Critical fluctuations at a many-body exceptional point

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Critical fluctuations at a many-body exceptional point

Ryo Hanai and Peter B. Littlewood

Phys. Rev. Research 2, 033018 – Published 6 July 2020

Abstract

Critical phenomena arise ubiquitously in various contexts of physics, from condensed matter, high-energy physics, cosmology, to biological systems, and consist of slow and long-distance fluctuations near a phase transition or critical point. Usually, these phenomena are associated with the softening of a massive mode. Here, we show that a non-Hermitian-induced mechanism of critical phenomena that does not fall into this class can arise in the steady state of generic driven-dissipative many-body systems with coupled binary order parameters such as exciton-polariton condensates and driven-dissipative Bose-Einstein condensates in a double-well potential. The criticality of this “critical exceptional point” is attributed to the coalescence of the collective eigenmodes that convert all the thermal-and-dissipative-noise-activated fluctuations to the Goldstone mode, leading to anomalously giant phase fluctuations that diverge at spatial dimensions $d \leq 4$ . Our dynamic renormalization group analysis shows that this gives rise to a strong-coupling fixed point at dimensions as high as $d < 8$ associated with a universality class beyond the classification by Hohenberg and Halperin, indicating how anomalously strong the many-body corrections are at this point. We find that this anomalous enhancement of many-body correlation is due to the appearance of a sound mode at the critical exceptional point despite the system's dissipative character.

Received 15 September 2019
Revised 14 June 2020
Accepted 17 June 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.033018

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Critical phenomena Exciton polariton PT-symmetric quantum mechanics

Bose-Einstein condensates Nonequilibrium systems Polariton condensate

Renormalization group

Condensed Matter, Materials & Applied PhysicsStatistical Physics & ThermodynamicsAtomic, Molecular & Optical

Authors & Affiliations

Ryo Hanai ^1,2,* and Peter B. Littlewood^1,3

¹James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA
²Department of Physics, Osaka University, Toyonaka 560-0043, Japan
³Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA

^*rhanai@uchicago.edu

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Vol. 2, Iss. 3 — July - September 2020

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Images

Figure 1
Difference between the conventional critical point (CP) and the critical exceptional point (CEP). (a)–(c) Schematic explanation of the criticality of the CP in conventional equilibrium cases [the figure describes the case of $U (1)$ -symmetry-breaking transition). As one sweeps the parameter $α$ to the CP, the massive longitudinal mode $δ φ_{∥}$ itself softens at the CP caused by the flattening of the free-energy landscape $V$ (where $φ$ is the order parameter). Note how the two eigenmodes, the Goldstone mode $δ φ_{⊥}$ and the longitudinal mode $δ φ_{∥}$ , are always orthogonal. (d)–(f) Schematic explanation of the criticality of the CEP. In the non-Hermitian case, the two eigenmodes are not necessarily orthogonal. As a result, while the Goldstone's theorem ensures the Goldstone mode $δ φ_{⊥}$ to be an eigenmode, the other eigenmode $δ φ_{+}$ is not pointing to the longitudinal direction. At the CEP, where the two eigenmodes coalesce to the Goldstone mode, anomalously giant phase fluctuations occur which diverge at $d \leq 4$ , as well as anomalously enhanced many-body correlation effect. [We briefly note that the free energy drawn in (d)–(f) is schematic.]
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Figure 2
Non-Hermitian phase transition and the critical exceptional point (CEP). (a) Schematic phase diagram of driven-dissipative many-body systems with coupled binary order parameters, in terms of the generic input parameters $(α_{1}, α_{2})$ . The solid line is the phase boundary of the non-Hermitian-induced phase transition, which exhibits an end point at the critical exceptional point (CEP). (b) Demonstration of the appearance of the CEP for systems described by the coupled driven-dissipative Gross-Pitaevskii equations. At a small decay rate $κ / g < 1$ , as the density of the loss component $n_{l 0} = {| Φ_{l}^{0} |}^{2}$ increases, the steady state exhibits a phase transition signaled by a discontinuity in the condensate emission energy $E$ . The CEP found at $κ / g = 1$ , represented by the star, marks the end point of the phase boundary. The discontinuity is absent at $κ / g > 1$ . We set $ω_{l} / g = - 1.9, ω_{g} / g = - 2$ , and $U_{g} / g = 0.1$ .
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Figure 3
Diagrammatic representation of nonlinear corrections. (a) The self-energy $Σ_{s_{1} s_{2}}$ . (b) The noise kernel correction $Δ σ_{s_{1} s_{2}}$ . (c) The three-point vertex correction to the KPZ-type coupling $Δ λ_{s_{2} s_{3}}^{s_{1}}$ and (d) the massive out-of-phase coupling $Δ t_{s}$ . Here, the solid line is the Green's function ${\tilde{G}}_{s s^{'}}^{0}$ , the open circle represents the longitudinal noise strength $σ_{∥ ∥}$ , the solid circle is the KPZ-type nonlinear coupling $λ_{s^{'} s^{''}}^{s}$ , and the solid square represents the massive out-of-phase coupling $t_{s}$ .
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