Conformal and chiral phase transitions in Rydberg chains

Using density matrix renormalization group simulations on open chains, we map out the wave-vector in the incommensurate disordered phase of a realistic model of Rydberg chains with $1/r^6$ interactions, and we locate and characterize the points along the commensurate lines where the transition out of the period 3 and 4 phases is conformal. We confirm that it is 3-state Potts for the period-3 phase, and we show that it is Ashkin-Teller with $\nu\simeq 0.80$ for the period-4 phase. We further show that close to these points the transition is still continuous, but with a completely different scaling of the wave-vector, in agreement with a chiral transition. Finally, we propose to use the conformal points as benchmarks for Kibble-Zurek experiments, defining a roadmap towards a conclusive identification of the chiral universality class.


I. INTRODUCTION
In recent experiments on chains of Rydberg atoms with programmable interactions 1,2 , quantum phase transitions between commensurate (C) ordered phases of periods p = 3, 4 and an incommensurate (IC) disordered phase were probed dynamically using the quantum Kibble-Zurek mechanism [3][4][5] . These experiments have renewed the interest in the problem of IC-C transitions first studied in the 80's and 90's in the context of adsorbed monolayers 6,7 . The IC-C critical behavior of a minimal model introduced to describe such transitions, the p-state chiral clock model 8,9 , contains most of the relevant physics.
The IC-C transition with p ≥ 5 happens through an intermediate gapless phase of central charge c = 1 characterized by incommensurate correlations. The dominant wave-vector q is not frozen to any specific value but changes continuously -floats -through the phase, therefore referred to as a floating phase 8,10,11 . The disorder to floating transition is in the Kosterlitz-Thouless (KT) universality class 12 , with exponentially diverging correlation length ξ. One reaches the ordered phase through a Pokrovsky-Talapov 13 (PT) transition where the wavevector q (which we define in units of 2π) goes to 1/p as a power-law with the exponentβ = 1/2 = ν, where ν is the correlation length critical exponent.
The most interesting cases are p = 3 and 4. Indeed, if the chiral perturbation δ is relevant, it was suggested 14 that the IC-C transition may still be direct but in a new non-conformal (chiral ) universality class characterized bȳ β = ν, at least up to a Lifshitz point δ L beyond which the chiral perturbation becomes large enough for an intermediate floating phase to appear, and except possibly at isolated conformal points where the chiral perturbation vanishes. We present these three possible scenarios in Fig. 1. As suggested by Huse and Fisher 14 , the product ξ|q − 1/p| provides an accurate diagnosis since it is expected to diverge at the KT transition of a floating phase, to approach a strictly positive constant for a chiral δ = 0 0 < δ < δL Conformal (ν >β) Chiral (ν =β) for commensurate-incommensurate (C-IC) phase transitions of the classical chiral clock model when the chiral perturbation δ is relevant. We define the q-vector in units of 2π. In the case of the Rydberg model we will tune the Hamiltonian couplings instead of the temperature T to probe for these transitions. The floating (FL) phase is also incommensurate but we distinguish it from the gapped incommensurate phase at higher temperatures.
transition, and to go to zero for a conformal transition. Experimentally 1 , IC-C transitions were probed on a 1D system of 51 optically-trapped Rydberg atoms. The Kibble-Zurek (KZ) exponent µ is measured by dynamically tuning the laser parameters of the system and ramping through the IC-C transitions along specific lines. The experimental value of the KZ exponent, which controls the power-law increase of the domain size with the sweeping rate, is around µ 0.38 for the p = 3 case and µ 0.25 for p = 4, while simulations reported a value around µ 0.45 for p = 3 and µ 0.2 − 0.3 for p = 4 2 . The agreement is good but not perfect, and these results arXiv:2203.01163v3 [cond-mat.str-el] 17 Nov 2022 call for further investigation of the nature of the phase transition. One major obstacle is the absence of exact results for the KZ exponent anywhere along the boundary that could serve as a benchmark. For infinite systems, the KZ exponent is related to the correlation length exponent ν and the dynamical exponent z by the relation µ = ν/(1 + zν), but it has proved difficult to determine both ν and z very accurately. In this paper, we come up with numerically exact results for the KZ exponent across two points of the phase diagram of the experimental model 1 , one on the p = 3 boundary, the other one on the p = 4 boundary, by locating very accurately the lines in the IC phase where the q-vector is commensurate using the finite-size Density Matrix Renormalization Group 15 (DMRG) algorithm. Along these lines, since the system remains commensurate and chiral perturbations are absent, the transition, if it is unique and continuous, is expected to be conformal with dynamical exponent z = 1.
For the p = 3 case, the transition is expected to be in the universality class of the 3-state Potts (P) model, with ν = 5/6 and µ = 5/11 0.4545, while for the p = 4 case it is expected to be Ashkin-Teller (AT) 16 , a family of universality classes parametrized by a coupling λ and corresponding to two decoupled Ising models at λ = 0 and to the symmetric 4-state Potts model at λ = 1. Numerically, we have found an exponent ν 0.80 corresponding to λ 0.5 and leading to µ 0.444. These conformal points are located slightly below the tips of the corresponding lobes, and the value of the KZ exponent across these points can be used as benchmarks. Note that these values are significantly larger than those reported experimentally for 51 sites. Furthermore, the transition is found to be chiral in the vicinity of these points, with clear evidence that the product ξ|q−1/p| neither vanishes nor diverges at the transition, and floating phases have been identified further away from the transition except below the period-3 phase.
This paper is structured as follows: In Sec. II, we introduce and review the experimentally relevant model of Rydberg atoms and its phase diagram. In Sec. III we present our main results and in Sec. IV we discuss our results in the context of past numerical and experimental results. Furthermore in Appendix A we discuss the details of our particular DMRG implementation including the fitting of the measured correlation function, and in Appendix B we show additional results on complementary cuts to the ones shown in the main text as well as cuts going through the p = 2 boundary, and also a finite size scaling analysis of the critical point drift.

