Tricritical Kibble-Zurek scaling in Rydberg atom ladders

Abstract

The Kibble-Zurek (KZ) mechanism has been extensively studied in various second-order phase transitions, yet the case of tricriticality-the point where second-order phase transition lines terminate-remains experimentally elusive. Here, we theoretically propose probing KZ scaling at tricritical points using Rydberg atom arrays arranged as two- and three-leg ladders, which realize the tricritical Ising and tricritical Potts universality classes. By slowly ramping the Rabi frequency and detuning, we extract two relevant tricritical exponents, $ν$ and $ν'$, both via conventional paths from the disordered to the ordered phase and via "tangential" paths confined entirely within the disordered phase. At faster speeds, ramping dynamics go beyond the standard KZ paradigm: data collapse analysis using the parent critical exponents (rather than the tricritical ones) reveals renormalization group flows toward the adjacent second-order critical line, and we identify it as a dynamical analog of Zamolodchikov's $c$-theorem. Our protocol is readily implementable on existing Rydberg quantum simulators. This provides a direct route to measuring distinct tricritical exponents which can reveal an emergent spacetime supersymmetry constraint $1/ν- 1/ν' = 1$. Moreover, this work deepens our theoretical understanding and opens new avenues for exploring beyond-KZ quantum dynamics with rich renormalization group structure.

Figures (9)

• Figure 1: Rydberg ladders and Kibble-Zurek protocols.a, b Two- and three-leg ladders that exhibit tricritical Ising and Potts points, respectively. Atoms of different colors correspond to different atomic species. The geometry is fixed (i.e., $r_{1,2}$), while the Rabi frequency $\Omega$ and detuning $\Delta$ can be tuned. c The schematic phase diagram and ramping protocols. The solid (dashed) blue line denotes the second (first) order Ising/Potts transition, and a tricritical point lies in between. In Protocol I, we keep $\Omega$ fixed and ramp $\Delta$ from the disordered phase to the symmetry breaking phase, both across and near the tricritical point. In Protocol II, we consider a tangential ramping direction. The orange dot arrow refers to ramping of higher speeds, where a dynamical analog to Zamolodchikov's $c$-theorem is proposed.
• Figure 2: KZ scaling of Binder ratio in the TCI case. TEBD simulations using Hamiltonian \ref{['eq:Hamiltonian']} with $N=2L$ atoms and $1/r^6$ interactions for (a--d). a Binder ratio $U$ as function of $g=\Delta-\Delta_\mathrm{TCI} \propto t$ for various ramping speed $s$ with $L=16$. The first peak is marked with a cross. The ground state Binder ratio for open boundary conditions (black dashed line) is included for reference. b KZ ramping across the TCI point, with data collapse achieved for $s^{\mu} L$. c KZ ramping near the TCI point, with data collapse achieved for $hL^{1/\nu'}$. d For tangential KZ, we achieve data collapse with $s^{\mu'} L$. In the insets, the data collapse figure of merit $\mathcal{D}$ is plotted against the corresponding critical exponents. e--h show analogous results using an effective spin-1 model with size $L$ for TEBD simulations. The gray dashed curves represent quadratic fits in (b,c,f,g) and cubic fits in (d,h).
• Figure 3: RG perspective of KZ in the TCI case.a RG flow (black lines) near the TCI. The blue line indicates the numerical phase boundary determined by Binder ratio and central charge (see Methods). b, c Ramping with a broad range of ramping speed $s$. Both figures are plotted with the same data but with different scalings of $s$. For slower ramping (upper right region in (b)), the TCI critical exponent allow the best data collapse. For faster ramping (lower left region in (c)), the Ising critical exponent allow the best data collapse.
• Figure 4: Central charge $c$ vs Rabi frequency $\Omega$ for two-leg (a) and three-leg (b) cases, using cesium 70S and sodium 69S. Insets show data collapse with horizontal axis $hL^{1/\nu'}$, validating the scaling proposal of Eq. \ref{['eq:scaling']}.
• Figure 5: The low-energy spectrum of TCI (a) and TCP (b). The states are identified by their conformal weights $(h,\bar{h})$, with conformal dimension (microscopically, energy) $\Delta_\phi=h+\bar{h}$ and conformal spin (microscopically, momentum) $s=h-\bar{h}$.