Vortex dynamics in strongly interacting fermionic superfluids from time-dependent density functional theory

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Vortex dynamics in strongly interacting fermionic superfluids from time-dependent density functional theory Gabriel Wlazłowski Warsaw University of Technology University of Washington Supported

Kelley (Cornell University) Piotr Magierski (WUT, Warsaw) Kenneth J.

1 Vortex dynamics in strongly interacting fermionic superfluids from time-dependent density functional theory Gabriel Wlazłowski Warsaw University of Technology University of Washington Supported by: Polish National Science Center (NCN) grant under decision No. DEC-2014/13/D/ST3/ Polish National Science Center (NCN) grant under decision No. DEC-2013/08/A/ST3/ U.S. DOE Office of Science Grant DE-FG02-97ER4101 In collaboration with: Aurel Bulgac (UW), Michael McNeil Forbes (WSU) Michelle M. Kelley (Cornell University) Piotr Magierski (WUT, Warsaw) Kenneth J. Roche (PNNL,UW) Kazuyuki Sekizawa (WUT, Warsaw) Wei Quan (University of Chicago) ECT* Workshops: 20 Mar 2017 to 24 Mar 2017 Superfluidity and Pairing Phenomena: from Cold Atomic Gases to Neutron Stars

2 See Seetalks talksby bya.a.bulgac Bulgacand andp.p.magierski... Magierski...

3 DFT: workhorse for electronic structure simulations The Hohenberg-Kohn theorem assures that the theory can reproduce exactly the ground state energy if the exact Energy Density Functional (EDF) is provided 1990 Often called as ab initio method Extension to Time-Dependent DFT is straightforward Can be extended to superfluid systems... (numerical cost increases dramatically) Very successful DFT industry (commercial codes for quantum chemistry and solid-state physics) 2012

5 Density Functional Theory Idea It can be shown that instead of wave function one may use a density distribution: contains vastly more information than the one needed reduced object - sufficient to extract one body observables In general: The energy is a functional of the density `

6 Universal part External field Kohn-Sham method: interacting system Formally rigorous way of approaching any interacting problem by mapping it exactly to a much easier-to-solve noninteracting system. non-interacting system Both systems described by the same density easy, if Energy Density Functional (EDF) is known... Fig. from: Prog.Part.Nucl.Phys.64: ,2010 More general: Interacting System System of non-interacting quasiparticles Note: There are easy and difficult observables in DFT. In general: easy observables are one-body observables. They are easily extracted and reliable.

7 Normal system Ultracold atoms are superfluid! pairing (anomalous) density BCS-like Here we already assumed that the paring field is local Δ(r,r') Δ(r) Otherwise: integro-differential equations... Note: kinetic density and Note: kinetic density and diagonal part of pairing diagonal part of pairing density is divergent density is divergent Regularization required! Regularization required! We use prescription given in: Bulgac, Yu, Phys. Rev. Lett. 88 (2002) Bulgac, Phys. Rev. C65 (2002)

8 Superfluid local density approximation (SLDA) Three dimensionless constants α, β, and γ determining the functional are extracted from QMC results for homogeneous systems (for example by fixing the total energy, the pairing gap and the effective mass.) NOTE: there is no fit to experimental results SLDA has been verified and validated against a large number of quantum Monte Carlo results for inhomogeneous systems and experimental data as well Forbes, Gandolfi, Gezerlis, PRL 106, (2011)

9 EDF for UFG: [polarized case] ASLDA Polarization dependent parameters QMC for polaron problem QMC for system forced to be in normal state Experimental data Figure from: A. Bulgac, M.M. Forbes, P. Magierski, Lecture Notes in Physics, Vol. 836, Chap. 9, p (2012) Unpolarized case: QMC available Small polarizations QMC doable (but hard) On Dy going pol namic work ar iz s o s : ed f gas es

10 Extension to time-dependent case and Runge & Gross theorem Up to arbitrary function α(t) and consequently density functional exists... NOTE: in general this functional: - is initial state dependent - at time t depends on densities in previous times t [t0,t](memory effects) Very little is known about the memory terms, but in principle it can be long ranged (see eg. Dobson, Brunner, Gross, Phys. Rev. Lett. 79 (1997) 1905) Time evolution of interacting system Memory effects are usually neglected = adiabatic approximation [Result: dissipation effects are not correctly taken into account except for one-body dissipation]...but our system is superfluid... Time evolution of non-interacting system