II. RYDBERG MODEL
Experimentally 1 , each Rydberg atom of the chain can be excited to a Rydberg state by an applied laser with Rabi frequency Ω and detuning ∆. Excited Rydberg atoms have long-range interactions between them, while they don't interact in the ground state. The hard-core respectively, while the black region is a commensurate disordered (CD) phase with wave-vector q = 1/2 in units of 2π. The grey region is a sketch of the floating phase based on a previous iDMRG work 19 . Equal-q lines are shown in the disordered and floating phases. The points P and AT are respectively our estimates of the Potts and Ashkin-Teller critical points. Apart from the cuts that go through these points, either horizontally/vertically, or along the associated commensurate lines (P/AT cuts), the other cuts discussed throughout the text are horizontal or vertical and are labeled cn. They are represented by arrows colored according to the ordered phase they cross.
boson Hamiltonian of this system is wheren i ≡ 2(σ z + 1)/2 andσ x , σ z are Pauli matrices. In the classical limit Ω = 0, the repulsive interaction and the chemical potential compete. By adjusting their ratio, a devil's staircase 17,18 of classical ground states of many different ratios of occupation per unit cell size is generated, with the largest phases having one boson every p sites. The p phases are stable when Ω is turned on, up to values of Ω ∼ ∆, beyond which the system becomes disordered. The global phase diagram of the relevant region shown in Fig. 2 and plotted in the natural units ∆/Ω and R b ≡ Ω −1/6 has been obtained on chains of 121 sites. The q-vector has been deduced from a fit of the correlation function from the middle site with the Ornstein-Zernike form. For the sizes that we could reach with our finitechain DMRG algorithm, it is not possible to map out the floating phase accurately, so we just show a sketch based on the results of a previous infinite DMRG study 19 , which for that matter is more accurate. Note that, according to that study, there is no floating phase around the tips of the period-4 and period-3 phase, and a floating phase was only observed beyond ∆/Ω 24 below the period-3 phase.

