Lee–Yang zeros in the Rydberg atoms

doi:10.1007/s11467-022-1226-6

Front. Phys.

2023, Vol. 18

Issue (2) : 22301 https://doi.org/10.1007/s11467-022-1226-6

RESEARCH ARTICLE

Lee–Yang zeros in the Rydberg atoms

Chengshu Li(

), Fan Yang

Institute for Advanced Study, Tsinghua University, Beijing 100084, China

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Abstract

Lee–Yang (LY) zeros play a fundamental role in the formulation of statistical physics in terms of (grand) partition functions, and assume theoretical significance for the phenomenon of phase transitions. In this paper, motivated by recent progress in cold Rydberg atom experiments, we explore the LY zeros in classical Rydberg blockade models. We find that the distribution of zeros of partition functions for these models in one dimension (1d) can be obtained analytically. We prove that all the LY zeros are real and negative for such models with arbitrary blockade radii. Therefore, no phase transitions happen in 1d classical Rydberg chains. We investigate how the zeros redistribute as one interpolates between different blockade radii. We also discuss possible experimental measurements of these zeros.

Keywords Lee–Yang zeros Rydberg atom statistical mechanics

Corresponding Author(s): Chengshu Li

Issue Date: 12 December 2022

Cite this article:

Chengshu Li,Fan Yang. Lee–Yang zeros in the Rydberg atoms[J]. Front. Phys. , 2023, 18(2): 22301.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-022-1226-6
https://academic.hep.com.cn/fop/EN/Y2023/V18/I2/22301

Fig.1 An illustration of the proof of Lemma 1 for the case

r = 2

. Blue dots denote the zeros and orange triangles (green squares) indicate that the polynomial is positive (negative) at that point. As a result of the recursion relation (5), the zeros in each group of three polynomials (e.g.,

y 1, 1, y 1, 2, y 1, 3

) determine the sign of the polynomials in the next group, which in turn guarantees the existence of zeros in between. A few examples are in order. We label the values of the polynomials with Greek letters as shown in the figure. First,

δ < 0

and

? = 0

give

γ < 0

, which together with

α > 0

fixes

y 2, 4

. Then,

ι = 0

and

κ > 0

give

θ < 0

, which together with

λ > 0

fixes

y 1, 4

. As a final example,

η < 0

and

ι = 0

give

ζ < 0

, which together with

β > 0

fixes

y 2, 6

. Note that we only mark the zeros and the points which are directly used to fix the zeros, e.g.,

α

β

γ

?

ζ

θ

ι

, and

λ

Fig.2 The zeros of the partition function (33) for

n = 36

. The largest

n / 3 = 12

of them converge to that of

r = 2

when

V

goes to

∞

, while the others diverge to

− ∞

exponentially, which are omitted in the lower panel.

Fig.3 The zeros of the partition function (35) for

n = 36

. The largest

n / 4 = 9

of them converge to that of

r = 3

when

V

goes to

∞

, while the others diverge to

− ∞

exponentially, which are omitted in the lower panel.

Fig.4 The experimental setup for (a) open boundary condition and (b) periodic boundary condition. Blue (red) balls denote atoms in the ground (Rydberg) state, and the green ball denotes the probe atom. Typical time dependence of

| L (t) |

as defined in Eq. (38) is shown in (c). Here we choose

n = 24

r = 1

. The difference between the zeros (

β Δ = − 1.37, 2.85

) and non-zeros (

β Δ = − 3, 2

) is clearly visible.

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