🔬 condensed matter

Entanglement spectrum and magic in higher-dimensional free fermionic systems

This paper investigates fermionic non-local magic in higher-dimensional free-fermion systems by linking it to the entanglement spectrum, revealing dimensional reduction effects, establishing bounds via entanglement capacity, distinguishing topological phases through entanglement-temperature responses, and characterizing its diffusive spreading in non-equilibrium dynamics.

Original authors: Beatrice Magni, Daniele Iannotti, Riccardo Travaglino

Published 2026-09-30

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CC BY 4.0

Original authors: Beatrice Magni, Daniele Iannotti, Riccardo Travaglino

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, the behavior of many particles together is often described by two distinct kinds of "resource." One is entanglement, a deep connection where particles lose their individual identity and act as a single unit, no matter how far apart they are. This concept is well understood and is the engine behind many proposed quantum technologies. The other resource is more subtle, often called "magic" by physicists, though it has nothing to do with sorcery. In this context, magic refers to a specific kind of complexity that makes a quantum state difficult to simulate on a classical computer. While entanglement tells us how much information is shared between parts of a system, magic tells us how hard it is to describe the system using simple, standard building blocks. Understanding how these two resources interact is crucial for figuring out which quantum states are truly powerful and which are merely complicated.

A team of researchers has now explored how this "magic" behaves in systems of free fermions—particles like electrons that do not interact with each other but move through space—specifically in two-dimensional grids. By looking at the mathematical structure known as the entanglement spectrum, which acts like a detailed map of how quantum information is distributed, they discovered that magic follows a very specific pattern in these systems. In a flat, two-dimensional grid, the amount of magic does not grow simply with the size of the boundary, as one might expect. Instead, it grows in a way that combines the length of the boundary with a logarithmic factor, a behavior that mirrors how entanglement entropy scales in similar systems. This finding suggests that the geometry of the system's energy levels, specifically the shape of the surface where particles can exist, directly dictates how much of this computational complexity is generated.

The researchers also found a powerful way to measure this magic by comparing it to another quantity called the capacity of entanglement, which measures how much the quantum connections fluctuate. They demonstrated that the amount of magic is always limited by this capacity, providing a stricter rule than previously known. To probe deeper, they introduced a method akin to turning a dial on a thermometer, allowing them to observe how the magic responds to different energy levels within the quantum state. When they applied this to a famous model of quantum matter known as the Kitaev model, the results were striking. In a phase where the system has a gap in its energy levels, the magic vanished almost completely at low energies. However, in a phase where the system is gapless and supports exotic, non-Abelian behavior, the magic persisted and followed a simple, universal rule. This distinction allowed the researchers to clearly separate two very different types of quantum phases based solely on how their magic behaves.

Beyond static systems, the team investigated what happens when these quantum grids are suddenly disturbed, a process known as a quench. They showed that the spread of magic over time can be understood through the movement of tiny, independent packets of information, much like how heat or sound travels through a material. In some cases, even when the entanglement between particles grows massively, the magic remains surprisingly small, indicating that the two resources can evolve independently. Finally, they looked at systems driven by random changes, simulating a chaotic environment. In these scenarios, the magic spreads out in a diffusive manner, slowly wandering across the grid rather than racing across it. These findings confirm that while entanglement and magic are related, they are distinct features of quantum matter, each revealing different aspects of how complexity emerges in the physical world.

Technical Summary: Entanglement spectrum and magic in higher-dimensional free fermionic systems

Problem and Context
This work investigates fermionic non-local magic (FNL) in higher-dimensional free-fermion systems, analyzing its relationship with the entanglement spectrum in both equilibrium and non-equilibrium settings. While entanglement entropy is a well-established measure of quantum correlations, "magic" (or non-stabilizerness) is a distinct resource crucial for quantum computational advantage. Total magic includes local contributions that are often irrelevant for many-body physics; FNL isolates the irreducible, non-local component tied to bipartite entanglement. However, calculating FNL in generic systems is computationally prohibitive due to the minimization over local unitaries. The authors focus on fermionic Gaussian states, where restricting the optimization to local Gaussian unitaries yields a computable, closed spectral expression for FNL based on the eigenvalues of the restricted Majorana covariance matrix. The central goal is to determine how FNL scales with spatial dimension, how it resolves low-energy features of the entanglement Hamiltonian, and how it spreads under unitary dynamics, comparing its behavior to entanglement entropy and the capacity of entanglement.

Methodology
The study employs the Majorana representation of free-fermion systems on lattice geometries, specifically strip geometries on cylinders.

