🔬 condensed matter

Entanglement entropy of fermions in a strange metal

This paper investigates the entanglement entropy of fermions in a solvable large- $N$ 1D strange metal model, demonstrating that the crossover from thermal to entanglement entropy follows a universal scaling law described by a standard conformal field theory formula with an effective central charge significantly enhanced by strong interactions.

Original authors: Santanu Singh, Surajit Bera, Chenyuan Li, Subir Sachdev, Sumilan Banerjee

Published 2026-08-06

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CC BY 4.0

Original authors: Santanu Singh, Surajit Bera, Chenyuan Li, Subir Sachdev, Sumilan Banerjee

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

Imagine the universe as a giant, invisible dance floor where tiny particles like electrons are constantly waltzing, jostling, and holding hands. In the world of physics, scientists love to study how these particles "entangle," a spooky quantum connection where two particles become so linked that what happens to one instantly affects the other, no matter how far apart they are. Usually, when particles dance in a calm, orderly way (like in a "Fermi liquid"), we can predict their steps perfectly. But sometimes, in materials called "strange metals," the dance gets chaotic. The particles lose their individual rhythm, and there are no clear "steps" or "partners" to follow. This is a mystery that has puzzled physicists for decades because these strange metals are found in superconductors—materials that could revolutionize how we transmit electricity. To solve this puzzle, scientists need a way to measure just how tangled these particles get. They use a mathematical tool called "entanglement entropy," which is basically a scorecard for how much information is shared between different parts of the system. If the score is high, the particles are deeply connected; if it's low, they are acting more independently. Understanding this score helps us figure out why strange metals behave so weirdly and whether we can ever control them.

In this study, a team of researchers decided to tackle this mystery by building a digital playground—a computer simulation of a one-dimensional chain of particles. They created a model called the "1D Yukawa-SYK model," which is like a simplified version of a strange metal where fermions (the dancing electrons) interact with bosons (the invisible force fields) through random, chaotic connections. Instead of trying to solve the impossible math of a real, messy metal, they used a clever trick called "large-N" theory, which simplifies the problem by imagining there are thousands of particle flavors instead of just a few, making the math solvable. They then asked a specific question: If you take a small slice of this digital chain and measure how entangled the electrons in that slice are with the rest of the world, what does the score look like?

The researchers found something fascinating and slightly surprising. In the chaotic "strange metal" state, the electrons don't just get entangled with distant neighbors across the chain; they get incredibly tangled with the force fields right next to them, inside the same small slice. It's as if the dancers are so busy holding hands with the person standing immediately next to them that they forget to look across the room. This "local" entanglement creates a massive amount of connection that grows with the size of the slice, a behavior known as a "volume law." However, the team also discovered that if you look at the bigger picture, the way the entanglement changes as you heat up the system or change the slice size follows a very specific, universal pattern. They found that all their messy data could be squashed down into a single, smooth curve using a formula that usually describes perfectly ordered systems (like a Conformal Field Theory), but with a twist: the "central charge" (a number that measures the complexity of the dance) was about 60% higher than expected for non-interacting particles. This suggests that even though the system is chaotic, it still follows a hidden, elegant rule, just one that is much more complex than we previously thought.

The paper explicitly rules out the idea that these strange metals behave like standard, well-behaved liquids where particles act independently or follow simple, predictable patterns. They show that the old rules don't apply here. Instead, the study suggests that the strange metal state is a unique phase where the electrons and the force fields are locked in a tight, local embrace that dominates the physics. The results are based on numerical simulations and mathematical solutions of their model, not on physical experiments with real materials, so while the findings are robust within the model, they are a theoretical prediction waiting for real-world verification. The authors show that this chaotic state isn't just random noise; it has a structure that can be described by a single scaling law, bridging the gap between the cold, quantum ground state and the warm, thermal state. This gives physicists a new, clearer lens to look at the enigmatic behavior of strange metals, hinting that the key to understanding them lies in the intense, local connections between particles and their environment.

Technical Summary: Entanglement Entropy of Fermions in a Strange Metal

Problem and Motivation
The scaling of ground-state entanglement entropy ( $S_A$ ) with subsystem size ( $L_A$ ) and its crossover to thermal entropy as a function of temperature ( $T$ ) are well-established for one-dimensional (1D) gapless systems described by Conformal Field Theory (CFT) (e.g., Tomonaga-Luttinger liquids) and for free fermions with a Fermi surface in any dimension. However, the behavior of entanglement entropy in strongly interacting, gapless fermionic systems lacking well-defined quasiparticles—such as strange metals and non-Fermi liquids (NFLs)—remains poorly understood. This paper addresses this gap by studying the entanglement entropy of fermions in a solvable large- $N$ 1D lattice model akin to the Yukawa-Sachdev-Ye-Kitaev (Yukawa-SYK) model. This model features two Fermi points coupled to scalar bosons via spatially random Yukawa interactions, providing a controlled framework to investigate a strange metal state arising at a quantum critical point (QCP).

