Proposal for Observing Yang-Lee Criticality in Rydberg Atomic Arrays

Ruizhe Shen, Tianqi Chen, Mohammad Mujahid Aliyu, Fang Qin (覃昉), Yin Zhong, Huanqian Loh, and Ching Hua Lee

Phys. Rev. Lett. 131, 080403 – Published 25 August 2023

Abstract

Yang-Lee edge singularities (YLES) are the edges of the partition function zeros of an interacting spin model in the space of complex control parameters. They play an important role in understanding non-Hermitian phase transitions in many-body physics, as well as characterizing the corresponding nonunitary criticality. Even though such partition function zeroes have been measured in dynamical experiments where time acts as the imaginary control field, experimentally demonstrating such YLES criticality with a physical imaginary field has remained elusive due to the difficulty of physically realizing non-Hermitian many-body models. We provide a protocol for observing the YLES by detecting kinked dynamical magnetization responses due to broken $P T$ symmetry, thus enabling the physical probing of nonunitary phase transitions in nonequilibrium settings. In particular, scaling analyses based on our nonunitary time evolution circuit with matrix product states accurately recover the exponents uniquely associated with the corresponding nonunitary CFT. We provide an explicit proposal for observing YLES criticality in Floquet quenched Rydberg atomic arrays with laser-induced loss, which paves the way towards a universal platform for simulating non-Hermitian many-body dynamical phenomena.

Received 24 February 2023
Revised 27 June 2023
Accepted 25 July 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.080403

Physics Subject Headings (PhySH)

Critical phenomena Phase transitions

Non-Hermitian systems Quantum many-body systems

Tensor network methods

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Ruizhe Shen ^1,*, Tianqi Chen ^2,†, Mohammad Mujahid Aliyu ^3,‡, Fang Qin (覃昉)¹, Yin Zhong ^4,5, Huanqian Loh ^1,3,§, and Ching Hua Lee ^1,6,∥

¹Department of Physics, National University of Singapore, Singapore 117551, Singapore
²School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 639798, Singapore
³Centre for Quantum Technologies, National University of Singapore, 117543 Singapore, Singapore
⁴School of Physical Science and Technology and Key Laboratory for Magnetism and Magnetic Materials of the MoE, Lanzhou University, Lanzhou 730000, China
⁵Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou 730000, China
⁶Joint School of National University of Singapore and Tianjin University, International Campus of Tianjin University, Binhai New City, Fuzhou 350207, China

^*e0554228@u.nus.edu
^†tianqi.chen@ntu.edu.sg
^‡e0382015@u.nus.edu
^§phylohh@nus.edu.sg
^∥phylch@nus.edu.sg

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 131, Iss. 8 — 25 August 2023

