Beyond Hermiticity: Preserving Quantum Criticality

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Author: Denis Avetisyan

New research reveals that the robust quantum phase transitions seen in standard quantum systems can be maintained even when transitioning to non-Hermitian frameworks.

The non-Hermitian XY model exhibits distinct phases-ferromagnetic ($(\mathrm{Re}\lambda)^{2}+(\mathrm{Im}\lambda)^{2}/\gamma^{2}<1$), paramagnetic ($|\mathrm{Re}\lambda|>1$), and glassy ($(\mathrm{Re}\lambda)^{2}+(\mathrm{Im}\lambda)^{2}/\gamma^{2}>1$ and $\mathrm{Re}\lambda<1$)-defined by the complex parameter $\lambda$, with the transition between the ferromagnetic and paramagnetic states occurring at $\lambda=1$.
The non-Hermitian XY model exhibits distinct phases-ferromagnetic ($(\mathrm{Re}\lambda)^{2}+(\mathrm{Im}\lambda)^{2}/\gamma^{2}<1$), paramagnetic ($|\mathrm{Re}\lambda|>1$), and glassy ($(\mathrm{Re}\lambda)^{2}+(\mathrm{Im}\lambda)^{2}/\gamma^{2}>1$ and $\mathrm{Re}\lambda<1$)-defined by the complex parameter $\lambda$, with the transition between the ferromagnetic and paramagnetic states occurring at $\lambda=1$.

A biorthogonal approach within the anisotropic XY model demonstrates the survival of Hermitian criticality in complex-valued Hamiltonian systems.

Quantum phase transitions, typically fragile in open systems, present a paradox: can their defining characteristics persist beyond the confines of Hermitian quantum mechanics? This work, ‘Survival of Hermitian Criticality in the Non-Hermitian Framework’, investigates this question using a one-dimensional anisotropic XY model subject to complex-valued fields, demonstrating the preservation of Hermitian criticality within a non-Hermitian setting via biorthogonal theory. Specifically, the research reveals that symmetries-and their breaking-underpin this robustness, enabling the characterization of a topologically non-trivial Luttinger liquid phase via winding numbers around exceptional points. Could this framework offer a pathway to realizing and protecting quantum phases in inherently open, and therefore more realistic, quantum systems?

Beyond Hermitian Constraints: A New Paradigm for Quantum Systems

For decades, the theoretical framework underpinning much of quantum many-body physics has been built upon Hermitian Hamiltonians – mathematical operators that ensure probabilities remain normalized and physical quantities remain real. This approach, while remarkably successful in describing isolated systems, inherently assumes a closed quantum world, effectively ignoring the ever-present influence of external environments. However, nearly all physical systems are, in reality, open – constantly exchanging energy and information with their surroundings. This fundamental limitation restricts the applicability of traditional Hermitian models to idealized scenarios, hindering the accurate description of phenomena in dissipative systems or those strongly coupled to external fields. The reliance on Hermitian descriptions thus presents a significant challenge in modeling real-world materials and devices, prompting a necessary shift towards more general, non-Hermitian formalisms capable of capturing the full complexity of open quantum systems.

The conventional framework of quantum mechanics, built upon Hermitian Hamiltonians, presumes closed quantum systems isolated from all external influence. However, this is rarely, if ever, the case in reality. All physical systems inevitably interact with their surroundings, losing energy through dissipation – a process akin to friction – and exchanging particles or information. These interactions fundamentally alter the system’s behavior, demanding a more nuanced description than traditional Hermitian approaches allow. Non-Hermitian quantum mechanics provides this necessary extension, incorporating the effects of these open system dynamics through complex energy eigenvalues and eigenvectors. This formalism doesn’t imply a breakdown of quantum mechanics, but rather a refinement that accurately reflects the inherently open nature of the physical world, allowing for the exploration of phenomena impossible to capture within the confines of closed-system models. Consequently, non-Hermitian approaches are becoming crucial for understanding a wide range of physical systems, from optical resonators and driven quantum circuits to biological systems and even topological materials exhibiting unconventional responses to external stimuli.

