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Journal of Physics A Mathematical and Theoretical

Hydrodynamics of low-dimensional quantum systems

Alexander G. Abanov, Benjamin Doyon, Jérôme Dubail, Alex Kamenev, Herbert E. Spohn

Published Aug 22, 2023• 3 citations
Publisher
DOI
10.1088/1751-8121/acecc8

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Describing emergent macroscopic phenomena in low spatial dimensions is known to be notoriously challenging, primarily due to strong interactions that render perturbative approaches inapplicable. On the other hand, low-dimensional systems host a wealth of unorthodox phenomena. A prominent example is the emergence of superdiffusive transport in one-dimensional interacting systems, traditionally studied in the framework of nonlinear fluctuating hydrodynamics. After identifying and discussing internal inconsistencies in the previous formulations, in this work we develop a general and systematic approach for constructing effective field theories of one-dimensional hydrodynamic systems in the form of coupled stochastic Langevin-type equations compatible with the physical requirements of local equilibrium states such as the thermodynamic Maxwell relation and fluctuation-dissipation symmetry. We implemented a general numerical integration scheme and exemplified our construction on a simple model of two interacting hydrodynamic modes with a non-Gaussian stationary equilibrium measure.
Low-dimensional systems are an important field of current theoretical and experimental research. Recent technological developments provide many possible realizations of effectively one-dimensional systems. These devices promise to give us access to a new range of phenomena. It is therefore very interesting to develop theoretical methods specific for such systems to model their behavior and calculate the correlators of the resulting theory. Incidentally, one such method exists and is known as Bosonization. It can be applied to one-dimensional systems and effectively describes low energy excitations in a universal way. We use the example of a correlator known as the Emptiness Formation Probability to show that Bosonization fails to describe some long range correlators corresponding to large disturbances (the EFP measures the probability for the ground state of the system to develop a region without particles). We trace this failure to the fact that Bosonization is constructed as a linear approximation of the full theory and we set up to develop a collective description with the required non-linearity. The resulting scheme is essentially a Hydrodynamic paradigm for quantum systems. We show how to construct such a hydrodynamic description for a variety of exactly integrable models and illustrate how it can be used to make new predictions. For the special case of the spin-1/2 XY model we take advantage of the structure of the model to express the EFP as a determinant of a very special type of matrix, known as Toeplitz Matrix. We use the theory of Toeplitz determinants to calculate the asymptotic behavior of the EFP in the XY model and discuss its relation with the criticality of the theory.
The dynamics of strongly interacting many-body quantum systems are notoriously complex and difficult to simulate. A recently proposed theory called generalized hydrodynamics (GHD) promises to efficiently accomplish such simulations for nearly integrable systems. We test GHD with bundles of ultracold one-dimensional (1D) Bose gases by performing large trap quenches in both the strong and intermediate coupling regimes. We find that theory and experiment agree well over dozens of trap oscillations, for average dimensionless coupling strengths that range from 0.3 to 9.3. Our results show that GHD can accurately describe the quantum dynamics of a 1D nearly integrable experimental system even when particle numbers are low and density changes are large and fast.