Boundary Perturbation Effects in Quantum Systems with Conserved Energy and Continuous Symmetry

This paper investigates one-dimensional quantum systems with conserved energy and continuous symmetry, revealing that boundary perturbations breaking the symmetry induce two distinct dynamical phases—charge conservation and extensive fluctuations—driven by a boundary-induced pumping mechanism that persists in free-fermion systems and finite-size interacting models under effective energy conservation, particularly in high-frequency Floquet settings.

The Gist

Imagine a long, crowded hallway filled with people (these are our particles or charges). In this hallway, there are two very strict rules:

1. The Energy Rule: The total amount of "energy" in the hallway never changes. People can move, but they can't just appear out of nowhere or vanish without a trace of energy.
2. The Symmetry Rule: The hallway has a specific "charge" (like a total number of people). Usually, this number is fixed.

Now, imagine we hire a boundary guard (the boundary perturbation) standing at one end of the hallway. This guard has a special job: they can secretly let people in or out, breaking the "Symmetry Rule."

The big question the authors asked is: If this guard tries to let people in or out, will the whole hallway fill up or empty out? Or will the hallway stay mostly the same?

The Two Surprising Outcomes

The paper discovers that the answer depends entirely on the "terrain" of the hallway (the bulk parameters). There are two distinct phases:

1. The "Frozen" Phase (The Impenetrable Wall)

In this scenario, even though the guard is trying to let people in, nothing happens. The total number of people in the hallway stays almost exactly the same.

• The Analogy: Imagine the hallway is a high-security vault. The guard tries to open the door, but the people inside are so tightly packed and energetic that moving one person requires a massive amount of energy. Because the "Energy Rule" forbids creating that extra energy, the guard is stuck. The door might wiggle, but no one actually crosses the threshold. The charge is "frozen."

2. The "Fluctuating" Phase (The Open Floodgate)

In this scenario, the guard opens the door, and chaos ensues. People flood in and out wildly. The total number of people in the hallway changes drastically.

• The Analogy: Imagine the hallway is a wide, flat, empty field. The guard opens the gate, and because the terrain is flat (no energy barrier), people can easily wander in and out. The system "thermalizes," meaning the charge spreads out and fluctuates wildly.

Why Does This Happen? (The "Pumping" Mechanism)

The authors explain this using a concept called "Effective Energy Conservation."

Think of the "Frozen" phase like trying to push a heavy boulder up a steep hill.

• To get a particle (the boulder) from the outside (the guard) into the hallway, it needs a specific amount of energy to climb the hill.
• If the hallway is already full of energy (the "gap" is open), there is no room for the boulder to land without breaking the energy rules. The "pump" fails.
• However, if the hill disappears (the energy gap closes), the boulder can roll right in. The "pump" works, and the charge fluctuates.

The "Floquet" Twist: The Rhythmic Kicker

The authors also tested what happens if the guard doesn't just stand there, but kicks the door open and shut rhythmically (this is Floquet dynamics).

• Slow Kicking (Low Frequency): If the guard kicks slowly, the system has time to heat up. The "Energy Rule" breaks down, the hallway gets chaotic, and the "Frozen" phase disappears. Everyone runs wild.
• Fast Kicking (High Frequency): If the guard kicks incredibly fast, something magical happens. The system is so busy reacting to the rapid kicks that it effectively "freezes" the energy. It's like spinning a fan so fast it looks like a solid disk. The "Frozen" phase reappears! The guard is too fast to actually let anyone in, so the charge stays protected.

The Big Picture

This research is important because it shows that energy conservation is a superpower.

• Even if you break the rules of symmetry (letting charge in/out), if you strictly conserve energy, the system can resist change and stay "frozen."
• This happens in simple systems (free electrons) and complex, interacting systems (people bumping into each other).
• It suggests that in quantum computers or new materials, we might be able to protect information (charge) from leaking out, not by building a stronger wall, but by carefully tuning the energy landscape so that "leaking" is energetically impossible.

In short: The paper reveals that a tiny disturbance at the edge of a quantum system can either do nothing (if the energy landscape is steep) or cause a massive flood (if the landscape is flat), and that shaking the system fast enough can magically restore the "nothing happens" state.

Technical Summary

1. Problem Statement

The paper investigates the dynamical response of one-dimensional quantum many-body systems possessing both energy conservation (governed by Hamiltonian evolution) and a global continuous $U(1)$ symmetry (conservation of total charge/particle number). Specifically, the authors study the impact of a boundary perturbation ($\hat{H}_B$) that explicitly breaks the $U(1)$ symmetry.

The central question is: How does the bulk charge evolve when the boundary allows charge to enter or leave the system, given that the total energy of the closed system is strictly conserved?

While one might expect the boundary perturbation to immediately induce a superposition of states across different charge sectors (leading to extensive charge fluctuations), the authors demonstrate that energy conservation imposes strict constraints, leading to a distinct phase where the bulk charge remains "frozen" despite the symmetry-breaking boundary.

2. Methodology

The study employs a combination of numerical simulations and analytical perturbation theory across three model classes:

1. Free-Fermion Chains: Solved exactly using Gaussian state simulations.
2. Interacting Fermionic Chains: Studied via Exact Diagonalization (ED) for small system sizes with nearest-neighbor interactions ($U \hat{n}_j \hat{n}_{j+1}$).
3. Interacting Spin Chains: Modeled as XXZ chains in a magnetic field, studied via ED.