11 Solving time-dependent problem... nonlinear coupled 3D Partial Differential Equations Supercomputing We simulate fermionic systems consisting of particles (cold atoms, neutron stars) also nuclear reactions (spin-orbit term required)

12 Solving... The system is placed on a large 3D spatial lattice of size Nx Ny Nz Discrete Variable Representation (DVR) solid framework (see for example: Bulgac, Forbes, Phys. Rev. C 87, (R) (2013)) Errors are well controlled exponential convergence No symmetry restrictions Number of PDEs is of the order of the number of spatial lattice points Typically (without spin-orbit term):

13 The spirit of SLDA is to exploit only local densities... Suitable for efficient parallelization (MPI) Excellent candidate for utilization multithreading computing units like GPUs 15 times Speed-up!!!

Validation against dynamical properties of the system Recent MIT

from Nature 499, 426 (2013) Li atoms near a Feshbach resonance (N 106)

14 Validation against dynamical properties of the system Recent MIT experiments: Nature 499, 426 (2013), PRL 113, (2014) Fig. from Nature 499, 426 (2013) Li atoms near a Feshbach resonance (N 106) cooled in harmonic trap 6 Step potential used to imprint a soliton (evolve to π phase shift) Let system evolve... Take picture (subtle imaging with tomography) NOTE: There were tests against other dynamical properties Fig. from PRL 113, (2014)

, Nature 499, 426 (2013) PRL 113, 065301(2014) Intertial

15 Experimental results RESULTS: In the final state: Observe an oscillating vortex line with long period Yefsah et al., Nature 499, 426 (2013) PRL 113, (2014) Intertial mass 200 times larger than the free fermion mass Precessional motion...

it/activities/fermi_school/mmxiv) School of Physics E.

16 Experimental results Cascade of Solitary Waves Figures taken from: M. Zwierlein talk, ( School of Physics E. Fermi Quantum Matter at Ultralow Temperatures Varenna, July 9th, 2014 See also: Mark J.H. Ku, et al., Phys. Rev. Lett. 116, (2016)

Experimental results Cascade of Solitary Waves Figures taken from: M. Zwierlein talk, (http://en.sif.

17 Experimental results Cascade of Solitary Waves Figures taken from: M. Zwierlein talk, ( School of Physics E. Fermi Quantum Matter at Ultralow Temperatures Varenna, July 9th, 2014 Challenge for theory to describe all stages of the cascade!

18 Mark J.H. Ku, et al., Phys. Rev. Lett. 116, (2016) Subtle imaging needed: - needed expansion - must ramp to specific value of magnetic field A. Munoz Mateo and J. Brand, PRL 113, (2014)? Cascade Experiment...

19 vs Simulations Phys. Rev. A 91, (2015) Movie 1 Crossing and reconnection! Movie 2

20 Phys. Rev. A 91, (2015)

21 Phys. Rev. A 91, (2015)

22 Dynamics of solitons and vortices GPE Fermionic simulations numerically expensive, cannot reach 106 particles... Other solution: use a bosonic theory for the dimer/cooper-pair wavefunction with DFT... Michael McNeil Forbes, Rishi Sharma, Phys. Rev. A 90, (2014) Note similarity to GPE... Accurate Equation of State state for a>0, speed of sound, phonon dispersion, static response, respects Galilean invariance Ambiguous role played by the wave function, as it describes at the same time both the number density and the order parameter. Density depletion at vortex/soliton core exaggerated! Systematically underestimates time scales by a factor of close to 2 What about cascade? Movie 1 - ETF Movie 2 - ETF

23 Phys. Rev. A 91, (2015)

24 Phys. Rev. A 91, (2015)

reconnection! Thus: possible of QT in cold atoms.

138, 741 (2005) Depletion of density Note: in ETF

reconnections in intermediate state (between vortex ring

25 Phys. Rev. A 91, (2015) Fermionic simulation: Crossing and reconnection! Thus: possible of QT in cold atoms... Aurel Bulgac, Yongle Yu, J. Low Temp. Phys. 138, 741 (2005) Depletion of density Note: in ETF (GPE-like approach) there is no crossings and reconnections in intermediate state (between vortex ring and vortex line)... Our observation: It is very hard to force GPE to get crossing and reconnection... by contrast to fermionic simulations!

