Non-Hermitian Numerical Renormalization Group: Solution of the non-Hermitian Kondo model

Imagine you have a tiny, lonely magnet (an "impurity") sitting in a sea of swirling, dancing electrons. In the real world, this setup is called the Kondo effect. Usually, the electrons dance around the magnet and eventually "hug" it, forming a tight, quiet pair that cancels out the magnet's noise. This is a classic physics problem that scientists have studied for decades.

But what happens if the world isn't perfectly closed? What if the electrons are leaking out, or the magnet is losing energy to its surroundings? In physics, we call these "open" systems. To describe them, we use Non-Hermitian math. Think of "Hermitian" as a perfect, closed box where energy is conserved, and "Non-Hermitian" as a leaky bucket where things can disappear or change in weird, complex ways.

This paper is about building a new, super-powerful tool to solve these "leaky bucket" problems, specifically for the Kondo effect.

The Problem: The Old Tools Broke

For a long time, physicists had a brilliant tool called the Numerical Renormalization Group (NRG). It's like a high-powered microscope that lets you zoom in on the Kondo effect, step-by-step, from the big picture down to the tiniest details. It worked perfectly for closed systems.

However, when scientists tried to apply this tool to "leaky" (Non-Hermitian) systems, it broke. Why?

Complex Numbers: In leaky systems, the energy levels aren't just simple numbers like 5 or 10; they are "complex" (involving imaginary numbers). It's like trying to sort a pile of apples by weight, but some apples are also glowing blue or red. You don't know how to sort them anymore.
The "Leak": Standard math assumes that if you have a list of states, you keep the ones with the lowest energy. But when energy is complex, "lowest" becomes a confusing concept. Do you pick the one with the lowest real number? The lowest imaginary number? The smallest total size?

The Solution: A New "Leaky" Microscope

The authors (Phillip Burke and Andrew Mitchell) invented a new version of the microscope called Non-Hermitian NRG (NH-NRG).

Here is how they fixed the broken parts using some clever tricks:

The Sorting Rule: They realized that even though the energy numbers are complex, the "real" part of the number (the part that acts like normal energy) is still the most important for stability. So, their new rule is: "Sort the states by their real energy, and ignore the imaginary glow for the sorting." This simple rule kept the calculation from crashing.
The Two-Sided Mirror: In normal physics, a mirror reflects the same image back. In this leaky world, the "left" reflection and the "right" reflection are different. The authors had to build a system that tracks both sides separately to make sure the math stays balanced.

What They Discovered: A New World of Physics

Once they turned on their new microscope, they saw things no one had seen before.

1. The "Re-entrant" Dance (The Kondo Effect comes back!)
In the old theories, if you increased the "leakiness" (dissipation) too much, the electrons would stop hugging the magnet. The magnet would stay lonely and noisy forever.

The Discovery: The authors found that if you make the leak really strong, the electrons suddenly decide to hug the magnet again! It's like a shy person who runs away when you look at them, but if you stare at them intensely enough, they finally come back and hold your hand. This is called re-entrant Kondo behavior.

2. The "Ghost" Phase (Complex Strong Coupling)
They also looked at a special type of Kondo model (the "pseudogap" model) where the electrons are sparse.

The Discovery: They found a completely new phase of matter. In this phase, the system settles into a stable state, but it's fundamentally different from anything in our normal world. The energy levels remain "complex" (glowing) forever; they never settle down to normal numbers. They call this the "Complex Strong Coupling" phase. It's a stable state that only exists because the system is leaking energy.

Why This Matters

This paper is a big deal because:

It's a Universal Tool: Their new code (which they made free for everyone to use) can solve these problems for any type of material, not just the simple ones.
It Connects Theory to Reality: These "leaky" systems aren't just math games. They describe real-world things like ultracold atoms in labs that are losing particles, or quantum computers that are losing information.
It Solves the Unsolvable: Before this, scientists could only guess what happened in these systems using approximations. Now, they have a precise, non-perturbative solution.

The Bottom Line

The authors built a new mathematical engine that can drive through the fog of "leaky" quantum systems. They discovered that when you push these systems hard enough, they don't just break; they transform into entirely new, stable forms of matter that were previously invisible to science. It's like discovering a new continent on a map that everyone thought was just empty ocean.

Here is a detailed technical summary of the paper "Non-Hermitian Numerical Renormalization Group: Solution of the non-Hermitian Kondo model" by Phillip C. Burke and Andrew K. Mitchell.

1. Problem Statement

The paper addresses the lack of non-perturbative numerical methods for studying strongly correlated non-Hermitian (NH) quantum systems. While single-particle NH physics (e.g., exceptional points, PT-symmetry) is well-explored, many-body phenomena in open quantum systems remain poorly understood due to the limitations of existing methods:

Perturbative scaling and Bethe ansatz are restricted to weak coupling regimes or specific integrable models (e.g., linear dispersion).
Exact diagonalization is limited by the exponential growth of the Hilbert space.
The Non-Hermitian Kondo model, a paradigm for strong correlation in open systems (relevant to ultracold atoms with inelastic scattering), has yielded conflicting results regarding its phase diagram and fixed points depending on the method used.

The authors aim to develop a robust, non-perturbative solver capable of handling arbitrary bath density of states (DOS) and strong coupling to resolve these discrepancies.

