Experimental observation of exact quantum critical states

This paper reports the unambiguous experimental realization of exact quantum critical states using a programmable superconducting qubit system, demonstrating that quasiperiodic zeros in hopping couplings protect these states and revealing anomalous mobility edges through the observation of coexisting delocalized dynamics and incommensurately distributed coupling zeros.

The Gist

Imagine a crowded dance floor where everyone is trying to move around. In physics, this "dance floor" is a material, and the "dancers" are electrons (or in this experiment, quantum bits).

Usually, there are two ways these dancers behave:

1. The Free-Flowing Crowd (Extended State): Everyone moves freely across the entire floor. If you drop a dancer in the middle, they can eventually reach any corner.
2. The Frozen Crowd (Localized State): Imagine the floor is covered in sticky glue or obstacles. If you drop a dancer, they get stuck in one spot and can't move anywhere else. This is called Anderson Localization.

But there is a mysterious, rare third state called the Critical State. This is the "Goldilocks" zone. The dancers aren't fully stuck, but they aren't fully free either. They move in a weird, fractal pattern—like a snowflake that looks the same no matter how much you zoom in. They are "stuck" in some places but "free" in others, creating a complex, self-similar dance.

The Big Problem:
For decades, scientists have been trying to prove these "Critical Dancers" actually exist. It's incredibly hard because in real experiments, the dance floor is too small. On a tiny floor, a "stuck" dancer might look like they are moving, and a "free" dancer might look stuck. It's like trying to judge the weather by looking at a single raindrop.

The Breakthrough:
This paper describes a team of scientists who finally caught these Critical Dancers in the act using a super-advanced quantum computer made of superconducting qubits (tiny artificial atoms). They didn't just guess; they built a custom "dance floor" with specific rules that guaranteed these critical states would appear.

Here is how they did it, using simple analogies:

1. The "Mosaic" Dance Floor

The scientists created a special pattern of connections between their qubits, called a Mosaic Model.

• The Rule: Imagine a row of dancers holding hands. Usually, they hold hands tightly. But in this experiment, they programmed the "hand-holding strength" (coupling) to vary in a specific, non-repeating pattern (quasiperiodic).
• The Magic Zeros: The most important part is that in this pattern, there are specific spots where the "hand-holding" strength drops to zero. These are the Incommensurately Distributed Zeros (IDZs).
• The Analogy: Imagine a line of people passing a ball. Usually, they pass it easily. But every few steps, there is a person who refuses to pass the ball (a zero). However, because the pattern of "refusers" is weird and non-repeating, the ball can still wiggle its way through the line, but it gets stuck in specific pockets. This creates the "Critical" state.

2. The "One-Way Street" Effect

The team observed something fascinating. When they started the "ball" (a quantum state) near one of these "refusers" (the zero), the ball would travel down the line, but it would never cross the zero.

• It was like a one-way street. The ball could move left or right, but the "zero" acted like an invisible wall that the ball could bounce off but never cross.
• This "one-sided" movement is the smoking gun evidence that the state is Critical. It proves the system is neither fully stuck nor fully free.

3. The "Long-Range" Test

To prove their theory, the scientists added a new twist: Long-Range Connections.

• Imagine the dancers are on a 2D grid. They can hold hands with their immediate neighbors, but the scientists also gave them the ability to reach across the room and hold hands with people far away.
• The Discovery: They found that as long as these "long-distance hand-holds" weren't too strong, the "Critical Dance" survived! The "refusers" (zeros) were still strong enough to keep the ball from crossing, even with the long-distance help.
• The Breaking Point: Only when the long-distance connections became very strong did the "refusers" get overwhelmed, the "one-way street" disappeared, and the dancers became fully free (Extended state).

4. The "Mobility Edges" (The Traffic Lights)

Finally, they looked at how energy affects this. They found that in this system, some dancers are stuck, some are critical, and some are free, all at the same time, depending on their "energy" (how fast they are dancing).

• They identified specific "traffic lights" (called Mobility Edges) that separate the stuck dancers from the critical ones. This is a rare and exotic phenomenon that was predicted by math but never clearly seen in an experiment until now.

Why Does This Matter?

Think of this like discovering a new type of matter.

• For Technology: Understanding these "Critical States" helps us design better materials for electronics that are more efficient or resistant to errors.
• For Physics: It solves a decades-old puzzle about how disorder affects quantum systems. It proves that there is a whole new "phase of matter" sitting right between order and chaos.

In a Nutshell:
The scientists built a quantum playground with a specific pattern of "roadblocks" (zeros). They showed that these roadblocks create a unique, fractal-like state of matter that is neither stuck nor free. They proved that this state is robust (hard to break) but can be destroyed if you add too many "long-distance bridges." This is the first time we have seen this exact quantum dance clearly, opening the door to new discoveries in quantum physics.

Technical Summary

1. Problem Statement

Anderson localization physics identifies three fundamental types of eigenstates: extended, localized, and critical. While extended and localized states are well-understood, critical states (existing at the transition point) are notoriously difficult to confirm experimentally.

• The Challenge: In finite-sized experimental systems, localized and extended states can exhibit fluctuations that mimic critical behavior, making rigorous identification impossible without reaching the thermodynamic limit or identifying a universal mechanism.
• The Gap: Although theoretical frameworks (such as Avila's global theory) predict that critical states in 1D quasiperiodic systems arise from Incommensurately Distributed Zeros (IDZs) in hopping couplings, experimental verification has been elusive due to finite-size effects and the lack of a controllable platform to isolate these mechanisms.

