Beyond Hermiticity: A Dissipative Spin Liquid Model Unveiled

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Author: Denis Avetisyan

Researchers have discovered an exactly solvable model demonstrating how non-Hermitian dynamics can give rise to exotic quantum phenomena in dissipative spin liquids.

The study demonstrates that the Liouvillian spectrum-analyzed for systems with linear sizes ranging from 4 to 64-reveals a phase transition point, $\gamma_{PT}$, which scales inversely with system size ($1/L$), while a critical value, $\gamma^*$, remains approximately constant at $3J/4$, and the fraction of broken $\mathcal{PT}$ symmetry smoothly transitions from 0 to 1 as the parameter $\gamma$ crosses these thresholds.
The study demonstrates that the Liouvillian spectrum-analyzed for systems with linear sizes ranging from 4 to 64-reveals a phase transition point, $\gamma_{PT}$, which scales inversely with system size ($1/L$), while a critical value, $\gamma^*$, remains approximately constant at $3J/4$, and the fraction of broken $\mathcal{PT}$ symmetry smoothly transitions from 0 to 1 as the parameter $\gamma$ crosses these thresholds.

This work reveals a rich interplay between symmetry, exceptional points, and Majorana fermions within a Liouvillian framework for the Yao-Lee spin-orbital model.

Understanding relaxation dynamics in open quantum systems remains a central challenge, yet analytical solutions are rare. This work introduces an exactly solvable model-the Dissipative Yao-Lee Spin-Orbital Model: Exact Solvability and $\mathcal{PT}$ Symmetry Breaking-that maps Liouvillian dynamics to a non-Hermitian fermionic system. We demonstrate the emergence of an exponentially large manifold of non-equilibrium steady states alongside an exceptional ring in the Liouvillian spectrum, revealing a characteristic parity-time symmetry breaking transition governing relaxation behavior. Does this framework offer a pathway toward realizing robust dissipative spin liquids and exploring the broader implications of Liouvillian spectral singularities in open quantum systems?

Beyond Equilibrium: Dissipative Quantum Systems

Conventional quantum mechanics, while remarkably successful, often operates under the simplification of isolated, or closed, systems – entities shielded from any interaction with their surroundings. However, this idealized scenario rarely reflects reality; virtually all quantum systems are, in fact, open, constantly exchanging energy and information with their environment. This interaction manifests as dissipation – the loss of energy – and decoherence – the destruction of quantum superposition and entanglement. Consequently, the behavior of real-world quantum systems deviates significantly from predictions based on closed-system assumptions, leading to emergent phenomena absent in the purely theoretical framework. Ignoring these environmental influences, so prevalent in nature, limits the capacity to accurately describe and ultimately harness the full potential of quantum mechanics, necessitating a shift towards a more comprehensive understanding of open quantum systems and their unique characteristics.

Unlike the isolated quantum systems traditionally considered, most physical systems are open, meaning they continually exchange energy and information with their surroundings. This constant interaction fundamentally alters their behavior, moving them away from predictable, equilibrium states and into the realm of dissipation and decoherence. Instead of evolving according to the simple Schrödinger equation, open quantum systems are described by master equations or other non-Hermitian Hamiltonians, accounting for the influence of the environment. This leads to phenomena not observed in closed systems, such as spontaneous emission, energy transfer, and the emergence of complex steady states – persistent, non-equilibrium configurations that are sustained by the ongoing energy flow. Consequently, understanding these open systems requires a departure from standard quantum mechanical approaches and the development of new theoretical tools to capture the richness of their dynamics.

The study of systems far from thermodynamic equilibrium demands a departure from traditional quantum mechanical approaches, which largely presume closed systems isolated from environmental influence. Existing analytical techniques, honed for equilibrium scenarios, often prove inadequate when addressing the continuous energy exchange and resulting decoherence inherent in open quantum systems. Consequently, researchers are developing new theoretical frameworks – such as non-equilibrium Green’s functions, quantum trajectory methods, and generalized master equations – alongside computational tools capable of handling the complex dynamics of these dissipative systems. These advancements aren’t merely mathematical exercises; they represent a fundamental shift in how physicists conceptualize and model the behavior of matter, paving the way for understanding phenomena ranging from quantum heat engines to the emergence of complex order in biological systems and offering potential pathways for novel quantum technologies.

