⚛️ quantum physics

Succession of Ising criticality and its threshold in critical quantum Ising model subject to $X+ZZ$ decoherence

This study demonstrates that the Ising criticality of a quantum system subject to $X+ZZ$ decoherence persists up to moderate noise levels with Ising-like scaling exponents, before eventually transitioning to a strong-to-weak spontaneous symmetry breaking phase at strong decoherence strengths.

Original authors: Yoshihito Kuno, Takahiro Orito, Ikuo Ichinose

Published 2026-10-26

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CC BY 4.0

Original authors: Yoshihito Kuno, Takahiro Orito, Ikuo Ichinose

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, the most delicate states are often the most powerful. Imagine a system of tiny magnets, each pointing either up or down, arranged in a line. When these magnets are perfectly isolated from the rest of the universe, they can settle into a special state called a critical point. At this precise moment, the system is balanced on a knife-edge between two different behaviors, and the particles become deeply connected to one another across the entire line, no matter how far apart they are. This state is described by a mathematical framework known as conformal field theory, which acts like a universal rulebook for how these connections behave. However, in the real world, nothing is ever truly isolated. The environment constantly nudges these quantum systems, introducing noise that scrambles their delicate connections and turns them into a messy, mixed state. Scientists have long wondered: if you take this perfect, balanced quantum state and let the environment interfere with it, does the special connection vanish completely, or does a ghost of it survive?

A team of researchers set out to answer this question by studying a specific type of quantum noise applied to a line of magnets. They focused on a scenario where the noise comes in two forms: one that flips the magnets randomly and another that twists the relationship between neighboring magnets. To understand what happens, the researchers used a clever mathematical trick. Instead of trying to track the messy, mixed state directly, they imagined the system as two identical copies of the original line of magnets, one stacked on top of the other. In this doubled view, the noise acts like a new kind of interaction between the two copies. This allowed them to map the messy, noisy situation onto a known model of physics called the quantum Ashkin-Teller model, which describes how two interacting lines of magnets behave. By using powerful computer simulations to follow the behavior of these doubled systems, they could see exactly how the critical connections changed as the noise grew stronger.

The researchers found that for a significant range of noise levels, the system did not lose its special character. Even though the state was no longer pure and was being disturbed by the environment, the way the particles connected to each other still followed the same rules as the original, perfect state. The simulations showed that the strength of these connections, and the way they spread across the system, matched the predictions for the Ising model, a classic description of magnetism. This was a surprising result because the mathematical model they mapped the system onto—the quantum Ashkin-Teller model—was expected to behave differently, following a different set of rules entirely. The data clearly showed that the system was not following those alternative rules. Instead, the "Ising" behavior, which is usually associated with perfectly isolated systems, persisted surprisingly well even when the system was mixed and noisy.

However, this resilience has a limit. As the researchers increased the strength of the noise beyond a certain point, the special connections began to break down. The simulations revealed a sharp transition where the system stopped behaving like the Ising model and entered a new, strange phase. In this new state, the system developed a different kind of order that was not present before. The researchers identified this as a phase where the system spontaneously breaks a symmetry in a way that is strong in one sense but weak in another. The delicate, long-range connections that defined the critical state disappeared, replaced by a different kind of behavior driven by the intense noise. This transition point acts as a threshold: below it, the quantum criticality survives the environment; above it, the environment wins, and the system changes its fundamental nature.

The study provides a clear picture of how quantum criticality can endure in the face of real-world interference. It demonstrates that the special properties of these balanced states are more robust than previously thought, surviving even when the system is no longer in a pure state. The findings suggest that there is a specific window of noise strength where the system retains its quantum character, but once the noise crosses a critical boundary, the system undergoes a fundamental transformation. This work helps scientists understand the limits of quantum stability, showing exactly how much disturbance a quantum system can take before its unique properties are washed away. By pinpointing where this line is drawn, the research offers a clearer understanding of how quantum systems behave when they are not perfectly isolated, a crucial step for building future quantum technologies that must operate in noisy environments.

Technical Summary: Succession of Ising Criticality and its Threshold in Critical Quantum Ising Model subject to X + ZZ Decoherence

Problem Statement
The paper investigates the fate of critical quantum many-body states when subjected to environmental decoherence, specifically focusing on the transverse-field Ising model (TFIM). While it is established that noise can destroy quantum orders, the behavior of critical states described by Conformal Field Theory (CFT) under decoherence remains an open issue. The authors address whether the universal properties of the Ising CFT (specifically the central charge $c=1/2$ and critical exponents) survive when the pure critical ground state is subjected to a specific $X + ZZ$ decoherence channel.

A key theoretical challenge is that the decohered state in the "doubled Hilbert space" formalism maps to the ground state of a quantum Ashkin-Teller (qAT) model. The qAT model possesses a critical line described by an orbifold boson CFT with central charge $c=1$ , which differs from the Ising CFT. The authors aim to determine if the mixed state criticality follows the $c=1$ qAT description or if the Ising criticality is partially preserved due to specific symmetries of the decoherence channel.

