Metric response of relative entropy: A universal indicator of quantum criticality

This paper proposes the metric response of quantum relative entropy as a universal indicator of quantum criticality, demonstrating that its susceptibility diverges at quantum critical points in the thermodynamic limit with distinct scaling behaviors for integrable and non-integrable spin chains, while also exhibiting finite-size divergence in classical limits due to the rank of reduced density matrices.

The Gist

The Big Idea: Measuring the "Stress" of a Quantum System

Imagine you have a long chain of tiny magnets (spins) that can point up or down. This chain is governed by a rulebook called a Hamiltonian. One of the rules in this book is a knob labeled $h$ (like a magnetic field).

Usually, if you turn this knob slightly, the magnets barely change their arrangement. But at a specific setting called a Quantum Critical Point (QCP), the whole chain suddenly wants to rearrange itself completely. It's like a calm lake suddenly turning into a stormy sea. Scientists want to find exactly where this "storm" happens and understand how wild it gets.

The authors of this paper propose a new, universal way to detect these storms. They call it the Metric Response of Quantum Relative Entropy (QRE).

The Analogy: The "Surprise" Meter

To understand their method, let's use an analogy of a Surprise Meter.

1. The Setup: Imagine you are looking at a small section of the magnet chain (say, 1, 2, or 3 magnets). You have a "map" (a density matrix) that tells you the probability of every possible arrangement of these magnets.
2. The Change: You turn the knob ($h$) just a tiny bit. The map changes slightly.
3. The Measurement: The authors ask: "How surprised would I be if I used the old map to predict the new reality?"
   • If the system is calm, the old map still works well. You aren't very surprised.
   • If the system is near a critical point (the storm), the old map becomes useless. You are extremely surprised.

This "surprise" is measured mathematically by Quantum Relative Entropy. The authors look at how fast this surprise grows as they turn the knob. They call the rate of this growth the Susceptibility (or the "Metric Response").

What They Found: Two Types of Storms

The researchers tested their "Surprise Meter" on two different types of magnet chains:

1. The "Predictable" Chain (Transverse Field Ising Model):

   • This is a well-known, solvable model.
   • The Result: As the chain gets longer, the "Surprise Meter" goes crazy, but it does so slowly. It grows like the square of a logarithm (think of it as a very slow, gentle explosion that gets bigger as the chain gets longer).
   • The Analogy: It's like a whisper that gets louder and louder as you add more people to the room, but it takes a huge room to hear it clearly.

2. The "Chaotic" Chain (Three-Spin Ising Model):

   • This model is harder to solve and involves magnets interacting with their neighbors' neighbors.
   • The Result: Here, the "Surprise Meter" explodes much faster. It grows as a power law (a steep, rapid climb).
   • The Analogy: This is like a fire that spreads instantly. As the chain gets longer, the signal of the storm becomes massive very quickly.

The Key Takeaway: The way the "Surprise Meter" explodes tells you exactly what kind of critical point you are looking at. It acts as a universal fingerprint for different types of quantum phase transitions.

The "Glitch" at the Extremes

The paper also noticed something weird when they turned the knob to the very extreme ends (making the magnetic field zero or infinite).

• The Problem: At these extremes, the "map" of the magnets becomes incomplete or "singular" (some probabilities become zero).
• The Glitch: When the map is incomplete, the "Surprise Meter" breaks down and shows a fake, infinite spike.
• The Distinction: The authors emphasize that this spike is not a real quantum storm (critical point). It's just a mathematical glitch because the system is too simple at those extremes. Real critical points happen in the middle, where the system is complex and the map is full.

Why This Matters (According to the Paper)

• It's Universal: You don't need to know the specific details of the material. Just look at how the "surprise" changes in a small piece of the system, and it will tell you if the whole system is critical.
• It Works for Small Pieces: You don't need to measure the whole infinite chain. Looking at just 1, 2, or 3 magnets is enough to see the signal of the whole system's criticality.
• It's Geometric: The authors describe this using "Information Geometry." Imagine the different settings of the knob as points on a map. Near a critical point, the distance between two settings becomes infinite. It's like trying to walk between two cities that are separated by a bottomless canyon; you can't take a finite step from one to the other.

Summary

The paper introduces a new tool to detect when a quantum system is about to undergo a massive change. By measuring how "surprised" a small part of the system is when the rules change slightly, they can detect the "storm" of a quantum phase transition. They showed that this tool works for both simple and complex systems, and the way the signal grows reveals the specific "personality" of the transition.

Technical Summary

Title: Metric Response of Relative Entropy: A Universal Indicator of Quantum Criticality

Problem Statement
The identification of quantum critical points (QCPs) in many-body systems has traditionally relied on order parameters, renormalization group (RG) theory, and fidelity susceptibility. While fidelity susceptibility and the geometric tensor have been shown to reflect the universality of quantum critical phenomena, there is a need for model-agnostic measures rooted in information geometry that can probe criticality across diverse systems, including non-integrable ones. Specifically, the paper addresses the behavior of the metric response of Quantum Relative Entropy (QRE) when a subsystem of a spin chain is traced out, and how this response scales with system size near critical points and classical limits.

