Density fluctuations and transport in normal and supercooled quantum liquids Eran Rabani School of Chemistry Tel Aviv University

Outline l Quantum Mode-Coupling Theory l Quantum generalized Langevin equation (QGLE) l Quantum mode-coupling approximations l Density fluctuations l Classical liquids l Quantum liquids l Quantum supercooled liquids l Quantum Kob-Andersen model l Preliminary results l Experimental realization

Literature l E. Rabani and D.R. Reichman, Phys. Rev. E 65, (2002) l E. Rabani and D.R. Reichman, J. Chem. Phys. 116, 6271 (2002) l D.R. Reichman and E. Rabani, J. Chem. Phys. 116, 6279 (2002) l E. Rabani and D.R. Reichman, Europhys. Lett. 60, 656 (2002) l E. Rabani and D.R. Reichman, J. Chem. Phys. 120, 1458 (2004) l E. Rabani and D.R. Reichman, Ann. Rev. Phys. Chem. 56, (2005) l E. Rabani, K. Miyazaki, and D.R. Reichman, J. Chem. Phys. 122, (2005) Collaboration: Kunimasa Miyazaki, Columbia University David Reichman, Columbia University Roi Baer, Hebrew University Daniel Neuhauser, UCLA Financial Support: Israel Science Foundation, EU-STERP, Ministry of Science, US-Israel Binational Science Foundation

Theory

Quantum mode coupling l Step 1: Formulation of an exact quantum generalized Langevin equation (QGLE) using Zwanzig- Mori projection operator technique, for the Kubo transform of the dynamical variable of interest: l Step 2: Approximate memory kernel for the QGLE using a quantum mode-coupling theory. l Step 3: Solution of the QGLE with the approximate memory kernel combined with exact static input generated from a suitable PIMC scheme.

Density fluctuations The exact QGLE for the Kubo transform of the intermediate scattering function is given by To study density fluctuations we need to specify the dynamical variable and the corresponding correlation function (the intermediate scattering function): The formal expression for the memory kernel is

Binary portion The time moments are given in terms of the density moments: Short time expansion to second order in time The formal expression for the memory kernel is

Mode coupling portion The projected dynamics is replaced with the full dynamics projected onto the slow decaying modes: In addition, four point correlation functions are replaced by a product of two point correlation functions: where the new projection operator projects onto the following slow modes:

Total memory kernel

PIMC scheme We need to calculate the following static Kubo transforms: where Using the coordinate representation of the matrix element: We obtain (to lowest order in e using P Trotter slices): Our result looks similar to the Barker energy estimator, however, it is numerically less noisy.

Classical liquids

Liquid lithium The normalized intermediate scattering function for liquid lithium. Theredcurves are results obtained from molecular dynamics simulations and the bluecurves are results obtained from a classical mode-coupling theory. The agreement between the theory and simulations is remarkable for all q values shown. The normalized intermediate scattering function for liquid lithium. The red curves are results obtained from molecular dynamics simulations and the blue curves are results obtained from a classical mode-coupling theory. The agreement between the theory and simulations is remarkable for all q values shown.

Quantum liquids

Quantum liquids p-H 2 The normalized intermediate scattering function for liquid para-hydrogen.Thered curves are results obtained from the QMCT and the greencurves are results obtained from an analytic continuation approach (MaxEnt). Left panels show the corresponding memory kernels computed from the QMCT. The normalized intermediate scattering function for liquid para-hydrogen. The red curves are results obtained from the QMCT and the green curves are results obtained from an analytic continuation approach (MaxEnt). Left panels show the corresponding memory kernels computed from the QMCT.

Dynamic structure factor p-H 2 The normalized dynamic structure factor for liquid para- hydrogen.Red– QMCT.Green- MaxEnt.Black– QVMassuming a single relaxation time.Blue circles- experimental results F. J. Bermejo, B. Fak, S. M. Bennington, R. Fernandez-Perea, C. Cabrillo,J. Dawidowski, M. T. Fernandez-Diaz, and P. Verkerk, Phys. Rev. B 60, (1999). The normalized dynamic structure factor for liquid para- hydrogen. Red – QMCT. Green - MaxEnt. Black – QVM assuming a single relaxation time. Blue circles - experimental results F. J. Bermejo, B. Fak, S. M. Bennington, R. Fernandez-Perea, C. Cabrillo,J. Dawidowski, M. T. Fernandez-Diaz, and P. Verkerk, Phys. Rev. B 60, (1999).

Quantum liquids o-D 2 The normalized intermediate scattering function for liquid ortho-deuterium.Thered curves are results obtained from the QMCT and the greencurves are results obtained from an analytic continuation approach (MaxEnt). Left panels show the corresponding memory kernels computed from the QMCT. The normalized intermediate scattering function for liquid ortho-deuterium. The red curves are results obtained from the QMCT and the green curves are results obtained from an analytic continuation approach (MaxEnt). Left panels show the corresponding memory kernels computed from the QMCT.

