A sine-square deformation approach to quantum critical points in one-dimensional systems

This paper proposes a sine-square deformation method to accurately determine quantum critical points in one-dimensional systems by identifying the transition to translational symmetry in local observables, demonstrating its effectiveness through density-matrix renormalization-group analysis of Ising chain models and suggesting a feasible experimental implementation using Rydberg atom arrays.

The Gist

The Big Idea: Smoothing Out the Rough Edges

Imagine you are trying to study the weather patterns of a vast, perfect ocean. But, you only have a small, rectangular swimming pool to work with. The problem is that the water in a pool behaves differently near the walls than it does in the middle. The walls create "ripples" and weird currents that don't exist in the real ocean. In physics, this is called a boundary effect.

Usually, to understand the "real" behavior of a system (like a quantum material), scientists want to simulate an infinite system with no walls. But computers can't handle infinite sizes. They have to use finite sizes, which means they have to deal with these annoying wall effects.

The Solution: The "Sine-Square Deformation" (SSD)
The authors of this paper propose a clever trick called Sine-Square Deformation (SSD). Think of this as a special "dimmer switch" for the energy of the system.

• Normal Open System: Imagine a chain of people holding hands. The people at the very ends (the edges) feel lonely and behave differently because they only have one neighbor. The people in the middle have two neighbors and feel stable.
• The SSD Trick: The authors suggest turning down the "strength" of the hand-holding for the people at the edges, gradually making it weaker until it's almost zero. Meanwhile, the people in the middle hold on just as tightly as usual.
• The Result: By gently fading out the edges, the "lonely" people at the ends stop acting weird. Suddenly, the whole chain behaves as if it were a perfect, endless loop, even though it's still a straight line with ends.

The Discovery: Finding the "Critical Point"

The main goal of the paper is to find the Quantum Critical Point (QCP).

• The Analogy: Imagine a crowd of people. If they are all calm, they are in a "Paramagnetic" phase (like a relaxed audience). If they are all shouting in a specific pattern, they are in an "Antiferromagnetic" phase (like a coordinated chant).
• The Critical Point: This is the exact moment the crowd switches from calm to chanting. It's a tipping point where the system is "gapless" (very sensitive and fluid).

The Paper's Claim:
The authors discovered a magical property of their "SSD trick." They found that if you tune your system to this exact Critical Point, the "ripples" caused by the edges disappear completely.

• Before the Critical Point: The people in the middle act differently than the people near the edges.
• At the Critical Point: Everyone in the chain, from the very first person to the very last, starts acting exactly the same. The system becomes perfectly uniform.

How they use this:
Instead of trying to calculate complex energy gaps (which is hard and requires huge computers), they simply look at a local measurement (like the "magnetization" or spin of a single atom). They ask: "Is the atom in the middle acting the same as the atom at the edge?"

• If No: You aren't at the critical point yet.
• If Yes: You have found the Critical Point!

Because this "uniformity" happens so clearly, they can find the exact tipping point using very small systems (only about 84 atoms), whereas other methods might need thousands of atoms to get the same accuracy.

The Experiments: Two Types of Chains

The authors tested this idea on two different types of "chains" (models):

1. The Nearest-Neighbor Chain: Atoms only talk to the person immediately next to them.
   • Result: Their method worked perfectly. They found the critical point with high precision, matching results from much larger, more expensive computer simulations.
2. The Long-Range Chain: Atoms can "whisper" to people far away down the line (like a long-range interaction).
   • Result: They found that the long-distance whispers slightly changed the rules. The critical point shifted a little bit, meaning the "tipping point" happens at a slightly different setting than in the simple chain.

The Real-World Application: Rydberg Atoms

The paper doesn't just stay in computer simulations. The authors propose a way to actually build this "SSD system" in a real lab using Rydberg atoms.

• The Setup: Imagine a row of atoms held in place by laser beams (optical tweezers).
• The Trick: By moving the atoms closer together or further apart in a specific zigzag pattern, the scientists can naturally create the "dimmer switch" effect. The atoms in the middle are close together (strong interaction), while the atoms at the edges are spaced out (weak interaction).
• The Claim: They showed that with current technology, you can arrange these atoms to mimic the SSD effect very accurately. This means real quantum computers (simulators) could use this method to find critical points without needing to build massive, perfect loops.

Summary

1. The Problem: Studying quantum systems is hard because the "edges" of the system mess up the results.
2. The Tool: The authors use a "Sine-Square Deformation" to gently fade out the edges, making the system behave like it has no edges at all.
3. The Method: They look for the moment when the "middle" and the "edge" of the system start acting exactly the same. This moment is the Quantum Critical Point.
4. The Benefit: This method is incredibly accurate and works even with small systems (like 84 atoms), saving a lot of computing power.
5. The Future: They showed this can be built in real labs using lasers and atoms, turning a theoretical math trick into a practical tool for quantum simulators.

