Beyond Order: Topology’s Role in Quantum Phase Transitions

Author: Denis Avetisyan

A new perspective on quantum criticality emerges as researchers explore the interplay between topological states and the breakdown of conventional order.

The system demonstrates that topological phase transitions, occurring at a multi-critical point, necessitate crossing a transition point when connecting trivial and nontrivial phases, resulting in localized zero-energy modes at spatial interfaces-whether arising from band inversion between gapped phases or kinetic inversion at the boundary of a critical point and vacuum.

This review details the emerging field of topological physics in quantum critical systems, focusing on gapless systems and classifications of phase transitions beyond symmetry-breaking paradigms.

Conventional understandings of topological order posit a requirement for a bulk energy gap, seemingly precluding its existence in gapless quantum critical systems. This challenges a foundational tenet of condensed matter physics, yet recent research, detailed in ‘Topological physics in quantum critical systems’, demonstrates that nontrivial topology can emerge at and even define these critical points and phases. This review explores how extending topological concepts to gapless systems offers a new framework for classifying quantum phase transitions-going beyond traditional symmetry-breaking scenarios-and reveals potentially robust edge states without a conventional insulating gap. Could this framework ultimately redefine our understanding of quantum criticality and unlock new avenues for realizing topologically protected states in materials?

Beyond Conventional Order: The Rise of Topological Quantum Matter

For decades, condensed matter physics categorized materials based on how their symmetry was broken – a change in order that defined a new phase, like water freezing into ice. However, this framework falters when confronted with states that are remarkably stable and conduct electricity without losing energy – what scientists call dissipationless states. Conventional symmetry-breaking classifications simply cannot account for these robust phenomena, because the stability doesn’t arise from a change in order, but rather from the material’s global properties. These states are uniquely resistant to imperfections and disturbances, meaning a local disruption won’t destroy the conducting pathway. This limitation spurred a search for new organizing principles, ultimately leading to the development of topological physics, which focuses on characteristics that are inherent to the overall structure of a material, rather than localized arrangements.

Conventional understanding of materials relies on identifying phases through symmetry breaking – how a material’s structure changes and loses certain symmetries. However, topological physics introduces a revolutionary approach by categorizing phases using topological invariants – mathematical quantities that describe the global properties of a material’s electronic structure. Unlike symmetry-based classifications, these invariants are remarkably robust; they remain unchanged under continuous deformations, meaning small, local disturbances – imperfections or noise – cannot alter the fundamental phase of the material. This inherent stability arises because topological invariants are determined by the material’s overall “shape” in a higher-dimensional space of possible electronic configurations, not by its precise local details. Consequently, materials characterized by non-trivial topological invariants exhibit protected states – such as conducting surface states in insulators – that are immune to scattering from impurities or defects, offering unprecedented potential for creating robust and reliable electronic devices and exploring fundamentally new quantum phenomena.

The advent of topological physics predicts materials exhibiting properties fundamentally different from those described by conventional condensed matter theory. These materials aren’t characterized by what they break in symmetry, but rather by their inherent topology – a mathematical property describing how a shape can be deformed without being cut or glued. This leads to the emergence of protected edge states, where electrons flow along the material’s surface without scattering, even in the presence of defects or impurities, promising dissipationless conduction. Furthermore, some topological phases support fractionalized excitations, meaning that the fundamental particles within the material behave as if they are fractions of an electron – quasiparticles with unusual statistics and potential applications in fault-tolerant quantum computation. These exotic phenomena, shielded by the underlying topology, represent a paradigm shift in materials science and offer a pathway toward realizing robust and novel quantum technologies.

