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arXiv:2001.06698 · Nature Communications

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

Natalia Chepiga, Frédéric Mila

Published Jan 18, 2021• 49 citations
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Abstract

Chains of Rydberg atoms have emerged as an amazing playground to study quantum physics in 1D. Playing with inter-atomic distances and laser detuning, one can in particular explore the commensurate-incommensurate transition out of density waves through the Kibble-Zurek mechanism, and the possible presence of a chiral transition with dynamical exponent z > 1. Here, we address this problem theoretically with effective blockade models where the short-distance repulsions are replaced by a constraint of no double occupancy. For the period-4 phase, we show that there is an Ashkin-Teller transition point with exponent ν = 0.78 surrounded by a direct chiral transition with a dynamical exponent z = 1.11 and a Kibble-Zurek exponent μ = 0.41. For Rydberg atoms with a van der Waals potential, we suggest that the experimental value μ = 0.25 is due to a chiral transition with z ≃ 1.9 and ν ≃ 0.47 surrounding an Ashkin-Teller transition close to the 4-state Potts universality.

DOI
10.1038/s41467-020-20641-y

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Chains of Rydberg atoms have emerged as an amazing playground to study quantum physics in 1D. Playing with inter-atomic distances and laser detuning, one can in particular explore the commensurate-incommensurate transition out of density waves through the Kibble-Zurek mechanism, and the possible presence of a chiral transition with dynamical exponent z > 1. Here, we address this problem theoretically with effective blockade models where the short-distance repulsions are replaced by a constraint of no double occupancy. For the period-4 phase, we show that there is an Ashkin-Teller transition point with exponent ν = 0.78 surrounded by a direct chiral transition with a dynamical exponent z = 1.11 and a Kibble-Zurek exponent μ = 0.41. For Rydberg atoms with a van der Waals potential, we suggest that the experimental value μ = 0.25 is due to a chiral transition with z ≃ 1.9 and ν ≃ 0.47 surrounding an Ashkin-Teller transition close to the 4-state Potts universality.
Using density matrix renormalization group simulations on open chains, we map out the wave vector in the incommensurate disordered phase of a realistic model of Rydberg chains with 1/r61/{r}^{6}1/r6 interactions, and we locate and characterize the points along the commensurate lines where the transition out of the period 3 and 4 phases is conformal. We confirm that it is three-state Potts for the period-3 phase, and we show that it is Ashkin-Teller with $\ensuremath{\nu}\ensuremath{\simeq}0.80$ for the period-4 phase. We further show that close to these points, the transition is still continuous, but with a completely different scaling of the wave vector, in agreement with a chiral transition. Finally, we propose to use the conformal points as benchmarks for Kibble-Zurek experiments, defining a roadmap towards a conclusive identification of the chiral universality class.
The experimental realization of the quantum Kibble-Zurek mechanism in arrays of trapped Rydberg atoms has brought the problem of commensurate-incommensurate transition back into the focus of active research. Relying on equilibrium simulations of finite intervals, direct chiral transitions at the boundary of the period-3 and period-4 phases have been predicted. Here, we study how these chiral transitions can be diagnosed experimentally with critical dynamics. We demonstrate that chiral transitions can be distinguished from the floating phases by comparing Kibble-Zurek dynamics on arrays with different numbers of atoms. Furthermore, by sweeping in the opposite direction and keeping track of the order parameter, we identify the location of conformal points. Finally, combining forward and backward sweeps, we extract all critical exponents characterizing the transition.
Using the corner-transfer matrix renormalization group approach, we revisit the three-state chiral Potts model on the square lattice, a model proposed in the eighties to describe commensurate-incommensurate transitions at surfaces, and with direct relevance to recent experiments on chains of Rydberg atoms. This model was suggested by Huse and Fisher to have a chiral transition in the vicinity of the Potts point, a possibility that turned out to be very difficult to definitely establish or refute numerically. Our results confirm that the transition changes character at a Lifshitz point that separates a line of Pokrosky-Talapov transitions far enough from the Potts point from a line of direct continuous order-disorder transition close to it. Thanks to the accuracy of the numerical results, we have been able to base the analysis entirely on effective exponents to deal with the crossovers that