Smearing of dynamical quantum phase transitions in dissipative free-fermion systems

This paper demonstrates that while nonanalyticities associated with dynamical quantum phase transitions in dissipative free-fermion systems can persist under purely gain or purely loss processes, they are completely smeared out as soon as both channels are simultaneously active, a phenomenon accompanied by the emergence of a nested lightcone structure in the reduced Loschmidt echo dynamics.

The Gist

The Big Picture: A Quantum "Snap" That Sometimes Fails

Imagine you have a complex machine made of tiny, invisible gears (quantum particles). You set it in motion, and it starts spinning in a very specific, rhythmic way. Suddenly, you hit a "reset" button (a quantum quench). The machine tries to snap back to its original state, but because it's moving so fast, it overshoots, wobbles, and eventually settles.

In the world of quantum physics, scientists look for specific moments during this wobble where the machine's behavior changes abruptly. They call these moments Dynamical Quantum Phase Transitions (DQPTs). Think of them like a sudden "click" or a "snap" in the rhythm of time.

The Problem: In the real world, nothing is perfectly isolated. Your quantum machine is always interacting with the air, the table, or the heat around it. This interaction is called dissipation (or noise). The big question this paper asks is: If we add noise to our machine, do those special "snap" moments still happen, or does the noise wash them out?

The Cast of Characters

To understand the answer, we need to meet the three main "actors" in this story:

1. The Perfect Vacuum (Unitary Dynamics): This is the ideal scenario where the machine is in a perfect vacuum. No air, no heat, no noise. The gears spin perfectly. In this world, the "snaps" (DQPTs) happen clearly and sharply.
2. The One-Way Door (Pure Gain or Pure Loss): Imagine the machine is in a room where particles can only enter (Gain) or only leave (Loss), but never both. It's like a leaky bucket that only leaks, or a bucket that only gets filled.
3. The Two-Way Chaos (Gain AND Loss): Now, imagine the bucket has a hole at the bottom and a hose pouring water in at the top simultaneously. This is the messy, realistic world where particles enter and leave at the same time.

The Main Discovery: The "Smearing" Effect

The authors of this paper discovered a very specific rule about how these "snaps" behave when noise is added:

• Scenario A: The Perfect Vacuum.
  The "snap" happens. The machine's rhythm breaks sharply at a specific time.

  • Analogy: A drum being hit perfectly. You hear a sharp CRACK.

• Scenario B: One-Way Door (Only Gain OR Only Loss).
  Surprisingly, the "snap" still happens. Even though the machine is leaking or filling up, the sharp rhythm remains. The "crack" is still there, though maybe slightly quieter.

  • Analogy: You are drumming while someone is slowly pouring sand on the drumhead. It's harder to hear, but if you pour sand only on one side (or only remove it), the rhythm can still keep its sharp beat.

• Scenario C: Two-Way Chaos (Gain AND Loss).
  This is the big surprise. As soon as you have both particles entering and leaving—even if the amount is tiny, like a single grain of sand—the sharp "snap" completely disappears.

  • Analogy: Imagine trying to hear a sharp drum crack while someone is simultaneously pouring water on the drum and sucking the sound out with a vacuum. The result isn't a quiet crack; it's a muddy, smooth "thud." The sharp edge is smeared out. The transition becomes smooth and boring.

The Takeaway: If you want to see these special quantum "snaps" in a noisy, real-world experiment, you must ensure that the noise is one-sided. If the noise is two-sided (particles going in and out), the magic moment vanishes.

The "Nested Lightcone" Surprise

The paper also found something cool about how information travels through the machine.

In a perfect vacuum, information spreads out like a ripple in a pond—a simple circle expanding outward. This is called a "lightcone."

However, when they added dissipation (the noise), they found a nested lightcone.

