Scaling of entanglement entropy and correlations in the variable-range extended Ising model

This paper investigates the ground state and quench dynamics of a one-dimensional variable-range extended Ising model, demonstrating that while two-point correlations transition from algebraic to exponential scaling beyond the coordination number $\mathcal{Z}$, the bipartite entanglement at the critical point follows a universal power-law decay with increasing $\mathcal{Z}$ for $\alpha > 1$.

The Gist

Imagine you are looking at a massive social network, like a giant digital web of people. In a "normal" social network, you mostly interact with your immediate neighbors—your family or roommates. If you want to send a message to someone across the world, it has to pass through a long chain of people. This is what physicists call "short-range" interaction.

But what if the rules changed? What if, as the network grew, everyone suddenly gained the ability to "shout" a little bit louder to people further away? This paper explores a mathematical model of a "social network" of quantum particles (qubits) where the range of their influence changes.

Here is the breakdown of their discovery using everyday analogies.

1. The "Shouting" Rule (Variable-Range Interaction)

The researchers studied a model called the Variable-Range Extended Ising Model.

• The "Short-Range" World: Imagine a line of people holding hands. You can only feel the person directly next to you. If you move, only your neighbor feels it.
• The "Long-Range" World: Now imagine those people aren't just holding hands; they are also shouting. The closer someone is, the louder they shout. The further away they are, the quieter the shout.

The scientists introduced a knob called $Z$ (the coordination number). Increasing $Z$ is like giving everyone a megaphone. It doesn't just change who they talk to; it changes the very "texture" of how information flows through the crowd.

2. The "Invisible Boundary" (Correlation Scaling)

In physics, "correlation" is a measure of how much one particle knows about another.

• The Discovery: The researchers found that there is a "magic distance" in this crowd, specifically at the distance $Z$.
• The Analogy: Imagine you are at a music festival. If you are standing within the "inner circle" (distance less than $Z$), the music feels like a continuous, powerful wave—everyone is in sync (this is algebraic decay). But once you step outside that inner circle (distance greater than $Z$), the music becomes a faint, muffled hum that disappears very quickly (this is exponential decay).

The $Z$ value acts like an invisible boundary that separates a "connected community" from a "disconnected crowd."

3. The "Social Anxiety" of Particles (Entanglement Entropy)

"Entanglement" is a spooky quantum phenomenon where two particles become so deeply linked that they act as one, no matter how far apart they are. "Entanglement Entropy" is basically a measure of how much "shared information" or "connection" exists between one group and the rest of the world.

• The Surprising Twist: Usually, when you increase the connections in a system, you’d expect the "connectedness" (entanglement) to go up. But these researchers found the opposite!
• The Analogy: Imagine a small group of friends at a party. If you suddenly make everyone in the room connected to everyone else via loud megaphones, the intimacy of the small group actually drops. Because everyone is shouting at everyone else, the unique, special bond of the small group gets "diluted" by the noise of the massive crowd.

As the coordination number $Z$ increases, the entanglement between a small block of particles and the rest of the system actually decreases following a specific mathematical pattern (a power law).

4. The "Sudden Shock" (Quantum Quench)

Finally, the researchers performed a "quench."

• The Analogy: Imagine a room full of people sitting in total silence (the ground state). Suddenly, someone slams a drum loudly (the quench), and everyone starts reacting to the rhythm.
• The Result: They looked at how the "connectedness" (entanglement) settles down after this shock. They found that even in this chaotic, moving state, the "dilution" effect remains. The way the connections settle down follows the same mathematical rules as when the system was sitting still.

Summary: Why does this matter?

In the quest to build Quantum Computers, we need to understand how information travels. If we build a computer where every part can talk to every other part (long-range), it might seem better, but this paper shows that it actually changes the "flavor" of the information and can actually "thin out" the quantum connections we rely on.

The researchers have essentially provided a map for how to navigate the transition from a quiet, local neighborhood to a loud, global village.

Technical Summary

Technical Summary: Scaling of Entanglement Entropy and Correlations in the Variable-Range Extended Ising Model

1. Problem Statement

Traditional statistical mechanics and quantum field theories are largely built upon short-range interaction models, where the onset of a Quantum Phase Transition (QPT) is characterized by a diverging correlation length and algebraically decaying correlation functions. However, in long-range interacting systems, the conventional narrative of correlation length often fails. These systems can exhibit persistent algebraic tails even away from criticality, making it difficult to apply standard frameworks of continuous QPTs.

