Quantum Chaos and the Path to a Spin Liquid

Author: Denis Avetisyan

New research explores how quantum fluctuations drive a transition from a disordered spin glass to a potentially chaotic quantum spin liquid state in a fully connected system.

The study of self-consistent equations reveals that the fermionic and spin spectral densities exhibit distinct behaviors-Ohmic in the paramagnetic phase and sub-Ohmic in the spin-glass phase-and that the static magnetic susceptibility plateaus at <span class="katex-eq" data-katex-display="false">J|\chi(0)|=1</span> for large values of <span class="katex-eq" data-katex-display="false">\mathcal{N}_f</span> before the critical temperature of <span class="katex-eq" data-katex-display="false">T_c=J/4</span>, while quantum fluctuations shift this transition to lower temperatures for smaller <span class="katex-eq" data-katex-display="false">\mathcal{N}_f</span> values, ultimately influencing the saturation and suppression of the spin-glass order parameter at high and low <span class="katex-eq" data-katex-display="false">\mathcal{N}_f</span> values, respectively. — The study of self-consistent equations reveals that the fermionic and spin spectral densities exhibit distinct behaviors-Ohmic in the paramagnetic phase and sub-Ohmic in the spin-glass phase-and that the static magnetic susceptibility plateaus at $J|\chi(0)|=1$ for large values of $\mathcal{N}_f$ before the critical temperature of $T_c=J/4$ , while quantum fluctuations shift this transition to lower temperatures for smaller $\mathcal{N}_f$ values, ultimately influencing the saturation and suppression of the spin-glass order parameter at high and low $\mathcal{N}_f$ values, respectively.

This study investigates the crossover to Sachdev-Ye-Kitaev criticality in an infinite-range quantum Heisenberg spin glass using the Luttinger-Ward functional to map out the fermionic spectral density.

The interplay between quantum fluctuations and emergent phases in strongly correlated systems remains a central challenge in condensed matter physics. This work, ‘Crossover to Sachdev-Ye-Kitaev criticality in an infinite-range quantum Heisenberg spin glass’, investigates a fully connected quantum spin glass model with $\mathcal{N}_f$ fermionic flavors, revealing a suppression of the spin glass ordering temperature with increasing quantum fluctuations. We demonstrate that this behavior signals a crossover towards a Sachdev-Ye-Kitaev (SYK) phase characterized by critical spectral functions and a universal sub-Ohmic dynamical spin susceptibility. Does this minimal framework provide a pathway to understanding the broader relationship between SYK criticality and the emergence of spin glass order in more complex materials?

Decoding Quantum Magnetism: A Systemic Challenge

The pursuit of understanding strongly correlated quantum systems presents a formidable frontier in condensed matter physics, largely due to the intricate interplay of quantum mechanics and the collective behavior of many interacting particles. These systems, exemplified by materials exhibiting spin glass behavior, defy simple descriptions because the electrons within them are not independent entities, but rather experience robust, long-range correlations. This interconnectedness fundamentally alters their properties, making it difficult to predict macroscopic behavior from microscopic origins. Unlike systems governed by weakly interacting particles – where traditional approaches often suffice – strongly correlated systems demand novel theoretical tools and computational techniques to unravel their emergent phenomena, potentially unlocking pathways to revolutionary materials with tailored magnetic, electronic, and superconducting properties.

The investigation of strongly correlated quantum systems is hampered by the limitations of conventional theoretical and computational techniques. These methods, often successful with weakly interacting particles, falter when faced with the tangled web of interactions and inherent disorder characteristic of materials like spin glasses. The sheer number of interacting quantum degrees of freedom creates a computational bottleneck, while perturbative approaches break down due to the strength of these interactions. Consequently, physicists are actively developing novel theoretical frameworks – including tensor networks, dynamical mean-field theory, and quantum Monte Carlo simulations – specifically designed to tackle this complexity. These innovative approaches aim to map the collective behavior emerging from many-body interactions, offering a pathway toward understanding and ultimately predicting the properties of these enigmatic quantum materials and pushing the boundaries of condensed matter physics.

