Quantum criticality beyond thermodynamic stability

This paper establishes that quantum criticality extends beyond thermodynamically stable systems by demonstrating that dynamically stable quadratic bosonic Hamiltonians exhibit critical behavior characterized by a unique quasiparticle vacuum and the closing of a spectral "Krein gap," which governs long-range correlations and entanglement scaling even in the absence of a ground state.

The Gist

The Big Picture: When "Stable" Doesn't Mean "Safe"

Imagine you are building a house of cards. In the world of standard physics, we usually only care about houses that are thermodynamically stable. This means the house has a solid floor; it won't collapse into a black hole, and it has a clear "lowest point" (the ground state) where the cards naturally want to rest.

For decades, physicists have studied what happens when you push these stable houses to their breaking point. This is called quantum criticality. It's like finding the exact moment a house of cards starts to wobble so much that the cards on the top are connected to the cards on the bottom, even if they are far apart. This "long-range connection" is a special state of matter.

The Problem:
The authors of this paper point out that nature has many "houses of cards" that don't have a floor. They are thermodynamically unstable. If you try to find the "lowest point" for these systems, you fall forever. Because they fall forever, traditional physics says they don't exist or can't be studied.

However, the authors argue that many of these unstable systems are actually dynamically stable.

• Thermodynamic Stability: "Does the house have a floor?" (No, it falls forever).
• Dynamical Stability: "If I nudge the cards, do they fly apart and explode, or do they just wobble in a controlled way?" (They wobble in a controlled way).

The paper asks: Can these "falling but wobbling" systems still have that special "long-range connection" (criticality)?

The New Tool: The "Krein Gap"

To answer this, the authors invented a new ruler called the Krein Gap.

Think of a standard quantum system like a set of stairs. The "energy gap" is the distance between the bottom step and the next one up. If the gap closes (the steps merge), the system becomes critical.

But for these unstable systems, the "stairs" are weird. Some steps go up, and some go down into a hole. The authors realized that instead of measuring the distance from the bottom, we should measure the distance between the upward-moving steps and the downward-moving steps.

• The Krein Gap: This is the smallest distance between a "particle" (moving up) and a "hole" (moving down).
• The Rule: As long as this gap is open (there is space between them), the system is calm, and connections between distant parts die out quickly (like a whisper that fades away).
• The Critical Moment: When the gap closes (the upward and downward steps touch), the system becomes critical. Suddenly, a whisper at one end of the room can be heard clearly at the other end.

The Key Character: The "Quasiparticle Vacuum"

In normal physics, we study the Ground State (the lowest energy state). But for these unstable systems, the Ground State doesn't exist.

The authors introduce a new character: the Quasiparticle Vacuum (QPV).

• Analogy: Imagine a calm lake. In a normal system, the lake has a bottom (the ground state). In an unstable system, the lake is infinite and has no bottom. However, the water can still be perfectly flat and calm.
• The QPV is this "perfectly flat water." It is the state where all the waves (quasiparticles) are gone.
• The paper proves that even without a "bottom," this flat water is a unique, well-defined state. And it is this state that becomes critical when the Krein Gap closes.

The Two Types of "Crashes"

When the gap closes, the system hits a "spectral singularity." The authors found two distinct ways this can happen, like two different types of traffic accidents:

1. The Exceptional Point (EP):

   • Analogy: Imagine two cars driving toward each other on a single-lane road. They merge into one car.
   • What happens: The system loses stability in a very specific way. The connections become long-range, and the system behaves like a standard critical point. It's a "clean" crash.

2. The Krein Collision (KC):

   • Analogy: Imagine a four-way intersection where two roads cross. You can approach the center from the North, South, East, or West.
   • What happens: This is a multicritical point. The behavior of the system depends entirely on how you approach the crash. If you come from the North, the connections might grow huge. If you come from the East, they might vanish. It's a messy, complex crash where the rules change based on your path.

