# Projects: Prescaling in and low-energy effective theory of non-thermal fixed points in a multicomponent Bose gas

Here we present supplementary online video material concerning the following publications:

## Prescaling in a far-from-equilibrium Bose gas

** C.-M. Schmied1,2, **A. N. Mikheev1, and Thomas Gasenzer1

1Kirchho*ff*-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany

2Dodd-Walls Centre for Photonic and Quantum Technologies, Department of Physics, University of Otago, Dunedin 9016, New Zealand

Non-equilibrium conditions give rise to a class of universally evolving low-energy configurations of fluctuating dilute Bose gases at a non-thermal fixed point. While the fixed point and thus full scaling in space and time is generically only reached at very long evolution times, we here propose that systems can show prescaling much earlier, on experimentally accessible time scales. During the prescaling evolution, some well-measurable short-distance properties of the spatial correlations already scale with the universal exponents of the fixed point while others still show scaling violations. Prescaling is characterized by the evolution obeying already, to a good approximation, the conservation laws which are associated with the asymptotically reached non-thermal fixed point, defining its belonging to a specific universality class. In our simulations, we consider *N* = 3 spa- tially uniform three-dimensional Bose gases of particles labeled, e.g., by different hyperfine magnetic quantum numbers, with identical inter- and intra-species interactions. In this system, the approach of a non-thermal fixed point is marked by low-energy phase excitations self-similarly redistributing towards smaller wave numbers. During prescaling, the full *U(N)* symmetry of the model is broken while the conserved transport, reflecting the remaining *U(1)* symmetries, leads to the buildup of a rescaling quasicondensate distribution.

## Low-energy effective theory of non-thermal fixed points in a multicomponent Bose gas

**A. N. Mikheev1, C.-M. Schmied1,2, and Thomas Gasenzer1 **

1Kirchho*ff*-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany

2Dodd-Walls Centre for Photonic and Quantum Technologies, Department of Physics, University of Otago, Dunedin 9016, New Zealand

Non-thermal fixed points in the evolution of a quantum many-body system quenched far out of equilibrium manifest themselves in a scaling evolution of correlations in space and time. We develop a low-energy effective theory of non-thermal fixed points in a bosonic quantum many-body system by integrating out long-wave-length density fluctuations. The system consists of *N* distinguishable spatially uniform Bose gases with *U(N)*-symmetric interactions. The effective theory describes interacting Goldstone modes of the total and relative- phase excitations. It is similar in character to the non-linear Luttinger-liquid description of low-energy phonons in a single dilute Bose gas, with the markable difference of a universal non-local coupling function depending, in the large-*N* limit, only on momentum, single-particle mass, and density of the gas. Our theory provides a perturbative description of the non-thermal fixed point, technically easy to apply to experimentally relevant cases with a small number of fields *N*. Numerical results for *N* = 3 allow us to characterize the analytical form of the scaling function and confirm the analytically predicted scaling exponents. The fixed point which is dominated by the relative phases is found to be Gaussian, while a non-Gaussian fixed point is anticipated to require scaling evolution with a distinctly lower power of time.

## Supplementary Material

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All videos by Markus Karl

## Videos: Prescaling in the evolution of a quenched U(3)-symmetric Bose gas towards an NTFP

The videos show the approach of the three-component 3D Bose gas towards the non-thermal fixed point. The right panels show a 2D cut through the center of the cubic volume. The upper and middle videos show the same evolution, with the second one entailing a higher resolution in time. In both videos, the panels show (from left to right, in columns upper, then lower) the single-particle radial momentum distribution and the corresponding first-order coherence function *g*_{1}^{(1)}(*r*) as a function of distance *r*, the radial momentum spectrum *C*_{12}(*k*) of the second-order correlation function *g*_{12}^{(2)}(*r*) of the relative phases between components 1 and 2, and *g*_{12}^{(2)}(*r*) itself, as well as 2D cuts in the plane *x* = 0 through the 3D distributions of the density *n*_{1}(*y,z*), the relative phase |*φ*_{1} - *φ*_{2}|, the phase *φ*_{1}, and the cosine of the relative phase cos|*φ*_{1} - *φ*_{2}|. In the third video, the rightmost column is replaced by the total density and phase distributions.