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Emergent Hydrodynamics in a 1D Bose-Fermi Mixture

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Emergent Hydrodynamics in a 1D Bose-Fermi Mixture Emergent Hydrodynamics in a 1D Bose-Fermi Mixture PACS numbers: INTRODUCTION OF MY LECTURE scale, all the densities and currents of local observables are the functions of ξ = x=t. Here the evolution can be Nonequilibrium dynamics of one dimensional (1D) intuitively understood by the transport of the quasipar- integrable systems is very insightful in understanding ticles in integrable systems. Within the light core the transport of many-body systems. Significantly differ- quasiparticles can arrive and leave that totally depends ent behaviours are observed from that of the higher di- on their local velocities. Moreover, the GH method has mensional systems since the existence of many conserved been adapted to study the evolution of 1D systems con- quantities in 1D. For example, 1D integrable systems do fined by the external field [8,9], also see experiment [10]. not thermalize to a usual state of thermodynamic equi- Beyond the Euler type of hydrodynamics, diffusion has librium [1], where after a long time evolution there is a also been studied by the GH [11, 12]. maximal entropy state constrained by all the conserved On the other hand, at low temperatures, universal phe- quantities[2]. When we consider such a process of time nomena emerge in quantum many-body systems. In high evolution, it is extremely hard to solve the Schrodinger dimensions, the low energy physics can be described by equation of the system with large number particles. Landau's Fermi liquids theory. However, for 1D sys- tems, the elementary excitations form collective motions of bosons. Thus the low energy physics of 1D systems is elegantly descried by the theory of Luttinger liquids [13{ 15]. The essence of the Luttinger liquid is the Haldane's macroscopic quantum hydrodynamics in which the ele- mentary excitations over the ground state of the 1D sys- tem can be treated as a free Gaussian field theory, i.e. a gas of noninteracting particles with a linear dispersion. Thanking the linear dispersions, the low energy physics of the 1D systems with spin internal degrees of freedom is universally described by the Luttinger liquid theory of spin charge separated degrees of freedom. The Luttinger liquid theory is particularly important in the unified de- scription of the properties in thermodynamic equilibrium states of 1D systems. It turns out that this theory can be applied to study the transport problems in 1D [16]. Based on this theory, within the light core, the transport FIG. 1: The light-core diagram of the of the energy current of the 1D systems at low temperatures is essentially cap- of the 1D BF mixtures at low temperatures. The initial state is set up as two semi-infinitely long 1D tubes joined together tured by the behaviors of the free collective excitations, with different temperatures. see Fig.1. However, in the region ξ ± v ∼ O(TL(R)), the nonlinear excitations should be also considered, where v Recently, a generalized Hydrodynamics (GH) approach is the velocity of collective excitations, and TL(R) are the [3,4] was introduced to study the evolution of macro- temperatures of the initial states of the left (right) halve. scopical systems in 1D. For integrable systems, there are However, the study of quantum hydrodynamics of the 1D infinite many conserved quantities in the thermodynamic systems with spin internal degrees of freedom is rather limit. With the help of local equilibrium approximation, preliminary [7]. whole states of the system are determined by the density In contrast to the spin charge separated Luttinger liq- distributions of the conserved quantities. These density uids in interacting Fermi gas [15, 17], the 1D degenerate distributions of all conserved quantities and the their con- Bose-Fermi (BF) mixtures display two decoupled Lut- tinuity equations are capable of determining the evolu- tinger liquids in bosonic and fermionic degrees of free- tions of the systems. Due to the existence of conserved dom at low temperatures [18{22]. In the former, the quantities, the steady state may admit nonzero station- effective antiferromagnetic spin-spin interaction leads to ary charge currents, indicating the presence of ballistic the Luttinger liquid of spinons, whereas in the latter the transport of integrable models. This elegant GH method Luttinger liquids separately emerge in the fermionic and has been applied to the study of the time evolution of two bosonic degrees of freedom. semi-infinite chains joined together with different tem- In this lecture, I will give an elementary introduction peratures [3{7]. The result shows that in a large time of the GH method and its applications in 1D integrable 2 models. In particular, I will present our recent study on the low-temperature