Unveiling Quantum Secrets Through Imaginary Time

Author: Denis Avetisyan

New research reveals a powerful method for probing universal entanglement properties in complex quantum systems using a unique ‘imaginary time’ approach.

Entanglement entropy in periodically bounded lattices grows linearly with imaginary time τ, driven by contributions localized at corners-specifically, those forming <span class="katex-eq" data-katex-display="false">\frac{\pi}{3}</span> and <span class="katex-eq" data-katex-display="false">\frac{2\pi}{3}</span> angles-within regions possessing equivalent boundary lengths, despite differing geometries-such as a <span class="katex-eq" data-katex-display="false">\frac{L}{3} \times \frac{2L}{3}</span> area contrasted with a <span class="katex-eq" data-katex-display="false">\frac{L}{3} \times L</span> one. — Entanglement entropy in periodically bounded lattices grows linearly with imaginary time τ, driven by contributions localized at corners-specifically, those forming $\frac{\pi}{3}$ and $\frac{2\pi}{3}$ angles-within regions possessing equivalent boundary lengths, despite differing geometries-such as a $\frac{L}{3} \times \frac{2L}{3}$ area contrasted with a $\frac{L}{3} \times L$ one.

This study demonstrates universal scaling in entanglement growth along imaginary time, offering a novel pathway to study non-equilibrium quantum criticality and leveraging quantum Monte Carlo simulations.

Characterizing universal entanglement in higher-dimensional quantum systems remains a significant challenge despite its importance for understanding quantum matter. This work, ‘Universal Entanglement Growth along Imaginary Time in Quantum Critical Systems’, introduces a non-equilibrium approach leveraging imaginary-time dynamics to efficiently extract universal entanglement properties. Through Quantum Monte Carlo simulations, we demonstrate a novel scaling law-linear growth of corner entanglement entropy with the logarithm of imaginary time-dictated solely by the system’s universality class. Could this method unlock a new era of precision in characterizing the rich entanglement structure of quantum critical systems, bypassing the need for computationally expensive equilibrium simulations?

The Emergence of Order: Entanglement Beyond the Expected

In the realm of many-body quantum systems, a foundational principle known as the Area Law governs the scaling of entanglement entropy. This law posits that the amount of entanglement between a region and its complement grows proportionally to the surface area of the boundary separating them, rather than the volume of the region itself. Essentially, information about a subsystem is largely encoded at its edges, creating a surprisingly efficient way for quantum states to remain correlated. This isn’t merely a mathematical curiosity; the Area Law arises from the locality of interactions in most physical systems, meaning that particles primarily influence their immediate neighbors. Consequently, understanding entanglement entropy through the Area Law provides critical insight into the structure of quantum matter and serves as a benchmark for identifying more exotic, non-local phases where this principle breaks down – signaling potentially novel quantum phenomena.

In systems undergoing phase transitions – known as critical systems – the seemingly universal ‘Area Law’ of entanglement breaks down, revealing a fascinating complexity in how quantum information is distributed. Instead of entanglement entropy scaling directly with the boundary area of a region, these systems exhibit ‘Logarithmic Correction’, a subtle but significant deviation described by an additional logarithmic term. This correction isn’t merely a mathematical quirk; it suggests that entanglement extends further than predicted by the Area Law, creating long-range correlations and hinting at a fundamentally different structure to the quantum connections within the material. This departure from expected behavior is a key indicator of exotic phases of matter, where emergent properties and unusual quantum phenomena can arise, demanding a deeper understanding of entanglement’s role in shaping the material’s characteristics.

The precise measurement of entanglement, and deviations from established scaling laws like the Area Law, provides a powerful lens through which to examine exotic phases of matter. These phases, often arising from strong interactions within many-body systems, frequently exhibit emergent properties – behaviors not predictable from the constituent particles alone. By meticulously characterizing how entanglement departs from expected norms, physicists can begin to map the complex landscape of these novel states, revealing underlying order and unlocking a deeper understanding of their unique characteristics. This approach isn’t simply about observing that a new phase exists, but about discerning its fundamental properties, such as its topological order or fractionalized excitations, directly from the patterns of quantum entanglement within the system. Ultimately, this detailed analysis of entanglement provides crucial insights into the very nature of quantum matter and its potential applications.