Methodology
The study employs a combination of the doubled Hilbert space formalism and Matrix Product States (MPS) to numerically analyze the mixed state.

Model and Decoherence: The system is the 1D TFIM at its critical point ( $J=h$ ). The decoherence channel consists of a combination of local $X$ and $ZZ$ operations ( $E_{ZZ} \circ E_X$ ). This specific choice ensures that the Kraus operators possess a weak Kramers-Wannier (KW) self-duality.
Doubled Hilbert Space Formalism: The mixed density matrix $\rho_D$ is vectorized into a state $|\rho_D\rangle\rangle$ in a doubled Hilbert space ( $\mathcal{H}_u \otimes \mathcal{H}_\ell$ ). In this representation, the decoherence acts as a filtering operator on two decoupled TFIM chains, effectively mapping the problem to the ground state of a quantum Ashkin-Teller (qAT) model on a two-leg ladder.
Numerical Simulation: The authors use the TeNPy library to generate the initial ground state and apply the decoherence filtering. They simulate system sizes up to $L=40$ (an 80-site ladder) with a maximum bond dimension $D=300$ .
Observables: To characterize the criticality, the study calculates:
- Rényi-2 Mutual Information (R2MI): Used to extract the effective central charge ( $c_{eff}$ ) by analyzing the subleading logarithmic term in the subsystem entanglement entropy, avoiding non-universal extensive terms.
- Susceptibilities ( $\chi_{II}$ and $\chi_{I}$ ): Derived from Rényi-2 and canonical $ZZ$ correlators to detect Strong-to-Weak Spontaneous Symmetry Breaking (SWSSB).
- Correlation Functions: Canonical $ZZ$ and $XX$ correlation functions are analyzed to extract critical exponents $\eta$ and $\eta_X$ .

Key Results
The numerical investigation reveals a distinct threshold for the preservation of Ising criticality under $X + ZZ$ decoherence:

Preservation of Ising Scaling (Moderate Decoherence): For decoherence strengths $p_{zz} \lesssim 0.3$ , the mixed state on the critical line exhibits scaling consistent with the Ising CFT, not the $c=1$ qAT CFT.
- The effective central charge extracted from R2MI remains close to $c_{eff} \approx 1/2$ .
- The critical exponent for the $ZZ$ correlation function, $\eta$ , remains close to the Ising value of $1/4$ .
- The critical exponent for the $XX$ correlation function, $\eta_X$ , remains close to the Ising value of $2$.
- This suggests that the weak KW self-duality of the specific decoherence channel allows the Ising criticality to survive, contradicting the expectation that the system would simply follow the $c=1$ qAT critical line.
Threshold and Transition (Strong Decoherence): As the decoherence strength increases beyond $p_{zz} \approx 0.3$ :
- The effective central charge $c_{eff}$ drops significantly from $1/2$ and vanishes for $p_{zz} \gtrsim 0.4$ .
- The system enters a phase characterized by Strong-to-Weak Spontaneous Symmetry Breaking (SWSSB), identified by the emergence of strong symmetry breaking ( $\chi_{II} \sim O(1)$ ) and the restoration of weak symmetry ( $\chi_{I} \sim 0$ ).
- In this strong decoherence regime, the Ising criticality is washed out, and the correlation amplitudes are strongly suppressed, although power-law decay forms persist with reduced amplitude.

Significance and Claims
The paper claims to provide a concrete numerical demonstration of how criticality evolves under decoherence, specifically challenging the assumption that the decohered TFIM simply maps to the $c=1$ qAT critical line.

Succession of Ising Criticality: The authors demonstrate that up to moderate decoherence strength, the mixed state retains Ising-like critical scaling ( $c=1/2, \eta=1/4, \eta_X=2$ ). This indicates that the specific symmetry properties of the $X+ZZ$ channel (weak KW duality) are sufficient to protect the Ising universality class against the noise that would otherwise drive the system toward the qAT universality class.
Threshold Identification: The study identifies a clear threshold ( $p_{zz} \approx 0.3$ ) where the Ising criticality breaks down and the system transitions into an SWSSB phase.
Methodological Validation: The work serves as a numerical validation of the doubled Hilbert space formalism for estimating critical exponents in mixed states, showing that independent observables (central charge, $\eta$ , $\eta_X$ ) yield mutually consistent Ising-like results within accessible system sizes.

The authors remain modest regarding the precision of their exponent extraction, noting that the results are based on finite-size consistency checks ( $L \le 40$ ) rather than a high-precision determination of asymptotic exponents in the thermodynamic limit. However, the consistency across multiple observables strongly supports the conclusion of Ising criticality succession up to the identified threshold.

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