🔬 condensed matter

Static vacancies as parametrized conformal defects in the critical $J_1$ -- $J_2$ transverse-field Ising chain

Using density-matrix renormalization-group calculations, this study demonstrates that two static nonmagnetic vacancies in the critical $J_1$ -- $J_2$ transverse-field Ising chain behave as a one-parameter family of partially transmissive conformal defects, characterized by algebraic interaction decay, a $J_2$ -dependent transmission ratio, and a nearly constant boundary entropy.

Original authors: R. L. Silva, R. C. Silva, R. J. C. Lopes, A. R. Pereira

Published 2026-07-08

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CC BY 4.0

Original authors: R. L. Silva, R. C. Silva, R. J. C. Lopes, A. R. Pereira

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

Imagine a long, thin line of tiny magnets (spins) that are all trying to align with their neighbors. This is a model physicists use to understand how materials behave at extremely low temperatures, right at the "tipping point" where they switch from being disordered to ordered. This tipping point is called quantum criticality.

In this paper, the researchers decided to poke holes in this line of magnets. Specifically, they removed two magnets (creating "vacancies") and watched how the rest of the line reacted. They wanted to see how these two empty spots "talked" to each other across the chain.

Here is the story of what they found, explained simply:

1. The Setup: A Chain with Holes

Think of the chain of magnets as a long rope.

The Normal Rope: In a perfect rope, if you wiggle one end, the whole rope moves.
The Rope with Holes: The researchers cut out two small sections of the rope. But here's the trick: they didn't just leave a gap. They left a tiny, invisible thread (a specific type of connection called a "second-neighbor coupling") that still connected the two sides of the hole.
The Goal: They wanted to see if the two holes could "feel" each other's presence and how easily a signal could pass through those holes.

2. The Big Surprise: From "Silence" to "Echo"

In a previous study (by the same authors), they looked at this system when it was not at the tipping point. There, the holes were like two people whispering in a soundproof room; the sound died out very quickly. The interaction between the holes vanished exponentially (very fast) as they moved apart.

But on the critical line (the tipping point), the rules changed completely.
Because the system is in a state of "criticality," the whole chain is hyper-sensitive. The interaction between the two holes didn't die out quickly. Instead, it faded away slowly, like an echo in a giant canyon. This is called an algebraic decay.

The Analogy: Imagine shouting across a valley. In the old (gapped) regime, the sound stops after a few miles. In this new (critical) regime, the sound travels for miles, getting quieter but never truly disappearing until it's very far away.

3. The "Transmission" of the Signal

The researchers also asked: "If a signal tries to jump over one of these holes, how much of it gets through?"

They found that the holes act like semi-transparent windows. They don't block the signal completely (which would be a solid wall), and they don't let it pass perfectly (which would be no hole at all).
Instead, they let a specific percentage of the signal through. This percentage depends on how strong that "invisible thread" (the second-neighbor coupling) is.
The Result: As they strengthened the connection, the holes became more "see-through." The signal transmission went from about 11% (mostly blocked) to 33% (more open).

4. The "Entropy" (The Confusion Factor)

In physics, "entropy" can be thought of as a measure of confusion or disorder. When you have a boundary (like a hole in the chain), it creates a little bit of extra "confusion" in the system.

The researchers measured this "boundary confusion."
They found it was small and steady. It was much closer to the value of a "free end" (where the rope just stops naturally) than to a "fixed end" (where the rope is nailed down tight).
The Takeaway: Even though there are holes, the system doesn't get very confused. It behaves almost like a normal, open chain, just with a few slightly leaky spots.

5. The Unified Picture: A Family of Defects

The most important conclusion is that these holes aren't random glitches. They form a smooth family.

By simply turning a dial (changing the strength of the connection $J_2$ ), the researchers could smoothly slide the system from a "mostly blocked" defect to a "more open" defect.
Everything changed smoothly and predictably. There were no sudden jumps or surprises. It's like tuning a radio: as you turn the knob, the station changes gradually, not in giant leaps.

Summary

The paper shows that when you have a quantum system right at its critical tipping point, removing magnets creates partially transparent windows rather than solid walls.

Interaction: The two holes "feel" each other over long distances (like an echo), not just short ones.
Transmission: The holes let some information through, and you can control exactly how much by adjusting the connections.
Stability: The system remains calm and predictable, behaving like a single, connected family of defects rather than a broken mess.

The researchers used powerful computer simulations (called DMRG) to watch this happen in chains of up to 300 magnets, confirming that these "static vacancies" act as conformal defects—a fancy way of saying they are special, scale-invariant objects that preserve the unique physics of the critical line while letting some things pass through.

Technical Summary: Static Vacancies as Parametrized Conformal Defects in the Critical J1–J2 Transverse-Field Ising Chain

Problem Statement
This work investigates the behavior of two static nonmagnetic vacancies within a transverse-field Ising chain featuring both first-neighbor ( $J_1$ ) and second-neighbor ( $J_2$ ) couplings. While previous research [1] established that in the gapped (ferromagnetic) regime, the interaction between such vacancies decays exponentially due to finite bulk correlation lengths, the behavior along the quantum critical line remains qualitatively different. In the critical regime, where the bulk correlation length diverges, vacancies act as boundary perturbations expected to interact via power-law Casimir forces. The central problem is to characterize these interactions and the nature of the vacancies as conformal defects across the critical line, specifically examining how the second-neighbor coupling $J_2$ tunes the defect properties.