• Analogy: Imagine dropping a stone in a pond. Usually, you see one big ripple. But in this noisy system, it's like dropping a stone that creates a ripple, which then creates a smaller ripple inside it, which creates another tiny ripple inside that. It's like a set of Russian nesting dolls made of waves.
• This structure appeared even in cases where the perfect vacuum machine didn't have it. The noise actually created a more complex, layered structure of how the system evolves.

Why Does This Matter?

1. For Scientists: It tells us exactly what conditions we need to create in the lab to see these quantum effects. If we want to study these "snaps," we need to control our environment carefully. We can't just have random noise; we need to know if that noise is "one-way" or "two-way."
2. For the Future: The authors suggest using a tool called the Reduced Loschmidt Echo (RLE). Think of this as a "stethoscope" for the quantum machine. Instead of listening to the whole machine (which is too hard), you listen to just a small part of it. This paper proves this stethoscope works even when the machine is noisy, making it a great tool for future experiments with cold atoms or trapped ions.

Summary in One Sentence

If you want to see a sharp quantum "snap" in a noisy system, the noise must be one-sided (only entering or only leaving); if the noise is two-sided (entering and leaving), the sharp snap gets washed away into a smooth blur.

Technical Summary

1. Problem Statement

The paper investigates the fate of Dynamical Quantum Phase Transitions (DQPTs) in open quantum systems. DQPTs are non-equilibrium phenomena characterized by non-analyticities (singularities) in the time evolution of the Loschmidt echo (LE), occurring when a system driven out of equilibrium by a quantum quench becomes orthogonal to its initial state at specific critical times.

While DQPTs are well-studied in isolated (unitary) systems, their behavior in dissipative open systems interacting with an environment remains less understood. The central question addressed is: Do the temporal non-analyticities (DQPTs) present in a unitary evolution persist when the system is subject to gain and loss dissipation?

2. Methodology

The authors employ the following theoretical framework and tools:

• Model System: They focus on dissipative free-fermion systems (quadratic fermionic models) interacting with an environment via gain and loss processes. The dynamics are governed by the Lindblad master equation.
• Dissipation Mechanism: The environment induces fermion gain ($\gamma_+$) and loss ($\gamma_-$) at each site. The Lindblad operators are chosen as $L_{j,+} = \sqrt{\gamma_+} c^\dagger_j$ and $L_{j,-} = \sqrt{\gamma_-} c_j$.
• Probe Observable: Instead of the full-system Loschmidt echo (which is exponentially small and difficult to measure), the authors utilize the Reduced Loschmidt Echo (RLE).
  • The RLE is defined as the fidelity between the initial reduced density matrix of a subsystem $A$ and its time-evolved version: $F_A(t) = \text{Tr}(\rho_A(0)\rho_A(t)) / \sqrt{\text{Tr}(\rho_A(0)^2)\text{Tr}(\rho_A(t)^2)}$.
  • They analyze the logarithmic RLE (rate function) $\zeta_A(t) = -\frac{1}{\ell} \log f_A(t)$, where $f_A(t)$ is the unnormalized overlap.
• Gaussian State Formalism: Since the initial state and the Hamiltonian/Lindblad operators are quadratic, the system remains in a Gaussian state throughout the evolution. The dynamics are fully characterized by the covariance matrix (or correlation matrix).
  • The RLE is expressed in terms of the eigenvalues of the product of the initial and time-evolved covariance matrices.
  • A DQPT occurs if an eigenvalue of this product approaches $-1$ in the thermodynamic limit (large subsystem size $\ell$).
• Analytical & Numerical Approach:
  • They derive general conditions for the persistence of DQPTs by relating the dissipative covariance matrix to the dissipationless (unitary) one.
  • They perform explicit calculations for two specific models: the tight-binding chain (starting from a Néel state) and the quantum transverse-field Ising chain (Kitaev chain formulation).
  • They verify analytical predictions using numerical diagonalization of the covariance matrices and scaling analysis of eigenvalues with system size.

