Quantum field theory for the chiral clock transition in one spatial dimension

Abstract

We describe the quantum phase transition in the N -state chiral clock model in spatial dimension d = 1. With couplings chosen to preserve time-reversal and spatial inversion symmetries, such a model is in the universality class of recent experimental studies of the ordering of pumped Rydberg states in a one-dimensional chain of trapped ultracold alkali atoms. For such couplings and N = 3, the clock model is expected to have a direct phase transition from a gapped phase with a broken global Z N symmetry, to a gapped phase with the Z N symmetry restored. The transition has dynamical critical exponent z = 1, and so cannot be described by a relativistic quantum field theory. We use a lattice duality transformation to map the transition onto that of a Bose gas in d = 1, involving the onset of a single boson condensate in the background of a higher-dimensional N -boson condensate.We present a renormalization group analysis of the strongly coupled field theory for the Bose gas transition in an expansion in 2 − d, with 4 − N chosen to be of order 2 − d. At two-loop order, we find a regime of parameters with a renormalization group fixed point which can describe a direct phase transition. We also present numerical density-matrix renormalization group studies of lattice chiral clock and Bose gas models for N = 3, finding good evidence for a direct phase transition, and obtain estimates for z and the correlation length exponent ν.arXiv:1808.07056v2 [cond-mat.str-el] 19 Oct 2018 42 4. I (M ) 4 44 C. Renormalization constants for the Z N dilute Bose gas 45 D. The self-dual phase boundary in the chiral clock model 46 E. Analysis as λ → 0 48

Asymmetric Particle Transport and Light-Cone Dynamics Induced by Anyonic Statistics

“…We note the intriguing possibility that a similar dynamical symmetry may exist in other models, such as the Z n chiral clock model [85,86], which has symmetry properties similar to the AHM. Finally, we point out that the inversion symmetry breaking associated with anyonic statistics is also present for non-Abelian anyons in quasi-1D systems [87][88][89]-for example, Majorana fermions (or, more generally, parafermions) at the edge of (fractional) quantum Hall systems, in deep connection with the underlying chirality.…”

Quantum Kibble–Zurek mechanism and critical dynamics on a programmable Rydberg simulator

“…The nature of the direct transition from disordered to Z 3 -ordered phases is discussed in Refs. [22][23][24]. Second, we have not explicitly identified the phase transition between disordered to Z 4 -ordered phases.…”

“…The exact nature of such phase transitions has been a subject of intense theoretical research for the past three decades [10,11,[22][23][24]28]. Only very recently, numerical studies of equilibrium scaling properties [22][23][24] provided evidence for a direct transition [24] along some paths across the phase boundary, where the expected range of values of the scaling exponent is µ < 0.45 [22], and µ > 0.25 [23]. Our experimental results are consistent with a direct CCM phase transition over a range of interaction strengths with µ ∼ 0.38, in agreement with the theoretical value obtained by combining the results of the most extensive numerical finite-size scaling studies [22,24] (dashed line in Fig.…”

“…Shaded areas indicate the regions consistent with Z2-(green), Z3-(purple), and Z4-ordered (orange) phases. The solid green line corresponds to µIsing, the purple dashed lines represent the upper[22], and lower[23] bounds of µCCM, while the purple dotted line is the value of µCCM obtained from the best numerical estimates of z[22] and ν[24]. Error bars represent the 68% confidence interval (b), and uncertainty of the power-law fit (c), which is dominated by systematic effects in the extraction of individual correlation lengths.…”

Kibble-Zurek exponent and chiral transition of the period-4 phase of Rydberg chains

“…For p = 3, the chiral perturbation is always relevant, and the question is whether it immediately opens a floating phase away from the Potts point, or whether the transition remains direct and continuous for a while, but in a new chiral universality class, as suggested by Huse and Fisher 18 , with a dynamical exponent z > 1. Numerical [19][20][21][22][23] and experimental evidence 14,15 in favor of this possibility has been obtained in the 1980s and early 1990s in the context of adsorbed layers, and very recently in the context of Rydberg atoms 9,12,[24][25][26] .…”