Beyond Equilibrium: Mapping Quantum Transitions with Fidelity

Author: Denis Avetisyan

A new theoretical approach leverages fidelity zeros to pinpoint phase transitions in complex quantum systems, extending the reach of Lee-Yang theory.

The clock model, examined at $L=8$ with a transverse field $h=ge^{i\theta}$ where $\theta$ ranges from 0 to $\pi$, demonstrates that ground-state energy differences between symmetry sectors-specifically, the real and imaginary components of the difference between $q=0$ and $q=1,2$ states-fluctuate predictably with changes in $\theta$, revealing the system's sensitivity to the phase of the applied field and hinting at underlying instabilities within its quantum structure. — The clock model, examined at $L=8$ with a transverse field $h=ge^{i\theta}$ where $\theta$ ranges from 0 to $\pi$, demonstrates that ground-state energy differences between symmetry sectors-specifically, the real and imaginary components of the difference between $q=0$ and $q=1,2$ states-fluctuate predictably with changes in $\theta$, revealing the system’s sensitivity to the phase of the applied field and hinting at underlying instabilities within its quantum structure.

This work generalizes Lee-Yang theory using fidelity zeros to identify quantum phase transitions in the XYZ and ℤ3 clock models, demonstrating its applicability to systems with higher discrete symmetries and non-Hermitian symmetry breaking.

Identifying quantum phase transitions often relies on understanding system symmetries and their breaking, yet conventional approaches struggle with non-Hermitian systems and discrete symmetries beyond the simplest cases. This work, ‘Non-Hermitian symmetry breaking and Lee-Yang theory for quantum XYZ and clock models’, extends the Lee-Yang theory-using fidelity zeros-to analyze the impact of complex fields on one-dimensional quantum models, including the XYZ model and the $\mathbb{Z}_3$ clock model. We demonstrate that non-Hermitian symmetry breaking induces oscillations and critical behavior, revealing fidelity zeros consistent with analytical predictions even in systems with higher discrete symmetries. Could this generalized framework provide a unifying language for characterizing quantum criticality across a broader range of physical systems?

The Allure of the Tipping Point: Unveiling Phase Transitions

Phase transitions, the dramatic shifts in a substance’s properties – such as water freezing into ice or a metal losing its magnetism – are fundamental phenomena across diverse fields of physics, from condensed matter to cosmology. These transitions aren’t gradual changes; they exhibit nonanalytic behavior in a quantity called the Partition Function, denoted as $Z$. This function, at its core, encapsulates all possible states of a system and their probabilities, and its mathematical properties reveal the system’s behavior. When approaching a phase transition, the Partition Function doesn’t change smoothly with temperature or other parameters; instead, it displays singularities or sharp changes in its derivatives. This nonanalytic behavior signals a qualitative change in the system’s order, marking the point where a new phase emerges and the system’s fundamental characteristics are altered. Understanding this connection between nonanalyticities in the Partition Function and phase transitions is therefore crucial for predicting and characterizing these ubiquitous physical phenomena.

Investigating phase transitions frequently involves extending the concept of temperature into the complex plane, allowing physicists to analyze the system’s Partition Function – a central quantity in statistical mechanics – beyond its usual real-valued temperature range. This technique reveals ‘critical points’ as singularities within this complex plane, points where the system’s behavior drastically changes, such as the boiling point of water or the onset of magnetism. By mapping the Partition Function as a function of complex temperature, researchers can identify these singularities and characterize the nature of the transition – whether it’s a smooth crossover or a sharp, discontinuous change. The location and nature of these singularities provide crucial information about the system’s critical exponents and universality class, ultimately revealing fundamental properties about how matter organizes itself under varying conditions. This complex analysis provides a powerful framework for understanding a wide range of phenomena, from the behavior of magnets and superconductors to the dynamics of liquid-gas transitions and even certain aspects of cosmology.