The Main Findings in Plain English

1. Stability is about movement, not energy: You don't need a system to have a "lowest energy" to study its critical behavior. You just need it to be dynamically stable (not exploding).
2. The Gap is the switch: The "Krein Gap" is the on/off switch for long-range connections. If the gap is open, connections are short. If the gap closes, connections stretch across the whole system.
3. Thermodynamics is a red herring: You can take a system that is thermodynamically unstable (no floor) and tweak it so it falls forever, but as long as the "Krein Gap" stays open, the connections between particles remain short and normal. The system only becomes "critical" when the gap closes, regardless of whether it has a floor or not.
4. Entanglement follows the rules: Even in these unstable systems, the amount of "quantum entanglement" (a spooky connection between particles) follows the same rules as normal systems. It scales with the size of the gap. If the gap gets tiny, the entanglement gets huge.

Why This Matters (According to the Paper)

The authors conclude that we have been looking at quantum criticality through the wrong lens. We were only looking at systems with a "floor" (thermodynamically stable).

This paper opens the door to studying a whole new class of systems found in:

• Photonics: Systems involving light.
• Opto-mechanics: Systems where light moves mechanical parts.
• Cavity-QED: Atoms trapped in mirrors.
• Magnonics: Systems involving magnetic waves.

Many of these real-world systems are "unstable" in the traditional sense (they pump energy in and out), but they are dynamically stable. This framework allows physicists to finally apply the powerful tools of "criticality" to these messy, real-world systems, treating them with the same mathematical rigor as the perfect, theoretical systems of the past.

Technical Summary

Technical Summary: Quantum Criticality Beyond Thermodynamic Stability

Problem Statement
Traditional quantum criticality in closed many-body systems is defined by structural changes in the many-body ground state (GS) as control parameters are tuned, typically associated with the closing of the many-body energy gap. This framework relies on the assumption of thermodynamic stability (the existence of a GS bounded from below). However, a significant class of physically relevant models, specifically quadratic bosonic Hamiltonians (QBHs), often lack a ground state due to being unbounded from below or above. Examples include magnon excitations in non-equilibrium magnetic systems, linearized cavity QED Hamiltonians, and the bosonic Kitaev chain. While these systems may be dynamically stable (exhibiting bounded time evolution), standard criticality theory based on the GS is inapplicable. The paper addresses the gap in understanding critical phenomena in these dynamically stable but thermodynamically unstable QBHs.

Methodology
The authors develop a theoretical framework for translationally invariant QBHs with finite-range couplings, shifting the focus from the ground state to the Quasiparticle Vacuum (QPV). The QPV is defined as the state annihilated by all quasiparticle normal modes of the Hamiltonian. For thermodynamically stable QBHs, the QPV coincides with the GS; for unstable ones, it remains a well-defined, unique, zero-mean Gaussian state.

Key methodological components include:

1. Dynamical Matrix Analysis: Utilizing the Bloch dynamical matrix $g(k)$, which is pseudo-Hermitian with respect to an indefinite metric $\tau_3$ arising from bosonic commutation relations.
2. Krein Theory: Applying Krein stability theory to characterize the stability phase boundaries. The authors distinguish between Exceptional Points (EPs), where eigenvectors merge and the matrix becomes non-diagonalizable, and Krein Collisions (KCs), where eigenvectors remain distinct but possess opposite Krein signatures (leading to degeneracy in the spectrum of the dynamical matrix).
3. Definition of the Krein Gap: Introducing a new spectral gap, the Krein gap ($\Delta_{\text{Krein}}$), defined as the minimal spectral separation between creation and annihilation operators (or equivalently, between particle and hole bands).
4. Correlation and Entanglement Analysis: Deriving analytic expressions for two-point correlation functions and the covariance matrix (CM) in the QPV. The authors utilize Fourier analysis to link the analyticity of the momentum-space CM to the exponential boundedness of real-space correlations. They also employ information-theoretic tools, specifically the Entanglement Entropy (EE) and the Quantum Metric Tensor (QMT), adapted for pseudo-Hermitian systems.

