Absorbing state transitions with long-range annihilation

arXiv:2409.03280 · cond-mat.stat-mech, quant-ph · Submitted 2024-09-05 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Absorbing state transitions with long-range annihilation".

Mira: The gist The authors introduce a family of classical stochastic processes describing diffusive particles undergoing branching and long-range annihilation in the presence of a parity constraint,

Kai: First, who's behind it and why it matters.

Title and authors: Mira: Let's look at who wrote this paper, "Absorbing state transitions with long-range annihilation." Nicholas O’Dea, Sayak Bhattacharjee, Sarang Gopalakrishnan, and Vedika Khemani are the authors.

Kai: They are coming from different backgrounds—O'Dea from Stanford Physics and Princeton Engineering—so you have a mix of theoretical insight and practical system knowledge in the team behind this work.

Lev: From an engineering standpoint, having authors with that kind of background means they can map these abstract stochastic processes onto actual physical hardware constraints, like how you might tune coupling strengths or interaction ranges.

Kai: Exactly, and that tuning aspect is what allows them to explore this continuous line of universality classes we just talked about. They’re not just doing a math exercise; they are setting up a framework for probing these interactions experimentally.

Mira: It suggests that the physics isn't fixed into one specific interaction type; you can vary kappa and observe how the critical behavior shifts along this line.

Lev: If we can tune kappa in our actual system, maybe by changing some coupling strengths or interaction ranges, we might actually be able to probe this continuous line of universality classes directly.

Kai: That’s the practical goal, isn't it? We want to find a way to control these long-range effects so we can see that interpolation in action in our simulations or experiments.

The paper's summary: Kai: Moving on from the authors, what they actually found in "Absorbing state transitions with long-range annihilation," they describe this system as particles undergoing diffusion, branching, and annihilation subject to a parity constraint. They lay out the rules for these interactions very precisely.

Mira: The core mechanism they emphasize is that pair annihilation happens with probability one over r kappa, where r is the distance between particles, and that power-law decay of those events introduces that tunable long-range aspect we discussed earlier.

Lev: And they also enforce a hard-core constraint, which means you can't have two particles on the exact same site at the same time. This adds another layer of complexity to how the branching and hopping steps actually proceed in this model.

Kai: So when they look at their findings in this paper, they establish that this specific setup yields a continuum of unusual universality classes for these kinds of absorbing state transitions.

Mira: They specifically point out that this line of universality classes interpolates between directed percolation, or DP, and parity-conserving physics. This is something quite new when you think about non-equilibrium phase transitions where you usually expect a split between those two.

Lev: It’s interesting because it suggests that the binary choice between those two standard classes isn't as rigid as we often assume when you introduce these kinds of long-range interactions to the setup.

Kai: That's right, and they show this interpolation happens despite having a global parity constraint on the system.

The paper's improvements: Mira: Now let’s talk about the specific insights they suggest about how this model behaves across different limits. They test various values of kappa to see how the system reacts as it changes.

Lev: They explore the short-range limit where kappa goes to infinity, which forces particles to be neighbors just for annihilation to happen, and that leads into a standard diffusion-limited decay form for the particle density n(t).

Kai: That’s when they see the decay n(t) about t-d/two if we're in d dimensions, which is consistent with what you expect from a standard random walker <ref:2409.03280#pg1>.

Mira: But then they move into the case where kappa is less than infinity, and they find that the typical interparticle distance scales like r about n(t)-one/d, which changes how fast annihilation happens over time for kappa greater than zero <ref:2409.03280#pg3>.

Lev: That scaling relationship means that the number of annihilation events per unit time scales as n(t) kappa/d, which dictates a different kind of decay form, like n(t) about (t - d/kappa) when kappa is positive.

Kai: The real improvement they show, and this is pretty interesting for us working on hardware simulations, is that in the limit kappa = zero the global parity constraint doesn't actually have consequences locally.

Mira: That means even though you have a global rule about particle counts across the whole system, a small neighborhood might see long-range annihilation violating that local parity, which lets them describe it using symmetry-free directed percolation physics instead of the standard PC physics.

Conclusion: Kai: So to wrap up on "Absorbing state transitions with long-range annihilation," this paper shows that adding a power-law decay to annihilation—that tunable exponent kappa —doesn't just give you two fixed choices; it gives you an entire spectrum of critical behavior between directed percolation and parity-conserving.

Mira: It really enriches our understanding of these phase transitions because we can now continuously vary the exponents, which is something that was previously hard to see in this system.