have hampered previous numerical investigations. The emerging picture is that of a new universality class with exponents that do not change between the Potts point and the Lifshitz point, and that appear to be consistent with those of a self-dual version of the model, namely correlation lengths exponents νx=2/3 in the direction of the asymmetry and νy=1 perpendicular to it, an incommensurability exponent β¯=2/3, a specific heat exponent that keeps the value α=1/3 of the three-state Potts model, and a dynamical exponent z=3/2. These results are in excellent agreement with experimental results obtained on reconstructed surfaces in the nineties, and shed light on recent Kibble-Zurek experiments on the period-3 phase of chains of Rydberg atoms.
Motivated by recent experiments on Rydberg atoms in a 1D trap, we study the phase transition out of the Z4\mathbb{Z}_4Z4​ crystalline phase in the context of a hard-boson model with two-site blockade. Using extensive DMRG simulations, we show that the transition along the commensurate line is in the Ashkin-Teller universality class with an exponent ν≃0.78\nu \simeq 0.78ν≃0.78, large enough for chiral pertubations to be relevant. This is further confirmed by our numerical results along the transition line which show that the transition immediately leaves the Ashkin-Teller universality class, and which are consistent with a small range of chiral transition followed by the opening of a floating phase.
Experiments on chains of Rydberg atoms appear as a playground to study quantum phase transitions in 1D. As a natural extension, we report a quantitative ground-state phase diagram of Rydberg atoms arranged in a two-leg ladder interacting via van der Waals potential. We address this problem numerically, using the density matrix renormalization group algorithm. Our results suggest that, quite remarkably, Z k crystalline phases, with the exception of the checkerboard phase, appear in pairs characterized by the same pattern of occupied rungs but distinguishable by a spontaneously broken Z ̃ 2 symmetry between the two legs of the ladder. Within each pair, the two phases are separated by a continuous transition in the Ising universality class, which eventually fuses with the Z k transition, whose nature depends on k . According to our results, the transition into the Z 2 ⊗ Z ̃ 2 phase changes its nature multiple times, including an Ashkin-Teller transition that is surprisingly stable over an extended interval, followed by the Z 4 -chiral transition, and finally in a two step-process mediated melting via the floating phase. The transition into the Z 3 phase with resonant states on the rungs belongs to the three-state Potts universality class at the commensurate point, to the Z 3 -chiral Huse-Fisher universality class away from it, and eventually it is through an intermediate floating phase. The Ising transition between Z 3 and Z 3 ⊗ Z ̃ 2 phases, entering the floating phase, opens the possibility to realize lattice supersymmetry in Rydberg quantum simulators.
We study the ground-state phase diagram of Rydberg atoms in a triangular-prism optical tweezer array using the density matrix renormalization group. By tuning the detuning-to-Rabi-frequency ratio and the Rydberg blockade radius, the system realizes several density-wave phases with spontaneous breaking of translational and leg-exchange symmetries. Unlike two-leg Rydberg ladders with Z2\mathbb{Z}_2Z2​ leg-exchange symmetry, the triangular prism has D3\mathbb{D}_3D3​ symmetry, leading to a richer set of ordered phases and transitions. For blockade radius moderately larger than the lattice spacing, a phase with alternating double and single Rydberg occupancy appears at large detuning. It breaks Z2\mathbb{Z}_2Z2​ translational and Z3\mathbb{Z}_3Z3​ rotational symmetry while preserving a rung reflection symmetry. Upon decreasing detuning, it melts through two Berezinskii-Kosterlitz-Thouless transitions with an intermediate critical phase described by a Z6\mathbb{Z}_6Z6​ clock model. At larger blockade radius, a phase with one Rydberg excitation per triangle and broken D3\mathbb{D}_3D3​ symmetry appears through a first-order transition. When double occupation of neighboring triangles is suppressed, rung-trimerized density waves develop as detuning increases from the disordered phase. Their melting follows the same structure as in Rydberg chains and two-leg ladders: the Z2\mathbb{Z}_2Z2​ case has Ising critical lines, while the Z3\mathbb{Z}_3Z3​ and Z4\mathbb{Z}_4Z4​ cases have chiral critical lines, with Potts and Ashkin-Teller points only on the corresponding commensurate lines. Inside the Z2\mathbb{Z}_2Z2​ rung-trimerized phase, an entanglement-entropy peak signals a crossover regime with enhanced period-2 density modulation before a first-order transition into a Z2×D3\mathbb{Z}_2\times\mathbb{D}_3Z2​×D3​ phase. Floating phases with incommensurate quasi-long-range order appear between trimerized states of different periods.