Technical Summary

Technical Summary: A Sine-Square Deformation Approach to Quantum Critical Points in One-Dimensional Systems

Problem Statement
Determining quantum critical points (QCPs) in one-dimensional systems typically requires finite-size scaling analysis, which relies heavily on the choice of boundary conditions. While periodic boundary conditions (PBCs) preserve translational symmetry and minimize boundary effects, they are computationally challenging for tensor-network algorithms like the density-matrix renormalization group (DMRG) and are physically difficult to realize in experimental quantum simulators (e.g., cold atoms, trapped ions, Rydberg arrays) which naturally possess open boundaries. Conversely, open boundary conditions (OBCs) break translational symmetry, introducing edge effects that obscure bulk properties and complicate the identification of phase transitions. The authors address the challenge of accurately locating QCPs in finite-size systems with open boundaries without relying on the difficult-to-measure excitation gap closing criterion.

Methodology
The paper proposes a method based on Sine-Square Deformation (SSD). SSD modulates the local energy scale of an open-boundary Hamiltonian using a site-dependent function $f_L(i) = \sin^2[\frac{\pi}{L}(i - 1/2)]$, which smoothly suppresses interactions near the edges while maintaining an open geometry.

The core proposition, supported by exactly solved cases and conformal field theory (CFT) arguments, is that for gapless systems, the ground state of an SSD-deformed Hamiltonian exhibits translational symmetry in the thermodynamic limit. Specifically, the expectation values of local observables become site-independent at the critical point.

The authors implement the following procedure:

1. Model Selection: They study two mixed-field antiferromagnetic spin-1/2 Ising chains: one with nearest-neighbor (NN) interactions and one with long-range (LR) interactions decaying as $1/|i-j|^6$.
2. Observables: Instead of using global order parameters (like staggered magnetization), they analyze the spatial variance of local transverse magnetization $\langle \hat{S}^x_i \rangle$. They define six difference quantities ($\Delta_1$ to $\Delta_6$) representing the differences in expectation values between specific site pairs (e.g., center vs. edge, or center vs. $L/3$).
3. Scaling Analysis: They calculate the ground states using DMRG for system sizes $L$ up to 84. They track the zero-crossings or local minima of the $\Delta_n$ quantities as a function of the transverse field $h_x$ for various longitudinal fields $h_z$.
4. Extrapolation: By fitting the convergence of these zero-crossing/minima points as $1/L \to 0$, they estimate the QCP in the thermodynamic limit.
5. Experimental Proposal: The authors propose a physical implementation of the SSD Hamiltonian using Rydberg atom arrays in optical tweezers. By programming atomic positions, they demonstrate how to engineer the required spatially dependent van der Waals interactions to approximate the SSD-modulated $J_1-J_2$ Ising couplings.

Key Results

• Nearest-Neighbor Model: The SSD-based approach accurately estimates the QCP for the NN model. For $h_z = 0.5$, the method yields $h_x^c = 0.40165(7)$ using system sizes up to $L=84$. This result agrees with literature values obtained via infinite-size DMRG and gap-closing criteria (which required $L \le 300$) within a relative error of ~0.1%. The authors note that gap-closing criteria fail to provide reliable estimates when restricted to the same small system sizes ($L \le 84$) due to the smooth variation of the gap near criticality.
• Long-Range Model: For the LR model, the phase boundary is found to be slightly shifted compared to the NN case, with a reduced region of antiferromagnetic order. At $h_z=0$, the QCP is estimated at $h_x^c = 0.488019(2)$, approximately 2.4% lower than the NN result, consistent with gap-closing calculations.
• Critical Exponents: The authors perform a finite-size scaling collapse of the observable $\Delta_1$. The data collapses onto a single curve using exponents consistent with the (1+1)-dimensional Ising universality class ($\nu=1, \beta=1/8$), suggesting that SSD observables can capture critical scaling behavior even though they are not conventional symmetry-breaking order parameters.
• Rydberg Implementation: The authors demonstrate that the geometric constraints of Rydberg atom arrays allow for the realization of SSD-modulated interactions. They show that for $L \gtrsim 100$, the ratio of next-nearest-neighbor to nearest-neighbor coupling ($J_2/J_1$) can be reduced to below 2%, effectively realizing the nearest-neighbor SSD model with high fidelity.

Significance and Claims
The paper claims that the SSD approach offers a robust diagnostic tool for quantum criticality that is particularly well-suited for current and near-future quantum platforms (NISQ devices and quantum simulators) where system sizes are limited and open boundaries are the norm.

Key claims regarding significance include:

• Precision with Small Systems: The method enables the determination of QCPs with high precision using relatively small system sizes ($L \le 84$), outperforming traditional gap-closing methods in this regime.
• Experimental Feasibility: Unlike gap-closing criteria which require access to excitation spectra (often difficult to measure), the SSD method relies solely on local observables, which are directly accessible in modern quantum simulators.
• Generality: The approach is applicable to transitions involving at least one gapped phase. The authors modestly note that its applicability to gapless-gapless transitions and higher-dimensional systems (where CFT descriptions may not hold) remains an open question for future investigation.
• Bridge between Theory and Experiment: The work bridges the gap between numerical boundary-conditioning techniques and experimental Hamiltonian engineering, suggesting that SSD is not just a numerical trick but a physically realizable state preparation protocol.

The authors conclude that while the theoretical equivalence between SSD and PBC ground states is rigorously proven only for specific solvable models, their numerical results suggest the approach is a practical and effective tool for identifying critical points and potentially extracting critical exponents in more general interacting systems.