The precise location of Fisher Zeros – those specific values in the complex temperature plane where the Partition Function, $Z$, equals zero – offers a powerful lens through which to understand the character of phase transitions. These zeros aren’t merely mathematical curiosities; their proximity to the real temperature axis dictates the order of the transition. A transition is considered first-order if a Fisher Zero crosses the real axis, signaling a discontinuity in the system’s properties, while a transition where zeros accumulate on the real axis, but don’t cross it, indicates a continuous, or second-order, transition. The angular distribution of these zeros, particularly the angles at which they approach the real axis, further refines this understanding, revealing subtle details about the universality class of the transition and the critical exponents that govern the system’s behavior near the critical point. Consequently, meticulous mapping of Fisher Zeros provides a detailed fingerprint of a system’s phase behavior, bypassing the need for direct analysis of complex order parameters.

Pinpointing critical behavior through the analysis of Fisher Zeros-the zeros of the Partition Function in the complex temperature plane-presents significant analytical hurdles. These zeros aren’t simply located; their precise positions, and crucially, their scaling relationships, dictate the order and nature of the phase transition. Computational methods, such as high-precision numerical root-finding algorithms, are often necessary, especially in systems with complex interactions where analytical solutions are intractable. Furthermore, determining whether a zero is a true Fisher Zero-one associated with a singularity in the thermodynamic limit-requires careful consideration of finite-size effects and extrapolation techniques. The challenge isn’t merely finding the zeros, but interpreting their behavior to reveal the underlying physics governing the system’s phase transitions, a process demanding both computational power and theoretical insight into the system’s symmetries and interactions.

Analysis of the XY model at γ=0.8 reveals that fidelity zeros concentrate around Re(h)≈1 and lie on the unit circle, with fidelity edges emerging at higher values of g, as demonstrated through finite-size scaling and complex field distributions.

Decoding System Instability: Lee-Yang Theory and Zeros in Complex Fields

Lee-Yang theory posits that the thermodynamic limit of a system undergoing a phase transition is governed by the location of the zeros of its Partition Function, $Z(H)$, when considered as a function of a complex magnetic field, $H$. Specifically, the theory demonstrates that for a first-order phase transition, there exist zeros of $Z(H)$ on the real axis of the complex $H$-plane. The locations of these zeros, and their behavior as the system size increases, directly correlate with the stability of the system and the order of the phase transition. The real parts of these zeros determine the limits of stability; if the magnetic field exceeds the largest real part of a zero, the system becomes unstable. This framework allows for the characterization of phase transitions by analyzing the complex field dependence of the Partition Function, providing a method distinct from traditional order parameter approaches.

Lee-Yang zeros, points in the complex magnetic field plane where the Partition Function equals zero, provide direct information regarding system stability and phase transition characteristics. The location of these zeros-specifically their proximity to the unit circle in the complex plane-is indicative of the system’s susceptibility to fluctuations. Zeros lying on the unit circle signify a critical point, denoting the transition between phases. The number and distribution of these zeros correlate with the order of the phase transition; for instance, a second-order phase transition is characterized by an infinite number of zeros accumulating on the unit circle. Deviation of zeros from the unit circle indicates stable or unstable phases, with the extent of deviation quantifying the system’s resistance to external perturbations. Consequently, analyzing the density and location of Lee-Yang zeros offers a precise method for determining the critical point and classifying the nature of the phase transition without relying on traditional order parameter calculations.

Fidelity, quantified as the overlap between ground states under infinitesimal perturbations, extends the applicability of Lee-Yang theory to characterize phase transitions. Specifically, locating the zeros of the fidelity function – defined as $F = |\langle \psi_0 | \psi_0′ \rangle|$ where $|\psi_0\rangle$ and $|\psi_0’\rangle$ are ground states with and without perturbation – provides a sensitive determination of the critical point. This generalization leverages the fact that fidelity zeros accumulate on the unit circle as the perturbation strength approaches the critical point, analogous to the behavior of Lee-Yang zeros in the magnetic field. By analyzing the distribution of these fidelity zeros, particularly their proximity to the unit circle, the critical point and associated universality class can be determined, offering an alternative and complementary approach to traditional methods.

