Numerical study of the chiral ${\mathbb{Z}}_3$ quantum phase transition in one spatial dimension

Recent experiments on a one-dimensional chain of trapped alkali-metal atoms [Bernien et al., Nature (London) 551, 579 (2017)] have observed a quantum transition associated with the onset of period-3 ordering of pumped Rydberg states. This spontaneous ${\mathbb{Z}}_3$ symmetry breaking is described by a constrained model of hard-core bosons proposed by Fendley et al. [Phys. Rev. B 69, 075106 (2004)]. By symmetry arguments, the transition is expected to be in the universality class of the ${\mathbb{Z}}_3$ chiral clock model with parameters preserving both time-reversal and spatial-inversion symmetries. We study the nature of the order–disorder transition in these models and numerically calculate its critical exponents with exact diagonalization and density-matrix renormalization-group techniques. We use finite-size scaling to determine the dynamical critical exponent $z$ and the correlation length exponent $\nu$. Our analysis presents the only known instance of a strongly coupled generic transition between gapped states with $z\ne 1$, implying an underlying nonconformal critical-field theory.

Figure 1

Schematic representation of (a),(b) the interactions and (c) a generic state of the ${\mathbb{Z}}_3$ chiral clock model. The arrows connote the eigenvalue of the operator $\sigma$ at each site with ${\sigma }_i=1,\omega ,{\omega }^2$ delineated by the arrow pointing at 12, 4, and 8 o'clock, respectively. Owing to the chirality of the couplings in Eq. (1), there are two distinct types of domain walls (DW) with their associated interaction strengths illustrated in (b). (d) The Rydberg and ground states of the two-level system defined by Eq. (4). The van der Waals interactions depend on the spacing between Rydberg excitations (e) and thus a representative state (f), once again, has two kinds of domain walls.

Figure 2

Three phases of the CCM, adapted from Ref. [16], are schematically limned in a cross section of the three-dimensional phase diagram for $\varphi =0$, $J=1-f$, and $L=100$. $P$ marks the transition point of the three-state Potts model. Blue diamonds indicate the quantum critical points obtained from finite-size DMRG, which are listed in Table 1. When overlaid with the previously found transition points (solid line), the two data sets agree perfectly.

Figure 3

(a) Scaling of the energy gap $\mathrm{\Delta }$ as a function of $f$ for individual system sizes with $\varphi =0$ and $\theta =\pi /8$. With $z=1.229$, all the curves intersect right at the critical point. The finesse of the crossing depends crucially on the $z$ chosen: on zooming in, the contrast in sharpness between $z=1$ (b) and $z=1.229$ (c) is vivid.

Figure 4

Log-log plot of the gap $\mathrm{\Delta }$ against the lattice size $L$ at the critical point $f=f_c$. As previously, $\varphi =0$, and the different values of $\theta$ are labeled. For sufficiently large systems, the dependence is exactly linear as expected. For reference, the curve for the Potts transition is indicated by a solid line.

Figure 5

Callan-Symanzik $\beta$ function plotted on a logarithmic scale against the system size for $\varphi =0$. The slopes of the curves in the linear region, corresponding to large lattices, give us the respective values of $-1/\nu$ for each choice of $\theta$.

Figure 6

Data collapse for six different chiral angles based on the critical exponents $z$ and $\nu$ recorded in Table 1. The data consists of three different sets, one for each value of $\delta f\equiv f-f_c$. Every data set is, in turn, comprised of 31 points for the differing system lengths $L$ ranging from 40 to 100, in steps of 2. All the data collapse perfectly onto a single line.

Figure 7

Numerical simulations using iDMRG. Diagrammatic representations of (a) two-site-translation invariant matrix product states and two-site transfer matrix $T$ and (b) the two-point correlation function. (c) Correlation lengths $\xi$ along the two lines $L_1$ ($\theta =\pi /16$) and $L_2$ ($\theta =\pi /8$), indicated in the inset of Fig. 2, for three different bond dimensions. The divergence of the correlation lengths signals the onset of the QPT. (d) The wave vector $k$, defined in Eq. (15), tends to zero as $\xi \to \infty$. However, the scaling form upon approaching the critical point is different for different chiral angles. Note that the points at the bottom, along the $x$ axis, correspond to values of ${\lambda }_1$ inside the symmetry-broken phase; the concomitant momenta are exactly zero.

Figure 8

(Upper panel) FSS analysis of the QPT from the disordered to the ${\mathbb{Z}}_3$-ordered phase, across the integrable line $w^2=UV+V^2$, based on exact diagonalization of the Hamiltonian (17). (a) Plot of the energy gap $\mathrm{\Delta }$, rescaled by $N^{-z}$ with $z=1$, as a function of $U/w$ for different system sizes $N$. The sharp intersection of the different curves at $U/w=-3.029$ determines the QCP and conveys that $z=1$. The location of the QCP agrees exactly with the analytically known result $U_c/w\approx -3.0299$ [Eq. (21)]. (b) Scaling plot of the energy gap for different system sizes; perfect data collapse is achieved for $\nu =5/6$. (Lower panel) Same as above but in the limit $V\to \infty$. FSS now betokens $U/w=-1.949$, $z\approx 4/3$, and $\nu \approx 5/7$.

Figure 9

(a) Phase boundary between the disordered and the ${\mathbb{Z}}_3$-ordered state above the Potts critical point [Eq. (21)] with nearest- and next-nearest-neighbor couplings only (U-V model, blue) and long-range Rydberg interactions (red). In the latter case, the interaction strength between two spins separated by $x$ sites is given by $V_x=C_6/x^6$. To plot the phase boundary on the same scale, we identify $V_2=V$. (b) The dynamical critical exponent $z$ when crossing this phase boundary in the U-V model, obtained from FSS with up to $N=21$ sites (blue). At the Potts point (green), we find $z=1$ as expected. Away from the Potts point, $z$ increases and, for $V\to \infty$, we reckon an asymptotic value of $z\approx 1.33$; the crossover in $z$ between these limits is likely a finite-size effect. The inset shows the analogously extracted value of $z$ for the Rydberg Hamiltonian. Here also we find that $z$ increases with the interaction strength, starting from a value close to one—this is consistent with the presence of a Potts point in the Rydberg model as well. Note that, in contrast to the U-V model, here, increasing $V$ would eventually lead to the appearance of the ${\mathbb{Z}}_4$ phase, which is not discussed in this work. The FSS analysis for the Rydberg model is limited to comparatively smaller systems, of up to 18 atoms, owing to the sizeably larger Hilbert space.