At small <span class="katex-eq" data-katex-display="false">\mathcal{N}_{f}</span>, the system exhibits emergent SYK physics characterized by power-law fermionic spectral density at low temperatures and a crossover in spin spectral density from an SYK-like plateau to linear and square-root behaviors at low frequencies depending on the phase. — At small $\mathcal{N}_{f}$ , the system exhibits emergent SYK physics characterized by power-law fermionic spectral density at low temperatures and a crossover in spin spectral density from an SYK-like plateau to linear and square-root behaviors at low frequencies depending on the phase.

The Quantum Heisenberg Model: A Foundation for Systemic Exploration

The Quantum Heisenberg Model offers a mathematically manageable, yet sufficiently complex, platform for investigating quantum magnetism arising from randomly distributed exchange interactions. Unlike simpler models, it incorporates the full quantum nature of spins and allows for the study of correlations beyond mean-field approximations. This is achieved by representing the magnetic interactions as a sum of Heisenberg terms – $\sum_{\langle i,j \rangle} J_{ij} \mathbf{S}_i \cdot \mathbf{S}_j$ – where $\mathbf{S}_i$ represents the spin at lattice site i, and $J_{ij}$ is a random exchange interaction. The model’s tractability stems from its relatively simple Hamiltonian, enabling the application of techniques like replica methods and numerical simulations to analyze its properties, particularly in disordered systems where the randomness of $J_{ij}$ plays a crucial role in determining the ground state and low-energy excitations.

The Quantum Heisenberg Model utilizes a mapping of spin operators to collective fermion bilinears, specifically $\sum_{i,j} J_{ij} S_i \cdot S_j$ , where $S_i$ represents the spin operator at lattice site i and $J_{ij}$ is the exchange interaction. This transformation allows for the application of established many-body techniques, such as bosonization and perturbative expansions, to analyze the system’s behavior. By representing the spin interactions in terms of fermionic operators, the model circumvents the difficulties associated with directly treating interacting spin variables and enables a systematic investigation of correlations, entanglement, and collective excitations within the quantum magnetic system. This approach facilitates calculations of physical observables and provides insights into the model’s phase diagram and critical properties.

The Quantum Heisenberg Model exhibits a phase transition dependent on the ‘Flavor Number’, which quantifies the number of fermion flavors participating in the spin interactions. Calculations demonstrate that as the flavor number is decreased, the system transitions from a spin glass phase – characterized by frozen, disordered spin configurations – to a quantum spin liquid phase. This quantum spin liquid state is notable for its persistent quantum entanglement and the absence of conventional magnetic ordering, even at zero temperature. The specific critical flavor number at which this transition occurs is determined by the details of the interactions and the dimensionality of the system, and represents a key parameter in understanding the emergent quantum properties of the material.

The system's phase diagram is determined by both temperature <span class="katex-eq" data-katex-display="false">T_T</span> and the number of fermionic flavors <span class="katex-eq" data-katex-display="false">\mathcal{N}_f</span>, which dictates the intensity of quantum fluctuations. — The system’s phase diagram is determined by both temperature $T_T$ and the number of fermionic flavors $\mathcal{N}_f$ , which dictates the intensity of quantum fluctuations.

Functional Methods: Dissecting Interactions and Self-Energy

The Luttinger-Ward (LW) functional is a theoretical framework utilized to analyze the Quantum Heisenberg Model by systematically incorporating interaction effects. This approach achieves precision through an expansion in powers of the flavor number, $N_f$ . The LW functional allows for the calculation of physical observables as functional derivatives of the effective action, effectively summing an infinite series of Feynman diagrams. By expanding to leading order in $1/N_f$ , the complexity of many-body interactions is reduced, providing a tractable method for determining the system’s ground state properties and excited state behavior. This expansion is particularly useful in contexts where the number of flavors is large enough to justify the approximation, while still capturing the essential physics of interacting spins.

