title: Quantum criticality in the 2d quasiperiodic Potts model
authors: Utkarsh Agrawal Sarang Gopalakrishnan Romain Vasseur
text: 
  
  
  
  
  
  
  
  
  
  
  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
  
  Department of Physics, University of Massachusetts, Amherst, MA 01003, USA
  
  
  
  Quantum critical points in quasiperiodic magnets can realize new universality classes, with critical properties distinct from those of clean or disordered systems. Here, we study quantum phase transitions separating ferromagnetic and paramagnetic phases in the quasiperiodic q-state Potts model in 2+1d. Using a controlled real-space renormalization group approach, we find that the critical behavior is largely independent of q, and is controlled by an infinite-quasiperiodicity fixed point. The correlation length exponent is found to be ν=1, saturating a modified version of the Harris-Luck criterion. 
  
  
  
  
  
  Quantum criticality in the 2d quasiperiodic Potts model
      Romain Vasseur
      December 30, 2023
  =======================================================
  
  
  
  Quenched disorder can dramatically affect the universality class of a quantum phase transition, and drive it to a new renormalization group (RG) fixed point if the correlation length exponent ν violates the Harris criterion ν≥ 2/d <cit.> with d the dimensionality of the system. As the effective randomness grows under renormalization, the new infrared fixed point can either be characterized by finite or infinite randomness. 
  Infinite-randomness fixed points can be analyzed using an asymptotically exact real space renormalization group (RSRG) approach <cit.> that yields exact predictions for critical exponents and scaling functions. The RSRG approach has been applied to many different quantum phase transitions in one and two dimensions, both at zero temperature and in the context of many-body localization <cit.>. 
  
  The structure of infinite-randomness critical points depends crucially on the assumption of spatially uncorrelated disorder. However, 
  
  many present-day experiments, involving, e.g., twisted bilayer graphene <cit.> and ultracold atoms in bichromatic laser potentials <cit.> 
  involve systems that are spatially inhomogeneous, but quasiperiodic rather than random. Quasiperiodic potentials are deterministic, with strong spatial correlations, so they do not lead to conventional infinite-randomness behavior <cit.>.
  
  Instead, when a clean critical point is unstable to quasiperiodicity, it flows to a new class of fixed points. Field theoretic methods <cit.> do not easily generalize to quasiperiodic systems <cit.>, because there is no disorder to average over. However, very recent results <cit.> have revealed the existence of “infinite-quasiperiodicity” quantum critical points <cit.> in one dimensional spin chains; at these critical points, RSRG yields exact predictions for exponents. 
  
  Despite their differences, infinite-quasiperiodicity and infinite-randomness critical points share the key feature that the dynamical critical scaling exponent z = ∞: thus, the characteristic timescale t_ξ associated with a length-scale ξ grows faster than any power law of ξ.
  So far, such infinite-quasiperiodicity fixed points have chiefly been studied in one dimension; higher-dimensional cases are poorly understood <cit.>. The z = ∞ dynamical scaling leads to a rapidly vanishing gap, which makes it hard to access the critical regime using Quantum Monte Carlo techniques <cit.>. Tensor network based approaches (see e.g. <cit.>) are also less suited to study 2d QP quantum criticality, due to large entanglement. 
  
  In this letter, we propose a general RSRG approach to study 2+1d quantum spin models with QP couplings. As in the implementations of RSRG for disordered systems in two dimensions,  the RG changes the underlying geometry of the system creating intricate and complex long range interactions <cit.>. Nevertheless the RG procedure can be efficiently implemented numerically. We focus on the 2d quantum Potts model, with q “colors” (q=2 corresponding to the Ising model). For clean systems, the phase transition separating paramagnetic and symmetry-broken phases is in the classical 3D Potts model universality class, which is a first-order for q≥ 3 <cit.>. Strong enough QP modulations should smoothe these first-order transitions <cit.>, driving them to a new strong quasiperiodicity fixed point that we describe using RSRG. Our results suggest that the critical properties do not depend on q>2, with the Ising case q=2 being special. 
  
  Beyond our numerical results for the critical exponents, we 
  
  propose a general argument for the correlation exponent ν=1 for these new infinite-quasiperiodicity transitions, based on the distribution of “defects” in the critical structure. Due to the deterministic and almost periodic nature of quasiperiodic potentials these defects form a definite pattern; in some special cases, the defects form a QP tiling with a length scale that defines the correlation length.
  Interestingly, the value of ν saturates a modified version of the Harris-Luck criterion <cit.>, namely ν≥ 1; the modifications are due to boundary fluctuations coming from correlations in boundaries of rectangular patches at all length scales. 
  