3. Key Contributions and Theoretical Results

A. General Conditions for DQPT Persistence

The authors derive a rigorous condition linking the dissipative dynamics to the unitary dynamics. Let $b(t) = e^{-(\gamma_+ + \gamma_-)t}$ and $\chi(t) = (2n_\infty - 1)(1 - b(t))$, where $n_\infty = \gamma_+ / (\gamma_+ + \gamma_-)$.

• Case 1: Both Gain and Loss Present ($\gamma_+ > 0, \gamma_- > 0$):
  The spectral norm of the relevant matrix product is strictly less than 1 for all $t > 0$. Consequently, DQPTs are completely smeared out. The rate function becomes analytic at all times, regardless of how small the dissipation rates are. Even an infinitesimal presence of both channels destroys the singularity.
• Case 2: Pure Gain or Pure Loss ($\gamma_+ > 0, \gamma_- = 0$ or vice versa):
  The spectral norm is bounded by 1. In this scenario, DQPTs can persist if and only if they exist in the corresponding unitary dynamics. The critical times remain the same ($t_c = \tilde{t}_c$). However, the presence of unitary DQPTs is a necessary but not sufficient condition; specific model details determine if the singularity survives.

B. Nested Lightcone Structure

The authors discover that dissipation induces a nested lightcone structure in the dynamics of the RLE.

• In unitary dynamics, lightcones arise from quasiparticle propagation with specific group velocities.
• In dissipative dynamics, the interplay between unitary evolution and dissipation generates contributions from quasiparticles with arbitrarily large group velocities (higher-order moments).
• This results in a hierarchy of lightcones (nested structure) that can appear even in models where the unitary evolution does not exhibit such a structure, due to coherent cancellations in the unitary phase.

4. Specific Model Results

Tight-Binding Chain (Néel Quench)

• Unitary Case: Exhibits DQPTs at times $\tilde{t}_c = (m + 1/2)\pi$.
• Dissipative Results:
  • Gain/Loss ($\gamma_+ = \gamma_-$): DQPTs are completely smeared out. The non-analyticities vanish.
  • Pure Gain/Loss: The DQPTs at the same critical times as the unitary case are preserved.
  • Scaling: Numerical scaling of the smallest eigenvalue $\nu_m$ confirms that it approaches $-1$ in the thermodynamic limit for pure gain/loss, but stays strictly greater than $-1$ when both channels are active.

Quantum Ising Chain (Transverse Field Quench)

• Unitary Case: Exhibits DQPTs at $t_c = (m + 1/2)t^*$.
• Dissipative Results:
  • Pure Gain/Loss: Interestingly, while the first DQPT peak (at $t^*/2$) remains visible (though smeared), the second peak (at $3t^*/2$) disappears. This illustrates that unitary DQPTs are necessary but not sufficient for dissipative DQPTs.
  • Gain + Loss: All non-analyticities are completely smeared out, consistent with the general theory.

5. Significance and Implications

• Robustness of DQPTs: The study establishes that DQPTs are extremely fragile in the presence of generic dissipation (simultaneous gain and loss). They are only robust in highly specific, idealized scenarios of pure gain or pure loss.
• Experimental Relevance: The Reduced Loschmidt Echo (RLE) is identified as a viable experimental probe for DQPTs in both isolated and open quantum systems (e.g., cold atoms, trapped ions).
• Theoretical Insight: The work provides a rigorous framework for understanding how environmental noise affects non-equilibrium quantum criticality. It highlights that the "smearing" of singularities is a generic feature of open systems, distinguishing them fundamentally from isolated unitary dynamics.
• New Phenomenology: The discovery of the nested lightcone structure induced by dissipation suggests new ways to control information spreading and entanglement dynamics in open quantum systems.

In summary, the paper concludes that while DQPTs can survive in open systems under very restrictive conditions (pure gain or pure loss), the simultaneous presence of gain and loss channels acts as a universal "smearing" mechanism that eliminates temporal non-analyticities, rendering the dynamics analytic.