26 Future direction (the latest review papers...) QT QT sosofar far studied studied with with superfluid superfluid Helium Helium

27 Kolmogorov scaling kinetic energy per unit mass If the energy is transferred to smaller scales without being dissipated: E kinetic energy per unit mass associated with the scale 1/k ε - energy rate (per unit mass) transfered to the system at large scales. k - wave number (from Fourier transformation of the velocity field). C dimensionless constant.

unit mass) transfered to the system at large scales.

28 Kolmogorov scaling kinetic energy per unit mass If the energy is transferred to smaller scales without being dissipated: E kinetic energy per unit mass associated with the scale 1/k ε - energy rate (per unit mass) transfered to the system at large scales. k - wave number (from Fourier transformation of the velocity field). C dimensionless constant. But how to generate small scales if vortices are quantized!

29 Crossing and reconnection (Feynman, 1955) Figs. from: Michał Wyszyński, Diploma thesis, WUT, 2016 energy proportional to the length small scale Movie 3 Movie 4 E. Fonda, et. al., Proc. Natl. Acad. Sci. U.S.A. 111, 4707 (2014).

30 Michał Wyszyński, Diploma thesis, WUT, 2016

31 E. Fonda, et. al., Proc. Natl. Acad. Sci. U.S.A. 111, 4707 (2014).

Crossing and reconnection (Feynman, 1955) Figs.

2016 energy proportional to the length small

Natl. Acad. Sci. U.S.A. 111, 4707 (2014).

32 Crossing and reconnection (Feynman, 1955) Figs. from: Michał Wyszyński, Diploma thesis, WUT, 2016 energy proportional to the length small scale Movie 3 Movie 4 E. Fonda, et. al., Proc. Natl. Acad. Sci. U.S.A. 111, 4707 (2014). Quantum turbulence: Chaotic dynamics of many quantized vortices Phys. Rev. Lett. 97 (2006) Problem: How to generate turbulent state in cloud of cold atoms?

33 Prediction of TDSLDA: Phase imprint of array of vortices! Phys. Rev. A 91, (2015) Movie 5

34 Phys. Rev. A 91, (2015)

35 Fig. from: Wlazlowski, et. al., Phys. Rev. A 92, (2015) SF PG N InI te aafentererestsing t tht e ferm rmioi niingfefeaatur heuni oncicga tuer o unta itaryr re gassini e of f y regim n gime e Ab-initio calculations of the shear viscosity: 1. Phys. Rev. Lett. 109, , (2012) 2. Phys. Rev. A 88, (2013) 3. Phys. Rev. A 92, (2015) Extraction of the shear viscosity for uniform system from non-uniform measurements (model dependent) Phys. Rev. Lett. 115, (2015)

36 What if the system is above critical temperature? No superfluid state: quantum turbulence not possible...

37 What if the system is above critical temperature? No superfluid state: quantum turbulence not possible... What about classical turbulence: Classical turbulence in three-dimensional systems is achieved for values of the Reynolds number of the order of 104 or higher.

38 What if the system is above critical temperature? No superfluid state: quantum turbulence not possible... What about classical turbulence: Classical turbulence in three-dimensional systems is achieved for values of the Reynolds number of the order of 104 or higher. For UFG:

39 What if the system is above critical temperature? The TheFermi Fermigas gasininthe theunitary unitaryregime regimeisisthus thusaarather rather No superfluid state: unique physical system; unique physical system; quantum turbulence not possible... below the critical temperature below the critical What about classical turbulence: temperature the and thesystem systemisis superfluid andcan cansustain sustain Classical turbulence insuperfluid three-dimensional systems is achieved for 4 turbulence in rather small clouds, values of quantum the Reynolds number of the order of 10 quantum turbulence in rather small clouds, or higher. while whileabove abovethe thecritical criticaltemperature temperature the theturbulent turbulentdynamics dynamicsisisstrongly stronglysuppressed suppressed for forany anycurrent currentexperimental experimentalrealizations realizationsfor for aavery verywide widerange rangeofofflow flowvelocities velocitiesand andcloud cloudsizes. sizes. For UFG:

