistics of conserved charges and currents, validating the "quasiparticle picture" for symmetry-resolved quantities.
4. Non-Asymptotic Results: The derivation includes results for the non-asymptotic regime (finite time and space), showing how the charged moments interpolate between short-time (linear growth) and long-time (saturation) behaviors.

4. Key Results

A. Equilibrium (GGE)

For a subsystem of size $x$ in a GGE characterized by occupation numbers $n_k$, the logarithm of the charged moments scales linearly with subsystem size:
$$\ln Z_m^{eq}(\alpha) = x \int \frac{dk}{2\pi} H_m^\alpha(k)$$
where the entropy function $H_m^\alpha(k)$ is given by:
$$H_m^\alpha(k) = \ln \left[ n_k^m e^{i\alpha} + (1-n_k)^m \right]$$
This result generalizes previous findings for standard Rényi entropies ($\alpha=0$) to the symmetry-resolved case.

B. Non-Equilibrium (Quantum Quench)

For a quantum quench from an integrable, pair-producing initial state (e.g., Néel or Dimer states), the charged moments depend on the ratio of time $t$ to subsystem size $x$. The result is:
$$\ln Z_m(\alpha) = i\frac{\alpha x}{2} + \int_{-\pi}^{\pi} \frac{dk}{2\pi} \min(2t|v_k|, x) \, \text{Re}\left[ H_m^\alpha(k) \right]$$
where $v_k = \partial_k E_k$ is the group velocity.

• Short-time regime ($t \ll x$): The growth is linear in time, driven by the ballistic propagation of quasiparticle pairs.
• Long-time regime ($t \gg x$): The system saturates to the equilibrium GGE result.
• Odd Cumulants: A significant finding is that for this class of initial states, the time-dependent part of the charged moments contains no odd cumulants (except the mean). This is attributed to the symmetric nature of the pair-produced excitations, which causes odd connected correlators of currents to vanish.

C. Symmetry-Resolved Entropies

The authors discuss the extraction of symmetry-resolved Rényi entropies $S_m(q)$ via Fourier transform of $Z_m(\alpha)$.

• In the Gaussian regime (sub-extensive charge fluctuations), the distribution of charge sectors is Gaussian.
• The symmetry-resolved entropy exhibits equipartition at leading order, with corrections scaling as $O((\Delta q)^2 / J)$, where $J$ is the integrated current.
• They identify a universal time delay in the growth of symmetry-resolved entanglement after a quench, determined by the maximum group velocity of the quasiparticles.

5. Significance

• Theoretical Validation: The work provides a rigorous derivation of the quasiparticle picture for symmetry-resolved entanglement, confirming conjectures based on phenomenological arguments.
• Methodological Advance: It demonstrates the power of BFT combined with twist field theory to handle complex non-equilibrium problems involving symmetry sectors, moving beyond exact free-fermion calculations to a hydrodynamic framework.
• Generality: While focused on free fermions, the framework suggests a pathway to study interacting integrable models (by replacing the entropy weight with Yang-Yang entropy) and potentially non-integrable systems via diffusive extensions.
• Experimental Relevance: The results offer concrete predictions for the distribution of entanglement in charge sectors, which are becoming accessible in experiments with ultracold atoms and quantum simulators.

In summary, this paper successfully bridges the gap between microscopic twist field formulations and macroscopic hydrodynamic fluctuation theory to solve for symmetry-resolved entanglement in both equilibrium and dynamic settings, providing a robust analytical tool for understanding quantum information scrambling in the presence of symmetries.