  
  
    
  Model. The q-state quantum Potts model is defined via the Hamiltonian
  
  
      H=-∑_⟨ i,j⟩ J_ijδ_n_i,n_j - ∑_i h_i/q∑_n_i,n_i'|n_i⟩⟨ n_i'|,
  
  
  defined on the square lattice with ⟨ i,j⟩ denoting nearest neighbor pairs, where n_i is a variable on site i that takes one of q possible values. The first term with J_ij>0 is a classical ferromagnetic interaction favoring aligned spins, while the second term is a quantum transverse field leading to a paramagnetic phase at large h_i's. For q=2 colors, this coincides with the familiar transverse field Ising model. The model is initially defined on a square lattice; however, we believe our results to be independent of the initial lattice geometry, as RSRG drastically changes the connectivity of the system.
  
  
  The couplings J_ij>0, h_i>0 are inhomogeneous, aperiodic but deterministic. Here, we consider J_ij = f_1(k_1 . r)+f_2(k_2.r), where r=(i_x,i_y)+1/2(j_x-i_x,j_y-i_y), k_1, and k_2 are two orthogonal unit vectors, and f_a(x)=f_a(x+φ^-1) for some irrational φ, which we take to be the golden ratio, φ = 1+√(5)/2. Similarly, the fields are taken from an initial potential of the form, h_i = g_1(i_x)+g_2(i_y) with g_a(x)=g_a(x+φ^-1). For concreteness, we focus on the following QP modulations throughout the paper,
  
  
      ij=   2+cos(2πφk_1. r+ϕ_1) + cos(2πφk_2 . r + ϕ_2)
      i =    (2+cos(2πφ i_x + ϕ_3) + cos(2πφ i_y + ϕ_4)),
  
  
  where g is a parameter driving the transition, ij=-ln J_ij and i=-ln h_i are defined so as to decrease the transient behavior in the RG (see below), and ϕ_i are some constant global phases which we average over. Unless otherwise stated, we take k⃗_1=(sinθ,cosθ), with the angle θ=√(2)π. Our results do not depend on the details of these distributions <cit.>.
  
  
  
  
    
  RG procedure.  We now describe the RSRG procedure we use to capture the critical properties of Eq. (<ref>). One step of the RG procedure consists of identifying the strongest coupling in the Hamiltonian (which sets the cutoff, Ω) and eliminating it, as follows <cit.>. If the strongest coupling is a bond J_ij, one merges the two spins connected by the bond into a new effective spin (or “cluster”) with magnetic moment μ_i' = μ_i + μ_j (μ_i =1 for initial physical spins). The effective transverse field acting on the cluster is given by second-order perturbation theory, h_i'≈h_i h_j/κ J_ij with κ=q/2; also, any other spin (or cluster) in the system that was connected to either i or j now picks up a bond to the new cluster, with coupling given by J_ik'=max(J_ik,J_jk). If instead the strongest spin is an effective field h_i, one eliminates the site i. Any other pair of sites j, k that were connected to i by bonds now pick up a new effective bond, which we estimate using 2nd order perturbation theory: J_jk' ≈ J_jk+J_ijJ_ik/κ h_i≈max(J_jk,J_ijJ_ik/κ h_i).
  This procedure correctly captures the low energy physics as long as Ω≫ J_ij,h_j (broadly distributed couplings) so that perturbation theory is controlled; we will see that for infinite-quasiperiodicity fixed points, the parameter controlling the error in perturbation theory flows to zero upon coarse-graining, leading to asymptotically exact predictions for universal properties. 
  
  
  
  
  We numerically run the RG procedure described above starting from a L× L square lattice.  We first focus on the q=3 Potts model – the critical behavior is largely independent of q ≥ 3. As the system moves along the RG flow, its geometry changes giving rise to graphs of increasingly intricate connectivity. Instead of implementing the RG in the naive sequence described above (i.e., always decimating a single largest coupling), we follow standard techniques <cit.> to optimize the decimation sequence. (We have checked that at the end of the RG procedure, the optimized and naive decimation sequences yield identical couplings, so this step is not an approximation.)
  
  
  
  
  
  
  
    
  Magnetization and fractal exponent.
  
