Quantum criticality in the 2d quasiperiodic Potts model
QQuantum criticality in the 2d quasiperiodic Potts model
Utkarsh Agrawal, Sarang Gopalakrishnan, and Romain Vasseur Department of Physics, University of Massachusetts, Amherst, MA 01003, USA Department of Physics and Astronomy, CUNY College of Staten Island,Staten Island, NY 10314; Physics Program and Initiative for the Theoretical Sciences,The Graduate Center, CUNY, New York, NY 10016, USA
Quantum critical points in quasiperiodic magnets can realize new universality classes, with criticalproperties distinct from those of clean or disordered systems. Here, we study quantum phasetransitions separating ferromagnetic and paramagnetic phases in the quasiperiodic q -state Pottsmodel in 2 + 1 d . Using a controlled real-space renormalization group approach, we find that thecritical behavior is largely independent of q , and is controlled by an infinite-quasiperiodicity fixedpoint. The correlation length exponent is found to be ν = 1, saturating a modified version of theHarris-Luck criterion. Quenched disorder can dramatically affect the univer-sality class of a quantum phase transition, and drive itto a new renormalization group (RG) fixed point if thecorrelation length exponent ν violates the Harris crite-rion ν ≥ /d [1, 2] with d the dimensionality of thesystem. As the effective randomness grows under renor-malization, the new infrared fixed point can either becharacterized by finite or infinite randomness. Infinite-randomness fixed points can be analyzed using an asymp-totically exact real space renormalization group (RSRG)approach [3–5] that yields exact predictions for criticalexponents and scaling functions. The RSRG approachhas been applied to many different quantum phase tran-sitions in one and two dimensions, both at zero tempera-ture and in the context of many-body localization [3–35].The structure of infinite-randomness critical points de-pends crucially on the assumption of spatially uncor-related disorder. However, many present-day experi-ments, involving, e.g., twisted bilayer graphene [36–38]and ultracold atoms in bichromatic laser potentials [39–43] involve systems that are spatially inhomogeneous, butquasiperiodic rather than random. Quasiperiodic poten-tials are deterministic, with strong spatial correlations,so they do not lead to conventional infinite-randomnessbehavior [44–50]. Instead, when a clean critical pointis unstable to quasiperiodicity, it flows to a new classof fixed points. Field theoretic methods [51–57] do noteasily generalize to quasiperiodic systems [58, 59], be-cause there is no disorder to average over. However,very recent results [60–62] have revealed the existenceof “infinite-quasiperiodicity” quantum critical points [62]in one dimensional spin chains; at these critical points,RSRG yields exact predictions for exponents. Despitetheir differences, infinite-quasiperiodicity and infinite-randomness critical points share the key feature that thedynamical critical scaling exponent z = ∞ : thus, thecharacteristic timescale t ξ associated with a length-scale ξ grows faster than any power law of ξ . So far, suchinfinite-quasiperiodicity fixed points have chiefly beenstudied in one dimension; higher-dimensional cases arepoorly understood [63–66]. The z = ∞ dynamical scal- ing leads to a rapidly vanishing gap, which makes it hardto access the critical regime using Quantum Monte Carlotechniques [67–70]. Tensor network based approaches(see e.g. [71]) are also less suited to study 2d QP quantumcriticality, due to large entanglement.In this letter, we propose a general RSRG approachto study 2+1d quantum spin models with QP couplings.As in the implementations of RSRG for disordered sys-tems in two dimensions, the RG changes the underly-ing geometry of the system creating intricate and com-plex long range interactions [9, 16, 19]. Nevertheless theRG procedure can be efficiently implemented numeri-cally. We focus on the 2d quantum Potts model, with q “colors” ( q = 2 corresponding to the Ising model). Forclean systems, the phase transition separating paramag-netic and symmetry-broken phases is in the classical 3DPotts model universality class, which is a first-order for q ≥ q >
2, with the Isingcase q = 2 being special. Beyond our numerical resultsfor the critical exponents, we propose a general argu-ment for the correlation exponent ν = 1 for these newinfinite-quasiperiodicity transitions, based on the distri-bution of “defects” in the critical structure. Due to thedeterministic and almost periodic nature of quasiperiodicpotentials these defects form a definite pattern; in somespecial cases, the defects form a QP tiling with a lengthscale that defines the correlation length. Interestingly,the value of ν saturates a modified version of the Harris-Luck criterion [76], namely ν ≥
1; the modifications aredue to boundary fluctuations coming from correlations inboundaries of rectangular patches at all length scales.
Model.