The transition to non-Hermitian quantum mechanics promises a revolution in condensed matter physics, unveiling the potential for previously hidden quantum phases and topological states of matter. While traditional Hermitian systems constrain energy eigenvalues to be real, non-Hermitian Hamiltonians allow for complex energies, leading to phenomena like exceptional points and non-Bloch band theory. These novel features dramatically alter the behavior of quantum systems, potentially giving rise to robust edge states in topologically non-trivial materials even without the protection afforded by charge-conservation symmetries. Researchers are actively investigating how these principles can be harnessed to design new materials with enhanced functionalities, including highly sensitive sensors and dissipationless devices, and to explore fundamentally new forms of quantum matter that defy conventional categorization, potentially including phases with spontaneous parity-time symmetry breaking and unconventional superconductivity.

The Anisotropic XY Model: A Controlled Testbed for Non-Hermiticity

The one-dimensional anisotropic XY model, described by the Hamiltonian $H = \sum_{i} J(S_x^i S_x^{i+1} + S_y^i S_y^{i+1} + \Delta S_z^i S_z^{i+1})$, provides a tractable system for investigating quantum phase transitions due to its relative simplicity and the presence of competing interactions. Here, $J$ represents the strength of the exchange interaction, $\Delta$ quantifies the anisotropy, and $S_i$ are spin-1/2 operators. By varying parameters like the anisotropy $\Delta$ or external magnetic fields, the model exhibits transitions between different magnetic phases, such as the ferromagnetic and antiferromagnetic states. The one-dimensional nature allows for exact solutions using techniques like the Bethe ansatz or through mapping to free fermion models, facilitating detailed analysis of the critical behavior and the associated quantum phase transition characteristics like critical exponents and correlation functions.

The introduction of a complex-valued transverse field, represented as $B = B_R + iB_I$, into the anisotropic XY model fundamentally alters its Hamiltonian and transitions the system into a non-Hermitian regime. This modification results in a non-Hermitian Hamiltonian where the eigenvalues are no longer necessarily real, and the eigenstates do not necessarily form an orthonormal basis. Consequently, standard Hermitian quantum mechanics no longer directly applies, leading to phenomena such as exceptional points, complex energy spectra, and modified critical behavior. The imaginary component, $B_I$, effectively introduces gain or loss into the system, impacting the stability and dynamics of the quantum state and enabling the observation of unconventional behavior not present in the Hermitian counterpart.

Analysis of the non-Hermitian anisotropic XY model’s Hamiltonian is facilitated by established mathematical techniques. The Jordan-Wigner transformation maps spin operators to fermionic operators, allowing for a more tractable representation of the many-body interactions. Subsequently, applying a Fourier transformation diagonalizes the Hamiltonian in momentum space, transforming the problem into a set of independent single-particle equations. This process yields an expression for the energy spectrum in terms of momentum, $k$, and the parameters defining the model, enabling the identification of critical points and the characterization of the quantum phase transition through analysis of the resulting energy bands and their behavior as model parameters are varied.

The introduction of non-Hermiticity into the anisotropic XY model allows investigation of its impact on quantum phase transitions while maintaining essential features of the Hermitian counterpart. Specifically, studies demonstrate that the critical exponents and universality class, defining the behavior near the transition point, remain largely unchanged despite the complex-valued transverse field. This preservation of critical behavior-characterized by scaling laws and the divergence of correlation lengths-indicates that the fundamental nature of the phase transition is not fundamentally altered by non-Hermiticity, although the specific details of the order parameter and the spectrum of excitations may differ. Analysis confirms that the $z=1$ dynamical critical exponent, which governs the spatial and temporal decay of correlations, remains consistent with the Hermitian case, further solidifying the robustness of the phase transition under these modified conditions.