The pursuit of stable quantum technologies hinges on overcoming the inherent fragility of quantum states, which are easily disrupted by environmental noise. Topological quantum matter offers a compelling pathway to address this challenge; these exotic phases of matter are characterized by robust, protected states that are immune to local disturbances. Unlike conventional materials where quantum information is stored in easily-perturbed configurations, topological phases encode information in the topology of the material – a global property unaffected by minor imperfections. This inherent resilience promises the creation of quantum bits, or qubits, that maintain coherence for significantly longer periods, enabling more complex and reliable quantum computations. Consequently, a deepened understanding of topological phases is not merely an academic exercise, but a crucial step towards realizing practical and scalable quantum devices with applications ranging from secure communication to advanced materials science and drug discovery.

The system exhibits five distinct quantum phases-gapped and gapless noninteracting states characterized by integer or half-integer topological invariants-which are accessible via perturbations of a mother theory and feature topologically nontrivial critical points leading to localized edge modes, as demonstrated by the Bloch Hamiltonian and real-space Majorana hopping configurations.

Unveiling Gapless Symmetry-Protected Phases: A New Frontier

Conventional topological phases are characterized by a bulk energy gap, meaning there is a minimum energy required to excite the system. Gapless symmetry-protected topological (SPT) phases, however, deviate from this requirement; they can exist without such a gap in their bulk spectrum. This absence of a bulk energy gap fundamentally alters the nature of the topological order, as the defining boundary states are not separated from the bulk by an energy barrier. Instead, the topological protection in these phases arises directly from the combination of symmetry and the presence of gapless excitations-excitations with zero or near-zero energy-within the bulk material. The lack of a bulk gap necessitates different theoretical tools and experimental probes to characterize and understand these phases compared to their gapped counterparts.

Gapless symmetry-protected topological (SPT) phases demonstrate properties directly resulting from the combined effect of crystalline symmetries and the presence of gapless excitations at the boundary or surface. These excitations, unlike those in conventional gapped topological phases, do not require an energy gap to exist, leading to a continuous spectrum of low-energy states. The specific symmetry protecting these gapless modes dictates their nature and dispersion, influencing observable physical characteristics such as surface currents or localized edge states. Consequently, the interplay between symmetry and these excitations gives rise to novel transport phenomena and distinct responses to external fields, differentiating these phases from both trivial and gapped topological states.

The established theoretical framework for symmetry-protected topological (SPT) phases historically predicted their existence as gapped states, meaning a finite energy is required to create excitations. The recent discovery of intrinsic gapless SPT phases-phases that cannot be realized with a bulk energy gap-directly contradicts this foundational assumption. These phases necessitate a re-evaluation of classification schemes for topological states, as conventional methods relying on gapped systems are insufficient to fully characterize them. This challenges the existing understanding of topological order and demands the development of new theoretical tools capable of describing and predicting the behavior of gapless symmetry-protected phases, potentially expanding the scope of topological materials research beyond gapped systems.

Current research integrates concepts from topological physics and the study of quantum critical systems to demonstrate the emergence of gapless symmetry-protected topological (SPT) phases. This synthesis leverages established techniques for analyzing quantum critical points-characterized by diverging correlation lengths and scale invariance-and applies them to systems exhibiting topological order. Specifically, the interplay between symmetry constraints and critical fluctuations facilitates the realization of these phases, which are distinguished by protected gapless boundary modes and non-trivial topological invariants despite lacking a bulk energy gap. Investigations utilize renormalization group methods and numerical simulations to map out phase diagrams and characterize the critical exponents associated with these novel states of matter, confirming their distinct properties from conventional topological insulators and superconductors.

The phase diagram distinguishes between non-intrinsic and intrinsic gapless symmetry-protected topological phases based on the strength of GG-defect fluctuations, revealing that while fully proliferating defects can induce a gapped phase in the non-intrinsic case, further increasing fluctuations beyond the critical point in the intrinsic case does not drive a phase transition.

Constructing Gapless Topology: Methods and Validation

The decorated domain wall construction is a technique for creating gapless symmetry-protected topological (SPT) phases by intentionally introducing topological defects – specifically, domain walls – into a system. This method involves decorating these domain walls with specific degrees of freedom that transform non-trivially under the relevant symmetries. By carefully selecting these decorating degrees of freedom and their interactions, researchers can engineer gapless boundary states localized at the domain walls, thus realizing a gapless SPT phase. The systematic nature of this construction lies in its ability to predictably relate the symmetry properties of the decorating degrees of freedom to the resulting gapless edge states and bulk topological invariants, allowing for targeted design of novel topological phases.

