Symmetry Resolved Entanglement Entropy: Equipartition under Driven and Non-unitary Evolution in a Compact Boson CFT

This paper investigates how symmetry-resolved entanglement entropy in a driven compact boson CFT deviates from equipartition due to an $\mathfrak{sl}^{(k)}(2,\mathbb{R})$ subalgebra-induced coupling between frequency modes, while also exploring the impact of non-unitary evolution on these entropies.

The Gist

The Big Picture: Entanglement as a "Shared Secret"

Imagine you have a massive, complex puzzle made of billions of tiny pieces. In quantum physics, these pieces are particles. When two groups of particles are "entangled," it's like they share a secret code. Even if you separate them by miles, they remain connected.

Physicists usually measure how much of this "secret" is shared using a number called Entanglement Entropy. Think of it as a "confusion score." The higher the score, the more mixed up and connected the two groups are.

But this paper asks a deeper question: What if the secret code has different "colors" or "flavors"?

In many quantum systems, there is a rule called a Symmetry (like a conservation law). For example, imagine the particles have a "charge" (like positive or negative electricity). The total charge must stay the same. The paper looks at how the "confusion score" is distributed among these different charge flavors. This is called Symmetry-Resolved Entanglement.

The Main Discovery: The "Equipartition" Rule vs. The "Breakdown"

Usually, physicists expect a rule called Equipartition.

• The Analogy: Imagine a large pizza (the total entanglement) being sliced into many pieces (different charge sectors). The rule of equipartition says that if the pizza is big enough, every slice gets exactly the same amount of cheese. It doesn't matter if the slice is "Charge 1" or "Charge 2"; they all get an equal share of the entanglement.

The Paper's Twist: The authors found that they can break this rule. They can make the pizza slices uneven, giving some charges way more cheese than others. They did this by "driving" the system with a special, rhythmic force.

The Two Experiments

The paper studies two different ways to mess with the quantum system:

1. The "Rhythmic Shaker" (Driven CFT)

Imagine you have a jar of marbles (the quantum system). Usually, they sit still. But here, the scientists shake the jar in a very specific, rhythmic pattern (a "Floquet drive").

• The Secret Ingredient (The $k$ parameter): They use a special shaking pattern that couples "low-frequency" marbles (slow, heavy ones) with "high-frequency" marbles (fast, light ones). Think of it like a DJ mixing a slow bass beat with a super-fast hi-hat.
• The Result:
  • Heating Phase: If they shake it just right, the system gets "hot" and chaotic. The entanglement grows wildly. In this chaos, the "pizza slices" (charge sectors) eventually get equal shares again, but it takes a long time.
  • Non-Heating Phase: If they shake it differently, the system stays calm and oscillates. Here, the "pizza slices" never get equal shares. The "Charge 1" slice might get a huge chunk of cheese, while "Charge 2" gets almost nothing.
• The Takeaway: By changing the rhythm of the shake (the parameter $k$), they can control how the entanglement is distributed. They found a "knob" that lets them decide whether the system shares its secrets equally or hoards them in specific categories.

2. The "Ghostly Measurement" (Non-Unitary Evolution)

Imagine you are watching a magic trick, but every time you look, the magician slightly changes the rules of reality. This is called a "non-unitary" evolution, often linked to "weak measurements" (peeking at the system without fully collapsing it).

• The Analogy: Imagine you are baking a cake (the quantum state). In a normal kitchen (unitary physics), you bake it perfectly. In this experiment, they are baking it in a "ghost kitchen" where time flows a bit differently (complex time).
• The Result: Even though the system is being "measured" and altered by this ghostly force, the long-term result is surprisingly similar to the shaking experiment. The entanglement still tries to distribute itself, but the "ghostly" nature of the measurement changes how fast it happens.
  • In a normal world, the cake bakes fast (linear growth).
  • In this ghost world, the cake bakes very slowly (logarithmic growth).
  • Crucially, the "charge distribution" (who gets the cheese) still follows the same mathematical rules as the shaking experiment, just on a different timeline.

Why Does This Matter?