Precise polynomial interpolation of numerically obtained second Rényi entropy corner contributions for Dirac fermions and free bosons allows for accurate determination of <span class="katex-eq" data-katex-display="false">a_{2}^{f/b}(\pi/3)</span> and <span class="katex-eq" data-katex-display="false">a_{2}^{f/b}(2\pi/3)</span> as required for further analysis. — Precise polynomial interpolation of numerically obtained second Rényi entropy corner contributions for Dirac fermions and free bosons allows for accurate determination of $a_{2}^{f/b}(\pi/3)$ and $a_{2}^{f/b}(2\pi/3)$ as required for further analysis.

Cornering Entanglement: A New Signature of Quantum Order

Entanglement entropy typically scales with the area of the boundary defining an entangling region. However, when this boundary includes geometric corners, an additional contribution to the total entanglement entropy arises, designated as the ‘Corner Contribution’. This contribution is not accounted for by simple area scaling laws and represents a deviation from the expected behavior at smooth boundaries. The magnitude of this corner contribution is dependent on the specific geometry of the corner and the underlying quantum system, but it consistently manifests as a logarithmic divergence as the corner is approached. This implies that the local geometry at corners plays a crucial role in determining the entanglement structure and necessitates specialized techniques to isolate and characterize this unique contribution to the total entanglement entropy.

Subtracted Corner Entanglement Entropy (SCEE) is a technique used to specifically quantify the contribution of geometric corners to the total entanglement entropy. Standard calculations of entanglement entropy often include contributions from the entire entangling region boundary, obscuring the unique signal originating from the corners themselves. SCEE addresses this by analytically or numerically removing the contributions from the straight edges of the boundary, effectively isolating the corner contribution. This is achieved through a subtraction process that relies on defining a reference geometry – typically a smoothed or rounded corner – and calculating the difference in entanglement entropy between the actual corner geometry and this reference. The resulting value represents the corner contribution, allowing for precise analysis of how these geometric features influence entanglement properties.

The corner contribution to entanglement entropy, while geometry-dependent, exhibits universal scaling behavior describable by a scaling function $\mathcal{S}(x)$ . This function encapsulates the essential physics of the corner, independent of specific microscopic details or the overall system size. Analysis of this scaling function reveals critical exponents and provides insights into the underlying conformal field theory governing the entanglement structure. Specifically, the form of $\mathcal{S}(x)$ dictates how the corner contribution scales with changes in system parameters, allowing for the characterization of universality classes and the determination of fundamental properties like the central charge of the corresponding conformal field theory.

Corner contributions to entanglement entropy serve as a sensitive probe of topological order in condensed matter systems and quantum field theories. Systems exhibiting non-trivial topology, such as those with topological phases or exhibiting fractionalized excitations, manifest distinct corner entanglement signatures compared to systems lacking these features. The magnitude and functional form of the corner contribution are directly related to the topological invariants characterizing the system, allowing for their determination via entanglement measurements. Specifically, these contributions are not merely geometric effects but reflect the presence of gapless boundary modes or zero-energy states localized at the corners of the entangling region, which are hallmarks of topological order. Analyzing these corner contributions provides a means to distinguish between different topological phases and characterize their properties, even in the absence of local order parameters.

Non-equilibrium dynamics analysis reveals that the variance <span class="katex-eq" data-katex-display="false">\Delta S_{2}</span> exhibits a size-independent plateau and scales linearly with τ, yielding a corner contribution of <span class="katex-eq" data-katex-display="false">s_{c}=0.069(10)</span> at <span class="katex-eq" data-katex-display="false">U=6</span>. — Non-equilibrium dynamics analysis reveals that the variance $\Delta S_{2}$ exhibits a size-independent plateau and scales linearly with τ, yielding a corner contribution of $s_{c}=0.069(10)$ at $U=6$ .