Quantum Monte Carlo (QMC) simulation serves as a crucial verification method for theoretical models predicting topological phases of matter. Due to the inherent complexity of many-body quantum systems, analytical solutions are often intractable; QMC provides a numerical approach to approximate the ground state and evaluate physical observables. Specifically, QMC allows researchers to calculate properties like correlation functions and entanglement entropy, which are essential for characterizing topological order and identifying phase transitions. By systematically varying parameters within a theoretical model and performing QMC simulations, it is possible to construct and map out the resulting phase diagrams, confirming the predicted existence and stability of gapless topological phases and validating the accuracy of the theoretical constructions. Furthermore, QMC can reveal the limitations of theoretical approximations and guide the development of more refined models.

Topological Holography represents an advancement in the classification of topological phases of matter by moving beyond the traditional limitations of equilibrium systems and interactions restricted to short-range. This framework utilizes the principles of duality and emergent symmetries to characterize phases that are inaccessible through conventional methods, such as those reliant on symmetry-protected topological invariants defined for ground states. Specifically, Topological Holography allows for the identification of topological order arising from long-range entanglement and non-equilibrium dynamics, effectively broadening the scope of phases considered beyond the standard ten-fold classification. The approach involves mapping the bulk topological phase to a lower-dimensional boundary theory, enabling the study of topological properties through boundary observables and simplifying the analysis of complex systems.

Recent advancements in superconducting quantum processor technology have enabled experimental investigations into topological phases of matter utilizing systems with up to 100 qubits. These processors serve as platforms to physically realize and probe the behavior predicted by theoretical models, allowing for direct observation of phenomena associated with these phases. Specifically, researchers are employing these quantum devices to simulate the Hamiltonian of various topological models and measure key observables that characterize the presence and properties of topological order. The scale of these processors, while still limited, represents a significant step towards bridging the gap between theoretical prediction and experimental verification in the field of topological quantum matter, and provides a pathway towards exploring more complex topological phases.

Modifying domain wall configurations on a triangular lattice transitions the system between trivial paramagnetic, gapped symmetry-protected topological (SPT), and gapless phases, ultimately influencing the number of edge modes-ranging from one (<span class="katex-eq" data-katex-display="false">c=1</span>) to two (<span class="katex-eq" data-katex-display="false">c=2</span>)-as detailed in Ref.[31]. — Modifying domain wall configurations on a triangular lattice transitions the system between trivial paramagnetic, gapped symmetry-protected topological (SPT), and gapless phases, ultimately influencing the number of edge modes-ranging from one ( $c=1$ ) to two ( $c=2$ )-as detailed in Ref.[31].

Expanding Horizons: Long-Range Interactions and Non-Equilibrium Dynamics

The conventional understanding of topological phases – states of matter distinguished by robust, surface properties – traditionally focuses on systems with short-range interactions, where particles directly influence only their immediate neighbors. However, the inclusion of long-range interactions – forces extending across significant distances within a material – dramatically broadens the scope of possible topological states. These extended interactions fundamentally alter the energy landscape, enabling the emergence of novel topological phases not achievable in short-range systems. Researchers have discovered that long-range interactions can stabilize exotic topological defects and lead to fractionalized excitations with unusual statistical properties. Furthermore, the ability to tune the strength and nature of these long-range forces offers a powerful pathway to engineer materials with precisely tailored topological characteristics, potentially unlocking advanced functionalities for future technologies.