2. Methodology: Non-Hermitian Numerical Renormalization Group (NH-NRG)

The authors generalize Wilson's Numerical Renormalization Group (NRG) to handle non-Hermitian Hamiltonians. The key technical adaptations include:

Logarithmic Discretization & Wilson Chain: The conduction electron bath is discretized logarithmically and mapped to a 1D Wilson chain, similar to standard NRG.
Bi-orthonormal Basis: Since NH Hamiltonians ( $\hat{H} \neq \hat{H}^\dagger$ $\hat{H} \neq = \hat{H}^{†}$ ) possess distinct left ( $\langle L|$ $⟨ L ∣$ ) and right ( $|R\rangle$ $∣ R ⟩$ ) eigenvectors with complex eigenvalues, the algorithm utilizes a bi-orthonormal basis ( $\langle L_i | R_j \rangle = \delta_{ij}$ $⟨ L_{i} ∣ R_{j} ⟩ = δ_{ij}$ ).
- The Hamiltonian is diagonalized to obtain complex eigenvalues $E_N$ and corresponding left/right eigenvectors.
- Symmetries (total charge $Q$ and spin projection $S_z$ ) are used to block-diagonalize the Hamiltonian to handle degeneracies.
Iterative Diagonalization: The system is built iteratively by adding Wilson chain sites. At each step $N$ , the Hamiltonian $\hat{H}_N$ is constructed from $\hat{H}_{N-1}$ and the new site.
Fock Space Truncation (Crucial Innovation): In Hermitian NRG, states with the lowest energy are retained. In the NH case, eigenvalues are complex. The authors determined that truncating based on the lowest real part of the eigenvalues ( $\text{Re}(E_N)$ ) yields the most stable and accurate results. This defines the "ground state" for the iteration.
Matrix Element Propagation: The algorithm explicitly tracks matrix elements for both creation and annihilation operators between left and right states, as they are not simple Hermitian conjugates in the NH context.
Validation: The method was validated against exact diagonalization for the non-interacting NH Resonant Level Model (RLM), showing perfect agreement.

3. Key Contributions

Development of NH-NRG: A fully non-perturbative numerical framework applicable to a wide range of quantum impurity models (Kondo, Anderson) with arbitrary bath DOS, overcoming the limitations of Bethe ansatz and perturbative RG.
Resolution of Phase Diagram Discrepancies: The method clarifies conflicting previous results regarding the NH Kondo model's phase diagram, particularly at intermediate-to-strong coupling.
Discovery of Novel Phases: Identification of a "re-entrant" Kondo phase and a completely new stable fixed point in the pseudogap regime that is intrinsically non-Hermitian.
Open Source Release: The authors provide their NH-NRG code as open-source software to facilitate future research in NH many-body physics.

4. Key Results

A. Metallic Non-Hermitian Kondo Model

Phase Diagram: The model exhibits two phases: a Strong Coupling (SC) phase (Kondo screened) and a Local Moment (LM) phase (unscreened).
Re-entrant Behavior: Contrary to earlier perturbative predictions suggesting a permanent transition to the LM phase with increasing dissipation ( $J_I$ ), NH-NRG reveals re-entrant Kondo physics. As dissipation increases, the system transitions from SC $\to$ LM $\to$ SC.
Termination of LM Phase: For sufficiently large real coupling ( $J_R \gtrsim 0.55$ ), the LM phase disappears entirely; the Kondo effect dominates regardless of dissipation strength.
RG Flow & Fixed Points:
- In the SC and LM phases, the RG flow converges to Hermitian fixed points (imaginary parts of eigenvalues vanish at large $N$ ), indicating "emergent Hermiticity" in the low-energy limit.
- At the critical point, the system flows to a non-Hermitian critical fixed point where the imaginary part of eigenvalues diverges exponentially.

B. Non-Hermitian Pseudogap Kondo Model

Shifted Critical Dimension: In the Hermitian pseudogap model, a lower-critical dimension exists at $r_c = 0.5$ (where $r$ is the power-law exponent of the DOS). Above this, Kondo screening is impossible.
Novel "Complex Strong Coupling" (CSC) Phase: With finite dissipation ( $J_I > 0$ ), the critical dimension shifts to higher values ( $r_c > 0.5$ ).
Intrinsically Non-Hermitian Fixed Point: In the region where the Hermitian model would have no screening ( $r > 0.5$ ), the NH model stabilizes a new phase with a Complex Strong Coupling (CSC) fixed point. This fixed point is fundamentally non-Hermitian, characterized by a persistent complex eigenspectrum where $\text{Im}(E_N)$ does not decay to zero.

C. Non-Hermitian Anderson Impurity Model (AIM)

Mapping Validation: The authors confirm the non-perturbative mapping between the NH Anderson model and the NH Kondo model. The phase diagrams in the complex hybridization plane ( $V$ ) mirror those of the Kondo model, including the re-entrant behavior and the termination of the LM phase.

5. Significance

Methodological Breakthrough: NH-NRG provides the first reliable tool to study strong correlations in open quantum systems beyond weak coupling and specific symmetries. It bridges the gap between single-particle NH physics and complex many-body phenomena.
New Physics: The discovery of the CSC fixed point demonstrates that strong dissipation can stabilize entirely new phases of matter that do not exist in Hermitian systems, challenging the notion that dissipation merely destroys coherence.
Broad Applicability: The method is applicable to diverse systems, including multi-impurity models, unconventional materials (e.g., graphene, topological insulators), and can be integrated into Dynamical Mean-Field Theory (DMFT) to study non-Hermitian lattice models.
Resolution of Controversies: The work resolves contradictions in the literature regarding the NH Kondo phase diagram, establishing that the "unscreened" phase is bounded and that re-entrant screening is a robust feature.

In summary, this paper establishes NH-NRG as the "gold standard" for solving non-Hermitian quantum impurity problems, revealing a rich and previously inaccessible landscape of non-perturbative many-body physics driven by dissipation.