The q -state quantum Potts model is definedvia the Hamiltonian H = − (cid:88) (cid:104) i,j (cid:105) J ij δ n i ,n j − (cid:88) i h i q (cid:88) n i ,n (cid:48) i | n i (cid:105)(cid:104) n (cid:48) i | , (1) a r X i v : . [ c ond - m a t . d i s - nn ] A ug defined on the square lattice with (cid:104) i, j (cid:105) denoting nearestneighbor pairs, where n i is a variable on site i that takesone of q possible values. The first term with J ij > h i ’s. For q = 2 col-ors, this coincides with the familiar transverse field Isingmodel. The model is initially defined on a square lattice;however, we believe our results to be independent of theinitial lattice geometry, as RSRG drastically changes theconnectivity of the system.The couplings J ij > , h i > J ij = f ( (cid:126)k .(cid:126)r ) + f ( (cid:126)k .(cid:126)r ), where (cid:126)r = ( i x , i y ) + ( j x − i x , j y − i y ), (cid:126)k , and (cid:126)k are two orthogonal unit vectors, and f a ( x ) = f a ( x + ϕ − ) for some irrational ϕ , which we take to be thegolden ratio, ϕ = √ . Similarly, the fields are takenfrom an initial potential of the form, h i = g ( i x ) + g ( i y )with g a ( x ) = g a ( x + ϕ − ). For concreteness, we focus onthe following QP modulations throughout the paper, (cid:96) J ij =2 + cos (2 πϕ(cid:126)k .(cid:126)r + φ ) + cos (2 πϕ(cid:126)k .(cid:126)r + φ ) (2) (cid:96) h i = g (2 + cos (2 πϕi x + φ ) + cos (2 πϕi y + φ )) , where g is a parameter driving the transition, (cid:96) J ij = − ln J ij and (cid:96) h i = − ln h i are defined so as to decreasethe transient behavior in the RG (see below), and φ i aresome constant global phases which we average over. Un-less otherwise stated, we take (cid:126)k = (sin θ, cos θ ), with theangle θ = √ π . Our results do not depend on the detailsof these distributions [77]. RG procedure.
We now describe the RSRG proce-dure we use to capture the critical properties of Eq. (1).One step of the RG procedure consists of identifying thestrongest coupling in the Hamiltonian (which sets thecutoff, Ω) and eliminating it, as follows [9, 15, 16, 78].If the strongest coupling is a bond J ij , one merges thetwo spins connected by the bond into a new effectivespin (or “cluster”) with magnetic moment µ (cid:48) i = µ i + µ j ( µ i = 1 for initial physical spins). The effective trans-verse field acting on the cluster is given by second-orderperturbation theory, h (cid:48) i ≈ h i h j κJ ij with κ = q/
2; also, anyother spin (or cluster) in the system that was connectedto either i or j now picks up a bond to the new clus-ter, with coupling given by J (cid:48) ik = max( J ik , J jk ). If in-stead the strongest spin is an effective field h i , one elim-inates the site i . Any other pair of sites j, k that wereconnected to i by bonds now pick up a new effectivebond, which we estimate using 2nd order perturbationtheory: J (cid:48) jk ≈ J jk + J ij J ik κh i ≈ max( J jk , J ij J ik κh i ). This pro-cedure correctly captures the low energy physics as longas Ω (cid:29) J ij , h j (broadly distributed couplings) so thatperturbation theory is controlled; we will see that forinfinite-quasiperiodicity fixed points, the parameter con-trolling the error in perturbation theory flows to zero • data -- 𝜇 ∼ 𝐿 FIG. 1.
Magnetization scaling.
Scaling collapse of themagnetization m ( L, g ) for q = 3 with the correlation lengthexponent ν = 1, critical coupling g c = 0 . x = 0 . Bottom inset:
Plot of theratio r ( L ) = m ( L ) m ( L/ vs g . In the para- and ferromagneticphases r ( L ) depends on L (large g corresponds to a ferro-magnet, small g to a paramagnet), while at the critical pointthis ratio is a constant. Defining the scaling dimension x via m ∼ L − x , we have 2 − x ≈ .
53 or x ≈ .
92. The critical pointis g c = 0 . Top inset:
Average magnetic moment µ M vs L giving µ M ∼ L d f with d f = 1 . ± . x + d f = 2. upon coarse-graining, leading to asymptotically exactpredictions for universal properties.We numerically run the RG procedure described abovestarting from a L × L square lattice. We first focus onthe q = 3 Potts model – the critical behavior is largelyindependent of q ≥
3. As the system moves along the RGflow, its geometry changes giving rise to graphs of increas-ingly intricate connectivity. Instead of implementing theRG in the naive sequence described above (i.e., alwaysdecimating a single largest coupling), we follow standardtechniques [16] to optimize the decimation sequence. (Wehave checked that at the end of the RG procedure, theoptimized and naive decimation sequences yield identicalcouplings, so this step is not an approximation.)
Magnetization and fractal exponent.