1. It's Not Just About Size: Usually, we think "big systems = equal sharing." This paper shows that even in a huge system, if you shake it the right way, you can force it to be unfair. You can create a situation where the "size" of the system doesn't matter as much as the "rhythm" of the shake.
2. Controlling Chaos: The authors found a way to control the breakdown of order. They showed that the "mixing" of slow and fast particles is the key to breaking the equal-sharing rule.
3. Real-World Applications: While this is theoretical physics, understanding how quantum information spreads (or gets stuck) in specific categories is crucial for building Quantum Computers. If you want to store information safely, you need to know how it distributes itself. If you want to process it fast, you need to know how to break that distribution.

Summary in One Sentence

The authors discovered that by rhythmically shaking a quantum system (or measuring it in a "ghostly" way), they can break the natural rule that says "all parts of a system share secrets equally," allowing them to control exactly how information is distributed among different particle types.

Technical Summary

1. Problem Statement

The paper investigates the dynamics of Symmetry-Resolved Entanglement Entropy (SREE) in two distinct non-equilibrium settings within 1+1 dimensional Conformal Field Theories (CFTs):

1. Bulk-driven Floquet CFTs: Systems subjected to periodic driving by spatially inhomogeneous Hamiltonians.
2. Non-unitary Quenches: Systems evolving under complex-time metrics (associated with post-selected weak measurements).

The central question is how entanglement equipartition—the phenomenon where entanglement is uniformly distributed across different charge sectors of a global symmetry (e.g., $U(1)$)—is maintained or broken in these dynamic regimes. While equipartition is known to hold in equilibrium and standard unitary quenches in the thermodynamic limit, the authors explore whether specific driving protocols or non-unitary evolutions can induce a breakdown of equipartition, even in the thermodynamic limit, by manipulating the effective length scales of the system.

2. Methodology

The authors utilize the Free Compact Boson CFT as a concrete solvable model, which possesses a global $U(1)$ symmetry generated by the conserved charge $Q = \int \partial \phi$.

• Theoretical Framework:

  • SREE Definition: They decompose the total entanglement entropy $S_A$ into a configurational part (average entanglement per sector) and a number entropy (fluctuations of the charge).
  • Charged Moments: They calculate the charged moments $Z_n(\alpha) = \text{Tr}(\rho_A^n e^{i\alpha Q_A})$ using composite twist fields ($\tau_{n, \alpha}$). These are twist fields dressed with vertex operators $V_\alpha$ that insert a $U(1)$ flux.
  • Replica Trick & Path Integrals: The calculation involves evaluating partition functions on $n$-sheeted Riemann surfaces (or cylinders in the non-unitary case) with specific boundary conditions and flux insertions.

• Specific Protocols:

  1. Driven CFT: The system is evolved using a sequence of Hamiltonians $H_i$ belonging to an $sl^{(k)}(2, \mathbb{R})$ subalgebra of the Virasoro algebra. The Hamiltonian takes the form $H \sim L_k + L_{-k} + \text{h.c.}$, where $k$ is an integer label. This introduces spatial inhomogeneity with a wavelength $\ell = L/k$.
  2. Non-unitary Quench: The system evolves via a complex-time operator $e^{-i H_{CFT}(1-i\mu)t}$, corresponding to a complex metric $ds^2 = -(1 \pm i\mu)dt^2 + dx^2$. This is mapped to a Euclidean path integral on a strip with specific boundary states.

3. Key Contributions

A. Control of Equipartition via the $sl^{(k)}(2, \mathbb{R})$ Label

The most significant theoretical contribution is the identification of the integer label $k$ (from the $sl^{(k)}(2, \mathbb{R})$ subalgebra) as a tunable parameter that controls the breakdown of equipartition.

• Mechanism: The Hamiltonian $H \sim L_k + L_{-k}$ couples low-frequency (IR) modes with high-frequency (UV) modes (specifically $\alpha_m$ couples to $\alpha_{k-m}$).
• Effect: This coupling creates an effective length scale determined by the driving wavelength $\ell = L/k$. Even in the thermodynamic limit ($L \to \infty$), if $k$ is scaled such that $\ell$ remains small (comparable to the UV cutoff), the system behaves as if the subsystem is small.
• Result: In this regime, the charge-dependent corrections to the entanglement entropy are not suppressed, leading to a visible breakdown of equipartition.