Researchers investigated a spin-orbital model constructed on the honeycomb lattice to provide a concrete setting for examining how energy dissipation affects quantum systems. This model, representing interacting quantum spins and orbital degrees of freedom, allows for the simulation of open systems constantly exchanging energy with their surroundings. Through detailed analysis, the study reveals that this system doesn’t simply relax to a single, predictable state; instead, it hosts an astonishingly large-specifically, an exponential-number of stable, non-equilibrium steady states. This proliferation of states arises from the interplay between dissipation and the unique topological properties of the honeycomb lattice, suggesting a rich landscape of potential quantum phenomena far from thermal equilibrium and offering possibilities for novel quantum technologies based on robust, non-equilibrium configurations.

A non-Hermitian bilayer Hamiltonian can model an open system via a mapping to a doubled Hilbert space, as illustrated by a hexagonal plaquette with x, y, and z bonds within the honeycomb lattice’s Brillouin zone.
A non-Hermitian bilayer Hamiltonian can model an open system via a mapping to a doubled Hilbert space, as illustrated by a hexagonal plaquette with x, y, and z bonds within the honeycomb lattice’s Brillouin zone.

Modeling Dissipation: From Master Equation to Non-Hermitian Hamiltonian

The Lindblad master equation is utilized to model the open quantum system dynamics of the spin-orbital model, specifically addressing dissipation arising from interactions with an external environment. This equation, a first-order differential equation for the density matrix $\rho$, accounts for both coherent and incoherent processes. It describes the time evolution of $\rho$ as $d\rho/dt = -\imath [H, \rho] + \mathcal{L}[\rho]$, where $H$ is the system Hamiltonian and $\mathcal{L}$ is the Lindblad superoperator. The Lindblad superoperator incorporates dissipation through a sum of jump operators ($L_i$) and their corresponding rates ($\Gamma_i$): $\mathcal{L}[\rho] = \sum_i \frac{1}{2} (L_i \rho L_i^\dagger – L_i^\dagger \rho L_i)$. This formulation ensures the complete positivity of the density matrix, a crucial requirement for physically realistic descriptions of open quantum systems.

The Liouvillian super-operator, which dictates the time evolution of the density matrix $\rho$ through the equation $\frac{d\rho}{dt} = \mathcal{L}\rho$, can be represented as a non-Hermitian Hamiltonian. This mapping is achieved by introducing a doubled Hilbert space, where each original state $|n\rangle$ is paired with a corresponding state $|n\rangle’$, and defining operators that act on this extended space. Specifically, the Liouvillian $\mathcal{L}$ is then isomorphic to a non-Hermitian Hamiltonian $\hat{H}_{NH}$ such that $\mathcal{L}\rho = -i[\hat{H}_{NH}, \rho]$. The non-Hermiticity arises because the Hamiltonian does not commute with its adjoint, reflecting the open nature of the system and the presence of dissipation or decoherence. This representation allows the use of standard Hamiltonian techniques to analyze the dynamics of the density matrix, particularly in determining eigenvalues that correspond to decay rates and relaxation times.

Mapping the Liouvillian super-operator to a non-Hermitian Hamiltonian necessitates a doubling of the Hilbert space. This is because the Liouvillian, acting on density matrices, describes the evolution of a system’s state, and its representation as a Hamiltonian requires an extended space to accommodate both the original states and their corresponding decay rates. Specifically, if the original Hilbert space is spanned by states $|n\rangle$, the extended space includes both $|n\rangle$ and $|n\rangle’$, where the primed states represent the decay pathways. This doubling allows for the treatment of dissipation as effective hopping between these doubled states, simplifying analytical calculations of system dynamics and enabling the identification of key parameters like the Liouvillian gap.

The Liouvillian gap, derived from the non-Hermitian Hamiltonian, represents the smallest eigenvalue of the Liouvillian super-operator and directly quantifies the slowest relaxation rate within the system. This gap dictates the long-time behavior of the density matrix, establishing the timescale over which the system returns to equilibrium following a perturbation. Specifically, the exponential decay of all excited state populations is governed by the Liouvillian gap; a smaller gap indicates slower relaxation and longer coherence times. Determining this gap is crucial for understanding the stability and dynamic properties of the open quantum system described by the spin-orbital model, as it provides a quantitative measure of dissipation strength and the efficiency of energy transfer to the environment.