Methodology
The authors employ a large- $N$ field theory approach combined with an imaginary-time path integral formalism recently developed for computing subsystem Rényi entropies.

Model Definition: The system consists of a 1D chain of Yukawa-SYK quantum dots with $N$ flavors of spinless fermions and real bosons. The fermions and bosons interact via random Yukawa couplings, and the bosons satisfy a spherical constraint.
Thermal Equilibrium: The authors first solve the large- $N$ saddle-point equations for the thermal state to determine the phase diagram, boson thermal mass, and fermionic/bosonic self-energies. This establishes the existence of a QCP separating a disordered Fermi liquid (FL) phase from an ordered non-Fermi liquid (NFL) phase.
Entanglement Calculation: To compute the second Rényi entropy ( $S^{(2)}_{Af}$ ) of a fermionic subsystem $A$ (tracing out bosons in $A$ and all degrees of freedom in the complement $B$ ), the authors utilize a path integral representation involving "entanglement replicas." This method introduces auxiliary fermionic source fields (a "kick" term) that couple two replicas at an imaginary time $\tau_0$ , breaking time-translation symmetry.
Numerical Implementation: The authors numerically solve the resulting large- $N$ saddle-point equations for the entangled system. They discretize imaginary time and use a recursive Green's function method to handle the matrix inversions efficiently. The ground-state entanglement is extracted by systematically extrapolating finite-temperature results to the $T \to 0$ limit using a scaling ansatz.

Key Contributions and Results
The paper presents exact numerical computations of the second Rényi entropy for the fermionic subsystem in the 1D Yukawa-SYK model. The main findings are:

Scaling Ansatz for Crossover: The crossover from thermal entropy (dominant at high $T$ ) to entanglement entropy (dominant at low $T$ ) is captured by a single scaling ansatz:
$S^{(2)}_{Af}(L_A, T) = f_L\left(\frac{L_A}{\ell(T)}\right) + \dots$
where $\ell(T)$ is a temperature-dependent length scale. Data for various subsystem sizes and temperatures collapse onto a single universal curve.
Intra-Subregion Entanglement and Volume Law: The analysis reveals a crucial role for intra-subregion entanglement between fermionic and bosonic degrees of freedom. The length scale $\ell(T)$ behaves as:
$\frac{1}{\ell(T)} = \frac{1}{\ell_1(T)} + \frac{1}{\ell_0}$
where $\ell_1(T) \sim 1/T$ represents the thermal length, and $\ell_0$ is a finite length scale at $T=0$ . The non-zero intercept $1/\ell_0$ implies a volume-law contribution to the entropy ( $S^{(2)}_{Af} \sim L_A/\ell_0$ ) even at zero temperature, arising from local fermion-boson entanglement within the subsystem.
Effective CFT Description: The universal scaling curve is well-described by a CFT-like formula:
$f\left(\frac{L_A}{\ell(T)}\right) = \frac{c_{\text{eff}}}{4} \ln \left[ \sinh \left( \frac{L_A}{\ell(T)} \right) \right] + \dots$
However, the effective central charge $c_{\text{eff}}$ is significantly enhanced compared to non-interacting systems.
- At the QCP (strange metal), $c_{\text{eff}} \approx 1.6$ , which is roughly 60% larger than the non-interacting value ( $c=1$ ).
- In the Fermi liquid phase ( $\gamma > \gamma_c$ ), $c_{\text{eff}}$ remains $>1$ but decreases as the system moves away from the critical point.
Competition of Entanglement Types: The results indicate that for small subsystems ( $L_A \lesssim \ell_0$ ), non-local inter-subsystem entanglement dominates, exhibiting logarithmic scaling ( $\sim \ln L_A$ ). However, for larger subsystems ( $L_A \gtrsim \ell_0$ ), this logarithmic behavior is cut off and obscured by the dominant volume-law entanglement between fermions and bosons within the same spatial region.

Significance
The paper demonstrates that in strongly coupled fermion-boson systems like the 1D Yukawa-SYK model, the ground state entanglement is not solely determined by non-local correlations characteristic of critical CFTs. Instead, local intra-subsystem entanglement between fermions and critical bosons plays a decisive role, leading to a volume-law contribution that masks the logarithmic scaling at larger length scales. The authors show that despite the absence of quasiparticles, the entanglement-to-thermal crossover can be described by a universal scaling function with an effective central charge that quantifies the strength of the fermion-boson coupling and entanglement. This provides a new perspective on characterizing strange metals and non-Fermi liquids through entanglement measures, highlighting the necessity of considering both local and non-local correlations in such systems.

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