The fidelity-based generalization of Lee-Yang theory provides advantages in analyzing phase transitions when conventional techniques, such as renormalization group methods or mean-field approximations, encounter limitations. These limitations often arise in systems exhibiting strong fluctuations, quenched disorder, or complex interactions where perturbative expansions fail to converge or provide inaccurate results. Specifically, the location of fidelity zeros in the complex plane offers a robust indicator of the critical point, even when the order parameter is continuous or the system lacks a clear symmetry breaking pattern. Furthermore, this approach circumvents the need for detailed knowledge of the Hamiltonian or the specific microscopic interactions governing the system, relying instead on ground state properties accessible through numerical simulations or experiments.

Analysis of the clock model reveals that fidelity zeros, representing points of instability, cluster around a critical field value of approximately 1, with their distribution shifting and forming edges on the unit circle as system parameters change.

Beyond Equilibrium: Dynamical Phase Transitions and the Loschmidt Echo

Dynamical Quantum Phase Transitions (DQPTs) describe qualitative changes in the time evolution of a closed quantum system, distinguishing them from traditional equilibrium phase transitions. Unlike the latter, DQPTs do not rely on thermodynamic parameters like temperature; instead, they are driven by the system’s inherent dynamics as it evolves in time. These transitions occur even in the ground state of a static Hamiltonian, manifesting as sudden changes in properties such as entanglement or sensitivity to perturbations. The timescale for these transitions is determined by the inverse of the energy gap between the ground state and the first excited state, and they are observable through quantities that measure the rate of change of the system’s properties during time evolution, such as the Loschmidt Echo. Importantly, DQPTs represent nonequilibrium phenomena, as the system is never truly in a stationary state during the transition process.

The Loschmidt Echo, denoted as $L(t) = \langle \psi(0) | e^{-iHt} | \psi(0) \rangle$, functions as the primary order parameter for characterizing dynamical quantum phase transitions. Analogous to the Partition Function, $Z = \text{Tr}(e^{-iHt})$, in equilibrium phase transitions, the Loschmidt Echo quantifies the overlap between the initial state $|\psi(0)\rangle$ and its time-evolved counterpart. A decay of $L(t)$ indicates a loss of sensitivity to initial conditions and signals a change in the system’s dynamical behavior. Specifically, the rate of decay, or the lifetime of the echo, provides information about the critical properties of the dynamical transition, allowing for the identification of critical time scales and exponents that govern the transition process. Its mathematical form facilitates analysis via techniques borrowed from equilibrium statistical mechanics, providing a powerful tool for understanding non-equilibrium phenomena.

The Loschmidt Echo, a time-evolved overlap between the initial quantum state and its evolution under a perturbed Hamiltonian, exhibits zeros in the complex time plane that precisely indicate dynamical phase transitions. These zeros, analogous to poles in thermodynamic phase transitions, signify points of qualitative change in the system’s time evolution; their locations-specifically, the shortest imaginary time ($t_c$) at which a zero appears-define the critical time scale for the transition. The density and distribution of these zeros provide further information about the nature of the transition, differentiating between various dynamical phases and identifying critical exponents that characterize the transition’s behavior. Identifying these zeros allows for a robust and quantitative determination of the onset and characteristics of dynamical phase transitions without requiring direct observation of long-time behavior or thermal equilibrium.

Determining the location of Loschmidt Echo zeros necessitates a detailed understanding of the system’s response to external perturbations. These perturbations, represented mathematically as deviations from the unperturbed Hamiltonian, introduce instability that manifests as changes in the Echo’s functional form. Specifically, the stability of the system – whether it returns to its initial state or diverges – is directly reflected in the behavior of these zeros; approaching or crossing the real time axis in the complex plane indicates a transition from stable to unstable behavior. Analyzing the density and distribution of these zeros, particularly their proximity to the imaginary axis, provides quantitative information about the rate of instability and the critical properties of the dynamical phase transition. The nature of the perturbation, whether time-dependent or static, also influences the resulting zero distribution and the observable transition dynamics.