Revealing Symmetries: Majorana Fermions and Complex Operators

Majorana decomposition is a mathematical technique employed to rewrite the Hamiltonian, $H$, of a fermionic system in terms of complex fermionic operators. This involves expressing the original fermionic operators in terms of linear combinations of Majorana operators, which are their own antiparticles. Specifically, a standard fermionic operator, $c_k$, can be represented as $c_k = \frac{1}{2}(d_k + i d_k^)$, where $d_k$ and $d_k^$ are Majorana operators. This transformation effectively replaces spinful electrons with spinless, complex fermions, reducing the number of independent fermionic degrees of freedom and simplifying the mathematical form of the Hamiltonian. The resulting Hamiltonian, expressed in terms of Majorana operators, facilitates the identification of symmetries and the analytical treatment of complex quantum systems.

Employing Majorana decomposition to recast the Hamiltonian in terms of complex fermionic operators reduces computational complexity by eliminating redundant degrees of freedom and facilitating algebraic simplification. Specifically, the transformation allows for the separation of operators into Hermitian and anti-Hermitian components, enabling the identification of conserved quantities and facilitating the construction of symmetry operators. This simplification is particularly effective in systems exhibiting parity-time ($\mathcal{PT}$) symmetry, where non-Hermitian Hamiltonians can still possess real eigenvalues under specific conditions, and the decomposition reveals the underlying $\mathcal{PT}$-symmetric structure. The resulting form of the Hamiltonian exposes hidden relationships between different parts of the system, making analytical calculations of key physical properties more tractable and revealing symmetries not readily apparent in the original formulation.

Employing a complex fermionic representation facilitates the analytical determination of system eigenstates and energy spectra by transforming the original Hamiltonian into a form more amenable to mathematical treatment. Specifically, this representation allows for the direct calculation of energy eigenvalues and corresponding eigenvectors without reliance on numerical approximations, provided the Hamiltonian exhibits sufficient symmetry. The resulting eigenstates are expressed as superpositions of complex fermionic operators, enabling precise prediction of observable quantities and a detailed characterization of the system’s quantum behavior. This analytical approach is particularly valuable for understanding the influence of specific Hamiltonian parameters on the energy spectrum and identifying potential topological phases or quantum critical points.

Analysis of the Liouvillian spectrum demonstrates the presence of a ring of exceptional points. These points signify a breakdown of $𝒫𝒯$-symmetry within the system, a condition where the symmetry between space and time reversal is no longer preserved. Exceptional points are characterized by coalescence of eigenvalues and loss of unitarity, resulting in non-Hermitian behavior. The observation of a ring structure indicates a specific topological arrangement of these symmetry-breaking points within the parameter space defining the Liouvillian operator, which has implications for the system’s stability and dynamics.

The model exhibits strong symmetries defined by hexagonal plaquettes and weaker symmetries represented by vertical square plaquettes.
The model exhibits strong symmetries defined by hexagonal plaquettes and weaker symmetries represented by vertical square plaquettes.

Non-Equilibrium Signatures: Exceptional Points and Steady State Manifolds

The system’s dynamics, governed by a non-Hermitian Hamiltonian, give rise to exceptional points – singularities in momentum space where the usual notion of eigenvalues breaks down. Unlike traditional bifurcations, exceptional points aren’t simply points of instability; they represent a coalescence of eigenstates, meaning that distinct initial conditions can evolve toward the same final state. This sensitivity to initial conditions and external perturbations is dramatically amplified near these points, as even infinitesimal changes can drastically alter the system’s trajectory. Consequently, the presence of exceptional points suggests a fundamentally different type of behavior, moving beyond the predictable responses associated with Hermitian systems and opening possibilities for novel control mechanisms and enhanced sensing capabilities, as the system becomes acutely responsive to external influences. The mathematical description involves eigenvalues colliding and losing their distinctness, represented by a vanishingly small separation between energy levels at the exceptional point, signaling a qualitative shift in the system’s stability and response.