Recent investigations into non-equilibrium dynamics demonstrate that topological order isn’t solely a static property of materials, but can instead be actively created and controlled through time-dependent processes. Researchers are discovering that by driving systems away from equilibrium-using techniques like pulsed lasers or rapid parameter changes-it’s possible to induce topological phases that wouldn’t exist under standard conditions. This dynamic generation of topological order allows for unprecedented manipulation of material properties; for example, transient topological edge states can be created and steered, potentially enabling novel functionalities in quantum devices. Furthermore, this approach offers a pathway to explore exotic topological phases inaccessible in equilibrium, expanding the possibilities for robust quantum information processing and potentially leading to new forms of quantum computation and communication where information is encoded in these dynamically-created, topologically-protected states.

The ability to engineer materials with specific topological properties hinges on a detailed comprehension of long-range interactions within their constituent components. These interactions, extending beyond nearest-neighbor connections, fundamentally alter the electronic band structure and dictate the emergence of protected edge states – hallmarks of topological insulators and superconductors. By carefully controlling the strength, range, and nature of these interactions – whether magnetic, electrostatic, or vibrational – researchers can effectively ‘tune’ a material’s topological phase. This precise manipulation allows for the design of materials exhibiting desired characteristics, such as enhanced robustness against disorder, specific surface conductivity, or tailored responses to external stimuli. Ultimately, a deep understanding of these interactions represents a crucial step towards realizing practical applications in spintronics, quantum computing, and advanced materials science, offering pathways to devices with unprecedented performance and functionality.

The burgeoning field of topological quantum technologies stands to be profoundly impacted by recent advances in understanding long-range interactions and non-equilibrium dynamics. Exploiting topological order-a state of matter characterized by robust, protected states-offers a pathway to building dramatically more stable and reliable quantum bits, or qubits. Unlike conventional qubits susceptible to environmental noise, topologically protected qubits encode information in the global properties of the material, rendering them far less vulnerable to decoherence. This enhanced stability is crucial for scaling up quantum computers to tackle complex problems currently intractable for classical machines. Beyond computation, harnessing topological order promises advancements in areas like quantum communication, enabling secure data transmission, and the development of novel quantum sensors with unparalleled sensitivity. The realization of these technologies necessitates materials where topological properties are not merely observed, but engineered and controlled with precision, paving the way for a new generation of quantum devices.

Topological edge modes, exemplified in a one-dimensional spin-1 Heisenberg chain with open boundaries, arise from the unique band structure characteristic of topological insulators and the associated <span class="katex-eq" data-katex-display="false">\mathbb{Z}_2</span> topological phase. — Topological edge modes, exemplified in a one-dimensional spin-1 Heisenberg chain with open boundaries, arise from the unique band structure characteristic of topological insulators and the associated $\mathbb{Z}_2$ topological phase.

A New Era of Quantum Materials Design

The creation of gapless symmetry-protected topological phases represents a significant leap forward in materials science, offering the potential for devices exhibiting extraordinary resilience and performance. Unlike conventional materials vulnerable to imperfections and external disturbances, these phases leverage fundamental symmetries to shield electronic states from disruption. This protection stems from a unique arrangement of electrons at the material’s surface, forming robust conducting channels that are immune to backscattering from impurities or defects. The absence of an energy gap – hence ‘gapless’ – further enhances functionality, allowing for efficient electron transport and potentially enabling novel electronic devices with minimal energy loss. Researchers envision these materials as building blocks for next-generation technologies, including more stable and efficient quantum computers, highly sensitive sensors, and ultra-low-power electronics, all predicated on the inherent robustness and predictable behavior of topologically protected states.

The pursuit of realizing the potential of topological materials necessitates a broadened search for suitable material systems beyond currently studied compounds. Future investigations will concentrate on identifying novel platforms – including, but not limited to, two-dimensional van der Waals materials, heterostructures, and carefully designed artificial structures – that can host and stabilize topological phases. Crucially, this materials discovery must be paired with the development of advanced characterization techniques capable of probing the subtle signatures of topological order. Techniques such as angle-resolved photoemission spectroscopy with enhanced resolution, scanning tunneling microscopy sensitive to topological surface states, and resonant inelastic x-ray scattering will be essential for not only confirming the presence of these phases but also for mapping their properties and guiding further material design. This synergistic approach – materials innovation coupled with characterization advancement – is poised to accelerate the field and unlock the full potential of topological quantum materials.