At the endof the RG, the surviving cluster with moment µ M de-termines the magnetization of the system, m ( L, g ) = µ M /L , where L is the linear size of the system. To locatethe critical point we plot r ( L, g ) = m ( L, g ) /m ( L/ , g )vs g for various L ; away from the criticality r ( L, g )changes with L, while being scale independent at thecritical point [78]. The critical magnetization scales as m ( L, g c ) ∼ L − x giving the crossing value r ( L, g c ) = 2 − x .The average moment of the cluster at the critical pointscales as µ M ∼ L d f with d f being the fractal dimensionof the spins in the cluster. Those two exponents satisfy b) a) FIG. 2.
Critical defects and quasiperiodic tiling struc-ture. a ) Geometry of the set S = { i : min { (cid:96) J ij } < (cid:96) h i } where (cid:96) h , (cid:96) J are defined in (2) with the angle θ = 0. We have taken g = 0 . S ,while white sites do not, and form single-site clusters. We seepockets of black sites separated by 1d section of white sites,marked by red lines. These red lines form a square QP till-ing. Large clusters in later steps of the RG are formed byjoining small clusters within different tiles/faces of the redlattice. Defects are breaks in the pattern of inter tile connec-tions away from the critical point. The number of breaks areproportional to the inverse of detuning parameter δ , giving ν = 1. b ) Geometry of S for g = g c = 0 .
425 and θ = √ π .The structure is not as clear and well defined as in the θ = 0case but we still see local puddles in S . the scaling relation d f + x = 2. Those quantities areplotted for the q = 3 Potts model in Fig. 1, and we find2 − x ≈ .
53 or x ≈ .
92 and d f = 1 . ± . d f + x = 2. Correlation length.
Assuming single parameterscaling with a diverging correlation length ξ ∼ | g − g c | − ν ,we expect the following scaling form for the magnetiza-tion m ( L, g ) = L − x f (( g − g c ) L /ν ), where f is a universalscaling function. Using the values of g c , and x obtainedfrom the plot of r ( L ), we find a nice collapse for ν ≈ ν = 1 holds exactly, atleast for some classes of quasiperiodic potentials.The argument for ν = 1 is as follows. Let us firstconsider the case where the quasiperiodic modulation isparallel to the lattice vectors, i.e., (cid:126)k = (1 , (cid:126)k = (0 , δ . We now look for “defects,” or points on the lattice where the two RG realizations be-gin to diverge (because one of them decimates fieldsand the other bonds). Defects occur when locally, fieldsare close ( (cid:46) δ ) in magnitude to the neighboring bonds;thus, a small detuning is enough to change the order ofdecimations. However, because the quasiperiodic struc-ture is approximated to precision ∼ δ by a rational ap-proximant with period ∼ /δ , each defect has an al-most perfect repeat at a distance ∼ /δ (along bothlattice directions). This can be seen by observing thatcos (2 πϕ ( x + F n ) + φ ) = cos(2 πϕx + φ ) + O ( ϕ − n ), where F n is the nth Fibonacci number: defects must repeatalong the vertical and horizontal axis, forming a QP till-ing, with a length scale ξ = F n ∼ ϕ n , with δ ∼ ϕ − n giv-ing ξ ∝ δ − (see [62] for a similar argument in quantumspin chains). Thus, when the RG reaches length scale1 /δ , defects will proliferate and drive the system awayfrom criticality, corresponding to ν = 1. To illustrate thistiling geometry, we plot the set S = { i : min { (cid:96) J ij } < (cid:96) h i } ,where the min is over nearest neighbors. This conditionis satisfied for couplings J that are decimated first in theRG, forming non-trivial clusters. The geometry of theset S is shown in Fig. 2.The geometry away from θ = 0 is less transparent, butnumerics once again suggests ν = 1; moreover, the modelremains strongly anisotropic under coarse-graining, withpreferred orientations (Fig. 2b). We now argue that, ifthis anisotropy persists under the RG, it leads to a modi-fication of the Harris-Luck bound on ν [76]. The standardargument for this criterion runs as follows. In a largepatch of the sample of linear dimension (cid:96) , the apparentlocal value of the critical point is δ (cid:96) ≡ (cid:104) g (cid:105) (cid:96) − g c ∼ (cid:96) w − d where w is the wandering exponent. Setting (cid:96) to the cor-relation length ξ ∼ δ ν , we get δ ξ ∼ δ ν ( d − w ) . When δ ξ issmall compared with the global detuning δ , the transitionis well-defined. This criterion amounts to ν > / ( d − w ).Generic patches of a quasiperiodic system have wander-ing exponent w = 0 in the bulk so the standard Luckcriterion reads ν > /d . However, this analysis ignores“boundary” terms due to lines or other sub-dimensionalregions of the sample where δ is locally away from itsaverage value. If one includes these boundary contri-butions, the deviation is δ (cid:96) ∼ (cid:96) ( d − − d ∼ /(cid:96) , so that ν ≥ ν = 1 (up to logarithmic corrections). Dynamical scaling and RG error.