Methodology
The authors employ Density-Matrix Renormalization-Group (DMRG) calculations on open chains of up to $N=300$ sites.

Model: The system is defined by the Hamiltonian $H = -J_1 \sum \sigma^x_i \sigma^x_{i+1} - J_2 \sum \sigma^x_i \sigma^x_{i+2} - b \sum \sigma^z_i$ .
Vacancy Implementation: A nonmagnetic vacancy is created by removing a spin at site $v$ . Crucially, while all $J_1$ and $J_2$ bonds touching $v$ are removed, the $J_2$ bond bridging across the vacancy (connecting $v-1$ and $v+1$ ) is retained. This ensures the chain remains a single connected system of $N-1$ spins, acting as a bulk defect rather than a cut.
Critical Line Calibration: The critical field $b_c(J_2)$ is located using the bulk spin-spin correlator exponent $\eta$ . The authors fit the correlator $\langle \sigma^x_i \sigma^x_j \rangle \sim |i-j|^{-\eta}$ in the central region of the chain, targeting the Ising universality class value $\eta = 1/4$ . They explicitly avoid using entanglement entropy central charge ( $c$ ) for calibration in this open geometry due to known boundary artifacts that overestimate $c$ .
Observables: Three key quantities are extracted:
1. Binding Energy ( $\Delta_b$ ): The interaction energy between two vacancies, defined as $\Delta_b(r) = E_{2v}(r) - 2E_{1v} + E_0$ .
2. Transmission Ratio ( $T$ ): The ratio of the spin correlator across a single vacancy to the clean chain correlator, $T(k) = |\langle \sigma^x_{v-k} \sigma^x_{v+k} \rangle_{vac}| / |\langle \sigma^x_{v-k} \sigma^x_{v+k} \rangle_{clean}|$ .
3. Boundary Entropy ( $\log g$ ): The Affleck-Ludwig boundary entropy extracted from the von Neumann entanglement entropy. To avoid spurious cusps, the authors perform a "matched-coordinate subtraction," accounting for the reduced chain length ( $N-1$ ) and the shift in lattice indices.

Key Results

Algebraic Casimir Interaction: Contrary to the exponential decay in the gapped phase, the vacancy-vacancy binding energy along the critical line decays algebraically: $|\Delta_b(r)| \sim r^{-\alpha}$ . The exponent $\alpha$ is close to the universal Casimir value of unity but exhibits a systematic dependence on $J_2$ . The extrapolated infinite-size exponent follows $\alpha_\infty \simeq 1.070 + 0.091 (J_2/J_1)$ for $J_2/J_1 \in [0.1, 1.0]$ .
Transmission Plateau: The transmission ratio $T(k)$ does not decay to zero but saturates to a finite, $J_2$ -dependent plateau $T_\infty$ . This value grows monotonically from approximately $0.11$ (at $J_2/J_1 = 0.1$ ) to $0.33$ (at $J_2/J_1 = 1.0$ ), indicating that the vacancy acts as a partially transmissive conformal defect rather than a perfect reflector or a trivial defect.
Boundary Entropy: The boundary entropy $\log g_\infty$ is found to be small, negative, and approximately constant across the studied range, with a value of $\log g_\infty \approx -0.073$ . This value is significantly closer to the free-boundary limit ( $\log g = 0$ ) than to the Ising fixed-boundary value ( $\log g = -\ln\sqrt{2} \approx -0.347$ ).
Unified Picture: The three observables ( $\alpha_\infty$ , $T_\infty$ , and $\log g_\infty$ ) vary smoothly and monotonically with the single parameter $J_2/J_1$ . The authors observe a strong correlation between the Casimir exponent and the transmission amplitude ( $\alpha_\infty \simeq 0.41 T_\infty + 1.014$ ).

Significance and Claims
The paper claims to establish that static vacancies in the critical $J_1-J_2$ Ising chain constitute a one-parameter family of conformal defects, parametrized by the second-neighbor coupling $J_2$ .

Defect Nature: The vacancies are identified as partially transmissive conformal defects. The retention of the bridging $J_2$ bond prevents the chain from severing, allowing for a continuous tuning of transmission from strongly reflective (low $J_2$ ) to more transmissive (high $J_2$ ).
Universality: The smooth variation of observables with $J_2$ is consistent with the known property of the Ising universality class ( $c=1/2$ ) where localized bond defects are exactly marginal, generating a line of conformal defects.
Robustness of $\eta=1/4$ : The study validates the use of the bulk order-parameter exponent $\eta=1/4$ as a robust diagnostic for locating the critical line in open geometries, avoiding the systematic errors associated with entanglement entropy fits in finite open chains.
Limitations: The authors modestly note that while they provide numerical evidence for this defect family, they do not provide an explicit analytic mapping between the lattice parameter $J_2/J_1$ and the theoretical parameters of known conformal defect families (e.g., the Roy-Saleur family). They suggest such a mapping would require constructing the lattice defect operator and projecting onto scaling fields, which is beyond the scope of the current work.

In summary, the work extends the understanding of impurity physics from gapped to critical regimes, demonstrating that quantum criticality transforms the vacancy interaction from screened exponential decay to algebraic Casimir forces, with the defect properties continuously tunable by the range of interactions.