Dissipation driven phase transition in the non-Hermitian Kondo model

Using the Bethe Ansatz, this paper reveals a novel $\widetilde{YSR}$ phase between the Kondo and unscreened phases in the non-Hermitian Kondo model, demonstrating that dissipation drives a phase transition characterized by distinct time scales and a generalized Kondo temperature.

The Gist

Imagine you have a tiny, stubborn magnet (an "impurity") sitting in a busy crowd of moving particles (an "electron sea"). In the classic world of physics, this magnet usually gets surrounded by the crowd, which arranges itself perfectly to cancel out the magnet's influence. This is called the Kondo effect, and it's like the crowd forming a protective shield around the magnet, making it invisible to the outside world.

Now, imagine this experiment isn't happening in a perfect, sealed vacuum. Instead, it's happening in a leaky room where particles are constantly escaping or being lost to the environment. This is dissipation. In the quantum world, this "leakiness" is described by something called a non-Hermitian model.

This paper discovers that when you add this "leakiness" to the Kondo effect, the story changes completely. Instead of just two outcomes (shielded or unshielded), there are actually three distinct phases, and the transition between them is driven by how fast the system is losing energy.

Here is the breakdown using a simple analogy:

The Three Phases of the "Leaky Magnet"

Think of the "loss strength" (how fast particles are leaking out) as the volume of a noisy fan in the room. As you turn the fan up (increasing the loss), the behavior of the magnet changes in three stages:

1. The Kondo Phase (The Quiet Room)

• The Fan: Off or very low.
• What happens: The crowd of particles gathers around the magnet and forms a perfect, tight hug. The magnet is completely "screened" (hidden).
• The Result: The system is stable. The magnet is effectively neutralized by its friends. This is the standard behavior we know from normal physics.

2. The YSR Phase (The "Ghost" Room)

• The Fan: Medium volume (The "Goldilocks" zone).
• What happens: This is the paper's big discovery. As the noise (loss) increases, the crowd can no longer form a perfect hug. Instead, a single particle from the crowd gets stuck to the magnet, forming a "bound state" (like a ghostly tether).
• The Twist: This bound state is unstable. It's like a balloon tied to the magnet that is slowly deflating.
  • The Energy Trick: At first, this "ghost tether" actually lowers the system's energy, making it the preferred state.
  • The Time Trick: However, because the tether is leaking energy, it eventually pops. If you wait long enough, the magnet ends up unshielded again, even though it started out shielded.
• The Result: A new, strange phase where the magnet is temporarily shielded by a "ghost" but destined to be exposed by time. The authors call this the Yu-Shiba-Rusinov (YSR) phase.

3. The Local Moment Phase (The Stormy Room)

• The Fan: Blasting at full volume.
• What happens: The noise is so loud that no particle can stick to the magnet, not even for a second. The "ghost tether" disappears entirely.
• The Result: The magnet is completely exposed and unscreened. It stands alone, screaming into the void. The crowd is too chaotic to form any kind of shield.

The Big Surprise: Time vs. Energy

In normal physics, we usually decide which state a system is in by looking at energy (which state is the "lowest" or most comfortable).

In this leaky, non-Hermitian world, time becomes just as important as energy.

• In the middle phase (the YSR phase), the "shielded" state might have lower energy, so it looks like the winner.
• BUT, because it is leaking energy, it dies out over time.
• The "unshielded" state might have higher energy, but it is more stable against the leak.

So, the system makes a phase transition not just because the energy changes, but because the rate of loss changes. It's like choosing between a delicious meal that spoils in an hour (low energy, high loss) and a dry cracker that lasts forever (higher energy, low loss). Depending on how hungry you are (energy) and how long you have to wait (time), your choice changes.

Why Does This Matter?

This isn't just a math puzzle. The authors suggest this could be tested in ultracold atom experiments (using lasers to trap atoms). By tuning the lasers to control how fast atoms are lost, scientists could watch the magnet switch from being shielded, to being "ghost-shielded," to being completely exposed.

In summary:
The paper shows that when you introduce "leaks" into a quantum system, you don't just get a messier version of the old physics. You get a brand new phase of matter where the battle between energy and time creates a temporary, unstable shield that eventually collapses, revealing a hidden complexity in how quantum systems behave in the real, imperfect world.

Technical Summary

1. Problem Statement

The paper investigates the behavior of the Kondo effect in an open quantum system where dissipation (energy loss) is significant. While the standard Kondo model describes the screening of a magnetic impurity by a cloud of conduction electrons in a closed (Hermitian) system, real-world quantum systems (such as ultracold atoms in optical lattices) are open and subject to inelastic scattering and particle loss.

Previous work (Nakagawa et al.) established that in a non-Hermitian Kondo model, a complex Kondo coupling $J = J_r + iJ_i$ (where the imaginary part arises from two-body loss rates) leads to two distinct phases:

1. Kondo Phase: The impurity is screened.
2. Local Moment (Unscreened) Phase: The impurity remains unscreened for large loss rates.

The authors identify a gap in this understanding: the renormalization group (RG) approach used in prior studies failed to detect an intermediate regime. The central problem is to determine if a new phase exists between the fully screened and fully unscreened regimes and to characterize its dynamical and energetic properties using exact methods.

2. Methodology

The authors employ a combination of Bethe Ansatz (BA) and perturbative calculations to solve the non-Hermitian Kondo model exactly.