We now turnbriefly to the dynamical scaling properties at this tran-sition. One can argue analytically that the timescale fora region of (cid:96) spins grows at least as ln t (cid:96) (cid:38) ln (cid:96) . Thisscaling follows naturally from the RG rules; recall thatthese rules involve a factor κ > (cid:96) to asingle spin, one picks up at least ln (cid:96) factors of κ inthe effective couplings [77], implying an energy scaling − ln E (cid:96) ∼ ln t (cid:96) (cid:38) ln (cid:96) . This scaling can be interpreted as 𝑞 = 3 a) b) c) 𝑞 = 10 ` • data -- Γ 𝑔 ∼ ln L FIG. 3.
RG errors and gap distribution. a ) Plot of the RG error, ∆ RG , vs RG time Γ( ≡ − ln Ω) at the critical point.Data from 9 different phase realizations are combined and averaged over windows of Γ of size 0.05. We see a trend of the errordecreasing with the RG, i.e increasing Γ (the black curve is a guide for the eye), whereas towards the end of the RG the databecomes more scattered and noisy. As we increase system sizes, the onset of the data scattering shifts towards latter stagesof the RG, consistent with the noisiness in the error at higher Γ being a finite size effect. b ) Distribution of logarithmic ofgap for q = 3, − ln ∆ E g ≡ Γ g . With increasing system size, the average is increasing with the distribution becoming broader,indicating a broadening of couplings and fields along the RG flow. Inset:
Scaling of the finite-size gap, showing Γ g vs L ; thefit is compatible with Γ g ∼ ln L . Binning window for Γ g was taken to be 0 . c ) Distribution of logarithmic of gap for q = 10with window size of 0 .
05. Unlike the q = 3 case, we see a systematic rise and fall in P (Γ g ), with the probability going to zerofor some values of the gap. This is reminiscent of the 1d case where a similar banding of couplings and gaps was observed [62]. the scaling of the finite size gap of a region of size (cid:96) . Thisdivergence might be subleading (as it is in the randomcase), but guarantees “activated” scaling, where t growsfaster than any power of (cid:96) . As we see in Fig. 3b, ournumerical results are consistent with ln t (cid:96) ∼ ln (cid:96) , i.e.,the same dynamical scaling as in one dimension [62]. Wenote that our data is also compatible with other types ofactivated scaling [77].A consequence of activated dynamical scaling is thatthe RG becomes increasingly accurate at late stages. Thetypical RG error (defined as log ∆ RG ≡ (cid:104) log( max J ij ,h i Ω ) (cid:105) ,where the max function is over all neighboring terms of Ω,with (cid:104)·(cid:105) denoting average over a small window of − log Ω,and several phase realizations) vs − ln Ω( ≡ Γ) at the crit-ical point is plotted in Fig. 3. a). We see that on aver-age, the RG error decreases along the RG flow, suggest-ing that the RG becomes asymptotically exact, as in therandom case [9]. While the system sizes we can accessremain away from the asymptotic regime where the RGis fully controlled, we observe very good quality criticaldata (Fig. 1) with no signs of finite-size drifts. Extrapo-lating these results, we expect the error of a typical RGstep to go to zero asymptotically with Γ.
Critical behavior vs q . We conclude this letter bybriefly discussing the case of q >
3. For q >
3, we ob-serve a similar behavior as for q = 3; there is a 2ndorder transition with the RG becoming more controlledwith the flow. The correlation exponent ν = 1 seems tohold, as expected from the general arguments discussedabove. Unsurprisingly, the location of critical point isnon-universal and changes with q and θ . The d f and x exponents appear to be same for all values of q > q ’s, though we cannot exclude small differences based on ournumerical data. Interestingly, for larger values of q , weobserve that the distributions of the gap and of couplingsform “bands”, with forbidden values in between the al-lowed bands (see Fig. 3.c). This is reminiscent of similarbanding properties that were observed in QP quantumspin chains [62]; it would be interesting to investigatewhether this can be leveraged to understand this RG an-alytically in the future.The case of q = 2 (the Ising model), is special. Inthis case, we find that the RG does not flow towards in-finite quasiperiodicity, and is therefore not controlled. Asimilar scenario occurs in 1d weak QP modulations aremarginally irrelevant [60–62, 79] at the clean fixed points.However, unlike the 1d case, we observe that even on in-troducing strong QP modulations, the RG does not flowto infinite quasiperiodicity. From the modified versionof the Luck criterion, we expect QP modulations to berelevant at the clean Ising transition, driving the systemto a finite quasiperiodicity fixed point that cannot be de-scribed using RSRG. It would be especially interestingto investigate the nature of this QP Ising transition, aswe expect it to be very different from the transitions de-scribed in this letter — in particular, it likely has a finitedynamical exponent z , as a consequence of the prefactor κ = 1 in the RG rules. Discussion.