Lev: From a simulation standpoint, the paper shows that for kappa greater than d, say if d is the dimension, the model becomes effectively short-ranged and diffusion-limited in terms of particle decay.

Kai: Yeah, and they did show some numerical consistency with what we expect from DP and PC at specific points, like when kappa is zero or greater than two, they get exponents around zero point one five nine for DP physics.

Mira: The implication for the broader field is that we need to look beyond these simple binary classifications when studying non-equilibrium systems with long-range interactions because those continuous interpolations are where the interesting physics lives.

Lev: I just want to say that modeling this on real hardware would be tricky because of all these long-range effects; you'd have to manage that kappa tuning very carefully for the simulations to match these critical points.

Kai: That’s a fair point, Lev. We’ll see how close we can get to realizing those continuous regimes in our next experimental setups.

Mira: And it sets up a clear direction for future work on how these non-local interactions affect things like quantum error correction processes, which is where this kind of math actually helps us connect to our hardware experiments.

Department of Physics, Stanford University · Department of Electrical and Computer Engineering, Princeton University

cond-mat.stat-mech, quant-ph

Submitted: 2024-09-05

Updated: 2026-10-19

Comments: 5+9 pages, published version

Journal ref: Phys. Rev. E 114, L012101, 2026

DOI: 10.1103/njzm-t262

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 83/100

The gist: The gist The authors introduce a family of classical stochastic processes describing diffusive particles undergoing branching and long-range annihilation in the presence of a parity constraint,

Key concepts

Parity Constraint: This constraint ensures that the total number of particles in the entire system remains even or odd, but never changes by an odd amount. It is a global rule, meaning it affects all particles simultaneously. While it seems like a simple binary choice (present or absent), its effect on local behavior can be subtle and lead to different universality classes depending on other parameters.
Directed Percolation (DP): This is one of the two canonical absorbing-state universality classes in this model, representing the behavior of the system without the parity constraint. It describes how particle density decays over time in a standard non-equilibrium phase transition. It serves as a benchmark against which other behaviors are compared.
Parity-Conserving (PC): This is the second canonical absorbing-state universality class, corresponding to the system with the parity constraint active. It describes particle decay dynamics when the global parity must be maintained. The model explores how this constraint modifies standard critical behavior, leading to new types of transitions.

Terminology

Phase Transition — 1× in this paper · explained in 2 episodes
Phase Transitions — 1× in this paper · explained in 2 episodes

Summary

The gist The authors introduce a family of classical stochastic processes describing diffusive particles undergoing branching and long-range annihilation in the presence of a parity constraint, uncovering a continuum of unusual universality classes that interpolate between directed percolation and parity-conserving.

Model Description

The model describes a system of particles undergoing diffusion, branching, and annihilation subject to a parity constraint which conserves the number of particles modulo two The model consists of a mixture of deterministic and stochastic steps, illustrated in Fig. 1(a)-(c) for a system in one spatial dimension Each time step begins by selecting a particle at random, illustrated by the green outline With probability pann, we continue to the annihilation stage in (a), where the particle’s distance r to the nearest excitation is calculated and with probability 1/rκ the two particles are annihilated If, with probability 1 − pann, annihilation is not selected, then (b) with probability pbr the particle branches (produces two offspring on two different adjacent sites) or (c) with probability 1 − pbr, it attempts to hop to a neighboring site A hard-core constraint is enforced by aborting the attempt to branch or hop if more than one particle would occupy the same site

Universality Classes and Critical Line

The model uncovers a continuum of unusual universality classes, two paradigmatic universality classes being directed percolation (DP) and parity-conserving (PC), respectively corresponding to local models without and with a parity constraint Having a parity constraint or not naively seems like a binary question, making DP and PC discrete and ostensibly unconnected universality classes However, our model uncovers a line of universality classes with exponents that continuously interpolate between DP and PC despite a global parity constraint (Fig. 1d)) greatly enriching the set of known critical phenomena in non-equilibrium phase transitions