Simulating the Quantum Realm: Quantum Monte Carlo in Action

Quantum Monte Carlo (QMC) methods are computational techniques used to study the behavior of interacting quantum many-body systems. These systems, often described by models such as the Hubbard model – a fundamental model in condensed matter physics addressing electron interactions – are analytically intractable due to the exponential growth of the Hilbert space with system size. QMC overcomes this limitation by employing stochastic methods, specifically Monte Carlo integration, to evaluate quantum mechanical properties. Unlike deterministic numerical methods, QMC relies on random sampling to approximate solutions, allowing for simulations of larger systems and more complex interactions. The accuracy of QMC results depends on the quality of the stochastic sampling and the control of statistical errors, but the approach provides a powerful tool for investigating phenomena inaccessible to other computational techniques.

Projector Quantum Monte Carlo (PQMC) is a stochastic method used to approximate the ground state wave function of many-body quantum systems. It operates by repeatedly applying a projection operator, $\exp(-\tau H)$ , to an initial trial wave function, effectively filtering out excited states and converging towards the ground state. This projection is performed using Markov Chain Monte Carlo (MCMC) techniques, enabling the calculation of ground state properties, including entanglement entropy. The efficiency of PQMC stems from its ability to directly sample the ground state, bypassing the need to solve the Schrödinger equation directly. Calculations of entanglement entropy, a measure of quantum correlations, are then performed on the sampled ground state wave function, providing insights into the system’s quantum properties and phases.

The Incremental Algorithm represents a refinement of Quantum Monte Carlo (QMC) methods specifically designed to improve the calculation of $S_2$ , the Second Rényi Entropy. Traditional QMC calculations of entanglement entropies can be computationally expensive due to the need to store and process large numbers of random walkers. The Incremental Algorithm addresses this by updating the walkers in smaller, iterative steps, reducing memory requirements and improving computational efficiency. This approach allows for more precise calculations of $S_2$ , particularly for larger systems where conventional methods become intractable, and enables the study of entanglement properties in complex quantum systems with greater accuracy and reduced computational cost.

Imaginary-time evolution, a core technique within Quantum Monte Carlo (QMC) simulations, is employed to isolate the ground state of a quantum many-body system. This method leverages the mathematical equivalence between the imaginary-time Schrödinger equation and the time-independent Schrödinger equation; by propagating an initial trial wavefunction in imaginary time τ, the wavefunction evolves towards the eigenstate with the lowest energy – the ground state. The process effectively filters out excited states, which decay exponentially with increasing τ, leaving the ground state as the dominant component. This allows for the accurate calculation of ground state properties, such as energy and entanglement, by sampling configurations weighted by the ground state wavefunction.

Recent research has established a universal non-equilibrium scaling law applicable to corner entanglement in two-dimensional systems. Utilizing Quantum Monte Carlo simulations, corner entanglement coefficients were calculated for both free Dirac fermions and the antiferromagnetic phase of the Hubbard model. Specifically, a coefficient of 0.3116 was obtained for free Dirac fermions, while the antiferromagnetic phase at an interaction strength of U=6 yielded a coefficient of 0.069(10). These results demonstrate the applicability of a consistent scaling law across differing quantum systems and provide benchmark values for future computational studies of correlated materials.

Non-equilibrium dynamics at the GNY quantum critical point (<span class="katex-eq" data-katex-display="false">\Delta S_{2}</span>) exhibit a size-independent plateau for short timescales (<span class="katex-eq" data-katex-display="false">\tau \ll L</span>), with a logarithmic dependence on time yielding a non-equilibrium coefficient of <span class="katex-eq" data-katex-display="false">s_{c}^{neq} = 0.345(7)</span> and validating universal scaling behavior as demonstrated by data collapse onto a master curve. — Non-equilibrium dynamics at the GNY quantum critical point ( $\Delta S_{2}$ ) exhibit a size-independent plateau for short timescales ( $\tau \ll L$ ), with a logarithmic dependence on time yielding a non-equilibrium coefficient of $s_{c}^{neq} = 0.345(7)$ and validating universal scaling behavior as demonstrated by data collapse onto a master curve.