Dimensional Reduction: For translationally invariant states, the authors utilize Fourier transformation along the periodic transverse direction to decompose the restricted Majorana covariance matrix into independent momentum sectors. This maps the higher-dimensional problem to a collection of independent lower-dimensional Gaussian problems.
Spectral Definitions: FNL is defined via the eigenvalues $\lambda_j$ of the restricted covariance matrix $i\Gamma_A$ using the function $m_\alpha(\lambda^2)$ . The capacity of entanglement ( $C_E$ ), defined as the variance of the entanglement Hamiltonian, is similarly expressed as a spectral sum.
Entanglement Temperature: The authors introduce an "entanglement temperature" deformation ( $T_E$ ) to probe the low-energy structure of the entanglement spectrum. This involves rescaling the entanglement energies $\epsilon_j$ to generate a family of states $|\psi(T_E)\rangle$ , allowing the study of FNL and $C_E$ as functions of $T_E$ .
Models: The equilibrium analysis is applied to the matter-Majorana sector of the Kitaev honeycomb model (comparing Abelian and non-Abelian phases) and a nearest-neighbor hopping model. Non-equilibrium dynamics are studied via quantum quenches from shift-invariant initial states and random Gaussian circuits.

Key Contributions and Results

Scaling in Higher Dimensions (Equilibrium):
For translationally invariant Gaussian states in strip geometries, the authors demonstrate that dimensional reduction leads to a multiplicative logarithmic violation of the boundary law for FNL in two dimensions. Specifically, when a finite fraction of transverse momenta correspond to critical (gapless) 1D chains, the FNL scales as $M^{FNL}_\alpha \sim L_y \log \ell$ (where $L_y$ is the transverse length and $\ell$ is the subsystem width). This scaling is analogous to the Gioev-Klich-Widom scaling observed in free-fermion entanglement entropy and arises whenever the Fermi surface has codimension one. The leading growth is controlled by the number of gapless transverse modes.
Relation to Capacity of Entanglement:
The paper establishes that the second Rényi FNL is bounded from above by the capacity of entanglement ( $M^{FNL}_2(A) \le C_E(A)$ ). This inequality is tighter than the bound provided by the Rényi entropy itself. This relationship motivates the use of $T_E$ -deformed FNL to probe the entanglement spectrum.
Kitaev Model and Phase Distinction:
Applied to the Kitaev model, the temperature-resolved FNL distinguishes between topological phases based on the gapped or gapless nature of the entanglement spectrum:
- Abelian Phase: The FNL is exponentially suppressed at low entanglement temperatures ( $M^{FNL}_2(T_E) \sim \sqrt{T_E} e^{-\Delta_E/T_E}$ ), reflecting the finite gap in the entanglement spectrum.
- Non-Abelian Phase: The FNL exhibits algebraic behavior ( $M^{FNL}_2(T_E) \propto L_y T_E$ ) due to protected gapless branches.
- Universal Ratio: In the non-Abelian phase, the ratio of FNL to the capacity of entanglement approaches a universal constant as $T_E \to 0$ , indicating that both quantities probe the same low-energy entanglement modes but with different spectral weights.
Non-Equilibrium Dynamics:
- Quenches: For quenches generated by number-conserving quadratic Hamiltonians, the authors extend the quasi-particle picture (QPP) to FNL in higher dimensions. They show that FNL and entanglement entropy share the same ballistic propagation mechanism but differ in the weighting functions applied to the transported modes. Notably, they identify scenarios (e.g., the collinear Mott state) where entanglement entropy grows extensively while FNL remains $O(1)$ , demonstrating that FNL captures complementary aspects of quantum complexity.
- Random Circuits: For random fermionic Gaussian circuits in two dimensions, the authors show that both entanglement entropy and FNL exhibit diffusive spreading ( $M^{FNL}_2(t) \propto L_y \sqrt{t}$ ) before saturation. This is derived from the diffusive behavior of individual Majorana modes and the upper bound provided by the 2-Rényi entropy.

Significance and Claims
The paper claims that FNL is a consistent and applicable quantity for studying quantum resources in higher-dimensional free-fermion systems. Its primary significance lies in providing a spectral quantity that, like the capacity of entanglement, probes features of the entanglement spectrum beyond what is captured by standard entanglement entropy. The work demonstrates that:

FNL scaling is governed by the geometry of the Fermi surface and the transport of single-particle entanglement degrees of freedom.
FNL can distinguish between gapped and gapless entanglement spectra (and thus topological phases) via its response to entanglement temperature.
FNL and entanglement entropy can decouple dynamically, with FNL remaining small even when entanglement is large, highlighting its role as a distinct measure of non-stabilizerness.
The framework allows for the characterization of diffusive spreading in random circuits, extending known 1D results to higher dimensions.

The authors conclude that FNL offers complementary information about quantum correlations, with its equilibrium scaling, low-energy response, and dynamical spreading all traceable to the underlying single-particle physics. They suggest future directions include extending the analysis to general subsystem geometries, exploring mixed-state extensions, and potentially testing these spectral functions against reconstructed correlation matrices in phononic platforms.

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