• Model Hamiltonian: The system is described by a non-Hermitian Hamiltonian (Eq. 1) representing a 1D chain of itinerant fermions interacting with a spin-1/2 impurity at the origin. The interaction coupling $J$ is complex, with the imaginary part $J_i$ proportional to the two-body loss rate.
• Integrability: The model is shown to be integrable. The Bethe Ansatz equations (BAE) for the non-Hermitian case are derived as the analytical continuation of the Hermitian BAE to complex coupling constants.
• Scaling Limit: The analysis focuses on the scaling limit ($D \to \infty$, $c \to 0$) where universal behavior emerges. The system is characterized by two RG invariants:
  • $T_K$: The generalized Kondo temperature.
  • $\alpha$: A parameter measuring the deviation from Hermiticity, related to the ratio of the imaginary to real parts of the coupling ($J_i/J_r^2$).
• Spectral Analysis: Since the Hamiltonian is non-Hermitian, eigenvalues are complex ($E = \mathcal{R}(E) + i\mathcal{I}(E)$). The authors analyze both the real part (energy) and the imaginary part (decay rate/lifetime) to determine ground state stability and phase transitions.

3. Key Contributions

The primary contribution of this work is the discovery and characterization of a novel intermediate phase, termed the Yu-Shiba-Rusinov (YSR)-like phase (denoted as $\tilde{Y}SR$), which exists between the Kondo and Local Moment phases.

Key theoretical advances include:

• Identification of the Impurity String (IS): The authors find a new isolated solution to the Bethe Ansatz equations (the "Impurity String") that appears specifically in the regime $\pi/2 < \alpha < 3\pi/2$. This solution corresponds to a bound mode localized near the impurity.
• Dynamical Phase Transition: The paper demonstrates that the phase transition is driven not just by energetics (minimizing real energy) but by dissipation and time scales. The competition between the energy of the bound state and its finite lifetime creates a complex phase diagram.
• Refutation of Purely Energetic Arguments: The authors show that while the ground state energy suggests a transition at $\alpha = \pi$, the dynamical stability (lifetime) dictates that the transition from the screened to the unscreened state actually occurs at $\alpha = \pi/2$.

4. Results and Phase Diagram

The system exhibits three distinct phases based on the parameter $\alpha$ (increasing order of losses):

A. Kondo Phase ($0 < \alpha < \pi/2$)

• State: $|K\rangle$.
• Screening: The impurity is fully screened ($S=0$) by a cloud of spinons (conduction electrons).
• Spectrum: No bound modes exist. The density of states (DOS) shows a Kondo resonance peak that shifts from $E=0$ to $E=T_K$ as $\alpha$ approaches $\pi/2$.
• Stability: The Kondo state is dynamically stable against depopulation of the screening cloud.

B. $\tilde{Y}SR Phase ($\pi/2 < \alpha < 3\pi/2$)

This is the novel phase characterized by the presence of an Impurity String (IS) solution with energy $E_{IS} = E_b + i\Gamma_b$.

• Bound Mode Energy: $E_b = -T_K \sin \alpha$ and decay rate $\Gamma_b = T_K \cos \alpha$.
• Sub-phases:
  • Screened Bound State ($|B\rangle$) for $\pi/2 < \alpha \le \pi$:
    • The IS is occupied. The impurity is screened by a single-particle bound mode.
    • Energetics: The real energy is lower than the unscreened state.
    • Dynamics: The bound mode has a negative imaginary part ($\Gamma_b < 0$), meaning it has a finite lifetime. Over long times ($t \gg |\Gamma_b|^{-1}$), the bound state decays, and the system eventually evolves into the unscreened state.
  • Unscreened State ($|U\rangle$) for $\pi \le \alpha < 3\pi/2$:
    • The IS is unoccupied. The impurity is partially screened or unscreened ($S=1/2$).
    • Energetics: The real energy is lower than the bound state in this range.
• Phase Transition Point: The transition from the Kondo phase to the $\tilde{Y}SR$ phase occurs at $\alpha = \pi/2$. At this point, the bound mode emerges from the continuum.

C. Local Moment Phase ($\alpha > 3\pi/2$)

• State: $|U\rangle$.
• Screening: The impurity is completely unscreened ($S=1/2$).
• Spectrum: The bound mode solution (IS) vanishes (its energy becomes zero or non-physical in the thermodynamic limit). The system behaves like a free local moment.

5. Significance and Implications

• Dissipation-Driven Transitions: The work establishes that dissipation can induce phase transitions that do not exist in Hermitian systems. The transition at $\alpha = \pi/2$ is a "dynamical phase transition" where the interplay between energy minimization and decay rates determines the physical state.
• Universality: The authors argue that the $\tilde{Y}SR$ phase is a universal feature of non-Hermitian systems with impurities, appearing in spin chains, superconductors, and PT-symmetric models.
• Experimental Relevance: The model is directly applicable to ultracold alkaline-earth atoms (e.g., $^{173}$Yb) in optical lattices. The parameter $\alpha$ can be tuned experimentally by adjusting the optical frequency to control the rate of two-body inelastic losses.
• Observability: The paper suggests that the transition from the Kondo phase to the $\tilde{Y}SR$ phase could be probed via Scanning Tunneling Microscopy (STM) measurements of the impurity density of states, which would show the emergence of a delta-function peak (the bound mode) at specific energies.

In summary, the paper resolves a discrepancy in the non-Hermitian Kondo model by revealing a hidden intermediate phase where the impurity is screened by a decaying bound mode, fundamentally altering the understanding of how dissipation affects quantum impurity physics.