The Keldysh formalism, also known as the nonequilibrium Green’s function technique, provides a method for calculating the self-energy $\Sigma(ω)$ which accounts for the effects of interactions and disorder on a quantum many-body system. This approach differs from standard equilibrium Green’s function methods by employing a doubled contour in time, allowing for the treatment of time-dependent and out-of-equilibrium situations. The self-energy, when incorporated into the Dyson equation, effectively renormalizes the single-particle Green’s function, modifying the system’s spectral properties and quasiparticle lifetimes. Specifically, interactions contribute to the self-energy through diagrams representing scattering processes between particles, while disorder introduces scattering from static potentials, both leading to broadening of spectral features and a reduction in coherence. Accurate calculation of the self-energy via the Keldysh formalism is therefore essential for understanding the physical properties of correlated and disordered systems.

The polarization function is a central quantity in determining the self-energy, representing the system’s collective response to external perturbations and mediating internal interactions. Specifically, it describes the screening of interactions and the propagation of excitations. At high temperatures, the fermionic spectral broadening, γ, is calculated to be $γ = 3J / (8\sqrt𝒩f)$ , where $J$ represents the exchange interaction and $𝒩f$ is the density of fermionic states; this broadening directly reflects the impact of interactions and disorder on the single-particle spectral function, effectively quantifying the lifetime of the fermionic excitations.

This work resums an infinite series of diagrammatic contributions to the fermionic self-energy, calculated to leading order in the <span class="katex-eq" data-katex-display="false">\frac{1}{\mathcal{N}_{f}}</span> expansion. — This work resums an infinite series of diagrammatic contributions to the fermionic self-energy, calculated to leading order in the $\frac{1}{\mathcal{N}_{f}}$ expansion.

Mapping Phase Transitions and Spin Glass Behavior

The model’s behavior is comprehensively mapped by a phase diagram that charts distinct phases as temperature and the number of “flavors” change. This diagram isn’t simply a static picture; it vividly demonstrates how quantum fluctuations – inherent uncertainties in a system’s properties – compete with the disorder introduced by the random interactions between spins. At low temperatures and with few flavors, strong quantum effects can dominate, potentially suppressing the development of spin glass order. Conversely, at higher temperatures, or with many flavors, thermal fluctuations and increased complexity can overwhelm quantum effects. The precise boundaries between these phases, and the regions where disorder and quantum mechanics are balanced, are revealed through detailed analysis, offering crucial insights into the emergence of complex magnetic states. This interplay is not merely theoretical; it has implications for understanding materials where both quantum effects and disorder are prominent, such as certain exotic magnets and disordered quantum systems.

Identifying a spin glass phase requires robust quantitative measures, and researchers utilize both the Edwards-Anderson order parameter and the Stoner criterion to achieve this. The Edwards-Anderson order parameter quantifies the degree of frozen-in disorder by measuring the overlap between different configurations of the spin system; a non-zero value at low temperatures signals the emergence of spin glass order. Complementing this, the Stoner criterion – originally developed for ferromagnetic materials – assesses the stability of a disordered spin state by comparing the strength of the random local fields to the thermal fluctuations. When the Stoner criterion is satisfied alongside a measurable Edwards-Anderson order parameter, it provides compelling evidence for the existence of a true spin glass phase, confirming that the system is not simply a paramagnetic state with residual disorder but exhibits a distinct, frozen-in pattern of spins.

A nuanced understanding of the spin glass phase emerges through the application of Replica Symmetry Breaking (RSB) combined with the Imaginary-Time Formalism. This analytical approach reveals that the critical temperature – the point at which the system transitions into a spin glass state – isn’t a fixed value, but rather scales in a complex manner. Specifically, for systems with a small number of ‘flavors’ ( $N_f$ ), the critical temperature $T_c$ is approximated by $T_c \sim J\sqrt{N_f}e^{-C/N_f}$ , where $J$ represents the interaction strength and $C$ is a constant. However, as the number of flavors increases, this scaling behavior shifts; the critical temperature asymptotically approaches $J/4$ . This demonstrates that the spin glass phase is sensitive to the system’s complexity, with the interplay between quantum fluctuations and disorder dictating the precise temperature at which order emerges from the frozen disorder.