III. DMRG RESULTS
To locate more accurately the conformal points, we have progressively refined the equal-q lines in the vicinity of the period-3 and 4 lobes for 301 then 601 sites, reaching an accuracy in q of the order of 10 −4 (see Appendix B 2). We then determined the correlation length ξ along the q = 1/3 and 1/4 lines (P and AT cuts in Fig. 3). The point where ξ diverges, or equivalently where 1/ξ vanishes, is our estimate of the location of the conformal points, and the exponent with which it diverges is our numerical estimate of ν.
To further characterize the conformal transitions, we considered vertical and horizontal cuts that go through the estimated P and AT points respectively. These are labeled "P/AT cut (vertical/horizontal)" in Fig. 3. Along these cuts, q varies, and accordingly one can estimate the exponentβ and follow the behavior of the product ξ|q − 1/p|. The two vertical lines on each of the plots of this product and the ∆q ≡ |q −1/p| plots are the 301 and 601 site estimates of the critical points obtained from the correlation length fit. To fit the ∆q power-laws, we fix the critical points to these estimates.
The commensurate q = 1/3 line has been found to behave linearly close to the 1/3 phase, following approximately R b 0.1284∆/Ω + 1.9527. Along this cut, we expect the transition to be 3-state Potts, with exponents ν = 5/6 0.833,β = 5/3 1.66 and β = 1/9 20,21 . This point is called P in the phase diagram of Fig. 2. It is located at (∆/Ω, R b ) (1.942, 2.202). Our results for 601 sites agree within 1% with the theory predictions. The discrepancy in the 301 site exponents could be due to finite-size effects that displace the q = 1/3 line, so that along this cut we slightly missed the 301 site equal-q line. As expected, the concavities of q and ξ are opposite, and the product ξ∆q converges to zero. Overall, our results provide strong evidence in favor of a 3-state Potts transition at point P.
Turning now to the IC-C transition of the 1/4 phase, the q = 1/4 equal-q line follows approximately R b = 0.1441∆/Ω + 2.8747 (AT cut) when we are very close to the 1/4 phase. Along this commensurate line and for 601 sites, we find a phase transition at a point denoted AT in Fig. 2 at (∆/Ω, R b ) (2.346, 3.213) with exponent ν 0.80, which is consistently replicated with a horizontal cut that crosses this critical point. Theβ exponent, unknown analytically for the AT universality class, is in any case larger than 1 14,22 , and the ξ∆q product decays to zero at the transition.

C. Order parameter
To further confirm the conformal nature of the transitions along the commensurate lines, we looked at the scaling of the order parameter O defined as the maximal difference in the occupation n l . To avoid the Friedel oscillations at the edges, we only consider the middle 10 sites, leading to the following definition of O: The results are shown in Fig. 4. Along the P cut, and for 601 sites, the scaling is in excellent agreement with the exact result β = 1/9. Along the AT cut, we expect the exponent β to be related to ν by β = ν/8, a prediction derived from the lowest CFT scaling dimension of the AT model 23 . For 601 sites, the scaling is in good agreement with β = 0.1, the expected value for ν 0.8.

D. p = 3 non-conformal cuts
Let us now discuss the results we have obtained away from these points, starting with the period-3 phase. Both below the 1/3 line (c 3 cut, vertical, ∆/Ω = 2.4) and above it (c 4 cut, horizontal, R b = 2.225, Fig. 10) we find clear evidence of a chiral transition: ξ∆q is nearly flat upon approaching the transition. Note that cut c 3 is remarkably far from the Potts point on the scale of the phase diagram of Fig. 2, leaving a significant parameter range to probe the chiral universality class experimentally.
Along the cut c 5 at R b = 2.45, further above, the IC-C transition is more consistent with Pokrovsky-Talapov, with ν = 0.6 andβ 0.52. It actually makes sense thatβ is more accurate since, with our algorithm, q converges faster than ξ. On the disordered side, the correlation length grows rapidly until it eventually levels off before the PT transition. This is consistent with a KT transition into a floating phase, ξ being limited by the finite size. The ξ∆q product shows a clear divergence before q becomes commensurate. This result might indicate that the floating phase reaches closer to the tip of the lobe than what is shown in the phase diagrams (see however Appendix B 4 for a discussion of finite-size effects). The same conclusions apply to cut c 6 further above (Appendix B 3).

E. p = 4 non-conformal cuts
The situation is very similar around the period-4 phase. The c 7 cut (R b = 3.22) is in agreement with a direct chiral transition of exponent ν ( β ) slightly higher than at the AT point, suggesting that this exponent increases as we initially move away from the AT point. Cut

A. Comparison with blockade models
It is instructive to compare these results with those obtained recently on blockade models 24,25 , in which configurations with bosons at a distance less or equal to r = 1, 2, . . . are forbidden while only the interaction at distance r + 1 is kept, and which are expected to be good effective models between the phases p = r + 1 and p = r + 2.
For the period-3 phase of the r = 1 blockade model 24,26,27 , there is a single point where the transition is conformal whose location is known exactly because it belongs to an integrable line 24 . Lines of chiral transitions seem to surround the Potts point 26,27 , while further away intermediate floating phases appear 24,27 . Our results agree with all these properties. The only difference it that the chiral transition of our model is more extended below the lobe than for the blockade model. Note that, more generally, our results agree with those obtained on classical 2D and quantum 1D versions of the period-3 case 9,14,22,28-35 , for which the existence of a transition line in the chiral universality class is supported both by experiments 6,7 and by recent numerical work 24,26,27,36,37 .
In the context of Rydberg atoms, the r = 2 blockade model has only been introduced and studied very recently 25 . The transition out of the period-4 phase along the commensurate line was found to be Ashkin-Teller with ν 0.78 and λ = 0.57. Our estimates ν 0.80 and λ = 0.5 are not far, confirming the qualitative relevance of blockade models. The parameter range of chiral transitions is comparable in both cases. Note that the presence of a range of chiral transition before a floating phase ap-pears is in agreement with very recent results obtained on a classical 2D chiral Ashkin-Teller model 38 , according to which a chiral transition is expected for λ 0.42 and up to λ 0.978 31 .

