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The quantum Kibble-Zurek mechanism: the role of boundary conditions, endpoints and kink types
by Jose Soto Garcia, Natalia Chepiga
Submission summary
| Authors (as registered SciPost users): | Natalia Chepiga · Jose Soto |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202501_00006v1 (pdf) |
| Date submitted: | 2025-01-07 22:29 |
| Submitted by: | Soto, Jose |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Computational |
Abstract
Quantum phase transitions are characterised by the universal scaling laws in the critical region surrounding the transitions. This universality is also manifested in the critical real-time dynamics through the quantum Kibble-Zurek mechanism. In recent experiments on a Rydberg atom quantum simulator, the Kibble-Zurek mechanism has been used to probe the nature of quantum phase transitions. In this paper, we analyze the caveats associated with this method and develop strategies to improve its accuracy. Focusing on two minimal models—transverse-field Ising and three-state Potts—we study the effect of boundary conditions, the location of the endpoints and some subtleties in the definition of the kink operators. In particular, we show that the critical scaling of the most intuitive types of kinks is extremely sensitive to the correct choice of endpoint, while more advanced types of kinks exhibit remarkably robust universal scaling. Furthermore, we show that when kinks are tracked over the entire chain, fixed boundary conditions improve the accuracy of the scaling. Surprisingly, the Kibble-Zurek critical scaling appears to be equally accurate whether the fixed boundary conditions are chosen to be symmetric or anti-symmetric. Finally, we show that the density of kinks extracted in the central part of long chains obeys the predicted universal scaling for all types of boundary conditions.
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This paper investigates the complexity of acquiring a precise scaling for the Kibble-Zurek mechanism in realistic experiment (or emulated experiment, like in the case analyzed by the authors). The authors highlight that the driven dynamics past the II-order transition point actually enhances local fluctuations that do not really disturb the ordered domain lenght but worsens the estimation of correct defect density. The authors propose a workaround of the problem by carefully redesigning what is actually considered a defect in an extended sense, and show numerically that delivers a more accurate and invariant scaling. The authors also analyze the role played by boundary conditions, and show that thr impact of a fixed boundary condition rule plays a marginal role, even more so when discarding a portion of the system near the edges as long as the final correlation length.
While none of the results is particularly surprising, I find it worth of praise that the authors took care of the problem of extracting clean scaling laws from noisy quantum measurement data. The message is simple and direct, and applying these techniques seems straightforward and intuitive (I will surely implement these in the future). For this reason, even though the research is not breakthrough, it is good and practically useful science, and worth to be published, in my opinion.
The authors should first consider to address a few comments from my side, however:
P2 - "Near the critical point of a continuous phase transition": Is not the phase transition supposed to be exactly of the second order? Please elaborate.
Eq4: I know that exact scaling law works in 1D. Is it equally valid in any spatial dimensions (and dimension of the defect, i.e. vortex vs string vs interface) or does it need adjustment? Please clarify.
P3 - "3-state Potts model in 1D": Technically, the model you consider is the Potts model with transverse field in 1D -or- the quantum Potts model in 1D -or- the Potts model in 1+1D. But the simple Potts model in 1D is a classical (not quantum) model, without frustration.
TFIM sounds terrible. I suggest to abbreviate "Transverse Field Ising Model" with the single word "Ising" like everybody does, whenever necessary (example "Ferromagnetic TFIM" => "Ferromagnetic Ising").
P4 - Sections 2.2 and 2.3 have exactly the same title. Confusing.
P4 "P_i is the projector": no it's not a projector if you make it traceless by subtracting 1/3.
P5 "relative energy difference between two successive sweeps, including an increase in the bond dimension, was not exceeding 10−9.": Relative to what? Please don't say relative to the actual ground state energy value, since the ground state energy is just a gauge. Relative to the first excitation energy gap, instead, is physically a good choice of baseline energyscale.
P5 "The time step was fixed at δ = 0.1" please justify this value. Did you verify that the integrated dynamics is already at convergence in the timestep?
P5 "second-order time-evolving block decimation (TEBD) algorithm" why did you use TEBD in OBC if you needed anyway to set up a TDVP for the PBC?
Figure 1 - "Kibble-Zurek scaling of the density of kinks nk": At this point in the document is not clear how you extract the density of kinks from the (I guess) Matrix Product State. Internal reference needed.
Figure 1 vs Figure 2: it took me a while to understand the difference between these two figures. Please state explicitly the system Length in figure 1, and the bond dimension used in figure 2.
Figure 2: It is uncldear to me why the Potts model panel shows only long system lengths (in fact you don't see Landau Zener at all). It is also confusing why the Potts has sweep rates in linear dispersion, unlike the Ising case which covers 6 orders of magnitude.
Page 6 "The standard protocol for extracting the KZ exponent μ consists of driving [...] and counting the number of kinks after the transition for various sweep rates": Actually that is the EXPERIMENTAL protocol, which you carry out when you have access to single shots of the state. For numerical simulations such as MPS, it is just as standard to actually acquire the information from the calculated correlation function. I am assuming here you are using a protocol to exctract single shot measurements from an MPS? What algorithm do you use for the local sampling?
Figure 3 and Figure 5 are excellent opportunities to illustrate what the authors consider an "isolated kink".
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