Key Contributions and Results

1. Generalized Criticality via the QPV:
   The paper establishes that criticality is a meaningful concept for the entire class of dynamically stable QBHs, regardless of thermodynamic stability. The relevant state for such systems is the QPV. The authors prove that a unique, translationally invariant QPV exists if and only if the indirect Krein gap is positive. If the direct Krein gap is positive, a unique translationally invariant QPV exists.

2. The Krein Gap as the Critical Indicator:
   The authors prove that for dynamically stable QBHs, the two-point spatial correlation functions in the QPV are exponentially bounded if and only if the Krein gap is positive ($\Delta_{\text{Krein}} > 0$). Consequently, long-range correlations (criticality) can only emerge when the Krein gap closes ($\Delta_{\text{Krein}} = 0$). This occurs precisely at the boundaries of dynamical stability, which are characterized by either EPs or KCs.

   • Thermodynamic vs. Dynamical Stability: The paper demonstrates that thermodynamic stability can be tuned independently of dynamical stability (by adjusting parameters that couple position and momentum quadratures). Crucially, the correlation functions in the QPV are "blind" to the loss of thermodynamic stability unless it coincides with a loss of dynamical stability. Thus, the critical phase boundary is determined solely by dynamical stability.

3. Distinct Critical Behaviors at EPs and KCs:

   • Exceptional Points (EPs): When the Krein gap closes at an EP, the system exhibits standard quantum critical behavior. The correlation length diverges with a critical exponent $\nu = 1/2$ and dynamic exponent $z=1$, leading to algebraic decay of correlations.
   • Krein Collisions (KCs): When the gap closes at a KC (an intersection of EP lines), the system exhibits multicritical behavior. The scaling of correlations and the divergence of the correlation length become highly dependent on the path of approach in parameter space. For instance, in a double harmonic chain model, the dynamic exponent $z$ varies continuously between 1 and 2 depending on the approach path. At the KC itself, real-space correlations can be trivially short-ranged (local), yet the system remains singular.

4. Information-Theoretic Indicators:

   • Entanglement Entropy (EE): The EE in the QPV obeys an area law for $\Delta_{\text{Krein}} > 0$ and scales inversely with the Krein gap as the gap closes, consistent with results for thermodynamically stable systems. However, near a KC, the EE exhibits path-dependent behavior, diverging along some paths and vanishing along others.
   • Quantum Metric Tensor (QMT): The pseudo-Hermitian QMT diverges at both EPs and KCs. Unlike correlation functions, which may appear trivial at the KC point itself, the QMT captures the singular nature of the KC directly, identifying it as a genuine multicritical point.

Significance and Claims
The paper claims to provide a unified framework for investigating critical phenomena in dynamically stable QBHs, extending the scope of criticality theory beyond the constraints of thermodynamic stability.

• Unification: It recovers known results for thermodynamically stable QBHs as corollaries, showing that the Krein gap generalizes the standard many-body energy gap (and the "symmetrized gap" for non-symmetric QBHs).
• Physical Relevance: The framework applies to models in photonics, opto-mechanics, cavity-QED, and magnonics, where thermodynamic instability is common but dynamical stability is preserved.
• Topological Implications: The authors note that dropping thermodynamic stability allows for the existence of topologically mandated boundary zero modes, which are forbidden in thermodynamically stable QBHs by no-go theorems. This aligns the spectral flow of bosonic systems more closely with fermionic systems.
• Computational Complexity: The work suggests a link between dynamical stability and the computational complexity of simulating quadratic bosonic systems, hinting that the onset of dynamical instability may correspond to a transition in complexity classes.

The authors conclude that the boundary of dynamical stability and the onset of criticality (or multicriticality) are one and the same for QBHs, with the Krein gap serving as the fundamental metric for proximity to these critical points.