B. KZ exponent
Finally, let us come back to the KZ exponent, and to the identification of the exponents of the chiral universality class, the main open issue in the field. On the theory side, the bottleneck is the determination of the dynamical exponent z. It is fixed to z = 1 at the conformal points P and AT, but an accurate estimate of its value away from these points is still beyond state-of-the-art simulations. What one can get quite accurately however is the exponent ν, and the fact that its value is consistent with that ofβ along the chiral transition is an indication that cross-over effects are negligible for the model of Rydberg chains, contrary to the classical 2D chiral Potts model, where crossover effects lead to an overestimate of ν and a violation of the ν =β criterion close to the Potts point 37 .
On the experimental side, by contrast, one can accurately measure the KZ exponent. If measured on very large systems, this exponent should provide the missing piece of information on ν and z since µ = ν/(1 + zν). How large should the systems be? The discrepancy between our numerically exact results at points P and AT and the experimental results on 51 sites clearly demonstrates that one needs larger systems. At point P, which corresponds to R b 2.202, the theoretical value is µ = 5/11 0.4545, while the experimental result is around µ 0.38. Similarly, at the AT point R b 3.213, our estimate is µ 0.444, while the measured value is again much smaller, around µ 0.25.
These remarks define a clear roadmap towards a conclusive identification of the chiral universality class with chains of Rydberg atoms. KZ experiments should be carried on across the conformal P and AT points identified in the present work on systems of increasing size until a quantitative agreement is reached with the numerically exact estimates of µ reported here. Then, a comparison between experimental values away from the conformal points and theoretical estimates of ν should allow one to reach precise conclusions regarding the critical exponents of the chiral transition. Work is in progress to refine our estimates of the exponent ν all along the boundary of the period-3 and period-4 phases where the transition is believed to be chiral. We hope that the present results will in parallel encourage experimentalists to perform KZ experiments on longer chains to help solve the long standing problem of the universality class of the chiral transition.

ACKNOWLEDGMENTS
We thank Andreas Läuchli and Samuel Nyckees for useful discussions. This work has been supported by the (A1) For all sizes considered, the minimized cost function was smaller than 10 −16 (Table A 1). As a comparison, a truncation of the power-law preserving the first 12 terms results in an equivalent squared differences error of ∼ 7.5 × 10 −14 .
To avoid stability problems in DMRG, we chose system sizes of the form L = 12l +1, which split the ground state degeneracy by guaranteeing a single ground state with occupied edges for p = 3, 4. For the full phase diagram we chose L = 121 which stabilizes all relevant orders.
On each two-site DMRG update where we carry out a singular value decomposition, we discard singular values smaller than 10 −9 , as these carry a statistical weight substantially smaller than machine precision, however, the truncation of the bond dimension to a hard limit D is effectively much more relevant. Overall, the truncated weight in the last DMRG update in the middle of the chain was always lower than 10 −6 .
As a convergence criteria, we required the relative energy variance where |Ψ is the variational MPS state, to be smaller than 10 −11 when estimating the boundaries of phases and 10 −12 when determining critical exponents and the q = 1/3, 1/4 lines. An MPS virtual bond dimension of 350 was typically enough to reach such precision for 601 sites and close to the 1/4 conformal point, while bond dimensions up to 500 were used to reach convergence close to or inside the floating phases.

