Extracting central charge from ground-state overlaps of spatially deformed Hamiltonians

This paper demonstrates that the central charge of (1+1)-dimensional conformal field theories can be directly extracted from ground-state overlaps of spatially deformed Hamiltonians, providing a robust, wave-function-based method to probe conformal data in both critical quantum chains and topological edge modes.

The Gist

Imagine you have a complex, vibrating string (like a guitar string, but made of quantum particles). In the world of physics, this string represents a "critical system"—a state of matter that is perfectly balanced between order and chaos, like water right at the boiling point.

Physicists want to know a specific number about this string, called the central charge. Think of this number as the string's "fingerprint" or its "ID card." It tells you exactly what kind of quantum world the string is living in. Usually, figuring out this ID card is like trying to solve a massive jigsaw puzzle by looking at every single piece (every particle) and how they wiggle. It's hard, slow, and requires complex math.

This paper introduces a much simpler trick: The "Stretch and Compare" Method.

The Big Idea: Stretching the String

The authors realized that if you gently "stretch" or "squeeze" the string in a very specific, mathematical way (called a q-Möbius deformation), the string changes shape, but its fundamental identity remains hidden inside the change.

Imagine you have a rubber band with a pattern on it.

1. The Original: You have the rubber band in its normal, relaxed state.
2. The Deformed: You stretch the rubber band so the pattern gets squished in the middle and stretched at the ends, but you do it using a very precise, smooth recipe.

The paper proves that if you take the "quantum wave" of the original rubber band and the "quantum wave" of the stretched rubber band and overlap them (like holding two transparent sheets on top of each other to see how much they match), the amount they don't match tells you the ID card number (the central charge) immediately.

The "Recipe" for the Stretch

The authors didn't just stretch it randomly. They used a special mathematical recipe involving a function called tanh (which looks like a smooth "S" curve).

• They applied this recipe to the energy of the system, making some parts of the string "heavier" and others "lighter" in a smooth wave pattern.
• They found a magic formula: The more the two states (original and stretched) fail to overlap, the higher the central charge. It's like a volume knob: the "loudness" of the mismatch is directly proportional to the fingerprint number.

Testing the Theory

To prove this wasn't just a pretty math trick, the authors tested it on four famous "quantum chains" (models of magnets and particles):

1. The Ising Chain: A simple model of a magnet.
2. The Three-State Potts Chain: A slightly more complex magnet model.
3. The Heisenberg Chain: A model where particles spin in all directions.
4. The SU(3) Chain: A very complex, high-level quantum model.

In all these cases, they used a powerful computer simulation (called DMRG) to calculate the overlap. The result? The "fingerprint" number they calculated matched the known, perfect theoretical values almost instantly. It was like guessing a person's height by looking at their shadow and getting it right every time.

What About the "Inside" of the String?

The paper also looked at what happens inside the stretched string. They checked the entanglement (a spooky quantum connection between particles).

• They found that even though the string was stretched, the "shape" of these quantum connections remained perfectly geometric and predictable.
• It's as if you stretched a rubber band, and the internal knots tied in the rubber band rearranged themselves perfectly to fit the new shape, keeping the same underlying logic. This confirmed that the "stretch" didn't break the physics; it just revealed it.

Going 2D: The Edge of a Topological Island

Finally, they took this idea and applied it to a 2D world (like a flat sheet of material). Imagine a sheet of paper that has a "gapless edge" (a special, active border) while the middle is quiet.

• They stretched the edge of this sheet.
• They found they could measure the fingerprint of the entire edge, or even just one side of the edge, by looking at the overlap.
• This is like being able to measure the heartbeat of a whole animal just by listening to its left ear, or just its right ear, without needing to listen to the whole body.

The Bottom Line

The paper claims that by simply deforming the shape of a quantum system's energy and comparing the before-and-after states, you can extract the most fundamental number that defines that system.

It's a new, simple, and robust way to read the "ID card" of quantum matter without needing to solve the entire universe's puzzle. It turns a complex, multi-step detective story into a single, elegant measurement.

Technical Summary

Technical Summary: Extracting Central Charge from Ground-State Overlaps of Spatially Deformed Hamiltonians

Problem Statement
The central charge $c$ is a fundamental universal datum of (1+1)-dimensional conformal field theories (CFTs), characterizing the Virasoro conformal anomaly, finite-size energy scaling, and entanglement entropy logarithmic coefficients. While established methods exist to extract $c$ from microscopic lattice models—such as finite-size spectral scaling, entanglement entropy scaling, and quantum quench return amplitudes—these approaches often rely on finite-size scaling analyses, dynamical protocols, or the resolution of excited states. The authors identify a need for a simpler, universal probe that directly encodes the conformal anomaly using only ground-state properties.

