Scale Invariance in the Lowest Landau Level

IF 1.3 4区 物理与天体物理 Q3 ASTRONOMY & ASTROPHYSICS Comptes Rendus Physique Pub Date : 2022-10-18 DOI:10.5802/crphys.137
J. Hofmann, W. Zwerger
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引用次数: 1

Abstract

We show that the discrete set of pair amplitudes $A_m$ introduced by Haldane are an angular-momentum resolved generalization of the Tan two-body contact, which parametrizes universal short-range correlations in atomic quantum gases. The pair amplitudes provide a complete description of translation-invariant and rotation-invariant states in the lowest Landau level (LLL), both compressible and incompressible. To leading nontrivial order beyond the non-interacting high-temperature limit, they are determined analytically in terms of the Haldane pseudopotential parameters $V_m$, which provides a qualitative description of the crossover towards incompressible ground states for different filling factors. Moreover, we show that for contact interactions $\sim g_2 \delta^{(2)}({\bf x})$, which are scale invariant at the classical level, the non-commutativity of the guiding center coordinates gives rise to a quantum anomaly in the commutator $i [\hat{H}_{\rm LLL}, \hat{D}_R] = (2 + \ell \partial_\ell) \hat{H}_{\rm LLL}$ with the dilatation operator $\hat{D}_R$ in the LLL, which replaces the trace anomaly in the absence of a magnetic field. The interaction-induced breaking of scale invariance gives rise to a finite frequency shift of the breathing mode in a harmonic trap, which describes transitions between different Landau levels, the strength of which is estimated in terms of the relevant dimensionless coupling constant $\tilde{g}_2$.
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最低朗道水平的尺度不变性
我们证明了Haldane引入的离散对振幅集$A_m$是Tan两体接触的角动量分辨推广,它参数化了原子量子气体中的普遍短程相关。对振幅提供了最低朗道能级(LLL)的平移不变和旋转不变状态的完整描述,包括可压缩和不可压缩。为了使非平凡序超越非相互作用的高温极限,它们被解析地确定为Haldane伪势参数$V_m$,它提供了对不同填充因子向不可压缩基态交叉的定性描述。此外,我们证明了接触相互作用$\sim g_2 \delta^{(2)}({\bf x})$在经典水平上是尺度不变的,引导中心坐标的非对易性导致了对易子$i [\hat{H}_{\rm LLL}, \hat{D}_R] = (2 + \ell \partial_\ell) \hat{H}_{\rm LLL}$与LLL中的扩张算子$\hat{D}_R$中的量子异常,取代了在没有磁场的情况下的迹异常。相互作用引起的尺度不变性的破坏引起了谐波陷阱中呼吸模式的有限频移,它描述了不同朗道能级之间的转换,其强度根据相关的无量纲耦合常数$\tilde{g}_2$进行估计。
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来源期刊
Comptes Rendus Physique
Comptes Rendus Physique 物理-天文与天体物理
CiteScore
2.80
自引率
0.00%
发文量
13
审稿时长
17.2 weeks
期刊介绍: The Comptes Rendus - Physique are an open acess and peer-reviewed electronic scientific journal publishing original research article. It is one of seven journals published by the Académie des sciences. Its objective is to enable researchers to quickly share their work with the international scientific community. The Comptes Rendus - Physique also publish journal articles, thematic issues and articles on the history of the Académie des sciences and its current scientific activity. From 2020 onwards, the journal''s policy is based on a diamond open access model: no fees are charged to authors to publish or to readers to access articles. Thus, articles are accessible immediately, free of charge and permanently after publication. The Comptes Rendus - Physique (8 issues per year) cover all fields of physics and astrophysics and propose dossiers. Thanks to this formula, readers of physics and astrophysics will find, in each issue, the presentation of a subject in particularly rapid development. The authors are chosen from among the most active researchers in the field and each file is coordinated by a guest editor, ensuring that the most recent and significant results are taken into account. In order to preserve the historical purpose of the Comptes Rendus, these issues also leave room for the usual notes and clarifications. The articles are written mainly in English.
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