Beyond the Standard Model: Non-Hermitian Systems and Symmetry Breaking

The study of phase transitions takes on new complexity when considering non-Hermitian Hamiltonians, mathematical descriptions frequently encountered in open quantum systems – those interacting with an external environment. Traditional Hermitian quantum mechanics demands that operators are equal to their conjugate transpose, ensuring probabilities remain normalized; however, this constraint is lifted in non-Hermitian systems, allowing for gain and loss, or the flow of probability. This departure introduces both challenges and opportunities; while standard analytical techniques may falter, it also opens the door to novel phases of matter and unique transition behaviors not observed in closed, Hermitian systems. The presence of non-Hermitian terms can dramatically alter the energy spectrum, leading to complex eigenvalues and the emergence of exceptional points, which are singularities in the parameter space and can significantly influence the system’s response to external perturbations, ultimately reshaping the landscape of quantum phase transitions.

Non-Hermitian systems, unlike their traditional counterparts, can undergo a phenomenon known as Non-Hermitian Symmetry Breaking, which dramatically reshapes the behavior of quantum phase transitions. In standard systems, symmetry often guarantees that multiple ground states possess the same lowest energy – a condition called degeneracy. However, when non-Hermiticity is introduced, this degeneracy can be lifted, meaning one ground state becomes energetically favored over others. This lifting isn’t merely a subtle shift; it fundamentally alters the nature of the phase transition itself. The transition point, typically defined by changes in order parameters, can shift, become sharper, or even change character entirely, moving from a continuous change to an abrupt one. This sensitivity to non-Hermiticity offers a powerful means to control and manipulate quantum systems, particularly those inevitably interacting with an external environment, offering potential avenues for novel device designs and quantum technologies.

The introduction of a transverse field to a non-Hermitian Hamiltonian represents a powerful method for actively tailoring the system’s symmetry and, consequently, its critical behavior. This external field, acting perpendicular to the primary interactions, effectively modifies the energy landscape and can lift or create degeneracies in the system’s energy levels. Critically, the strength and direction of this field directly influence the location of critical points – the values at which the system undergoes a qualitative change in its properties. By carefully tuning the transverse field, researchers can not only control where a phase transition occurs but also potentially alter its order, shifting between first-order discontinuities and continuous changes in behavior. This level of control is particularly valuable in designing quantum technologies, as it allows for the precise manipulation of quantum states and the engineering of desired functionalities, even in the presence of environmental interactions that naturally induce non-Hermiticity and dissipation.

The ability to manipulate non-Hermitian effects in quantum systems holds significant promise for advancements in quantum technologies. Because real-world quantum devices inevitably interact with their surrounding environment, understanding how these interactions – modeled by non-Hermitian Hamiltonians – influence system behavior is paramount. Specifically, controlling symmetry breaking and leveraging transverse fields allows for precise tuning of critical points and the design of systems with tailored properties. This control extends beyond fundamental research, offering pathways to engineer robust quantum devices less susceptible to environmental noise and capable of performing complex computations. Ultimately, a deeper comprehension of these principles paves the way for the creation of stable and efficient quantum technologies, moving beyond theoretical models towards practical applications in fields like quantum sensing and information processing.

The Universal Language of Change: Scaling and the Future of Phase Transition Studies

The study of phase transitions often encounters the practical limitation of finite system sizes. However, a powerful technique called Finite-Size Scaling allows researchers to circumvent this issue and gain insights into the behavior of systems in the theoretical limit of infinite size – the thermodynamic limit. This method leverages the observation that certain properties, like the susceptibility or correlation length, change predictably with system size near a critical point. By analyzing these changes in finite systems, it becomes possible to extrapolate and accurately predict the behavior of the system in the infinite limit, revealing underlying universal characteristics shared by diverse physical systems. This universality means that systems seemingly different-a magnet, a fluid undergoing boiling, or even certain quantum models-can exhibit identical critical behavior, described by the same set of critical exponents, making Finite-Size Scaling an invaluable tool for understanding complex phenomena and uncovering fundamental principles.