transport of one-dimensional fermi the transport properties of the 1D Bose-Fermi mixtures gases. Physical Review B, 99(1):014305, 2019. with delta-function interactions by means of the GH ap- [8] Benjamin Doyon and Takato Yoshimura. A note on gen- proach. Building on the Bethe anssatz solution, I rig- eralized hydrodynamics: inhomogeneous fields and other concepts. SciPost Physics, 2(2):014, 2017. orously prove the existence of conserved charges in both [9] Jean-S´ebastienCaux, Benjamin Doyon, J´er^omeDubail, the bosonic and fermionic degrees of freedom, providing Robert Konik, and Takato Yoshimura. Hydrodynamics the Euler type of hydrodynamics. The velocities of el- of the interacting bose gas in the quantum newton cradle ementary excitations in the two degrees of freedom are setup. arXiv preprint arXiv:1711.00873, 2017. significantly different due to different quantum statisti- [10] Max Schemmer, Isabelle Bouchoule, Benjamin Doyon, cal effects. The charge-charge separation emerges in the and J´er^omeDubail. Generalized hydrodynamics on an steady state of the systems within the light core. The dis- atom chip. Physical Review Letters, 122(9):090601, 2019. tributions of the densities and currents of local conserved [11] Jacopo De Nardis, Denis Bernard, and Benjamin Doyon. Hydrodynamic diffusion in integrable systems. Physical quantities in bosonic and fermionic degrees of freedom review letters, 121(16):160603, 2018. show the ballistic transport of the quasiparticles with two [12] Jacopo De Nardis, Denis Bernard, and Benjamin Doyon. different velocities of the excitations in the two degrees Diffusion in generalized hydrodynamics and quasiparticle of freedom. The obtained results reveal subtle differences scattering. arXiv preprint arXiv:1812.00767, 2018. in densities and currents of bosons and fermions, showing [13] F. D. M. Haldane. Effective harmonic-fluid approach to interaction and quantum statistic effects in transport. low-energy properties of one-dimensional quantum fluids. To students: please read the key reference [3] in ad- Phys. Rev. Lett., 47:1840{1843, Dec 1981. [14] F. D. M. Haldane. 'luttinger liquid theory' of one- vance. dimensionalquantum fluids: I. properties of the luttinger model and their extension to the general 1d interacting spinless fermi gas. J. Phys. C: Solid State Physics, 14. [15] Thierry Giamarchi. Quantum physics in one dimension. [16] Bruno Bertini, Lorenzo Piroli, and Pasquale Calabrese. [1] Toshiya Kinoshita, Trevor Wenger, and David S Weiss. A Universal broadening of the light cone in low-temperature quantum newton's cradle. Nature, 440(7086):900, 2006. transport. Physical review letters, 120(17):176801, 2018. [2] Marcos Rigol, Vanja Dunjko, Vladimir Yurovsky, and [17] J. Y. Lee, X. W. Guan, K. Sakai, and M. T. Batche- Maxim Olshanii. Relaxation in a completely integrable lor. Thermodynamics, spin-charge separation, and cor- many-body quantum system: an ab initio study of the relation functions of spin-1=2 fermions with repulsive in- dynamics of the highly excited states of 1d lattice hard- teraction. Phys. Rev. B, 85:085414, Feb 2012. core bosons. Physical review letters, 98(5):050405, 2007. [18] C. K. Lai and C. N. Yang. Ground-state energy of a [3] Olalla A Castro-Alvaredo, Benjamin Doyon, and Takato mixture of fermions and bosons in one dimension with a Yoshimura. Emergent hydrodynamics in integrable quan- repulsive δ-function interaction. Phys. Rev. A, 3:393{399, tum systems out of equilibrium. Physical Review X, Jan 1971. 6(4):041065, 2016. [19] M. T. Batchelor, M. Bortz, X. W. Guan, and N. Oelk- [4] Bruno Bertini, Mario Collura, Jacopo De Nardis, and ers. Exact results for the one-dimensional mixed boson- Maurizio Fagotti. Transport in out-of-equilibrium x x z fermion interacting gas. Phys. Rev. A, 72:061603, Dec chains: Exact profiles of charges and currents. Physical 2005. review letters, 117(20):207201, 2016. [20] Holger Frahm and Guillaume Palacios. Correlation func- [5] Denis Bernard and Benjamin Doyon. Conformal field tions of one-dimensional bose-fermi mixtures. Phys. Rev. theory out of equilibrium: a review. Journal of Statisti- A, 72:061604, Dec 2005. cal Mechanics: Theory and Experiment, 2016(6):064005, [21] Adilet Imambekov and Eugene Demler. Exactly solvable 2016. case of a one-dimensional bose{fermi mixture. Physical [6] Benjamin Doyon, J´er^omeDubail, Robert Konik, and Review A, 73(2):021602, 2006. Takato Yoshimura. Large-scale description of interacting [22] Adilet Imambekov and Eugene Demler. Applications of one-dimensional bose gases: generalized hydrodynamics exact solution for strongly interacting one-dimensional supersedes conventional hydrodynamics. Physical review bose{fermi mixture: Low-temperature correlation func- letters, 119(19):195301, 2017. tions, density profiles, and collective modes. Annals of [7] M´arton Mesty´an, Bruno
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