Methodology
The authors propose a measure defined as the susceptibility of the Quantum Relative Entropy (QRE), denoted as $\Sigma_{hh}$. This quantity is derived from the information-geometric origin of the QRE, defined as $S(\hat{\rho} || \hat{\sigma}) = \text{tr}[\hat{\rho}(\ln \hat{\rho} - \ln \hat{\sigma})]$.

1. Definition: For a ground state $|\psi_0(\vec{\lambda})\rangle$ of a Hamiltonian with parameters $\vec{\lambda}$, a Reduced Density Matrix (RDM) $\hat{\rho}_\lambda$ is obtained by tracing out all but $n$ adjacent sites. The metric response $\Sigma_{ij}(\vec{\lambda})$ is defined as the second derivative of the QRE between infinitesimally separated parameter values:
   $$\Sigma_{ij}(\vec{\lambda}) = \frac{1}{2} \text{tr}[\hat{\rho}^{-1} \cdot \partial_i \hat{\rho} \cdot \partial_j \hat{\rho}]$$
   For a single parameter $h$, the diagonal component $\Sigma_{hh}$ is analyzed.
2. Theoretical Framework: The authors utilize Renormalization Group (RG) theory and Conformal Field Theory (CFT) to derive scaling laws for $\Sigma_{hh}$ near QCPs. They relate $\Sigma_{hh}$ to the uncertainty of the gradient of the entanglement Hamiltonian ($\hat{H}_E = -\ln \hat{\rho}$).
3. Models Studied:
   • Transverse Field Ising Model (TFIM): An integrable model solvable via Jordan-Wigner transformation to free fermions. Analytical calculations are performed for subsystem sizes $n=1$ and $n=2$.
   • Three-Spin Ising Chain: A non-integrable model with three-spin interactions, described by a Hamiltonian with a self-dual critical point belonging to the four-state Potts universality class. Numerical analysis is performed using Exact Diagonalization (ED) with a maximally reduced basis (using bitmasking and symmetry constraints) for subsystem sizes $n=1, 2, 3$.
4. Analysis: The study examines the Finite-Size Scaling (FSS) of the peak values of $\Sigma_{hh}$ and the location of their turning points as the system size $N$ increases. It also investigates the behavior of $\Sigma_{hh}$ near classical limits ($h \to 0$ and $h \to \infty$).

Key Results

1. Divergence at QCPs: The susceptibility $\Sigma_{hh}$ diverges at QCPs in the thermodynamic limit ($N \to \infty$) even for small subsystem sizes ($n \le 3$). The nature of this divergence depends on the universality class of the transition:
   • TFIM (Ising Universality, $z=\nu=1$): The peak of $\Sigma_{hh}$ exhibits a square-logarithmic divergence ($\sim \ln^2 N$). The turning points approach the critical point $h_c = \pm 1$ asymptotically.
   • Three-Spin Chain (Four-State Potts Universality, $z=1, \nu=2/3$): The peak of $\Sigma_{hh}$ exhibits a power-law divergence ($\sim N^2$).
2. Universality: The scaling behavior is governed by the critical exponents of the QCP and is independent of the subsystem size $n$ (provided $n$ is small enough to be invertible). The results confirm that $\Sigma_{hh}$ captures the universal scaling of the most relevant operator driving the phase transition.
3. Classical Limits and Rank Deficiency: Unlike the divergence at QCPs which requires $N \to \infty$, the paper finds that $\Sigma_{hh}$ can diverge at finite $N$ when the system is tuned to classical limits ($h \to 0$ or $h \to \infty$) if the subsystem size $n$ exceeds a certain threshold. This non-universal divergence arises because the RDMs become rank-deficient (singular) due to the degeneracy of symmetry-broken ground states, making the inverse $\hat{\rho}^{-1}$ ill-defined.
   • For TFIM, $\Sigma_2$ diverges as $h^{-4}$ near $h=0$.
   • For the three-spin chain, $\Sigma_3$ diverges as $h^{-4}$ near $h=0$.
   • This pathology is distinct from the critical divergence, as it is related to the specific degeneracy of the ground state manifold rather than gapless excitations.

Significance and Claims
The paper claims that the metric response of QRE serves as a universal indicator of quantum criticality that is:

• Model-Agnostic: Applicable to both integrable and non-integrable systems without requiring knowledge of specific order parameters.
• Information-Theoretic: Rooted in the geometry of the parameter space (Fisher-Rao metric generalization) and the fluctuations of the entanglement Hamiltonian.
• Robust: The divergence at QCPs is a thermodynamic property linked to critical exponents, whereas divergences at classical limits are finite-size artifacts related to rank deficiency.
• Geometric Interpretation: The singularity at a QCP implies the absence of a finite-length geodesic connecting distinct quantum phases in the parameter space, effectively characterizing the phase transition as a geometric singularity in the relative entropy landscape.

The authors conclude that this measure provides a direct link between information geometry and the universality classes of quantum phase transitions, offering a potential tool for characterizing complex quantum systems and topological phase transitions in future investigations.