We analyzed the critical behavior ofquantum phase transitions separating ferromagnetic andparamagnetic phases in the quasiperiodic q -state Pottsmodel in two dimensions. Using a controlled real-spacerenormalization group approach, we found that the crit-ical behavior is independent of q , and is controlled bya new RG fixed point providing the first example of“infinite-quasiperiodicity” behavior in two dimensions.We argued on general grounds that such QP quantumphase transitions have correlation length exponent ν = 1,saturating a modified version of the Harris-Luck crite-rion. It would be interesting to find other examplesof infinite-quasiperiodicity transitions, both in two andthree dimensions. The case of the 2d QP Ising model alsodeserves more attention, as it should provide a differenttype of QP transition with finite dynamical exponent.We leave these questions for future works. Acknowledgments:
The authors thank Snir Gazit, SidParameswaran and Jed Pixley for useful discussions.This work was supported by the National Science Foun-dation under NSF Grant No. DMR-1653271 (S.G.),the US Department of Energy, Office of Science, BasicEnergy Sciences, under Early Career Award No. DE-SC0019168 (U.A. and R.V.), and the Alfred P. SloanFoundation through a Sloan Research Fellowship (R.V.) [1] A. B. Harris, Journal of Physics C: Solid State Physics , 1671 (1974).[2] J. T. Chayes, L. Chayes, D. S. Fisher, and T. Spencer,Physical Review Letters , 2999 (1986).[3] S. K. Ma, C. Dasgupta, and C. H. Hu, Physical ReviewLetters , 1434 (1979).[4] D. S. Fisher, Physical Review Letters , 534 (1992).[5] D. S. Fisher, Phys. Rev. B , 3799 (1994).[6] D. S. Fisher, Phys. Rev. B , 6411 (1995).[7] D. S. Fisher, Physica A: Statistical Mechanics and itsApplications , 222 (1999), proceedings of the 20thIUPAP International Conference on Statistical Physics.[8] O. Motrunich, K. Damle, and D. A. Huse, arXiv:cond-mat/0005543 (2000), 10.1103/PhysRevB.63.134424.[9] O. Motrunich, S.-C. Mau, D. A. Huse, and D. S. Fisher,Physical Review B , 1160 (2000).[10] T. Senthil and S. N. Majumdar, Physical Review Letters , 3001 (1996).[11] R. A. Hyman and K. Yang, Phys. Rev. Lett. , 1783(1997).[12] E. Altman, Y. Kafri, A. Polkovnikov, and G. Refael,Phys. Rev. Lett. , 150402 (2004).[13] G. Refael and J. E. Moore, Phys. Rev. Lett. , 260602(2004).[14] F. Igl´oi and C. Monthus, Physics reports , 277 (2005).[15] I. A. Kov´acs and F. Igl´oi, Physical Review B , 214416(2009).[16] I. A. Kovacs and F. Igloi, arXiv:1005.4740 [cond-mat](2010), 10.1103/PhysRevB.82.054437.[17] K. Damle and D. A. Huse, Phys. Rev. Lett. , 277203(2002).[18] C. R. Laumann, D. A. Huse, A. W. Ludwig, G. Refael,S. Trebst, and M. Troyer, Physical Review B - Con-densed Matter and Materials Physics , 224201 (2012).[19] Y.-C. Lin, F. Igl´oi, and H. Rieger, Physical Review Let-ters , 147202 (2007).[20] L. Fidkowski, G. Refael, N. E. Bonesteel, and J. E.Moore, Physical Review B - Condensed Matter and Ma-terials Physics , 224204 (2008). [21] N. E. Bonesteel and K. Yang, Physical Review Letters , 140405 (2007), arXiv:0612503 [cond-mat].[22] L. Zhang, B. Zhao, T. Devakul, and D. A. Huse, Phys.Rev. B , 224201 (2016).[23] A. Goremykina, R. Vasseur, and M. Serbyn, Phys. Rev.Lett. , 040601 (2019).[24] P. T. Dumitrescu, A. Goremykina, S. A. Parameswaran,M. Serbyn, and R. Vasseur, Phys. Rev. B , 094205(2019).[25] A. Morningstar and D. A. Huse, Phys. Rev. B , 224205(2019).[26] A. Morningstar, D. A. Huse, and J. Z. Imbrie, arXivpreprint arXiv:2006.04825 (2020).[27] R. Vosk, D. A. Huse, and E. Altman, Phys. Rev. X ,031032 (2015).[28] A. C. Potter, R. Vasseur, and S. A. Parameswaran, Phys.Rev. X , 031033 (2015).[29] P. T. Dumitrescu, R. Vasseur, and A. C. Potter, Phys.Rev. Lett. , 110604 (2017).[30] T. Thiery, F. m. c. Huveneers, M. M¨uller, andW. De Roeck, Phys. Rev. Lett. , 140601 (2018).[31] D. Pekker, G. Refael, E. Altman, E. Demler, andV. Oganesyan, Phys. Rev. X , 011052 (2014).[32] R. Vosk and E. Altman, Phys. Rev. Lett. , 067204(2013).[33] R. Vasseur, A. C. Potter, and S. A. Parameswaran, Phys.Rev. Lett. , 217201 (2015).[34] Y.-Z. You, X.-L. Qi, and C. Xu, Phys. Rev. B , 104205(2016).