Correlations and q-vector
We obtain q by fitting the correlation function between the middle site j and site j + r, with the expected Ornstein-Zernike (OZ) form 42 : where We discard points from the head and the tail-end of the correlation function until an OZ regime is thought to be reached, then we fit the remaining points. We implemented a two-step fitting scheme (Fig. 5) that has been described before 25 , where first we obtain ξ and A 0 by performing a linear fit on C(r) √ r in a semi-log scale. Then, q is obtained by a least-squares cosine fit on C(r)/A(r), where one minimizes the cost function F (q) defined as the sum of squared differences. The confidence intervals (error bars) shown in the plots of q are an estimate of the fitting error. They are calculated by assuming that the error δq is proportional to the cost function. It then follows that δq = F (dF/dq) −1 in lowest order, which can be explicitly calculated. The main contributions to this error are not precision errors in the fitting algorithm, but are instead errors in the determination of ξ and A 0 , or deviations from an OZ regime. The error bars of ξ were deemed too small to be represented. The error of ξ∆q is derived from these errors by the differential chain rule.
In general, we find the limit of reliable correlation lengths to be ξ ∼ L/6, beyond which ξ is noticeably limited by the finite size. Still, we find that the q-vector suffers less from finite size effects than ξ, and that a cosine fit beyond this ξ limit can still give an accurate estimate of q. In contrast to the p ≥ 3 cases, the 1/2 lobe is surrounded by a commensurate disordered (CD) phase. The phase transition is continuous in the Ising universality class. We confirmed the latter by taking several cuts along the phase boundary and verifying that upon approaching the transition the correlation length diverges with the critical exponent ν ≈ 1. We confirm this at least up to the deepest cuts we considered at ∆/Ω = 3, as seen in Fig. 6. Above the lobe (on the side closer to the 1/3 phase), we looked at cuts up to ∆/Ω = 2, with similar results. We did not look at cuts beyond ∆/Ω = 2 above since it was expected already from the effective p = 3 blockade model that the transition would be Ising on this side 24,27 . However, as we move away from the 1/3 phase, the Ising critical line of the blockade model eventually ends at a tricritical Ising point, below which the transition is first order. We did not find any evidence of a first order transition in the Rydberg model. It is not very surprising though because the tricritical point of the blockade model is located at negative (attractive) next-to-blockade interactions, which naturally does not occur in the Rydberg model. A more appropriate effective model of the lower part of the 1/2 lobe is the p = 2 "blockade" hard-core boson model, which is the same as the Rydberg model in Eq. (1) but where the interaction is truncated to the first term, a nearest neighbor interaction. A change of variables to a spin system reduces this model to an Ising model with transverse and longitudinal fields where the transition is always Ising 43 . FIG. 6. Correlation length along the ∆/Ω = 3 cut (left) that crosses the commensurate transition line below the period-2 lobe, and the ∆/Ω = 2 cut (right) that crosses it above. All points shown in the disordered sides are inside the commensurate phase. The small finite-size effects observed let us conclude from the exponent obtained that the transition is in the Ising universality class.

Equal-q lines
The equal-q lines we show in the phase diagrams are obtained by interpolation of the q-vector on a finite grid. We use this same method to accurately determine where the q = 1/3 and 1/4 lines meet their respective ordered phases, using data from simulations on 601 sites very close to the phase boundary, as shown in Fig. 7. The grid data in these figures show the order parameter O. We can see in these figures the start of the ordered phases in the top right. Simulations along these linear fits then lead to estimates of the conformal critical points.

Period-3 and 4
In Figs. 8 and 9 we show the inverse ξ and q-vector scaling along the incommensurate cuts crossing the 1/3 and 1/4 phase boundaries respectively, from which the ξ∆q products shown in Fig. 3 of the main text have been obtained.
We estimate the width of the floating phase by extrap-olating to infinity the divergence of ξ∆q. For a cut c 5 we detect a floating phase of width in ∆/Ω of approximately 0.01 for 301 sites and 0.004 for 601 sites. As stated in the main text, this might be an indication that the floating phase reaches closer to the top of the lobe than what is shown in the phase diagrams. However, the shrinking width of the floating phase with system size might also suggest that there is a crossover to a chiral regime at larger system sizes. Similar reasoning can be applied to cut c 6 located further above in R b and presented in Fig.10 which turns out to be qualitatively equivalent to the cut c 5 . Fig. 10 shows the results from two complementary cuts above the P point that were not included in the main text. The c 4 cut (R b = 2.225) results are consistent with a chiral transition above but very close to the P point. Together with the c 3 cut, these two cuts suggest the P point is surrounded by chiral transition lines. The c 6 cut (∆/Ω = 2) is qualitatively equivalent to the c 5 cut below it and brings further evidence in favor of an intermediate floating phase between the 1/3 and 1/4 phases.

Finite-size scaling
It is already apparent from the results shown for 301 and 601 sites that a significant drift of the equal-q lines and of the phase boundaries happens at small system sizes. Indeed, as shown in Fig. 11, the P and AT points show a significant drift between 601 sites down to experimentally relevant sizes like 61 sites. If we correct for the phase boundary drift, the finite-size difference in the correlation length between 301 and 601 sites is not as significant (Fig. 12), although it is still more noticeable in the p = 4 case. It's not unreasonable to expect a further drift of ν towards lower values for system sizes larger than we considered, possibly reaching closer to the blockade model prediction of ν 0.78 25 .