Dynamic structure factor o-D 2 The normalized dynamic structure factor for liquid ortho- deuterium.Red– QMCT.Green MaxEnt.Black– QVM assuming a single relaxation time.Blue circles- experimental results from M. Mukherjee, F. J. Bermejo, B. Fak, and S. M. Bennington, Europhys. The normalized dynamic structure factor for liquid ortho- deuterium. Red – QMCT. Green - MaxEnt. Black – QVM assuming a single relaxation time. Blue circles - experimental results from M. Mukherjee, F. J. Bermejo, B. Fak, and S. M. Bennington, Europhys. Lett. 40, 153 (1997).

Quantum Transport

QGLE for VACF We need to obtain a QGLE for the We need to obtain a QGLE for the velocity autocorrelation function (v is the velocity of a tagged liquid particle along an arbitrary direction): Following similar lines to those sketched for the classical theory, we obtain an exact quantum generalized Langevin equation (QGLE): and the memory kernel is formally given by where we have used the following projection operator

Quantum Mode Coupling Theory The Kernel is approximated by Fast decaying quantum binary term: The slow decaying quantum mode-coupling term: The vertex:

MC Memory Kernel for VACF In addition, four point correlation functions are replaced by a product of two point correlation functions: The slow decaying quantum mode-coupling term is obtained using a set of approximations. The projected dynamics is replaced with the full dynamics projected onto the slow decaying modes: where the new projection operator is given by:

Static input from PIMC The static input for the memory kernel of the velocity autocorrelation function generated from a PIMC simulation method for liquid para-hydrogen at T=14K (red curve) and T=25K (blue curve).

Velocity autocorrelation function The normalized velocity autocorrelation function calculated from the quantum mode-coupling theory (blue curve) and from an analytic continuation of imaginary- time PIMC data (blue curve) for liquid para-hydrogen at T=14K (lower panel) and T=25K (upper panel). The good agreement between the two methods is a strong support for the accuracy of the quantum mode-coupling approach for liquid para- hydrogen.

Memory kernel for VACF The Kubo transform of the memory kernel for the velocity autocorrelation function for liquid para-hydrogen at T=14K (upper panel) and T=25K (lower panel). Shown are the fast- decaying binary term (red curve), the slow-decaying mode-coupling term (green curve) and the total memory kernel (blue curve). The contribution of the slow mode- coupling portion of the memory kernel is significant at the low temperature, while at the high temperature, the kernel can be approximated by only the fast binary portion.

Self-Diffusion - Liquid para-H 2 The frequency dependent diffusion constant for liquid para- hydrogen at T=14K and T=25K. The self- diffusion obtained from the Green-Kubo relation is 0.30 and 1.69 (Å 2 /ps) for T=14K and T=25K, respectively. These results are in good agreement with the experimental results (0.40 and 1.60) and with the maximum entropy analytic continuation method (0.28 and 1.47).

VACF – Liquid ortho-D 2 The normalized velocity autocorrelation function and its Kubo transform calculated from the quantum mode-coupling theory for liquid ortho- deuterium (upper panel) and liquid para-hydrogen (lower panel) at T=20.7K

Self-Diffusion - Liquid ortho-D 2 The frequency dependent diffusion constant for liquid para-hydrogen (green curve) and ortho-deuterium (red curve) at T=20.7K. The self- diffusion obtained from the Green-Kubo relation is 0.49 and 0.64 (Å 2 /ps) for para- hydrogen and ortho- deuterium, respectively. The result for ortho-deuterium is in reasonable agreement with the experimental results (0.36 Å 2 /ps).

Normal Liquid Helium The normalized velocity autocorrelation function calculated from the quantum mode-coupling theory (blue curve), from an analytic continuation of imaginary-time PIMC data (red curve), and from a semiclassical approach (Makri – green curve) for liquid helium above the transition.

Self-Diffusion – Normal Liquid Helium analytic continuation of imaginary-time PIMC data (red curve), from a semiclassical approach (Makri – green curve), and from the CMD method (black curve) The frequency dependent diffusion constant for normal liquid helium at T=4K. The results shown were calculated from the quantum mode-coupling theory (blue curve), from an

Quantum glasses

l Can we form a structural quantum glass (onset of quantum fluctuations, super fluidity)? l Are there anythermodynamicsignatures that are different for a quantum glass? l Are there any thermodynamic signatures that are different for a quantum glass? l Are there anydynamicsignatures that are different for a quantum glass? l Are there any dynamic signatures that are different for a quantum glass?