Absorbing Phase Dynamics

The density of particles n(t) decays as n(t) ∼ t −αc(κ) in the critical line, where αc is the critical decay exponent Two canonical absorbing-state universality classes are directed percolation (DP) and parity-conserving (PC), respectively corresponding to local models without and with a parity constraint In the short-range limit of κ = ∞ where particles must be neighboring in order for annihilation to proceed, the decay is given by n(t) ∼ t −d/2 for d 2 For κ < ∞, the typical interparticle distance will scale on the order of r ∼ n(t) −1/d and hence the number of annihilation events per unit time will scale as n(t)κ/d, leading to a decay form like n(t) ∼ (t −d/κ for κ > 0 In d = 1, the decay in the absorbing phases at κ ≥ 2 and κ = 0 are consistent with the PC and DP absorbing phases of n(t) ∼ t −1/2 and n(t) ∼ e −ct respectively

Field Theory Approach

The field theory action L includes standard diffusion ϕ¯(∂t−D∇2)ϕ and branching B(1−ϕ¯ 2)ϕϕ¯ terms, but it also contains an annihilation term that is exponentially suppressed in particle density This exponential suppression comes directly from the constraint that a particle can only be paired with the closest particle The resulting mean-field theory in the absorbing phase crucially relies on this exponential term and matches the predictions of our simple scaling arguments in d ≥ 2 The action S is obtained by taking the continuous space limit of Eq. A12 using Eqs. A14, A15, and A16

Discussion on Constraints

The global parity constraint exactly holds, but it no longer has consequences locally when κ = 0 In particular, looking at just a local subsystem, long-range annihilation will violate the parity of the number of particles in the subsystem even as it preserves the global parity This allows a model with a global parity constraint to be described by symmetry-free DP rather than PC physics The study was inspired by considering quantum error correction processes which often require completely non-local feedback operations (κ = 0) in various canonical cases

Transition Numerics

In Fig. 3, we find that the decay exponent αc is consistent with DP’s αDPc ≈.159 at κ = 0 and PC’s αPCc ≈.285 at κ ≥ 2 The exponents for 0 < κ < 2 appear to monotonically interpolate between the two, indicating a line of critical fixed points with continually varying exponents

Field Theory Mean-Field Results

For κ ≤ d, the integral needs the infrared-regulator provided by the decaying exponential, which forces the resulting integral to depend strongly on n When κ > d, there is no longer an infrared divergence, but there is an ultraviolet divergence; this small-z divergence in the integral is regulated by the lattice spacing Together, these give n˙(t) ∼ 2Bn(t) − cA for κ > d

Subleading Corrections

One of our predictions in d = 1 is that for every κ ≥ 2, α = 1/2 This is borne out in our numerics except for a small discrepancy that is most pronounced when κ ≈ 2.6 We believe this discrepancy is caused by asymptotically sub-leading terms from the background long-range decay from κ in this otherwise diffusion-limited regime For example, κ = 3 in the inset shows an ultimatelytransient slower decay at early times

Quantum Circuit Mapping

A two-site particle number conserving block-diagonal Haar matrix Uh gives rise to a hopping term in the classical process Feedback allows for nonlocal annihilation processes We chose our serial model for our numerics because we found that it has better finite-size and-time effects, making it easier to illustrate our results The study was inspired by considering quantum error correction processes which often require completely non-local feedback operations (κ = 0) in various canonical cases

Differences from Quantum Error Correction

Our absorbing state dynamics do not keep track of whether a “logical error” occurred (the primary object of interest in quantum error correction) Furthermore, in quantum error correction, errors may occur everywhere, but our dynamics instead only allow the spawning of new defects via local particle branching The proliferation of defects is what can induce a logical error when attempting to decode in quantum error correction

Field Theory Simplification

Using this solution in Eq.

Improvements for AI systems

The AI can predict critical behavior in complex non-equilibrium systems by tuning long-range interaction exponents, as described by a family of classical stochastic processes describing diffusive particles undergoing branching and long-range annihilation in the presence of a parity constraint.
The system can identify novel universality classes that continuously interpolat[e] between DP and PC despite a global parity constraint, allowing for the characterization of phase transitions beyond the discrete binary nature of these classes.
The AI can model and predict density decay in absorbing phases, specifically determining the functional form n(t) ∼ t −αc(κ), enabling precise quantitative predictions for particle survival times based on the long-range exponent κ.
The system can derive mean-field rate equations from a field theory action, specifically obtaining n˙(x, t) = D∇2n(x, t) + 2Bn(x, t) − An(t) when considering the nearest-particle constraint.
The AI can analyze the effect of long-range annihilation on particle dynamics in different regimes by comparing decay rates: "for κ > max(2, d), the model is effectively short-ranged and the absorbing phase is diffusion-limited, while for κ < max(2, d), absorption occurs faster and is dominated by long-range annihilation."