  At the end of the RG, the surviving cluster with moment  determines the magnetization of the system, m(L,g)= /L^2, where L is the linear size of the system. To locate the critical point we plot r(L,g)=m(L,g)/m(L/2,g) vs g for various L; away from the criticality r(L,g) changes with L, while being scale independent at the critical point <cit.>. 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 point scales as ∼ L^d_f with d_f being the fractal dimension of the spins in the cluster. Those two exponents satisfy the scaling relation d_f + x = 2.
  Those quantities are plotted for the q=3 Potts model in Fig. <ref>, and we find 2^-x≈ 0.53 or x≈ 0.92 and d_f = 1.085 ± 0.024, consistent with the relation d_f + x = 2.
  
  
  
    
  Correlation length. Assuming single parameter scaling with a diverging correlation length ξ∼ |g - g_c|^-ν, we expect the following scaling form for the magnetization m(L,g)=L^-xf((g-g_c)L^1/ν), where f is a universal scaling function. Using the values of g_c, and x obtained from the plot of r(L), we find a nice collapse for ν≈ 1. We now argue that this result ν = 1 holds exactly, at least for some classes of quasiperiodic potentials. 
  
  The argument for ν = 1 is as follows. Let us first consider the case where the quasiperiodic modulation is parallel to the lattice vectors, i.e., k_1=(1,0), k_2=(0,1) in (<ref>). We now consider running the RG for two realizations of the lattice, one at criticality and one detuned by a distance δ. We now look for “defects,” or points on the lattice where the two RG realizations begin to diverge (because one of them decimates fields and the other bonds). Defects occur when locally, fields are close (δ) in magnitude to the neighboring bonds; thus, a small detuning is enough to change the order of decimations. However, because the quasiperiodic structure is approximated to precision ∼δ by a rational approximant with period ∼ 1/δ, each defect has an almost perfect repeat at a distance ∼ 1/δ (along both lattice directions). This can be seen by observing that cos(2πφ (x+F_n)+ϕ) = cos (2πφ x + ϕ) + 𝒪(φ^-n), where F_n is the nth Fibonacci number: defects must repeat along the vertical and horizontal axis, forming a QP tilling, with a length scale ξ=F_n∼φ^n, with δ∼φ^-n giving ξ∝δ^-1 (see <cit.> for a similar argument in quantum spin chains). 
  Thus, when the RG reaches length scale 1/δ, defects will proliferate and drive the system away from criticality, corresponding to ν=1. To illustrate this tiling geometry, we plot the set S = { i:  min{ij} < i}, where the min is over nearest neighbors. This condition is satisfied for couplings J that are decimated first in the RG, forming non-trivial clusters. The geometry of the set S is shown in Fig. <ref>. 
  
  The geometry away from θ = 0 is less transparent, but numerics once again suggests ν = 1; moreover, the model remains strongly anisotropic under coarse-graining, with preferred orientations (Fig. <ref>b). We now argue that, if this anisotropy persists under the RG, it leads to a modification of the Harris-Luck bound on ν <cit.>. The standard argument for this criterion runs as follows. In a large patch of the sample of linear dimension ℓ, the apparent local value of the critical point is δ_ℓ≡⟨ g ⟩_ℓ - g_c ∼ℓ^w-d where w is the wandering exponent. Setting ℓ to the correlation length ξ∼δ^ν, we get δ_ξ∼δ^ν(d-w). When δ_ξ is small compared with the global detuning δ, the transition is well-defined. This criterion amounts to ν > 1/(d - w). Generic patches of a quasiperiodic system have wandering exponent w = 0 in the bulk so the standard Luck criterion reads ν > 1/d. However, this analysis ignores “boundary” terms due to lines or other sub-dimensional regions of the sample where δ is locally away from its average value. If one includes these boundary contributions, the deviation is δ_ℓ∼ℓ^(d-1)-d∼ 1/ℓ, so that ν≥ 1 regardless of dimensionality. The quasiperiodic Potts model appears to saturate this modified bound,  with ν = 1 (up to logarithmic corrections). 
  