Kob-Andersen model The Kob-Andersen model is based on a binary mixture of Lennard-Jones (BMLJ) particles with the following parameters: The system undergoes anergodic-to-nonergodictransition at T= Classical mode-coupling theory predicts a transition at aboutT=0.92. The system undergoes an ergodic-to-nonergodic transition at T= Classical mode-coupling theory predicts a transition at about T=0.92. Kob and Andersen, Phys. Rev. E 51, 4626 (1995) Kob and Andersen, Phys. Rev. E 52, 4134 (1995) Nauroth and Kob, Phys. Rev. E 55, 657 (1997)

Classical results for the Kob-Andersen model

Intermediate Scattering Function Self intermediate scattering function versus time for A and B particles at two wave length for several temperatures. Kob and Andersen, Phys. Rev. E 52, 4134 (1995). MCT predictions Early  Late   regime

Self-diffusion Left: Mean square displacement versus time forAparticles at different temperatures. Right: Diffusion constant versus temperature forAandBparticles.Solidlines are best fits to power-law anddashed lines are best fits to Vogel-Fulcher law. Left: Mean square displacement versus time for A particles at different temperatures. Right: Diffusion constant versus temperature for A and B particles. Solid lines are best fits to power-law and dashed lines are best fits to Vogel-Fulcher law.

Quantum results for the Kob-Andersen model

Average potential energy Average potential energy per particle for the quantum Kob- Andersen model. Simulation were done forN=500and P=100. There is a clear change near T=1 (similar to the classical case). Is there any slowing down near this temperature? Average potential energy per particle for the quantum Kob- Andersen model. Simulation were done for N=500 and P=100. There is a clear change near T=1 (similar to the classical case). Is there any slowing down near this temperature?

Intermediate scattering function Intermediate scattering function at  =1.2for several values of the temperature atq=q max. No significant slowing down is observed far below the classicalT c =0.92 Intermediate scattering function at  =1.2 for several values of the temperature at q=q max. No significant slowing down is observed far below the classical T c =0.92.

Static structure factor

Pair correlation function

Low wave vector results Intermediate scattering function at  =1.2 for several values of the temperature at q=q max /2. The A particles show coherent fluctuation not observed classically at this value of q.

And even lower Intermediate scattering function at  =1.2 for several values of the temperature at q=q max /4.

Experimental realization

Mixtures of p-H 2 and o-D 2 T=10

T=8

Centroid configurations T=10, TP density T=8, TP density T=6, < TP density At the triple-point (TP) density the mixture freezes into an ordered crystals below T=10. But at a slightly lower temperature, the system can be supercooled and even at T=6 is still disordered.

Classical Results – Kob-Andersen Time dependence of the coherent and incoherent intermediate scattering function for two wave vectors at T = 2. The dashed line with the symbols are the results from the simulation and the solid lines are the prediction of the classical MCT theory. From Kob and collaborators (Phys. Rev. E 55, 657 (1997), J. Non-Cryst. Solids 307, 181 (2002)).

Classical Results – Kob-Andersen Time dependence of the coherent and incoherent intermediate scattering function for two wave vectors at T = The dashed line with the symbols are the results from the simulation and the solid lines are the prediction of the classical MCT theory. From Kob and collaborators (Phys. Rev. E 55, 657 (1997), J. Non-Cryst. Solids 307, 181 (2002)).

Simple Example - VACF Now, lets make asimple Gaussianapproximation to the memory kernel: Now, lets make a simple Gaussian approximation to the memory kernel: Even this simple approximation (short time expansion) captures some of the hallmarks of normal monoatomic liquids. The reason is that the approximation is done at the level of the memory kernel, and thus better results are obtained for the correlation function itself. However, this approximation completely neglects the long time decay of the memory kernel.

Structure factor and nonergodic parameter

l E. Rabani and D.R. Reichman, J. Phys. Chem. B 105, 6550 (2001) l D.R. Reichman and E. Rabani, Phys. Rev. Lett. 87, (2001) l E. Rabani and D.R. Reichman, Phys. Rev. E 65, (2002) l E. Rabani and D.R. Reichman, J. Chem. Phys. 116, 6271 (2002) l D.R. Reichman and E. Rabani, J. Chem. Phys. 116, 6279 (2002) l E. Rabani, D.R. Reichman, G. Krilov, and B.J. Berne, PNAS USA 99, 1129 (2002) l E. Rabani and D.R. Reichman, Europhys. Lett. 60, 656 (2002) l E. Rabani, in Proceedings of "The Monte Carlo Method in the Physical Sciences: Celebrating the 50th anniversary of the Metropolis algorithm", AIP Conference Proceedings, vol. 690, 281 (2003) l E. Rabani and D.R. Reichman, J. Chem. Phys. 120, 1458 (2004) l E. Rabani and D.R. Reichman, Ann. Rev. Phys. Chem. 56, (2005) l E. Rabani, K. Miyazaki, and D.R. Reichman, J. Chem. Phys. 122, (2005)