Entangled States, Topological Order, and the Fabric of Reality

Author: Denis Avetisyan

New research reveals a fundamental link between the complex entanglement of many particles and the underlying topological order governing their long-range behavior.

This review establishes a concrete correspondence between multipartite entanglement and the partition function of the topological quantum field theory describing a gapped system, validated through Levin-Wen string-net models.

A persistent challenge in condensed matter physics is bridging the gap between many-body entanglement and emergent topological order. This paper, ‘From Multipartite Entanglement to TQFT’, proposes a concrete relationship between genuine multipartite entanglement-characterized by a $d$ -dimensional manifold $M$ -in the ground state of a gapped system and the partition function of its low-energy topological quantum field theory (TQFT). Specifically, we demonstrate that ground state entanglement directly constrains the modular tensor category defining the TQFT, verified through analysis of (2+1)-dimensional Levin-Wen string-net models. Does this entanglement-TQFT correspondence offer a new route to classifying and understanding phases of matter beyond traditional symmetry-breaking paradigms?

Fragility and the Promise of Topological Order

Conventional quantum states, while powerful in theory, frequently suffer from a critical vulnerability: fragility. These states are often defined by local properties, meaning a disturbance – even a minor one – at a specific point can disrupt the entire system and lead to decoherence, the loss of quantum information. This susceptibility stems from the fact that information is encoded in easily perturbed degrees of freedom, like the precise energy or position of a particle. Imagine a delicately balanced tower of blocks; a single nudge can cause it to collapse. This inherent instability poses a significant obstacle to building practical quantum technologies, as maintaining the coherence of qubits – the quantum equivalent of bits – requires extreme isolation and control, shielding them from any external influence. The challenge lies in finding ways to encode and protect quantum information from these inevitable local perturbations, paving the way for more robust and reliable quantum systems.

Unlike conventional quantum states where information resides in local properties – easily disrupted by environmental noise – topological phases exhibit remarkable resilience. These exotic states encode information not in the presence or absence of particles at specific locations, but in the global arrangement and connectivity of the system as a whole. This means the crucial data is distributed across the material in a non-local manner, akin to the pattern of knots in a rope rather than the individual strands. Disturbances affecting a small region have minimal impact because the essential information is protected by the system’s overall topology – its fundamental shape and connectivity. Consequently, topological phases offer a pathway to building robust quantum technologies, less vulnerable to the decoherence that plagues traditional quantum systems, and potentially enabling more stable quantum computation and information storage.

The exploration of topological phases necessitates a departure from traditional frameworks of quantum description, demanding novel theoretical tools and concepts. Conventional methods, focused on local order parameters and symmetry breaking, prove inadequate in characterizing these states where information is encoded non-locally. Physicists are increasingly turning to concepts borrowed from fields like topology and geometry – areas of mathematics concerned with properties preserved under continuous deformations – to develop a more complete understanding. This includes utilizing concepts like $k\$-theory$ and $homotopy groups$ to classify topological phases and predict their unique properties. The pursuit of this new theoretical landscape not only deepens the understanding of condensed matter systems but also fosters cross-disciplinary innovation, challenging the foundations of quantum mechanics and pushing the boundaries of theoretical physics itself.

Anyons and the Mathematical Language of Braids

Modular tensor categories (MTCs) formalize the behavior of anyons, quasiparticles exhibiting exchange statistics differing from the bosonic or fermionic cases. In traditional particle physics, the wavefunction acquires a phase of +1 for bosons and -1 for fermions upon particle exchange. Anyons, described within the MTC framework, acquire an arbitrary complex phase $e^{i\theta}$ upon exchange, where θ is not necessarily a multiple of π. This non-trivial exchange statistic is a direct consequence of the category’s mathematical structure, specifically the braiding relations governing how anyonic worldlines can be interwoven. The rigorous nature of MTCs ensures the consistency of physical predictions derived from these exotic exchange statistics, allowing for a mathematically sound treatment of anyonic systems.

Quantum dimension, denoted as $q$ , is a complex number assigned to each simple object, or irreducible representation, within a modular tensor category. It is not a physical dimension in the conventional sense, but rather a numerical property characterizing the ‘size’ or ‘weight’ of that representation in the category’s fusion rules. Specifically, for a simple object $X$ , its quantum dimension satisfies the equation $q(X)q(X') = \sum_{Y} N_{XX'}^Y q(Y)$ , where $N_{XX'}^Y$ are the fusion multiplicities and the sum is over all simple objects $Y$ . The square of the quantum dimension, $|q(X)|^2$ , gives the multiplicity of the trivial representation in the fusion of $X$ with its dual, and is therefore always a positive integer. A crucial property is that the total quantum dimension of the category, the product of the quantum dimensions of all simple objects, must be a root of unity to ensure the category admits a braiding structure and defines a consistent topological quantum field theory.