[35] S. Gopalakrishnan and S. Parameswaran, Physics Re-ports , 1 (2020), dynamics and transport at thethreshold of many-body localization.[36] R. Bistritzer and A. H. MacDonald, Proceedings of theNational Academy of Sciences of the United States ofAmerica , 12233 (2011), arXiv:1009.4203.[37] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken,J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe,T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero, Nature , 80 (2018).[38] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi,E. Kaxiras, and P. Jarillo-Herrero, Nature , 43(2018).[39] P. Bordia, H. L¨uschen, S. Scherg, S. Gopalakrishnan,M. Knap, U. Schneider, and I. Bloch, Physical ReviewX , 041047 (2017).[40] B. Deissler, M. Zaccanti, G. Roati, C. D’Errico, M. Fat-tori, M. Modugno, G. Modugno, and M. Inguscio, Na-ture Physics , 354 (2010), arXiv:0910.5062.[41] H. P. L¨uschen, P. Bordia, S. S. Hodgman, M. Schreiber,S. Sarkar, A. J. Daley, M. H. Fischer, E. Altman, I. Bloch,and U. Schneider, Physical Review X , 011034 (2017),arXiv:1610.01613.[42] G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort,M. Zaccanti, G. Modugno, M. Modugno, and M. Ingus-cio, Nature , 895 (2008).[43] M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L¨uschen,M. H. Fischer, R. Vosk, E. Altman, U. Schneider, andI. Bloch, Science , 842 (2015), arXiv:1501.05661.[44] Y. Yoo, J. Lee, and B. Swingle, (2020),arXiv:2005.10835.[45] H. Yao, T. Giamarchi, and L. Sanchez-Palencia, (2020),arXiv:2002.06559.[46] F. Baboux, E. Levy, A. Lemaˆıtre, C. G´omez, E. Ga-lopin, L. Le Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, Physical Review B , 161114 (2017),arXiv:1607.03813.[47] M. Verbin, O. Zilberberg, Y. Lahini, Y. E. Kraus,and Y. Silberberg, Physical Review B - CondensedMatter and Materials Physics , 064201 (2015),arXiv:1403.7124.[48] Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti,N. Davidson, and Y. Silberberg, Physical Review Letters , 013901 (2009).[49] D. Tanese, E. Gurevich, F. Baboux, T. Jacqmin,A. Lemaˆıtre, E. Galopin, I. Sagnes, A. Amo, J. Bloch,and E. Akkermans, Physical Review Letters , 146404(2014).[50] K. Deguchi, S. Matsukawa, N. K. Sato, T. Hattori,K. Ishida, H. Takakura, and T. Ishimasa, Nature Mate-rials , 1013 (2012).[51] S. Wiseman and E. Domany, Physical Review Letters ,22 (1998).[52] A. Aharony and A. B. Harris, Physical Review Letters , 3700 (1996).[53] K. H. Fischer, Physica B+C , 813 (1977).[54] D. Boyanovsky and J. L. Cardy, Physical Review B ,154 (1982).[55] Y. B. Kim and X. G. Wen, Physical Review B , 4043(1994).[56] J. Ye, S. Sachdev, and N. Read, Physical Review Letters , 4011 (1993).[57] C. A. Doty and D. S. Fisher, Phys. Rev. B , 2167(1992).[58] J. Vidal, D. Mouhanna, and T. Giamarchi, Physical Re-view Letters , 3908 (1999).[59] J. Vidal, D. Mouhanna, and T. Giamarchi, Physical Re-view B , 014201 (2001).[60] P. J. D. Crowley, A. Chandran, and C. R. Laumann,arXiv:1812.01660 [cond-mat] (2018), arXiv:1812.01660.[61] P. J. Crowley, A. Chandran, and C. R. Laumann, Phys-ical Review Letters , 175702 (2018). [62] U. Agrawal, S. Gopalakrishnan, and R. Vasseur,Nature Communications 2020 11:1 , 1 (2020),arXiv:1908.02774.[63] M. Sbroscia, K. Viebahn, E. Carter, J.-C. Yu, A. Gaunt,and U. Schneider, (2020), arXiv:2001.10912.[64] T. Inoue and S. Yamamoto, (2020), arXiv:2004.09850.[65] A. Szab´o and U. Schneider, Physical Review B ,014205 (2020), arXiv:1909.02048.[66] A. Jagannathan, Phys. Rev. Lett. , 047202 (2004).[67] C. Pich, A. P. Young, H. Rieger, and N. Kawashima,Physical Review Letters , 5916 (1998).[68] M. Guo, R. N. Bhatt, and D. A. Huse, Physical ReviewLetters , 4137 (1994).[69] H. Rieger and A. P. Young, Physical Review Letters ,4141 (1994).[70] B. Kang, S. A. Parameswaran, A. C. Potter, R. Vasseur,and S. Gazit, arXiv e-prints , arXiv:2008.09617 (2020),arXiv:2008.09617 [cond-mat.str-el].[71] R. Or´us, Annals of Physics , 117 (2014).[72] A. Bazavov, B. A. Berg, and S. Dubey, Phase Transi-tion Properties of 3D Potts Models , Tech. Rep. (2008)arXiv:0804.1402v2.[73] M. Hellmund and W. Janke, 10.1103/Phys-RevE.74.051113.[74] F. Y. Wu, Reviews of Modern Physics , 235 (1982).[75] K. Hui and A. N. Berker, Physical Review Letters ,2507 (1989).[76] J. M. Luck, Epl , 359 (1993).[77] See Supplemental Material for additional numerical re-sults, and a bound on the scaling of the finite-size gap.[78] R. Yu, H. Saleur, and S. Haas, Physical Review B - Con-densed Matter and Materials Physics , 140402 (2008).[79] A. Chandran and C. R. Laumann, Phys. Rev. X ,031061 (2017). upplemental material for “Quantum criticality in the 2d quasiperiodic Potts model” Utkarsh Agrawal and Romain Vasseur