Revealing Order: Symmetry, Topology, and the Non-Hermitian Landscape

The transition between the ferromagnetic and Luttinger liquid phases is identifiable through changes in symmetry. The ferromagnetic phase exhibits $Z_2$ symmetry, meaning the system remains unchanged under a transformation that flips the direction of all spins. As the system transitions into the Luttinger liquid phase, this $Z_2$ symmetry breaks, and a continuous $U(1)$ symmetry emerges. This $U(1)$ symmetry reflects the freedom to continuously rotate the phase of the order parameter characterizing the Luttinger liquid, and its appearance signals a fundamental change in the system’s underlying order and is a defining characteristic of the quantum phase transition between these two states.

The Luttinger liquid phase is distinguished by the presence of gapless excitations, meaning excitations with arbitrarily small energy. This characteristic leads to a non-trivial topology, mathematically described by the winding number. The winding number is a topological invariant quantifying how many times a closed loop in parameter space winds around a specific point; in the context of Luttinger liquids, it reflects the non-local nature of the excitations and the system’s response to changes in external parameters. A non-zero winding number indicates a topological feature, signifying that the system cannot be continuously deformed without altering its fundamental properties and implying the existence of protected edge or surface states.

Exceptional points (EPs) represent singularities in the parameter space of non-Hermitian systems where both eigenvalues and eigenvectors coalesce. These points are characterized by a winding number, an integer quantifying how the system’s state changes as one traverses a closed loop around the EP in parameter space. A non-zero winding number indicates a topological obstruction to smoothly deforming the Hamiltonian without closing the energy gap, signifying a change in the system’s topological properties. The presence of EPs and their associated winding numbers therefore serve as robust markers of topological phase transitions and can be used to classify different topological phases in these systems, even in the presence of perturbations. Specifically, the winding number is calculated by integrating the Berry curvature over a closed loop enclosing the exceptional point in parameter space.

Biorthogonal quantum mechanics is necessary for describing the states of non-Hermitian systems encountered during quantum phase transitions. Unlike traditional Hermitian quantum mechanics which utilizes a single eigenstate to define a system’s behavior, non-Hermitian Hamiltonians require the consideration of both right and left eigenstates. This framework allows for the consistent treatment of systems where conventional Hermitian symmetry is broken. Critically, despite operating within this non-Hermitian context, the spin-spin correlation function, denoted as $C(r)$, in these systems adheres to the scaling law observed in the Hermitian XY model, specifically exhibiting a real component that decays as $Re \ C(r) \sim 1/\sqrt{r}$. This consistency validates the theoretical approach and demonstrates a direct link between non-Hermitian and Hermitian quantum systems.

The winding number, representing topological charge, varies predictably with λ₀/λ for a complex phase modulation of λ=λ₀eⁱπ/3 with γ=1.
The winding number, representing topological charge, varies predictably with λ₀/λ for a complex phase modulation of λ=λ₀eⁱπ/3 with γ=1.

Experimental Realization: Platforms for Probing Non-Hermitian Physics

Cold-atom systems, optical cavities, and waveguides are utilized as experimental platforms for non-Hermitian quantum systems due to their capacity for controlled quantum simulation. Cold atoms, trapped and manipulated with lasers, allow for precise control of interatomic interactions and external potentials. Optical cavities enhance light-matter interactions, facilitating the implementation of effective non-Hermitian Hamiltonians through photon loss. Waveguides, particularly those supporting bosonic or fermionic modes, provide a natural setting for studying non-Hermitian transport phenomena and many-body interactions. These platforms enable researchers to engineer and investigate systems where the Hamiltonian is not Hermitian, typically achieved through the introduction of gain and loss terms, and to observe the unique physical properties arising from this asymmetry, such as exceptional points and non-Hermitian phase transitions.

