The logarithm of the transfer matrix of the two-dimensional, three state chiral clock model is shown to be equivalent to a one-dimensional non-Hermitian quantum Hamiltonian. In the region of small chirality the critical behaviour of the quantum model is investigated by a series expansion and by finite-size scaling. A single phase boundary between a modulated paramagnetic and a ferromagnetic phase is located with good accuracy. Both methods yield comparable results for the critical exponent beta of the wavevector of the modulated high temperature phase. The critical exponents nu x , nu tau for the correlation lengths obtained from finite-size scaling point towards the existence of a Lifshitz point at finite chirality. The results of the series expansion for the exponent nu x , however, are inconsistent with finite-size scaling. For larger chirality the authors' finite-size results clearly reveal the occurrence of an incommensurate phase with algebraically decaying correlations between the ferromagnetic and the modulated paramagnetic phase. They analyse the critical behaviour of this incommensurate phase at its melting line and determine the exponent eta that controls the algebraic decay of correlations.
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
Z3 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
Z3 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 $\ensuremath{\nu}$. Our analysis presents the only known instance of a strongly coupled generic transition between gapped states with $z\ensuremath{\ne}1$, implying an underlying nonconformal critical-field theory.
A Monte Carlo renormalization-group scheme based on a ``floating sublattice'' blocking rule is constructed for the two-dimensional three-state chiral clock model. The presence of a critical phase is established by matching certain wave-vector-dependent correlation functions derived from lattices differing in linear size by a factor of 2. The wave vector, which in a function of temperature and the asymmetry parameter \ensuremath{\Delta}, is determined from the simulations. For \ensuremath{\Delta}=0.45, a critical phase is established in a temperature range. For \ensuremath{\Delta}=0.35, no critical phase is observed, suggesting that the phase diagram has a Lifschitz point in the interval 0.35<\ensuremath{\Delta}<0.45. Alternatively, the wave vector and the width of the critical phase at \ensuremath{\Delta}=0.35 are much smaller than for \ensuremath{\Delta}=0.45..AE
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
ZN symmetry, to a gapped phase with the
ZN symmetry restored. The transition has dynamical critical exponent $z\ensuremath{\ne}1$, 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 $2\ensuremath{-}d$, with $4\ensuremath{-}N$ chosen to be of order $2\ensuremath{-}d$. 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 $\ensuremath{\nu}$.
Using the tensor renormalization group method based on the higher-order singular value decomposition, we have studied the phase transitions of the five-state clock model on the square lattice. The temperature dependence of the specific heat indicates the system has two phase transitions, as verified clearly by the correlation function at three representative temperatures. By calculating the magnetic susceptibility, we obtained only the upper critical temperature as T c2 = 0.9565(7). Investigating the fixed-point tensor, we precisely locate the transition temperatures at T c1 = 0.9029(1) and T c2 = 0.9520(1), consistent well with the Monte Carlo and the density matrix renormalization group results.