Honeycomb Lattices and Beyond: Materials Shaped by Entanglement

The honeycomb lattice, a repeating pattern of hexagonal cells, provides a remarkably versatile framework for investigating the fundamental relationship between strong interactions and quantum entanglement. This structure isn’t merely a geometric curiosity; it serves as a simplified, yet powerful, model for understanding complex quantum phenomena occurring in real materials. Researchers leverage this lattice to explore how strongly correlated electrons – those exhibiting significant interactions with each other – give rise to emergent behaviors and novel quantum phases. By studying these interactions within the constrained geometry of the honeycomb, scientists can gain insights into how entanglement, a uniquely quantum property linking particles even at a distance, dictates the material’s properties. The relative simplicity of the lattice allows for detailed theoretical calculations and numerical simulations, providing a crucial testing ground for understanding the complex interplay between correlation and entanglement in condensed matter systems – a connection vital for designing materials with tailored quantum characteristics.

Dirac semimetals represent a fascinating class of materials where the conduction and valence bands meet at specific points in momentum space, known as Dirac points, resulting in unique electronic behaviors. These materials often exhibit a honeycomb lattice structure – a repeating pattern of hexagonal cells – which is crucial to their unusual properties. The arrangement of atoms within this lattice creates a linear dispersion relation near the Dirac points, meaning electrons behave as massless Dirac fermions, similar to particles in high-energy physics. This allows for exceptionally high electron mobility and opens possibilities for novel electronic devices, as well as exhibiting topologically protected surface states that are robust against imperfections. The interplay between the lattice geometry and the electronic band structure defines the material’s conductivity and response to external stimuli, making Dirac semimetals a vibrant area of condensed matter research.

Investigations employing the Hubbard Model on a honeycomb lattice, coupled with Quantum Monte Carlo (QMC) simulations, have definitively demonstrated the emergence of an antiferromagnetic phase. This phase arises from the strong interactions between electrons within the material, leading to a spontaneous alignment of electron spins in an alternating pattern. The simulations reveal that as the strength of electron-electron interactions increases, the system undergoes a phase transition, shifting from a paramagnetic state to this ordered antiferromagnetic arrangement. This finding is significant because it provides a concrete example of how microscopic interactions can give rise to macroscopic magnetic order, offering insights into the behavior of correlated electron systems and potentially paving the way for the design of novel magnetic materials. Further analysis of this phase confirms the presence of collective spin excitations, indicative of a robust and well-defined magnetic ground state.

The emergence of an antiferromagnetic phase within the honeycomb lattice isn’t merely a shift in magnetic order, but a profound alteration of the material’s fundamental excitations. This broken symmetry, where opposing magnetic moments align, gives rise to $\text{Goldstone modes}$ – gapless excitations representing collective fluctuations of the ordered moments. These modes are intrinsically linked to the spontaneous breaking of continuous symmetry, and their existence confirms a deep connection between the material’s magnetic order, its underlying entanglement structure, and its observable physical properties. The presence of these excitations isn’t simply a consequence of the antiferromagnetic phase; it defines it, indicating a system capable of responding to external stimuli in a unique and collective manner, and offering a pathway to control and manipulate material behavior through entanglement-driven phenomena.

Quantum Monte Carlo simulations of the Hubbard model on a honeycomb lattice reveal a compelling connection between material properties and fundamental physics. Specifically, calculations focusing on the antiferromagnetic phase at an interaction strength of U=7 demonstrate a corner entanglement coefficient of 0.070. This value provides strong validation of theoretical predictions concerning gapless bosonic excitations, indicating a robust link between the system’s entanglement structure and its low-energy behavior. The precise measurement of this coefficient not only confirms the accuracy of the simulations but also provides a benchmark for understanding the emergence of collective phenomena in strongly correlated materials, furthering insight into the interplay of quantum entanglement and symmetry breaking.

Entanglement and Topology: A Future Shaped by Quantum Order

The concept of ‘Topological Entanglement Entropy’ offers a powerful diagnostic for identifying topological order within a quantum system. Unlike conventional entanglement, which diminishes with distance, topological entanglement exhibits a constant, non-zero value even when considering spatially separated regions. This persistent entanglement isn’t tied to local correlations but rather to the global, topological properties of the system’s wave function. Specifically, this constant term appears as a contribution to the entanglement entropy – a measure of quantum correlations – signaling the presence of long-range quantum entanglement protected by the system’s topology. The magnitude of this topological entanglement entropy is directly related to the number of degenerate ground states, providing a quantifiable link between entanglement and the underlying topological order, and ultimately offering insights into novel quantum phases of matter.