Exceptional points represent a fundamental departure from conventional quantum mechanics, where eigenvalues – the solutions defining a system’s allowed energies – are typically distinct. At these points, however, eigenvalues coalesce, and the corresponding eigenvectors become indistinguishable. This confluence isn’t merely a mathematical curiosity; it signifies a qualitative change in the system’s behavior. The system becomes extraordinarily sensitive to even infinitesimal perturbations, meaning a tiny change in external conditions can trigger a dramatic shift in its properties. This enhanced sensitivity arises because the usual protection offered by distinct energy levels is lost, and the system essentially exists at a ‘bifurcation’ point in its parameter space. Consequently, the presence of exceptional points suggests potential applications in areas like sensing and switching, where precise control and responsiveness are paramount, though it also implies a vulnerability to noise and instability that must be carefully considered.

The system’s inherent dissipation doesn’t drive it towards a single, stable equilibrium, but rather supports an unexpectedly rich landscape of non-equilibrium steady states. This isn’t a mere handful of possibilities; the analysis demonstrates the existence of an exponentially large manifold of such states, meaning the number of stable configurations grows dramatically with even minor increases in system complexity. These steady states aren’t fleeting moments of balance, but robust attractors sustained by the continuous flow of energy through the system. The proliferation of these states suggests a remarkable capacity for the system to maintain diverse configurations despite ongoing energy loss, hinting at complex internal dynamics and a surprising resilience against external perturbations. This manifold represents a fundamentally different kind of stability than typically observed in conservative systems, one built not on minimizing energy, but on balancing dissipation and internal driving forces.

The exponentially large manifold of non-equilibrium steady states isn’t a random assortment, but rather a direct consequence of the system’s inherent symmetries and the unique features of exceptional points within its momentum space. These exceptional points, where eigenvalues and eigenvectors coalesce, act as organizing centers, dictating the dimensionality and topology of the steady state manifold. Specifically, each symmetry of the system effectively protects a corresponding subspace within this manifold, ensuring the persistence of certain steady states even under perturbations. The proximity of an exceptional point to a given steady state enhances its sensitivity, but also fundamentally shapes its properties, influencing the system’s response to external stimuli and governing the pathways between different non-equilibrium configurations. Consequently, understanding the interplay between symmetry, exceptional points, and the resulting steady state manifold is crucial for predicting and controlling the behavior of this non-Hermitian system, revealing a rich landscape of possibilities beyond traditional equilibrium physics.

The energy gap, characterized by its real and imaginary components, reveals exceptional points forming a ring in momentum space under parameters γ = 0.4 and J = 1.
The energy gap, characterized by its real and imaginary components, reveals exceptional points forming a ring in momentum space under parameters γ = 0.4 and J = 1.

The pursuit of exact solvability, as demonstrated in this work on the dissipative Yao-Lee spin-orbital model, echoes a fundamental principle of mathematical physics. The model’s ability to reveal the interplay between symmetry and non-Hermitian dynamics highlights the importance of rigorous mathematical frameworks in understanding complex physical systems. As Albert Einstein once stated, “God does not play dice with the universe.” This sentiment resonates with the desire for deterministic, provable solutions, as opposed to relying on probabilistic approximations. The identification of exceptional points and the associated symmetry breaking within the Liouvillian spectrum underscores the necessity for a consistent and predictable foundation, a boundary where the mathematics, and therefore the physics, remains elegant and unyielding.

Beyond the Dissipation

The exact solvability demonstrated by this work is, of course, a temporary reprieve. Let N approach infinity – what remains invariant? The model’s reliance on a specific dissipative mechanism, while mathematically tractable, begs the question of robustness. Does the observed symmetry breaking, and the attendant emergence of exceptional points, hold under more general, physically realistic dissipation? The proliferation of solvable models often obscures the fact that nature rarely conforms to convenient Hamiltonians.

A crucial direction lies in extending this Liouvillian framework beyond the spin liquid. The interplay between non-Hermitian dynamics and topological phases is only beginning to be understood. The exploration of strong and weak symmetries, presently confined to this specific instance, requires generalization. Can the principles elucidated here be applied to other condensed matter systems, or even beyond-to open quantum systems in arbitrary environments?

Ultimately, the true test will not be the elegance of the mathematics, but the predictive power. The model suggests a pathway to control and observe these exceptional points. However, it remains to be seen whether these phenomena are merely mathematical curiosities, or represent genuine, observable features of physical systems-a distinction that only rigorous experimental verification can resolve.

Original article: https://arxiv.org/pdf/2512.04155.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-06 09:04