Department of Physics, University of Massachusetts, Amherst, MA 01003, USA
Sarang Gopalakrishnan
Department of Physics and Astronomy, CUNY College of Staten Island,Staten Island, NY 10314; Physics Program and Initiative for the Theoretical Sciences,The Graduate Center, CUNY, New York, NY 10016, USA
I. CRITICAL BEHAVIOR FOR DIFFERENT VALUES OF q AND QUASIPERIODIC POTENTIALS
In the main text we focused on the critical properties for a given quasiperiodic (QP) distribution (eq 2 in the maintext) for q = 3. Here we present some numerical results for different distributions and values of q , illustrating theuniversality of our results. A. q > Fig. 1 shows the collapse of the magnetization for q = 6 and q = 10, for the QP potentials studied in the main text.The exponents quoted in the main text also lead to very good collapses in these cases, strongly suggesting a singleuniversality class for all values of q . We also find that the gap scales as − ln ∆ E ∼ ln L (not shown), as for q = 3. q = 6 q = 10 FIG. 1: Magnetization collapses for q = 6 and q = 10. The critical exponents ν = 1 and x = 0 .
92 used in these plotsare the same as for q = 3. B. Different angle in the QP potentials
In the main text we focused on the angle θ = √ π in the quasiperiodic potentials. Fig. 2 shows the magnetizationcollapse for θ = √ π , again with the same critical exponents. C. Different QP potentials
In the main text, we defined the QP potentials using the logarithmic variables − ln J ij and − ln h i to decrease thetransient nature of the RG. We show here that a more natural choice for the the QP potentials expressed directly interms of J ij , h i leads to the same critical behavior, albeit with stronger finite size effects. This is expected from ourFIG. 2: Collapse for θ = √ π for q = 3 with the critical exponents ν = 1 , x = 0 . g := − ln λ . The critical exponents are ν = 1and x = 0 .
92 as in the main text.argument for ν = 1, that relies on the repetition of defects which should scale universally for most functions withfrequency ϕ . We consider J ij =4 + cos (2 πϕ~k .~r + φ ) + cos (2 πϕ~k .~r + φ ) (1) h i = λ (4 + cos (2 πϕi x + φ ) + cos (2 πϕi y + φ )) , with the various variables defined in the main text below eq 2. The parameter tuning the transition is defined as g ≡ − ln λ . Fig. 3 shows a good magnetization collapse using the same critical exponents as in the main text.FIG. 4: The set of points( x, y ) such that 0 < f ( x, y ) < (cid:15) (with f ( x, y ) = 2 + cos (2 πϕ~k .~r ) + cos (2 πϕ~k .~r ), see text)for various values of (cid:15) . The dashed lines represent the orientation of ~k and ~k . As (cid:15) is decreased, the typical lengthscale separating points in this set increases. II. REPETITION PATTERNS IN 2D QUASIPERIODIC FUNCTIONS
To complement the discussion in the main text regarding the repetition of “defects”, we consider a simplifiedquestion that illustrates this pattern. Given a 2d QP function f ( x, y ) ≥
0, we consider the pattern of the points( x, y ) such that 0 < f ( x, y ) < (cid:15) , as a function of (cid:15) . We choose f ( x, y ) = 2 + cos (2 πϕ~k .~r ) + cos (2 πϕ~k .~r ) and k = (sin θ, cos θ ), with θ = √ π/
2; and plot the set M (cid:15) = { ( x, y ) | < f ( x, y ) < (cid:15) } in Fig. 4 for various valuesof (cid:15) . We see that the set M (cid:15) has some clear structure, with the typical length scale separating points in the set M (cid:15) increasing with decreasing (cid:15) . We find similar behavior for different angles θ , phases, and frequencies; however, thepattern of the set M (cid:15) is not universal. L g g =5.45 * ln L + c data FIG. 5: Linear fit of the (logarithm of the) gap Γ g = − ln ∆ E vs ln L , possibly consistent with a large but finitedynamical exponent z ≈ .