The pursuit of novel quantum phases of matter is fundamentally intertwined with understanding the connection between quantum entanglement and topology. Unlike traditional phases distinguished by symmetry breaking, topological phases are characterized by robust properties stemming from the global properties of the wavefunction, and these properties are often revealed through measurements of entanglement. Specifically, the way quantum particles become entangled – linked in such a way that they share the same fate, no matter how far apart – can act as a fingerprint for topological order. By quantifying entanglement patterns, physicists can identify materials exhibiting these exotic states, even when conventional methods fail. This approach transcends simple observation; it allows for the characterization of these phases, detailing their unique properties and potential for technological application, promising advancements in areas like quantum computing and materials science.

Continued advancements in materials science hinge on the development of computational methods capable of accurately modeling complex quantum systems. Current techniques often struggle with the exponential growth in computational demand as system size increases, limiting the scope of materials that can be effectively studied. Future research prioritizes the creation of more efficient algorithms and the leveraging of high-performance computing resources to overcome these limitations. This will enable scientists to explore a broader range of materials-including those with strong correlations and topological properties-and predict their behavior with greater precision. The ultimate goal is to accelerate the discovery of novel quantum phases of matter and design materials tailored for specific applications in areas like superconductivity, quantum computing, and energy storage.

The potential to engineer materials exhibiting previously unattainable characteristics represents a significant leap forward, driven by a deeper understanding of entanglement and topology. This isn’t merely about incremental improvements; researchers envision crafting substances with tailored electronic behavior, superconductivity at higher temperatures, or entirely new forms of quantum computation. Exploiting topological principles allows for the creation of robust quantum states, protected from local disturbances – a critical requirement for stable and scalable quantum technologies. Such advancements promise to reshape fields ranging from energy transmission and storage to advanced sensing and secure communication, potentially ushering in an era of unprecedented technological innovation and fundamentally altering the landscape of materials science and quantum engineering.

The research illuminates how universal entanglement growth, particularly the corner contribution to entanglement entropy, emerges not from imposed structure but from the interplay of local rules governing quantum many-body systems. This mirrors the observation that order doesn’t require architects; it arises organically. As Bertrand Russell noted, “The good life is one inspired by love and guided by knowledge.” This pursuit of knowledge, as demonstrated by the imaginary-time dynamics protocol, reveals fundamental scaling laws inherent in quantum criticality, showcasing how self-organization-the system revealing its properties through internal dynamics-is ultimately stronger than any attempt at forced design. The study confirms that constraints-like those present in quantum systems-stimulate inventiveness, leading to a deeper understanding of these complex phenomena.

The Road Ahead

The demonstrated protocol, leveraging imaginary-time dynamics to probe entanglement growth, subtly shifts the focus from seeking top-down control of quantum systems to understanding emergent behavior. The system is a living organism where every local connection matters; forcing a narrative of predetermined outcomes feels increasingly… quaint. This work doesn’t solve quantum criticality, of course. It reveals a tool, a method to dissect the scaling laws governing entanglement as a proxy for the underlying physics, but the true complexity lies in extending this approach to genuinely disordered systems, where the very notion of a ‘universal’ scaling law may be an oversimplification.

A pressing challenge remains in bridging the gap between these carefully constructed, controlled simulations and the messy reality of non-equilibrium phenomena. The corner contribution to entanglement, so elegantly extracted here, feels like a beacon, but the signal weakens as one considers more complex geometries or interactions. Future investigations will likely need to embrace tensor network methods, or even machine learning techniques, not to replace analytical understanding, but to navigate the exponentially growing parameter space.

Ultimately, this research suggests that top-down control often suppresses creative adaptation. The path forward isn’t about dictating quantum behavior, but about crafting environments where the inherent self-organization of these systems can be observed and understood. The true prize won’t be predicting the future, but learning to read the present – the subtle whispers of entanglement as it propagates through the quantum landscape.

Original article: https://arxiv.org/pdf/2512.23361.pdf

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