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Theoretical characterization of the Yang-Lee edge singularities of ${\hat{H}}_{TFI}$ [Eq. (1)] and their associated ground state discontinuities: (a) Phase diagram of ${\hat{H}}_{TFI}$ in the space of real and complex fields $h_{x}$ and $γ$ . The YLES (dashed curve) is the phase boundary demarcating the paramagnetic (PM) $Im E_{g} = 0$ phase and the ferromagnetic (FM) $| Im E_{g} | > 0$ phase, where $E_{g}$ is the ground state eigenenergy with the minimal real part. (b) Plots of $Im E_{g}$ vs $γ$ at the four values of $h_{x}$ indicated in (a), revealing that $E_{g}$ is nonanalytic at YLES $γ_{Y L}$ . (c) Identification of Yang-Lee phase transitions by the ground state magnetization order parameter $M_{x} = | ⟨ \sum_{j} {\hat{σ}}_{j}^{x} / L ⟩ |$ . Magnetization kinks (dashed lines) occur at the same $γ_{YL}$ as in (b). (d) The critical point (at $γ_{YL} = 0.1837$ for $h_{x} = 1.5$ ) can be equivalently extracted from divergences in either the derivative of the imaginary ground state energy ${d | Im [E_{g} (γ)] | / d γ}$ (blue), or that of the ground state $x$ magnetization $(d M_{x} / d γ)$ (red). All results are obtained from exact diagonalization (ED) with open boundary conditions, with interaction strength $J = 1$ and system size $L = 8$ .
Reuse & Permissions
Figure 2
(a) Flow of the complex spectrum of ${\hat{H}}_{TFI}$ [Eq. (1)] as $γ$ is tuned across $γ_{YL} \approx 0.1837$ . The ground state eigenenergies (boxed) undergo spontaneous $P T$ symmetry breaking and rapidly acquire imaginary parts (green to yellow, red arrow) as $γ$ increases slightly above $γ_{YL}$ , much more so than other eigenenergies. (b) and (c): Evolution of the overlap $| ⟨ ψ (T) | ψ_{i} ⟩ |$ between the dynamically evolved state $| ψ (T) ⟩ = e^{- i T {\hat{H}}_{TFI}} | ψ (0) ⟩ / ∥ e^{- i T {\hat{H}}_{TFI}} | ψ (0) ⟩ ∥$ and all eigenstates $| ψ_{i} ⟩$ (blue) of ${\hat{H}}_{TFI}$ , with initial state being $| ψ (0) ⟩ = | ↓ ↓ \dots \dots ⟩$ . For $γ$ below the YLES $γ_{YL}$ (b), the ground state overlap (red dashed) decreases rapidly due to mixing. But for $γ > γ_{YL}$ , the ground state component becomes dominant beyond time $T J \sim O (10)$ due to the large $Im E_{g}$ of the ground state, leading to kinked magnetization responses detectable in our proposed Rydberg atom system of Fig. 4 below. All results are obtained from ED with $J = 1$ , $h_{x} = 1.5$ , and $L = 8$ .
Reuse & Permissions
Figure 3
Anomalous critical scaling of dynamically determined Yang-Lee edge singularities of ${\hat{H}}_{TFI}$ , with $J = 1$ and $h_{x} = 1.5$ . (a) Size-dependent YLES $γ_{YL}^{L}$ as kinks in the derivative $[d M_{x} (T) / d γ]$ . The dynamical magnetization $M_{x} (T) = | ⟨ ψ (T) | \sum_{j} {\hat{σ}}_{j}^{x} | ψ (T) ⟩ | / L$ is measured at time $T = 20 / J$ for different system sizes $L = 8$ to 16 using our tMPS algorithm from (d). (b) Extrapolation of $γ_{YL}^{L}$ with $1 / L$ gives the thermodynamic limit YLES value $γ_{YL}^{\infty} \approx 0.139$ through polynomial fitting. (c) Plotting our data for $\log (γ_{YL}^{L} - γ_{YL}^{\infty})$ against $- \log L$ yields the critical exponent $α \approx 2.423$ as the gradient [Eq. (2)], which agrees closely with the CFT result $α = 12 / 5$ [75, 76]. (d) The nonunitary circuit for implementing the time evolution of our model [6], where each step is decomposed into odd-bond $U_{odd}$ (light green) and even-bond $U_{even}$ (dark green) parts, with normalization (red solid line) performed at the end.
Reuse & Permissions
Figure 4
Proposed Rydberg atomic array for realizing the Floquet drives $U_{1}^{Floq}$ and $U_{2}^{Floq}$ in Eq. (3). Optical tweezers (red, left) trap Cesium atoms (green), with their two hyperfine ground states defining pseudospin- $1 / 2$ states $| ↓ ⟩ = | 6 S_{1 / 2}, F = 3, m_{F} = + 3 ⟩$ and $| ↑ ⟩ = | 6 S_{1 / 2}, F = 4, m_{F} = + 4 ⟩$ (right). In $U_{1}^{Floq}$ (Step 1), $Ω$ driving (blue dashed arrow) couples $| ↑ ⟩$ and the Rydberg state $| 60 P_{3 / 2}, m_{J} = + 3 / 2 ⟩$ , resulting in a Rydberg-dressed nearest-neighbor $| ↑ ↑ ⟩$ interaction ${\hat{H}}_{int}$ (purple arrow) that can be converted to a ferromagnetic interaction via spin echo (red loop). In $U_{2}^{Floq}$ (Step 2), a microwave field (brown loop) generates the transverse field $F {\hat{σ}}^{x}$ , while a laser couples $| ↑ ⟩ \to | 6 P_{3 / 2}, F = 5, m_{F} = + 5 ⟩$ , resulting in the imaginary field $i g {\hat{σ}}^{z}$ [6]. The quenching durations $τ_{J}$ , $τ_{x}$ and $τ_{γ}$ are chosen such that $(τ_{x} F, τ_{γ} g, τ_{J} J_{0}) = (T / N) (h_{x}, γ, J)$ .
Reuse & Permissions

Physical Review Letters