Modular tensor categories (MTCs) facilitate the construction of consistent physical models for topological phases of matter by providing a framework to define and analyze their emergent excitations and properties. Specifically, MTCs ensure that the resulting theories satisfy the necessary consistency conditions for describing physically realizable systems, such as the positivity of probabilities and the absence of unphysical states. These categories allow for the calculation of topological invariants, like the $\mathbb{Z}_k$ invariants associated with fractional quantum Hall states, and predict the existence of protected edge states and non-local operators characteristic of these phases. Furthermore, the mathematical structure of MTCs directly relates to the braiding statistics of anyons, providing a tool to compute fusion rules and predict the behavior of anyonic systems in response to external stimuli.

Levin-Wen Models: Constructing Robust Quantum States on a Lattice

Levin-Wen string-net models are a class of explicitly defined lattice models constructed to provide a microscopic, or “UV complete,” description of topological quantum field theories (TQFTs). Unlike traditional approaches to TQFTs which often begin with an abstract mathematical formulation, string-net models define the theory in terms of local interactions on a lattice, allowing for computational study and a clear connection to physical systems. These models achieve topological order through the restriction of allowed configurations to those representing closed braids of strings, effectively enforcing long-range entanglement and protecting the system from local perturbations. The parameters defining the interactions within the lattice model directly correspond to the parameters of the TQFT it completes, allowing for a systematic derivation of topological invariants from the microscopic dynamics.

Levin-Wen models utilize string-net configurations as the basis for defining their ground state. These configurations are graphical representations consisting of strings and nodes on a lattice, where the allowed configurations are dictated by fusion rules associated with the underlying topological quantum field theory. Specifically, each node is assigned a quantum dimension and can connect to neighboring nodes via strings representing the exchange of anyons. The connectivity and types of strings are constrained by the fusion rules, ensuring that the ground state is robust to local perturbations and exhibits topological order, characterized by degenerate ground states that depend on the non-contractible loops within the lattice. The string-net configurations effectively encode the topological degrees of freedom of the system, providing a physical realization of abstract topological concepts on a lattice structure.

This research demonstrates a quantifiable correspondence between multipartite entanglement present in the ground state of Levin-Wen 3-dimensional string-net models and the partition function of the associated topological quantum field theory. Specifically, the study proves that certain entanglement measures, calculated directly from the ground state wave function, accurately reproduce the topological invariants encoded within the partition function $Z$ . This verification strengthens the assertion that Levin-Wen models serve as valid ultraviolet completions for topological phases, and provides a computational method for accessing topological properties via entanglement calculations on the lattice.

Fusion Rules, the S-Matrix, and the Signature of Topological Order

The behavior of anyonic excitations – particles exhibiting neither purely bosonic nor fermionic statistics – is governed by fusion coefficients, which fundamentally define how these exotic entities combine and decompose when brought together. These coefficients aren’t simply numerical values; they act as rules specifying the allowed outcomes of anyonic interactions. For example, fusing two anyons of type ‘a’ might yield another anyon of type ‘b’, or a superposition of several types, with the precise probabilities dictated by the corresponding fusion coefficient. Importantly, these coefficients also determine which processes are forbidden, effectively shaping the landscape of possible interactions within the system. This dictates the system’s response to external stimuli and ultimately manifests as observable physical properties, making the precise determination of fusion rules crucial for understanding and predicting the behavior of topologically ordered phases of matter. $N_{ab}^{c}$ represents the multiplicity of how anyon $c$ appears when fusing anyons $a$ and $b$ .

The Verlinde formula establishes a profound link between the seemingly disparate concepts of scattering and fusion rules in two-dimensional systems exhibiting topological order. This mathematical relationship reveals that the $S$ -matrix, which fully characterizes the scattering amplitudes of particles, is uniquely determined by the fusion rules – those governing how anyonic excitations combine. Consequently, knowing how these particles fuse is sufficient to predict their scattering behavior, and vice versa. This isn’t merely a theoretical curiosity; it allows physicists to calculate measurable quantities from a system’s topological properties, providing a powerful tool for verifying the presence of exotic states of matter and connecting abstract theoretical models to concrete experimental observations. The formula essentially translates the algebraic rules governing particle combinations into the physical language of scattering, opening a pathway to understand and characterize topological phases through scattering experiments.

