⚛️ quantum physics

Scaling of entanglement with distance between subsystems: detection of phase transitions in the $t'$ - $\pi$ -flux model

This paper introduces a method to detect phase transitions in strongly correlated electron models, specifically the $t'$ - $\pi$ -flux Hubbard model, by analyzing the sharp signatures in the logarithmic derivatives of entanglement negativity between small clusters at varying distances, thereby overcoming the limitations of large system sizes and quantum Monte Carlo noise.

Original authors: Maksim Ulybyshev, Gergő Roósz, Gabriele Bellomia, Karsten Held, Fakher Assaad, Frederic Bippus

Published 2026-10-19

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CC BY 4.0

Original authors: Maksim Ulybyshev, Gergő Roósz, Gabriele Bellomia, Karsten Held, Fakher Assaad, Frederic Bippus

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 world of materials science, the most dramatic changes often happen not when a substance melts or freezes, but when it undergoes a subtle shift in how its internal parts are connected. These shifts, known as phase transitions, can turn a material from a conductor of electricity into an insulator, or from a disordered state into a magnet. For decades, physicists have relied on specific markers, called order parameters, to spot these changes. However, many of the most interesting materials defy these simple markers, hiding their transformations in complex patterns of quantum connection. To understand these hidden shifts, scientists have turned to a concept called entanglement. In the quantum realm, entanglement is a deep link between particles where the state of one instantly influences the other, regardless of the distance between them. While this phenomenon is often discussed in the context of tiny, isolated particles, it also exists in the vast, crowded lattices of atoms that make up solid matter. Detecting these connections in large, messy systems has been a major hurdle, as the mathematical tools required to map them usually break down when the system gets too big or too complex.

A team of researchers has now developed a new way to spot these hidden phase transitions by looking at how entanglement changes as the distance between two small groups of atoms increases. Instead of trying to measure the entire material at once, which is often impossible with current computer power, they focused on tiny clusters of atoms and moved them further and further apart. They tested this approach on a specific model of electrons moving through a grid, a system known as the t-prime-pi-flux Hubbard model. This model is particularly rich because it can exist in three distinct states: a semi-metal where electrons flow freely, a band insulator where they are trapped by the structure of the material, and a magnetic insulator where they are trapped by their own interactions. The researchers wanted to see if the way entanglement fades as the distance between two clusters grows could reveal exactly where the material switches from one state to another.

To do this, the scientists used a powerful computer simulation technique called quantum Monte Carlo. This method allows them to track the behavior of electrons in a material by simulating millions of random configurations, effectively averaging out the noise to find the true physical properties. They calculated a specific measure of entanglement called negativity, which quantifies how strongly two separate regions of the material are linked. In their simulations, they placed two small groups of atoms at varying distances from each other and measured the negativity between them while slowly changing the conditions of the material, such as the strength of the electron repulsion or the way they hop between sites. They found that as they moved the groups apart, the entanglement did not just fade away smoothly. Instead, at the precise moment the material underwent a phase transition, the rate at which this entanglement changed with distance showed a sharp, distinct signature.

The results were striking. When the material was in a semi-metal state, the entanglement between distant groups decayed in a predictable, slow manner. But as the material crossed into an insulating state, the decay pattern changed dramatically, shifting to a much faster drop-off. By analyzing the mathematical slope of this change, the researchers could pinpoint the exact moment of the transition. This method worked for all three types of transitions they studied, including those where the material became magnetic and those where it became a standard insulator. Crucially, they demonstrated that this technique works even when the system is too large to be solved exactly, a limitation that has plagued previous attempts to use entanglement as a diagnostic tool. They also showed how to filter out the statistical noise inherent in their computer simulations, ensuring that the sharp peaks they observed were real physical effects and not just calculation errors.

One of the most interesting findings was the behavior of spin entanglement, which measures the connection between the magnetic orientations of the electrons. In some cases, this specific type of connection vanished completely once the groups of atoms were separated by just a few atomic spacings, a phenomenon known as sudden death. However, the broader measure of entanglement they used, which includes the movement of the electrons themselves, remained detectable even at larger distances. This allowed them to see the transition clearly where other methods might have failed. The study suggests that by simply watching how entanglement stretches and breaks as distance increases, scientists can identify the critical points of complex materials without needing to know the specific order parameter that defines the phase. This approach offers a robust, universal way to map the landscape of quantum matter, revealing the hidden boundaries between different states of existence in the solid world.

Technical Summary: Scaling of Entanglement with Distance for Phase Transition Detection

Problem Statement
Detecting and classifying quantum phase transitions in strongly correlated fermionic systems remains a significant challenge, particularly when standard Landau-Ginzburg order parameters are absent or difficult to define. While entanglement-based approaches, such as system-size scaling of entanglement entropy, have proven effective for theoretical models, they face severe limitations in interacting fermionic systems. Specifically, exact solutions are rare, and numerical methods like Quantum Monte Carlo (QMC) are often restricted to small system sizes. Furthermore, constructing and storing the reduced density matrix (RDM) for extensive subsystems is computationally prohibitive. Although entanglement between neighboring small subsystems has shown signatures of transitions in specific cases (e.g., spin systems, Kitaev models), these signatures are not guaranteed to persist or be detectable in all scenarios, particularly in the presence of strong interactions and at finite temperatures.

Methodology
The authors propose a "scaling with distance" approach to detect phase transitions without varying the total system size. Instead of scaling the system size $L$ , they vary the distance $d$ between two small, disconnected subsystems (either single sites or dimers) and analyze the behavior of entanglement measures.