41 Glitch: a sudden increase of the rotational frequency Vortex model (P. W. Anderson and N. Itoh, Nature 256 (1975)) Glitches in the Vela pulsar Period (sec) 1970 year Presently the standard picture for pulsar glitches Can explain: post-glitch relaxation, statistics of the glitching populations... Idea: Superfluid interior contains quantized vortices pinned to the crustal lattice Glitches are believed to occur when a large number of vortices simultaneously unpin and move outward Open problem: Nuclear or Interstitial pinning Time Figs from: P. Donati et al., Nuclear Physics A 742 (2004) 363 V.B. Bhatia, A Textbook of Astronomy and Astrophysics with Elements of Cosmology, Alpha Science, First observed in 1969: V. Radhakrishnan and R. N. Manchester, Nature 222, (1969); P. E. Reichley and G. S. Downs, Nature 222, (1969);

42 HFB: Avogadro et al. Interstitial pinning Nuclear pinning TF+LDA: Donati, Pizzochero Fig. from: P. Avogadro et al., Phys. Rev. C 75, (R) (2007) Figs from: P. Donati et al., Nuclear Physics A 742 (2004) 363

43 Movie 6 attracted by nucleus repelled by nucleus What is response of the gyroscope when pushed?

44 Phys. Rev. Lett. 117, (2016)

45 Our strategy... Instead of solving static problem... we solve time-dependent problem... and observing dynamics of the system we determine the forces. Unambiguous determination of the force sign...

46 We performed 3D, dynamical simulations by TDDFT with superfluidity TDSLDA equations Energy density functional (EDF) : Fayans EDF (FaNDF0) w/o LS S.A. Fayans, JETP Letters 68, 169 (1998); FP81: B. Friedman and V. R. Pandharipande, Nucl. Phys. A 361, 502 (1981) WFF88: R. B. Wiringa, V. Fiks, and A. Fabrocini, Phys. Rev. C 38, 1010 (1988). Potentials

47 We performed 3D, dynamical simulations by TDDFT with superfluidity TDSLDA equations (similar to TDHFB, TD-BdG) Energy density functional (EDF) The coupling constant g is chosen to reproduce the neutron pairing gap in pure neutron matter. Potentials

48 Target: extract vortex-nucleus force (microscopic input for glitch models) We use Newton's 3rd law and extract the force from motion of the nucleus... Vortex-nucleus interaction We construct Fext dynamically: We use uniform electric field as external force...

49 Initial state Self-consistent solution of static problem J. Negele and D. Vautherin, Nucl. Phys. A 207, 298 (1973) 3D lattice no symmetry restrictions Simulations for background density: kf=0.75 fm-1 and kf=0.97 fm-1

50 Static solutions: Eunpinned < Epinned

51 G. Wlazłowski, K. Sekizawa, P. Magierski, A. Bulgac, M.M. Forbes Phys. Rev. Lett. 117, (2016) Results Blue line: vortex-core Red dot: c.m. of protons v0: along x-axis Force exerts on the nucleus will be shown by a black arrow or? Movie 7 7/9 Microscopic Calculation of Vortex-Nucleus Interaction for Neutron Star Glitches Thu., June 23, 2016

52 Phys. Rev. Lett. 117, (2016)

53 Force decomposition centripetal Fc(R) nucleus F(R) Ft(R) R vortex tangential component vanishes Force is central At close separation the force is not a function of distance R only Range: fm

54 Force per unit length Inspired by the vortex filament model (describes vortex-vortex Interaction) Total force acting on nucleus

55 Force per unit length Inspired by the vortex filament model (describes vortex-vortex Interaction) Total force acting on nucleus Ansatz for f(r) - Pade approximant We minimize chi2 with respect of {ak,bk} Measured force frames of movie Predicted for given set {ak,bk} from irrotational and incompressible hydrodynamics

56 Force per unit length n=0.014fm-3 Movie 8 Only part close to nucleus surface contributes...

57 Phys. Rev. Lett. 117, (2016)

58 Figures from: K. Sekizawa, G. Wlazłowski, P. Magierski, A. Bulgac, M.M. Forbes, JPS Conf. Proc. 14, (2017) Deformation induced by the mutual interaction

Nuclear Fission and Fusion Reactions within Superfluid TDDFT

Nuclear Fission and Fusion Reactions within Superfluid TDDFT Piotr Magierski (Warsaw University of Technology) Collaborators: Aurel Bulgac (Univ. of Washington) Kenneth J. Roche (PNNL) Ionel Stetcu (LANL)