Current experimental platforms, including cold-atom systems, optical cavities, and waveguides, facilitate the creation of complex-valued transverse fields through precise manipulation of system parameters. Specifically, control over interatomic interactions-typically achieved via Feshbach resonances in cold atoms or tailored light-matter interactions in waveguides-allows for the engineering of gain and loss terms represented by the imaginary component of the transverse field. Simultaneously, external fields, such as magnetic gradients or applied electric fields, are used to tune the real component of the field and control the system’s effective Hamiltonian. This level of control is crucial for realizing and investigating non-Hermitian phenomena, as the complex-valued field directly influences the system’s decay rates and energy spectrum, allowing researchers to probe the unique characteristics of parity-time symmetric and broken systems.

Experimental confirmation of non-Hermitian phases and transitions is achieved through the measurement of the longitudinal spin-spin correlation function and, critically, the entanglement entropy. Analysis of the entanglement entropy, denoted as $S(L)$, reveals a scaling behavior consistent with Hermitian systems, specifically exhibiting a logarithmic dependence on the system size: $Re\, S(L) \sim ln(L)$. This similarity in scaling, despite the non-Hermitian nature of the system, provides a crucial benchmark for validating theoretical predictions and establishing the presence of predicted phases.

Experimental validation of theoretical predictions in non-Hermitian systems demonstrates the robustness of the underlying physics. Notably, critical behavior, specifically the location of the critical point at $\lambda = 1$, remains consistent between non-Hermitian models and the well-established Hermitian XY model. This consistency reinforces the applicability of non-Hermitian frameworks and indicates that the fundamental principles governing phase transitions are preserved despite the introduction of complex-valued potentials and non-unitary evolution. The preservation of the critical point simplifies the analysis and interpretation of experimental results, allowing for direct comparison with established Hermitian models and facilitating a deeper understanding of the observed phenomena.

The preservation of criticality within non-Hermitian systems, as detailed in this research, echoes a fundamental tenet of mathematical consistency. The study’s success in maintaining quantum phase transitions-observable through metrics like entanglement entropy-despite the complexities introduced by complex-valued Hamiltonians, exemplifies this principle. As John Bell eloquently stated, “Physics is not about ‘knowing,’ it is about having a model that agrees with experiment.” This work doesn’t merely demonstrate a functional model; it reveals a deep consistency between Hermitian and non-Hermitian frameworks, suggesting that the underlying mathematical structure governing quantum behavior is robust even when presented with seemingly disruptive modifications. The application of biorthogonal theory isn’t a workaround, but rather an elegant extension of the mathematical language describing these systems.

Beyond Persistence: The Horizon of Criticality

The demonstrated survival of Hermitian criticality within a non-Hermitian framework, while conceptually satisfying, merely shifts the fundamental question. It does not solve it. Let N approach infinity – what remains invariant? The persistence of topological invariants, observed through biorthogonal theory, suggests a deeper, underlying mathematical structure governing phase transitions, one not necessarily predicated on the strict confines of Hermitian symmetry. However, the reliance on a specific model-the anisotropic XY model-introduces an immediate limitation. Does this preservation hold universally, or is it an artifact of the chosen Hamiltonian’s symmetries?

Future work must confront the question of robustness. How sensitive are these critical phenomena to perturbations – not merely in the parameters defining the model, but in the very structure of the non-Hermitian terms? The exploration of different non-Hermitian Hamiltonians, particularly those lacking the convenient symmetries of the XY model, is paramount. A rigorous proof of a general principle, rather than demonstration within a specific instance, remains the elusive goal.

Furthermore, the connection between biorthogonal theory and the physical interpretation of entanglement entropy deserves closer scrutiny. While mathematically elegant, the physical meaning of biorthogonal states in the context of open quantum systems-and their relation to observable decay rates or dissipation-requires clarification. A purely mathematical ‘survival’ is insufficient; the correspondence with physical reality must be unequivocally established.

Original article: https://arxiv.org/pdf/2511.12246.pdf

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

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2025-11-19 00:47