45. Note the upward curvature trend indicating a faster than any power law decay of ∆ E . III. SCALING OF THE GAPA. Additional numerical data
In the main text we argued that the logarithmic of the gap, − ln ∆ E ≡ Γ g , scales with system size as Γ g ∼ ln L .Here we try to fit the finite-size gap ∆ E as a power law of system size, ∆ E ∼ L − z (i.e finite dynamical exponent). Theresults are shown in Fig. 5, with the best fit giving ∆ E ∼ L − . , corresponding to a fairly large dynamical exponent z ≈ .
45. Although we cannot definitely rule out such a finite, large dynamical exponent, we do observe a slight upwardcurvature in Fig. 5, consistent with a logarithmically-diverging apparent dynamical exponent z ( L ) ∼ ln L → ∞ . B. Bound on the scaling of the gap
In fact, we can rule out a finite dynamical exponents based on the general structure of the RG rules, assumingsome self-similar structure at criticality. The banded structure of the couplings suggests that there is some fractionof initial couplings and fields (defined as − ln J and − ln h as in the main text) which are local maxima and can bedecimated in one go, in any order. Let us refer to the new set of bonds after decimating the local maxima as thefirst generation. After the above set of decimations we can again identify local maxima and decimate them to get2nd generation couplings, fields and so on. Each generation eliminates certain fraction of spins. We assume that atthe critical point this fraction is a constant in the sense that it is of same order for all generations. This implies thatdecimating a region of linear size L to a single spin will require in general ∼ log L generations. This structure can bemade rigorous in one spatial dimension, but the banded structure of the couplings in our numerics strongly suggeststhat the same holds true in 2D as well.We study the scaling of the gap with L or equivalently with the number of generations m . After m generations thetypical magnetic fields h ( m ) will be given by h ( m ) = Q M +1 h κ nm Q M J , where h (0) , J (0) are values of fields and couplings inthe initial distribution or the 0th generation. We show that the term κ n m in the above expression gives a bound onthe gap scaling and is enough to give an “infinite” dynamical exponent. Since we are only interested in the factorsof κ , let us introduce the notation f ] as the power of the κ in the expression for f , where f is a product/ratio offields and couplings. We then prove that at the critical point we have h ( m ) ] & m or in other words after runningthe RG upto length scale L , the typical magnetic field would decrease at least by a factor of ∼ κ − log L .Moving on to the proof, we work with logarithmic variables ‘ [ J,h ] ≡ − ln[ J, h ]. We have ‘ ( m +1) h = ‘ ( m ) h + P n ( ‘ ( m ) h − ‘ ( m ) J ) + n ln κ , or ‘ ( m +1) h ] = n + ‘ ( m ) h ] + P n ( ‘ ( m ) h − ‘ ( m ) J )], where n is some constant not depending on m . Notethat each ‘ ( m ) h in the sum is greater than ‘ ( m ) J by construction. For each term in the sum we have ‘ ( m ) h − ‘ ( m ) J ] = P ( ‘ ( m − h − ‘ ( m − J )] + n , where n > h and J ). The value of n is notimportant other than the fact that at the critical point we expect it to not depend on m , though it might show somefluctuations. This immediately gives h ( m ) − ‘ ( m ) J ] ≥ n m . Putting this in the expression for ‘ ( m +1) h ] we get ‘ ( m +1) h ] ≥ n + ‘ ( m ) h ] + nn m , or in other words, the above recursion implies ‘ ( m ) h ] & m .This argument indicates that the gap at the critical point of the RSRG studied in this paper is bounded by κ − log L ,which comes from keeping track of the factors of κ along the RG. For the Ising model ( κ = 1) we do not see signs of“infinite” dynamical constant or of infinite quasiperiodic behavior, suggesting that the above lower bound is saturated.This suggest the saturation of the log L scaling for the Potts model ( κ >κ >