While much research has focused on varying the power-law exponent $\alpha$ (where interaction $J(r) \sim r^{-\alpha}$), there is a lack of systematic analysis regarding how the range of interaction—controlled by the coordination number $Z$—affects classical correlations and bipartite entanglement.

2. Methodology

The authors investigate the Variable-Range Extended Ising (VREI) model on a one-dimensional lattice of $N$ qubits with periodic boundary conditions.

• Model Definition: The Hamiltonian includes a transverse field $h$ and ferromagnetic interactions $J_r$ between qubits at distance $r$. The range is tuned by the coordination number $Z$, with interaction strength $J_r = r^{-\alpha}/A$, where $A$ is the Kac normalization constant.
• Mathematical Approach:
  • The model is solved exactly using the Jordan-Wigner transformation, mapping the spin system to a fermionic Hamiltonian.
  • The Hamiltonian is diagonalized via Fourier transformation and a Bogoliubov transformation, yielding the dispersion relation $\omega_k$ and quasi-particle velocity $v_{\alpha,Z}$.
  • Correlation Functions: Two-point fermionic correlation functions ($\alpha_r$ and $\beta_r$) are calculated in the thermodynamic limit.
  • Entanglement: Bipartite entanglement (von Neumann entropy $S_M$) is quantified for a block of $M$ qubits. For $M=1$, the authors use analytical approximations; for $M>1$, they employ the correlation matrix method.
  • Dynamics: The authors simulate a sudden quench from the infinite-field limit to the critical Hamiltonian to study the time-evolution of entanglement.

3. Key Contributions and Results

A. Emergence of a Natural Length Scale ($r = Z$)
The study identifies $Z$ as a critical crossover length scale.

• For $r < Z$: The system exhibits effective long-range behavior, characterized by algebraic decay in correlation functions.
• For $r > Z$: The system exhibits effective short-range behavior, where correlations decay exponentially (away from criticality) or algebraically (at criticality).

B. Scaling of Bipartite Entanglement at Criticality
The authors demonstrate that at the critical point, the bipartite entanglement $S_M$ does not remain constant or grow logarithmically with $M$ in the traditional sense, but rather decreases as a power law with the coordination number $Z$:
$$S_M \sim Z^{-\gamma}$$

• For the specific case of $\alpha = 2$, the exponent is $\gamma = 1$.
• For $\alpha > 1$, the exponent $\gamma$ is a quadratic function of $\alpha$.
• This scaling holds for both single-site ($M=1$) and multi-site ($M>1$) partitions, indicating that increasing the connectivity of the system actually suppresses the bipartite entanglement at the critical point.

C. Quasi-particle Velocity and Dispersion
The authors derive a $Z$-dependent quasi-particle velocity $v_{\alpha,Z} \approx Z^\eta$. This velocity distinguishes the weak long-range regime ($1 < \alpha < 2$) from the effective short-range regime ($\alpha > 2$), providing a bridge between the two behaviors on the $\alpha$ axis.

D. Quench Dynamics
In the dynamical regime (sudden quench):

• Transient Growth: Entanglement grows linearly with time $t$ until it reaches a saturation time $\tau_{\alpha,Z} \propto Z^{-\eta}$.
• Steady State: The long-time averaged entanglement follows the same power-law scaling with $Z$ ($S \sim Z^{-\gamma}$) as the static ground-state entanglement, confirming the robustness of this scaling behavior.

4. Significance

This work provides a new framework for understanding the transition from short-range to long-range physics by using the coordination number $Z$ as a control parameter rather than just the exponent $\alpha$.

The discovery that increasing the coordination number $Z$ leads to a power-law decrease in bipartite entanglement is a counter-intuitive result that challenges standard expectations in quantum information theory. This research is highly relevant for experimental platforms like Rydberg gases and programmable quantum computers, where variable-range interactions can be physically realized and manipulated. It offers a theoretical basis for predicting how connectivity affects information spreading and entanglement in complex quantum networks.