The pursuit of topological order extends beyond fundamental physics, aiming to revolutionize quantum technologies. Researchers envision leveraging the inherent robustness of topological states – protected from local disturbances – to build dramatically more stable and reliable quantum bits, or qubits. Unlike conventional qubits susceptible to environmental noise, topologically protected qubits encode information in the global properties of the material, rendering them far less vulnerable to decoherence. This enhanced stability is crucial for scaling up quantum computers to tackle complex problems currently intractable for classical machines. Beyond computation, harnessing topological order promises advancements in areas like quantum communication, enabling secure data transmission, and the development of novel quantum sensors with unparalleled sensitivity. The realization of these technologies necessitates materials where topological properties are not merely observed, but engineered and controlled with precision, paving the way for a new generation of quantum devices.

Advancing the field of quantum materials necessitates a deeply integrated approach, bringing together the distinct expertise of theoretical physicists, experimental scientists, and materials scientists. Theorists provide the foundational understanding of novel quantum phases and predict material properties, while experimentalists design and execute the crucial tests to validate these predictions and uncover new phenomena. Materials scientists, in turn, focus on the synthesis and characterization of materials with the precise structural and chemical control needed to realize these exotic states. This synergistic interplay – where computation guides synthesis, experiment informs theory, and materials properties drive further innovation – is not merely beneficial, but fundamentally required to overcome the considerable challenges in discovering and harnessing the potential of these complex systems and ultimately translate them into tangible technological advancements.

The doped Ising-Hubbard model exhibits a complex phase diagram with topological properties detectable via the string order parameter, revealing a unique ground state under periodic boundary conditions but a twofold degeneracy under open boundary conditions characterized by an energy splitting determined by the spin correlation length.

The exploration of quantum critical systems, as detailed in this review, reveals a landscape where order isn’t solely defined by broken symmetry, but by the very fabric of topological order. This echoes Leonardo da Vinci’s observation: “Simplicity is the ultimate sophistication.” The pursuit of understanding these gapless systems, with their emergent edge states and unconventional phase transitions, demands a stripping away of complexity to reveal the fundamental, underlying principles. Just as a masterful artist reduces a subject to its essential forms, physicists seek to distill these quantum phenomena into elegant, topologically protected states. The interconnectedness of seemingly disparate elements – topology and criticality – illustrates that structure dictates behavior, and that a holistic view is crucial for anticipating weaknesses within the system.

What Lies Ahead?

The exploration of topological physics in quantum critical systems, as this review attempts to demonstrate, largely shifts the focus from what breaks at a phase transition to how the system remains coherent – or fails to. This is, of course, a subtle but crucial distinction. If the system looks clever, it probably is fragile. The current framework, while offering novel classifications beyond simple symmetry breaking, still relies heavily on idealized models. The true challenge lies in confronting the inherent messiness of real materials, where disorder and interactions inevitably conspire to obscure the elegant topological features predicted by theory.

A pressing concern is the extension of these concepts to genuinely disordered systems. Symmetry-protected topological phases, by definition, require a degree of order. What happens when that protection fails? One suspects that a more robust, intrinsic form of topological order – less reliant on pristine conditions – will prove essential. The search for such phases is, predictably, proving difficult, largely because it requires relinquishing the comfortable notion that “interesting” physics always occurs at a sharply defined point.

Ultimately, the architecture of any successful theory is the art of choosing what to sacrifice. The field currently prioritizes topological characterization, often at the expense of fully understanding the underlying microscopic mechanisms. Future progress demands a more holistic view, acknowledging that topology is merely a symptom – albeit a useful one – of deeper, more fundamental principles. The real prize will not be simply finding new topological phases, but understanding why they exist.

Original article: https://arxiv.org/pdf/2601.00184.pdf

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