  
  
  
  
  
  
    
  Dynamical scaling and RG error. We now turn briefly to the dynamical scaling properties at this transition. One can argue analytically that the timescale for a region of ℓ spins grows at least as ln t_ℓln^2ℓ. This scaling follows naturally from the RG rules; recall that these rules involve a factor κ > 1 at each step. One can check that upon decimating a region of size ℓ to a single spin, one picks up at least ln^2 ℓ factors of κ in the effective couplings <cit.>, implying an energy scaling - ln E_ℓ∼ln t_ℓ≳ln^2 ℓ. This scaling can be interpreted as the scaling of the finite size gap of a region of size ℓ. 
   This divergence might be subleading (as it is in the random case), but guarantees “activated” scaling, where t grows faster than any power of ℓ. As we see in Fig. <ref>b, our numerical results are consistent with ln t_ℓ∼ln^2 ℓ, i.e., the same dynamical scaling as in one dimension <cit.>. We note that our data is also compatible with other types of activated scaling <cit.>.
  
  A consequence of activated dynamical scaling is that the RG becomes increasingly accurate at late stages. The typical RG error (defined as log≡⟨log(max J_ij,h_i/Ω) ⟩, where the max function is over all neighboring terms of Ω, with ⟨·⟩ denoting average over a small window of -logΩ, and several phase realizations) vs -lnΩ(≡Γ) at the critical point is plotted in Fig. <ref>. a). We see that on average, the RG error decreases along the RG flow, suggesting that the RG becomes asymptotically exact, as in the random case <cit.>. While the system sizes we can access remain away from the asymptotic regime where the RG is fully controlled, we observe very good quality critical data (Fig. <ref>) with no signs of finite-size drifts.
  Extrapolating these results, we expect the error of a typical RG step to go to zero asymptotically with Γ. 
  
  
  
    
  Critical behavior vs q.
  
  We conclude this letter by briefly discussing the case of q>3. 
  For q>3, we observe a similar behavior as for q=3; there is a 2nd order transition with the RG becoming more controlled with the flow. The correlation exponent ν=1 seems to hold, as expected from the general arguments discussed above. Unsurprisingly, the location of critical point is non-universal and changes with q and θ. The d_f and x exponents appear to be same for all values of q>2, suggesting the same universality class for different q's, though we cannot exclude small differences based on our numerical data.  Interestingly, for larger values of q, we observe that the distributions of the gap and of couplings form “bands”, with forbidden values in between the allowed bands (see Fig. <ref>.c). This is reminiscent of similar banding properties that were observed in QP quantum spin chains <cit.>; it would be interesting to investigate whether this can be leveraged to understand this RG analytically in the future. 
  
  
  The case of q=2 (the Ising model), is special. In this case, we find that the RG does not flow towards infinite quasiperiodicity, and is therefore not controlled. A similar scenario occurs in 1d weak QP modulations are marginally irrelevant <cit.> at the clean fixed points. However, unlike the 1d case, we observe that even on introducing strong QP modulations, the RG does not flow to infinite quasiperiodicity. From the modified version of the Luck criterion, we expect QP modulations to be relevant at the clean Ising transition, driving the system to a finite quasiperiodicity fixed point that cannot be described using RSRG. It would be especially interesting to investigate the nature of this QP Ising transition, as we expect it to be very different from the transitions described in this letter — in particular, it likely has a finite dynamical exponent z, as a consequence of the prefactor κ=1 in the RG rules.  
  
  
  
    
  Discussion.
  
  We analyzed the critical behavior of quantum phase transitions separating ferromagnetic and paramagnetic phases in the quasiperiodic q-state Potts model in two dimensions. Using a controlled real-space renormalization group approach, we found that the critical behavior is independent of q, and is controlled by a new RG fixed point providing the first example of “infinite-quasiperiodicity” behavior in two dimensions. We argued on general grounds that such QP quantum phase transitions have correlation length exponent ν=1, saturating a modified version of the Harris-Luck criterion. It would be interesting to find other examples of infinite-quasiperiodicity transitions, both in two and three dimensions. The case of the 2d QP Ising model also deserves more attention, as it should provide a different type of QP transition with finite dynamical exponent.  We leave these questions for future works.
  
  
  
    
  Acknowledgments: The authors thank Snir Gazit, Sid Parameswaran and Jed Pixley for useful discussions. This work was supported by the National Science Foundation under NSF Grant No. DMR-1653271 (S.G.), the US Department of Energy, Office of Science, Basic Energy Sciences, under Early Career Award No. DE-SC0019168 (U.A. and R.V.), and the Alfred P. Sloan Foundation through a Sloan Research Fellowship (R.V.)
  
  
  
  
  [pages=1]sup_mat.pdf
  [pages=2]sup_mat.pdf
  [pages=3]sup_mat.pdf
  [pages=4]sup_mat.pdf
  
  