The profound connection between fusion rules and the S-matrix isn’t merely a mathematical curiosity; it establishes a crucial pathway for validating theoretical predictions about topological order through observable phenomena. Topological order, characterized by exotic properties like fractional statistics – where particles exchange in a way that differs from the familiar bosons and fermions – leaves specific imprints on scattering experiments. The Verlinde formula, derived from the fusion rules, precisely dictates the allowed scattering amplitudes encoded within the S-matrix. Consequently, by meticulously analyzing experimental scattering data, researchers can effectively verify the predicted fusion rules and, in turn, confirm the presence of these elusive topological phases of matter. This interplay allows for a rigorous test of theoretical frameworks and provides a means to characterize and classify novel quantum states exhibiting fractional statistics and other non-abelian properties.

Multi-invariants and Entanglement: Probing the Quantum Fabric

Multi-invariants represent a powerful means of characterizing the subtle order present within a quantum system’s ground state – its lowest energy configuration. These mathematical functions, calculated directly from the wave function that describes the system, possess a unique stability; they remain constant even when the system undergoes local unitary transformations, which are operations that rotate the quantum state at individual locations without affecting the overall system properties. This invariance isn’t merely a mathematical curiosity; it signals that the multi-invariant captures a fundamental, intrinsic property of the ground state, independent of specific measurement choices. Consequently, multi-invariants effectively distill the essential topological information encoded within the quantum state, offering a robust tool for classifying and understanding phases of matter that are distinguished by global, rather than local, order – a crucial aspect in the search for novel quantum materials and the development of fault-tolerant quantum computation schemes.

Recent research establishes a remarkable connection between ground state entanglement and topological quantum field theories (TQFTs). Specifically, the work demonstrates that the partition function – a central object in TQFTs describing the system’s overall state – can be precisely recovered by analyzing the entanglement present within the ground state of certain quantum systems. This recovery is not merely theoretical; it has been explicitly verified for Levin-Wen string-net models, a class of systems known for hosting exotic topological phases of matter. The findings suggest that measurable quantities related to entanglement can serve as a powerful tool for characterizing and understanding these complex quantum states, potentially unlocking new avenues for designing materials with tailored topological properties and advancing the development of robust quantum computation schemes.

The established connection between ground state entanglement and topological quantum field theories suggests a powerful pathway toward designing materials with intrinsically protected quantum properties. This theoretical framework promises to move beyond conventional materials science by enabling the creation of novel phases of matter where information is encoded not in local degrees of freedom, but in the global entanglement structure of the system. Crucially, this approach holds significant implications for fault-tolerant quantum computation, as topological protection inherently shields quantum information from local disturbances – a major obstacle in building practical quantum computers. By harnessing these principles, researchers envision constructing quantum devices where computations are robust against errors, potentially unlocking the full potential of quantum information processing and ushering in a new era of technological advancement.

The pursuit of establishing concrete links between multipartite entanglement and Topological Quantum Field Theory, as demonstrated in this work, echoes a fundamental tenet of rigorous inquiry. It isn’t enough to simply observe correlation; one must strive for demonstrable, predictive relationships. As Jean-Paul Sartre noted, “Existence precedes essence.” This resonates with the article’s core concept: the observed entanglement is the foundation upon which the low-energy TQFT-the ‘essence’-is built. The Levin-Wen models aren’t merely a convenient mathematical tool, but a manifestation of this underlying reality. Predictive power is not causality, of course, but this research moves closer to establishing a causal link, verifying the relationship through concrete models and multi-invariants.

Where Do the Threads Lead?

The demonstrated correspondence between multipartite entanglement and the low-energy TQFT, while rigorously established for string-net models, does not constitute a universal proof. It reveals a relationship, certainly, but the extent to which this relationship generalizes beyond the confines of explicitly constructible models remains an open, and critical, question. The devil, as always, resides in the gap between mathematical elegance and physical reality. Determining the conditions under which this correspondence fails – identifying the types of entanglement or topological order that lie outside its purview – will likely prove as informative as confirming its continued validity.

Furthermore, the current framework primarily addresses gapped systems. The behavior of gapless topological phases, or those exhibiting more subtle forms of long-range entanglement, demands further investigation. Can the established machinery be extended, perhaps through renormalization group techniques or alternative mathematical formalisms, to accommodate these more complex scenarios? Any claim of success in this area will require, at minimum, careful consideration of infrared divergences and the precise definition of entanglement measures in the gapless limit.

Ultimately, the true test lies not in confirming a theoretical connection, but in leveraging it. Can this entanglement-TQFT correspondence be used to predict novel topological phases, or to design materials with specific topological properties? Until the theoretical insights translate into demonstrable experimental control, they remain, however beautiful, just another layer of abstraction. Anything less would be, to put it mildly, insufficiently grounded.

Original article: https://arxiv.org/pdf/2602.16770.pdf

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