Methodology
The paper proposes a method based on spatially deformed Hamiltonians, specifically utilizing the $q$-Möbius deformation. The approach proceeds in two stages:

1. Continuum CFT Derivation:
   The authors consider a unitary CFT on a circle of circumference $L$. They introduce a spatially modulated Hamiltonian density $H_q(\theta)$ by multiplying the uniform stress tensor density $T(x) + \bar{T}(x)$ by an envelope function $f_{q,\theta}(x) = 1 - \tanh(2\theta)\cos(2q\pi x/L)$.

   • Using the Virasoro algebra, they demonstrate that this deformation is generated by a unitary transformation $U_q(\theta) = e^{-\frac{\theta}{q}(L_q - L_{-q})}e^{-\frac{\theta}{q}(\bar{L}_q - \bar{L}_{-q})}$.
   • The deformed Hamiltonian is related to the uniform Hamiltonian $H$ by a rescaling of the spectrum and a constant energy shift. Crucially, the ground state of the deformed Hamiltonian, $|0_q(\theta)\rangle$, is obtained by acting on the conformal vacuum $|0\rangle$ with $U_q(\theta)$.
   • By disentangling the transformation, the authors derive an exact overlap formula between the uniform ground state and the deformed ground state:
     $$\langle 0 | 0_q(\theta) \rangle = [\cosh \theta]^{-\frac{c(q^2-1)}{6q}}$$
     This result shows that the decay of the overlap depends exclusively on the central charge $c$, the deformation mode $q$, and the strength $\theta$.

2. Lattice Realization and Numerical Implementation:
   To apply this to microscopic models, the authors define a lattice $q$-Möbius deformation by applying the same envelope function $f_{q,\theta}(X)$ to local Hamiltonian terms $h_X$.

   • They define an effective central charge estimator based on the lattice ground-state overlap:
     $$c_{\text{eff}}(L; q, \theta) = -\frac{6q}{q^2-1} \frac{\ln |\langle \Omega_0 | \Omega_q(\theta) \rangle|}{\ln[\cosh \theta]}$$
   • Using the Density Matrix Renormalization Group (DMRG) method, they compute the ground states of both uniform and deformed Hamiltonians for various critical chains, approximating them as Matrix Product States (MPS) to calculate overlaps.
   • They also extend the construction to a 2D Chern insulator (Qi-Wu-Zhang model) on a cylinder, where they deform the Hamiltonian terms based on spatial position to probe gapless edge modes.

Key Contributions and Results

• Exact Overlap Formula: The paper derives a rigorous analytical formula linking the ground-state overlap of a $q$-Möbius deformed CFT directly to its central charge. This provides a theoretical foundation for using wave-function overlaps as a probe of conformal data.
• Numerical Validation in 1D: The authors apply the estimator to four paradigmatic critical chains: the Ising model ($c=1/2$), the three-state Potts model ($c=4/5$), the spin-1/2 Heisenberg chain ($c=1$), and the SU(3) Uimin-Lai-Sutherland chain ($c=2$).
  • Numerical results show that $c_{\text{eff}}$ rapidly converges to the expected CFT values as the system size $L$ increases.
  • The method proves robust against variations in the deformation strength $\theta$.
• Entanglement Structure: The authors demonstrate that the deformed ground states retain universal geometric structures.
  • Entanglement Spectrum: The spectrum retains the tower structure of the boundary CFT, with gaps scaled by a geometric factor $W_A(q, \theta)$ determined by the deformation.
  • Entanglement Entropy: The entropy follows the analytical prediction derived from the conformal map, confirming that the deformed states remain moderately entangled and numerically tractable via DMRG.
• Extension to 2D Topological Phases: The method is successfully applied to the gapless edge modes of a 2D Chern insulator.
  • Deforming both edges yields an effective central charge $c=1$ (representing the chiral and anti-chiral sectors).
  • By localizing the deformation to a single edge, the authors extract $c=1/2$, corresponding to a single chiral sector. This demonstrates the ability to probe chiral central charges in 2D systems using ground-state overlaps.

Significance and Claims
The paper claims to provide a simple and robust wave-function-based route to probing conformal data, specifically the central charge, in both critical systems and topological edge modes.

• Simplicity: Unlike methods requiring excited state resolution or complex dynamical protocols, this approach relies solely on ground-state overlaps.
• Universality: The estimator converges to universal CFT values across different microscopic models.
• Geometric Insight: The work highlights that spatial deformations preserve the underlying conformal geometry in entanglement properties, justifying the use of standard tensor network methods for these deformed states.
• Experimental Potential: The authors note that the unitary origin of the deformation suggests a possible experimental direction. If deformed ground states can be prepared (e.g., via quantum simulators), the overlap could be accessed through interferometric or randomized-measurement protocols, offering a potential pathway for experimental verification.

The authors remain modest regarding immediate experimental realization, framing it as a "possible experimental direction" contingent on the ability to prepare states with sufficient fidelity. The primary contribution remains the theoretical derivation and numerical validation of a new, overlap-based diagnostic tool for conformal anomalies.