Model System: The study focuses on the $t'$ - $\pi$ -flux Hubbard model on a square lattice. This model is chosen for its rich phase diagram, exhibiting three distinct transitions:
- Semimetal (SM) to Band Insulator (BI) (Gaussian universality class).
- Semimetal (SM) to Antiferromagnetic (AFM) Mott Insulator (MI) (Gross-Neveu universality class).
- Magnetic order-disorder transition (MI to BI) (O(3) Wilson-Fisher universality class).
Entanglement Measures:
- Fermionic Negativity ( $N^F$ ): Used as the primary measure of bipartite entanglement. It extends the positive partial transpose (PPT) criterion to fermionic modes via a partial time-reversal operation, ensuring it captures entanglement even in the presence of local parity violation. The authors utilize the logarithmic negativity $\mathcal{N}^F = \log_2(2N^F + 1)$ .
- Spin Entanglement ( $N^S$ ): A filtered version of the negativity where off-diagonal elements of the RDM that do not preserve local charge are removed. This isolates entanglement arising purely from spin fluctuations.
- Entanglement Entropy: Calculated for the non-interacting limit ( $U=0$ ) to provide a reference for the SM-BI transition.
Computational Techniques:
- Auxiliary-Field QMC: Simulations are performed using the ALF code framework at finite temperature ( $T = 1/16t$ ).
- RDM Reconstruction: The authors introduce a general methodology to compute arbitrary equal-time and time-displaced correlators from QMC data. This involves a computer algebra system for linear operators acting on Fock space vectors, which translates observables into products of fermionic Green's functions.
- Noise Mitigation: To handle QMC noise at large distances, the authors derive a method to extract the negativity from noisy data by extending Vidal et al.'s formula. They also employ a strategy of selecting the Hubbard-Stratonovich channel (spin or charge) with smaller fluctuations for each observable.
- Analysis: The core diagnostic tool is the first and second-order logarithmic derivatives of the negativity with respect to the tuning parameter (either $t'$ or $U$ ).

Key Contributions

Distance Scaling Protocol: The paper establishes a protocol where the distance $d$ between small subsystems acts analogously to system size $L$ in finite-size scaling analysis. This allows for the detection of phase transitions in regimes where large-system scaling is inaccessible.
Algorithmic Advancement: The development of an automated Wick decomposition within QMC codes to compute multi-site reduced density matrices from Green's functions. This enables the calculation of entanglement measures beyond Rényi entropy and negativity for arbitrary subsystem sizes.
Noise Handling: A specific technique to mitigate QMC noise when computing negativity at large separations, ensuring the reliability of the distance scaling analysis.

Results
The study demonstrates that sharp signatures of phase transitions appear in the logarithmic derivatives of the fermionic negativity as a function of distance:

SM-BI Transition (Non-interacting, $U=0$ ):
- The transition is marked by a change in the decay of correlations from a power law ( $d^{-2}$ ) in the semimetal to an exponential decay ( $e^{-d/\xi}$ ) in the band insulator.
- The logarithmic derivative of the negativity, $\partial \log N^F / \partial t'$ , exhibits a peak that grows linearly with distance $d$ at the critical point ( $t'=3$ ). The second derivative shows a peak whose height scales linearly with $d$ , consistent with a correlation length exponent $\nu=1$ .
- Entanglement entropy calculations for large systems confirm the area law and subleading scaling behaviors.
MI-BI Transition (Interacting, $U=7$ ):
- This transition belongs to the O(3) Wilson-Fisher universality class ( $\nu \approx 0.711$ ).
- While spin entanglement ( $N^S$ ) suffers "sudden death" beyond nearest neighbors, the fermionic negativity ( $N^F$ ) remains non-zero.
- The logarithmic derivatives of $N^F$ show a peak growing with distance at the transition. The authors attribute this to the indirect influence of critical spin fluctuations on the charge sector, which drives the divergence in the correlation length of the charge correlations despite the charge sector being gapped.
SM-MI Transition (Interacting, varying $U$ ):
- This transition belongs to the Gross-Neveu-Heisenberg universality class. Recent large-scale simulations suggest $\nu \approx 1.11$ .
- Consistent with the theoretical expectation that $1 < \nu < 2$ , the first logarithmic derivative does not show a divergent peak, but the second logarithmic derivative exhibits a clear peak that scales with distance.
- Spin entanglement again shows sudden death at larger distances, limiting its utility for distance scaling in this regime.

Significance and Claims
The paper claims that the scaling of the fermionic negativity with the distance between small subsystems serves as a robust, universal diagnostic tool for detecting quantum phase transitions in strongly correlated fermionic systems.

Universality: The method successfully detects transitions across different universality classes (Gaussian, Gross-Neveu, O(3)) by analyzing the scaling of the logarithmic derivatives with distance. The behavior of these derivatives is directly linked to the correlation length exponent $\nu$ of the transition.
Applicability to Interacting Systems: Unlike previous methods requiring large system sizes or exact diagonalization, this approach works within the constraints of QMC simulations for interacting models where large system sizes are inaccessible.
Beyond Order Parameters: The technique is effective even for transitions lacking conventional order parameters, provided the correlation length diverges.
Robustness: The fermionic negativity is shown to be superior to spin entanglement in this context, as it does not suffer from sudden death at large distances and captures the full physics of the model, including charge fluctuations.

The authors conclude that this "distance scaling" of entanglement measures offers a practical and theoretically grounded method for identifying critical points in complex many-body systems where traditional scaling analyses are computationally infeasible.

Technical Summary: Scaling of Entanglement with Distance for Phase Transition Detection

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