Key Analytical Tools:

• Degenerate Perturbation Theory: The authors construct an effective Hamiltonian ($\hat{H}_{\text{eff}}$) within a subspace of degenerate eigenstates of the unperturbed bulk Hamiltonian ($\hat{H}_0$) sharing the same energy $E_0$.
• Pumping Mechanism Analysis: They analyze the matrix elements of the boundary operator connecting states with different charges but identical energies.
• Floquet Dynamics: The system is subjected to periodic driving (alternating bulk and boundary kicks) to test the robustness of the phases against energy non-conservation.
• BCH Expansion & Path Counting: Used in the appendices to rigorously bound energy absorption and analyze virtual processes in the thermodynamic limit.

3. Key Contributions and Results

A. Discovery of Two Distinct Dynamical Phases

The system exhibits a phase transition controlled by bulk parameters (e.g., chemical potential $\mu_0$ or interaction strength $U$):

1. Charge-Fluctuating Phase:
   • The boundary perturbation successfully pumps charge into/out of the bulk.
   • The variance of the total charge, $\delta N^2$, scales extensively with system size ($L$).
   • The final state is a superposition of wavefunctions spanning multiple charge sectors.
2. Charge-Frozen Phase:
   • Despite the symmetry-breaking boundary, the bulk charge remains nearly conserved.
   • The charge variance $\delta N^2$ remains $O(1)$ (finite) in the thermodynamic limit, regardless of system size.
   • The system effectively resists charge flow.

B. Mechanism: Effective Energy Conservation and Spectral Gaps

The authors attribute the "frozen" phase to effective energy conservation.

• The Logic: Moving charge into or out of the system requires changing the particle number. In the presence of a spectral gap in the quasiparticle spectrum, adding/removing particles at the boundary inevitably changes the total energy of the state.
• The Constraint: Since the total energy is conserved by the Hamiltonian evolution, transitions to states with different particle numbers are energetically forbidden if they do not share the same energy eigenvalue.
• The Criterion: The transition occurs when the single-particle spectral gap closes.
  • Gap Open ($|\mu_0| > 2t_0$): No states with different particle numbers share the same energy. The effective Hamiltonian $\hat{P}\hat{H}\hat{P}$ (projected onto the degenerate subspace) becomes purely diagonal. Charge sectors are disconnected.
  • Gap Closed ($|\mu_0| < 2t_0$): Many states with different particle numbers become degenerate. The boundary operator connects these states, creating off-diagonal terms in $\hat{P}\hat{H}\hat{P}$, allowing charge pumping.

C. The "Pumping" Criterion

The authors propose a general criterion for the phase transition based on the effective Hamiltonian derived from degenerate perturbation theory:
$$\hat{H}_{\text{eff}} \approx \hat{P} \hat{H} \hat{P}$$
where $\hat{P}$ projects onto the subspace of states with energy $E_0$.

• If the matrix element $\langle N+2 | \hat{H}_B | N \rangle$ vanishes (or is exponentially suppressed) within the degenerate subspace, the system is in the Frozen Phase.
• If these matrix elements are $O(1)$ (due to a large number of available final states compensating for small individual amplitudes), the system is in the Fluctuating Phase.
• This criterion holds for both free and interacting systems, provided the Eigenstate Thermalization Hypothesis (ETH) holds within charge sectors.

D. Floquet Dynamics and High-Frequency Stabilization

The study extends to periodically driven (Floquet) systems where energy is not strictly conserved:

• Low Frequency: The system heats up to infinite temperature, breaking effective energy conservation. The charge-frozen phase disappears; extensive fluctuations occur.
• High Frequency:
  • Free Systems: A strict threshold frequency $f_c$ exists above which heating is suppressed, and the charge-frozen phase survives exactly in the thermodynamic limit.
  • Interacting Systems: The frozen phase survives only at finite system sizes. The non-heating window shrinks as $L \to \infty$ (consistent with prethermalization theory), but for practical finite sizes, the phase is observable.
• Symmetry Replacement: When the authors replaced energy conservation with $z$-spin conservation in a spinful model, the frozen phase vanished, confirming that energy conservation is the unique and necessary condition for this phenomenon.

4. Significance

• Fundamental Physics: The work reveals a novel mechanism where energy conservation alone can protect a global symmetry (charge) from being broken by a local boundary perturbation, even in the absence of integrability. This challenges the intuition that local symmetry breaking inevitably leads to global symmetry breaking in the bulk.
• Phase Transition without Order Parameter: It identifies a dynamical phase transition driven by the spectral properties of the bulk (gap closing) rather than a standard Landau symmetry-breaking order parameter.
• Robustness in Non-Equilibrium: The findings on Floquet systems highlight the existence of "prethermal" regimes where effective conservation laws can stabilize distinct phases in driven quantum matter, with specific implications for finite-size stabilization in interacting systems.
• General Applicability: The proposed "pumping mechanism" and the $\hat{P}\hat{H}\hat{P}$ criterion provide a general framework for analyzing boundary effects in quantum systems with multiple conservation laws.

In summary, Gao and Chen demonstrate that in 1D quantum systems, a boundary perturbation breaking a continuous symmetry can fail to affect the bulk charge if the bulk spectrum possesses a gap that prevents energy-conserving transitions between different charge sectors. This leads to a robust "charge-frozen" phase, a phenomenon strictly tied to energy conservation and observable in both static and high-frequency driven systems.