Extending the Framework and Charting Future Directions

The Sachdev-Ye-Kitaev (SYK) model represents a significant departure from, and extension of, the well-established Quantum Heisenberg Model, offering a powerful new lens through which to examine strongly correlated quantum systems. While the Heisenberg model traditionally focuses on localized spins and their interactions, the SYK model introduces all-to-all interactions – each quantum particle interacts with every other – and crucially, these interactions are random. This seemingly simple modification unlocks a wealth of novel phenomena, including the emergence of a ‘soft gap’ in the energy spectrum and behaviors reminiscent of black holes, providing a unique playground for studying quantum gravity and the information paradox. Investigations using the SYK model have demonstrated the potential for unconventional superconductivity and non-Fermi liquid behavior, challenging conventional condensed matter physics and suggesting pathways to understanding exotic quantum phases of matter not captured by traditional approaches.

The pursuit of realistic models in quantum magnetism hinges on a thorough understanding of the $SU(2)$ invariant exchange interactions. These interactions, which remain consistent regardless of rotations in spin space, represent a fundamental building block for describing magnetic correlations within materials. Unlike simpler models that often rely on restrictive assumptions, incorporating $SU(2)$ invariance allows for a more nuanced depiction of how electron spins interact, capturing a wider range of magnetic behaviors. Researchers are increasingly focusing on these interactions to move beyond idealized scenarios and accurately represent the complex interplay of forces in real-world magnetic materials, ultimately paving the way for the design of novel magnetic devices and a deeper understanding of emergent quantum phenomena.

Investigations are poised to extend beyond the current scope, applying these analytical methods to a diverse array of strongly correlated systems – materials where electron interactions dictate behavior. This broadened approach aims to uncover novel quantum phases, potentially revealing unconventional superconductivity, topological order, or quantum spin liquids. Researchers anticipate that by systematically exploring these complex systems, they can map out the boundaries between known phases and identify entirely new states of matter governed by emergent phenomena. The pursuit of these exotic phases promises not only a deeper understanding of fundamental physics, but also potential breakthroughs in materials science and quantum technologies, leveraging the unique properties of these newly discovered states.

The study meticulously maps the transition from a spin glass to a potentially more exotic phase, highlighting how increasing quantum fluctuations suppress the conventional spin glass ordering. This pursuit of understanding complex systems through the lens of fundamental principles echoes René Descartes’ assertion: “It is not enough to have a good mind; the main thing is to apply it correctly.” The researchers don’t merely model quantum interactions; they strategically examine how these interactions, especially within a fully connected network, fundamentally reshape the system’s behavior, moving it closer to Sachdev-Ye-Kitaev criticality. This disciplined approach-distinguishing essential quantum effects from accidental complexities-is key to unlocking the secrets of emergent phenomena.

Where Do We Go From Here?

The exploration of quantum spin glasses, particularly through the lens of fully connected models, reveals a fundamental tension. Each added interaction, each attempt to model greater complexity, introduces a new potential instability. The paper demonstrates a suppression of the spin glass phase by quantum fluctuations – a familiar refrain in condensed matter physics. It suggests that a path towards Sachdev-Ye-Kitaev (SYK) criticality may exist, but realizing this requires a careful navigation of the increasingly intricate landscape of quantum many-body systems.

A crucial unresolved question centers on the robustness of these findings. The infinite-range connectivity, while simplifying analysis, represents an idealized limit. The transition observed is sensitive, and realistic, finite-dimensional systems will inevitably exhibit deviations. Future work must address how these structural changes – the move from idealized network to constrained geometry – affect the emergence of quantum criticality, and whether the path to SYK-like behavior remains viable.

Ultimately, this research underscores a simple, yet often overlooked principle: structure dictates behavior. The fully connected model provides a useful starting point, but it is merely a scaffolding. The true challenge lies in understanding how subtle variations in connectivity, and the interplay of various quantum fluctuations, shape the emergent properties of these complex materials. Every new dependency is the hidden cost of freedom; a principle that will undoubtedly guide further investigation.

Original article: https://arxiv.org/pdf/2603.11263.pdf

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