B. Phase-Dependent Dynamics in Driven CFTs

The authors analyze the SREE across the three phases of Floquet dynamics defined by the trace of the evolution matrix:

1. Heating Phase ($|\text{Tr}(M)| > 2$): Entanglement grows linearly with time. Equipartition is asymptotically restored only after a very long time as the effective scale $B_1(N) \propto N$ becomes large.
2. Phase Transition ($|\text{Tr}(M)| = 2$): Entanglement grows logarithmically. Restoration of equipartition is slower ($B_1(N) \propto \log N$).
3. Non-heating Phase ($|\text{Tr}(M)| < 2$): Entanglement oscillates. The effective scale $B_1(N)$ remains bounded, meaning equipartition is never restored dynamically; the charge sectors remain distinct indefinitely.

C. Non-Unitary Evolution and Complex Metrics

The paper extends the analysis to non-unitary evolution (post-selected weak measurements).

• They map the non-unitary quench to a path integral on a cylinder of length $W(t)$.
• Scaling: For $\mu > 0$ (non-unitary), $W(t)$ grows logarithmically with time ($W(t) \sim \log t$), whereas for unitary evolution ($\mu=0$), it grows linearly ($W(t) \sim t$).
• Implication: The Number Entropy (Shannon entropy of charge fluctuations) scales as $\log(\log t)$ in the non-unitary case, contrasting with $\log t$ in the unitary case. This demonstrates that non-unitary dynamics fundamentally alter the statistics of charge fluctuations.

4. Key Results

• Breakdown of Equipartition: The SREE for a charge sector $q$ is given by:
  $$S_A(q) \approx S_A - \frac{1}{2}\log(4\pi B_1) + \frac{B}{2B_1} - \frac{B+B_1}{4B_1^2}q^2$$
  where $B_1$ is a width parameter proportional to the effective system size (or driving scale).

  • When $B_1$ is large (standard thermodynamic limit), the $q^2$ term vanishes, and equipartition holds.
  • When $B_1$ is moderate (controlled by the driving parameter $k$), the $q^2$ term is significant, and equipartition breaks down. The entanglement is no longer uniform across charge sectors.

• Oscillatory Behavior in Charge Distribution: For large charges $|q| \gtrsim 2\pi B_1$, the Gaussian approximation fails, and the charge distribution exhibits oscillatory behavior ($(-1)^q$) due to the finite integration range of the flux parameter $\alpha$. This implies a maximum valid charge $q_{max}$ beyond which the Gaussian description fails.

• Number Entropy Scaling:

  • Heating Phase: $S_{num} \sim \log N$
  • Phase Transition: $S_{num} \sim \log(\log N)$
  • Non-heating Phase: $S_{num}$ is bounded/oscillatory.
  • Non-unitary Quench: $S_{num} \sim \log(\log t)$ (distinct from unitary $\log t$).

5. Significance

• Algebraic Control of Entanglement: The paper provides a novel mechanism to control entanglement structure using the algebraic properties of the Virasoro subalgebra ($sl^{(k)}(2, \mathbb{R})$). It shows that "effective size" is not just a geometric property but can be engineered via the driving frequency and mode coupling.
• Beyond Thermodynamic Limits: It challenges the standard assumption that equipartition is an inevitable feature of the thermodynamic limit. By tuning the driving parameter $k$, one can observe non-trivial charge-dependent entanglement even in infinite systems.
• Non-Unitary Physics: The results offer new insights into measurement-induced phases and non-unitary CFTs, showing that complex-time evolution leads to distinct scaling laws for charge fluctuations compared to standard unitary dynamics.
• Universality: The authors argue that the mechanism of coupling IR and UV modes via the Virasoro algebra is general and likely persists beyond the free boson CFT, suggesting broad applicability to other CFTs and potentially holographic systems (AdS/CFT).

In summary, this work establishes a rigorous link between the algebraic structure of driven CFTs, the effective length scales of the system, and the detailed structure of symmetry-resolved entanglement, revealing regimes where the fundamental assumption of equipartition fails.