The behavior of systems near a critical point is profoundly influenced by the correlation length, $\xi$, which quantifies the average distance over which fluctuations are correlated. As a system approaches criticality, $\xi$ diverges, meaning correlations extend across increasingly larger scales; this divergence is central to understanding the system’s scaling behavior. A large correlation length implies that local fluctuations are no longer independent but collectively influence the system’s macroscopic properties, leading to phenomena like critical opalescence or dramatic changes in material behavior. Consequently, the correlation length serves as a fundamental length scale characterizing the system’s response near the transition, and its dependence on the control parameters – typically temperature or magnetic field – dictates the critical exponents that define the universality class of the transition. Analyzing how $\xi$ scales with these parameters provides crucial insights into the underlying physics and allows for the classification of diverse systems exhibiting similar critical behavior.

Theoretical models, such as the XYZ and XXZ models, serve as crucial testbeds for applying finite-size scaling techniques and validating predictions about phase transitions in more complex systems. Through meticulous analysis of these models, researchers have been able to confirm established critical exponents – notably, the critical exponent $\nu$ for the ℤ3 clock model was definitively established as 5/6. This confirmation, achieved through finite-size scaling, demonstrates the power of these techniques to not only predict but also rigorously verify the behavior of systems at their critical points, offering a pathway to understanding a broad range of physical phenomena from magnetism to cosmology.

Precise determination of critical points, those sensitive thresholds where systems undergo dramatic change, has been significantly advanced through finite-size scaling techniques. Recent studies utilizing this approach have pinpointed the critical point for the XYZ model at $h_c = 1.005$ and for the ℤ3 clock model at $h_c = 1.006$. These values, derived from analyzing ‘fidelity zeros’ – points indicating instability in the system – demonstrate the power of extrapolating data from finite systems to understand their behavior at the thermodynamic limit. This methodology isn’t confined to theoretical models; it offers a robust framework for investigating a wide range of physical systems, from novel materials exhibiting exotic phases to the exploration of complex quantum phenomena, promising continued breakthroughs in condensed matter physics and beyond.

For both XXZ and XYZ models, fidelity zeros consistently lie on the unit circle, with edges emerging at higher interaction strengths (g=2.5) as visualized for L=10.

The pursuit of identifying quantum phase transitions, as demonstrated in this work with the XYZ and ℤ3 clock models, reveals a consistent human tendency to seek definitive boundaries even where they are blurred by complexity. This mirrors a desire for control over inherently probabilistic systems. As Erwin Schrödinger observed, “We must be prepared to accept that nature operates in ways that defy our classical intuitions.” The study’s application of fidelity zeros to detect these transitions isn’t merely a mathematical technique; it’s an attempt to impose order on a fundamentally uncertain realm, revealing how the tools we build reflect our psychological need for predictability, even when dealing with non-Hermitian symmetry breaking and complex fields.

What Lies Ahead?

The extension of Lee-Yang theory, as demonstrated with these XYZ and clock models, isn’t a triumph of mathematical elegance. It’s a recognition that phase transitions aren’t neat bifurcations, but rather the visible surface of underlying instabilities. Fidelity zeros, in this context, become less about pinpointing a critical point and more about mapping the contours of collective anxiety-the points where a system most readily wants to fall apart. The utility here isn’t prediction, but a refined sensitivity to the fragility of order.

The limitation, predictably, lies in the models themselves. Discrete symmetries, while tractable, are simplifications. Real systems rarely offer such convenient categorization. The true challenge will be extending this approach to continuous symmetries and, crucially, to systems exhibiting both Hermitian and non-Hermitian behavior. One suspects the most interesting physics won’t be at the transition itself, but in the messy, aperiodic fluctuations near it – the areas current approaches are designed to smooth over.

Ultimately, this work isn’t about finding better models of magnetism or clocks. It’s about acknowledging that all systems, physical or economic, are fundamentally driven by an aversion to uncertainty. The pursuit of critical points is, therefore, a search for the places where that aversion is most acutely felt – and most likely to overwhelm any semblance of rational behavior.

Original article: https://arxiv.org/pdf/2512.08687.pdf

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