2. Methodology

The authors utilized a programmable superconducting quantum processor to simulate a quasiperiodic mosaic model with tunable parameters.

• Hardware: A 2D array of 66 frequency-tunable transmon qubits and 110 tunable couplers. The experiment utilized a 24-site subarray configured to emulate a 1D spin chain.
• Hamiltonian: The system implements a 1D mosaic model with nearest-neighbor (NN) exchange couplings ($J_j$) and on-site potentials ($V_j$), plus tunable long-range couplings ($J^L_{m,n}$):
  $$H/\hbar = -\sum J_j (\sigma^+_j \sigma^-_{j+1} + h.c.) + \sum V_j \sigma^+_j \sigma^-_j - \sum J^L_{m,n} (\sigma^+_m \sigma^-_n + h.c.)$$
  • Mosaic Pattern: The NN coupling $J_j$ alternates between a constant $\lambda$ (odd sites) and a quasiperiodic term $2J \cos(2\pi\alpha j + \theta)$ (even sites), where $\alpha = (\sqrt{5}-1)/2$.
  • IDZs: The quasiperiodic term naturally creates zeros in the coupling strength at specific sites in the thermodynamic limit.
• Experimental Control:
  • Tunability: Independent control of coupling strengths ($J, \lambda, J^L_{m,n}$) and on-site potentials ($V_0$).
  • Long-Range Coupling: By activating specific couplers in the 2D array, the team introduced long-range interactions to test the robustness of critical states.
  • Dynamics: The system was initialized in specific states (e.g., single-site excitations or superpositions near IDZs), and time-evolved observables were measured.
• Observables:
  • Fractal Dimension ($D$): Quantifies the spatial extent of the wavefunction (0 for localized, 1 for extended, $0<D<1$ for critical).
  • Integrated Width ($M$): Measures the expansion of the wave packet.
  • Uni-side Dynamics: Observing propagation restricted to one side of an IDZ bond.

3. Key Contributions

1. Unambiguous Experimental Realization: The first clear experimental observation of exact quantum critical states governed by a rigorous theoretical mechanism.
2. Identification of the Universal Mechanism: Confirmed that Incommensurately Distributed Zeros (IDZs) in hopping couplings are the essential ingredient for stabilizing critical states.
3. Discovery of a Generalized Mechanism: Demonstrated that critical states are robust against weak long-range couplings. The IDZs are "dressed" but not removed by weak interactions, preserving the critical phase. The phase only transitions to extended when long-range couplings exceed a specific threshold that eliminates the IDZs.
4. Resolution of Anomalous Mobility Edges (MEs): Mapped the energy-dependent transition between localized and critical states, revealing the existence of exact mobility edges ($E_c = \pm \lambda$) predicted by theory.

4. Key Results

• Phase Transitions:
  • Localized Phase ($\lambda < J$): Wavefunctions remain confined; density profiles do not spread.
  • Critical Phase ($\lambda > J$): Wavefunctions exhibit delocalized dynamics but preserve local configurations. Crucially, the experiment observed uni-side quantum dynamics: wave packets initialized near an IDZ propagate only to one side of the bond, a signature of the IDZ protection.
  • Extended Phase: When long-range couplings ($J^L_{m,n}$) were increased beyond a threshold ($\approx 0.3J - 0.4J$), the IDZs were effectively removed, and the system transitioned to a fully extended phase with rapid delocalization.
• Robustness of IDZs: The critical phase persisted even with moderate long-range couplings, confirming that the IDZ mechanism is robust. The transition to the extended phase only occurred when the long-range terms were strong enough to fully suppress the zeros.
• Mobility Edges: By tuning the on-site potential ($V_0 = J$), the system became exactly solvable. The experiment observed a sharp transition in dynamics at energy $E = \lambda$:
  • States with $|E| < \lambda$ showed critical behavior (high fractal dimension).
  • States with $|E| > \lambda$ showed localized behavior (low fractal dimension).
  • This confirmed the existence of anomalous mobility edges separating localized and critical zones.
• Finite-Size Scaling: Theoretical analysis and numerical simulations confirmed that the experimental observations align with the thermodynamic limit predictions, validating the use of the 24-site system to probe critical phenomena.

5. Significance

• Fundamental Physics: This work resolves a decades-old challenge in condensed matter physics by providing the first rigorous experimental proof of exact critical states and their underlying mechanism (IDZs).
• New Mechanism: It establishes a "generalized mechanism" where critical states are protected by IDZs against perturbations, a finding that extends beyond idealized 1D models to more complex, realistic systems with long-range interactions.
• Platform for Quantum Simulation: The programmable superconducting platform demonstrates high controllability for studying disordered systems, offering a pathway to explore:
  • Many-body critical phases (interacting systems).
  • Higher-dimensional critical phenomena.
  • The interplay between criticality, noise, and dissipation.
• Standard for Detection: The paper sets a new standard for characterizing quantum critical states, utilizing time-averaged fractal dimensions and uni-side dynamics as "smoking gun" signatures, applicable to other platforms like trapped ions and neutral atoms.

In summary, this paper bridges the gap between rigorous mathematical theory and experimental reality, confirming that quasiperiodic zeros in hopping couplings are the key to stabilizing the elusive quantum critical phase.