Probing dynamical properties of Fermi-Hubbard systems with a quantum gas microscope Peter Brown Bakr Lab Solvay workshop, February 19 th 2019
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1 Probing dynamical properties of Fermi-Hubbard systems with a quantum gas microscope Peter Brown Bakr Lab Solvay workshop, February 19 th 2019
2 Outline 1. Studying strongly-interacting quantum matter with ultracold atoms 2. The Fermi-Hubbard Model 3. Measuring diffusion and conductivity in the repulsive Hubbard model 4. Measuring spectral functions and the pseudogap in the attractive Hubbard model
3 High temperature superconductors Heavy fermion metals Strongly correlated quantum matter Topological phases Spintronic materials
4 High temperature superconductors
5 High temperature superconductors Yttrium Barium Copper Oxide
6 High temperature superconductors Yttrium Barium Copper Oxide Hubbard model
7 Use a synthetic quantum system of ultracold atoms - Feynman (paraphrased)
8 Interacting systems of ultracold atoms enlarged model for condensed matter physics Why ultracold atoms? Understood from first principles Complete control of microscopic parameters Clean systems, no impurities Dynamics on observable timescales Large interparticle spacing makes optical imaging/manipulation possible
9 Microscopy of ultracold atoms in optical lattices Similar fermion microscopes at: Harvard, MIT, MPQ, Toronto, Strathclyde
10 Outline 1. Studying strongly-interacting quantum matter with ultracold atoms 2. The Fermi-Hubbard Model 3. Measuring diffusion and conductivity in the repulsive Hubbard model 4. Measuring spectral functions and the pseudogap in the attractive Hubbard model
11 The Fermi-Hubbard model Hopping (kinetic energy) On-site interaction t U How much of the phenomenology of the cuprates does the Hubbard model reproduce?
12 temperature Cuprate phase diagram doping
13 temperature Cuprate phase diagram doping
14 Antiferromagnetic spin correlations Detection of AFMs with microscopes: Parsons Greiner, Science 353, 1253 (2016) Boll... Bloch, Gross, Science 353, 1257 (2016) Cheuk Zwierlein, Science 353, 1260 (2016) Previous work without microscopes: Grief Esslinger, Science 340, 1307 (2013) Hart Hulet, Nature 519, 211 (2015) Drewes Köhl, PRL 118, (2017)
15 temperature Cuprate phase diagram doping
16 Outline 1. Studying strongly-interacting quantum matter with ultracold atoms 2. The Fermi-Hubbard Model 3. Measuring diffusion and conductivity in the repulsive Hubbard model 4. Measuring spectral functions and the pseudogap in the attractive Hubbard model
17 temperature Cuprate phase diagram doping
18 Conventional (weakly interacting) Unconventional (strongly correlated)
19 Previous Work Mass transport experiments with Fermions Mesoscopic systems: Brantut et al. Science 337, 1069 (2012) (ETH Zurich) Lebrat et al. PRX 8, (2018) (ETH Zurich) Valtolina et al. Science 350, 1505 (2015) (Florence) Bulk systems: Ott et al. PRL 92, (2004) (Florence) Strohmaier et al. PRL 99, (2007) (ETH Zurich) Schnedier et al. Nat. Phys 8, 213 (2012) (Munich) Xu et al. arxiv: (2016) (UIUC) Anderson et al. arxiv: (2017) (Toronto)
20 Measurement Protocol Brown et. al., Science 363, 379 (2019) Spin transport: Nichols Zwierlein, Science 363, 383 (2019)
21 Measurement Protocol Brown et. al., Science 363, 379 (2019)
22 Measurement Protocol Brown et. al., Science 363, 379 (2019)
23 Measurement Protocol Brown et. al., Science 363, 379 (2019)
24 Measurement Protocol Brown et. al., Science 363, 379 (2019)
25 Decaying density modulation
26 Decaying density modulation Not explained by diffusion alone!
27 Decaying density modulation
28 Decaying density modulation
29 Decaying density modulation
30 Hydrodynamic Model Diffusion (Fick s Law) neglects finite time to establish current. D, diffusion constant Γ, current relaxation. rate. Crossover from diffusive mode to sound mode. charge conservation Fick s Law
31 Hydrodynamic Parameters
32 Compressibility High-T limit DQMC
33 Resistivity Versus Temperature Brown et. al., Science 363, 379 (2019)
34 Resistivity Versus Temperature Brown et. al., Science 363, 379 (2019) DMFT FTLM
35 Outline 1. Studying strongly-interacting quantum matter with ultracold atoms 2. The Fermi-Hubbard Model 3. Measuring diffusion and conductivity in the repulsive Hubbard model 4. Measuring spectral functions and the pseudogap in the attractive Hubbard model
36 Photoemission spectroscopy Using a photon, excite a particle from an interacting system Measure the energy and momentum of the ejected particle single-particle excitations of a many-body system Rev. Mod. Phys. 75, 473 (2003)
37 What does ARPES measure? Remove particle (emission) How does an excitation propagate in a manybody system? Momentum resolved density of states ARPES particle current gives access to emission Remove hole (injection) Emission + injection Emission only
38 What does ARPES measure? Remove particle (emission) How does an excitation propagate in a manybody system? Momentum resolved density of states ARPES particle current gives access to emission Remove hole (injection) Emission + injection Emission only
39 What does ARPES measure? Remove particle (emission) How does an excitation propagate in a manybody system? Momentum resolved density of states ARPES particle current gives access to emission Remove hole (injection) Emission + injection Emission only
40 The BCS limit Fermi gas, excitations have definite momentum and energy BCS, pairing appears as a gap Dispersion exhibits backbending Fermi gas
41 The BCS limit Fermi gas, excitations have definite momentum and energy BCS, pairing appears as a gap Dispersion exhibits backbending BCS
42 3D Fermi Gas Pseudogaps Depression in the spectral function at the Fermi energy. Cold atom experiments: backbending in dispersion above T c. Observed in High-T c superconductors and unitary Fermi gas HTSC, PG origin controversial: precursor to SC or indicative of a competing order. Stewart Jin, Nature 454, 744 (2008) Gaebler Jin, Nature Phys. 6, 569 (2010) 2D Fermi Gas Pseudogap reviews: Low Temp. Phys. 41, 319 (2015) Rep. Prog. Phys. 80, (2017) Feld Kohl, Nature 480, (2011)
43 Pseudogap in the attractive Accessible model: on a lattice and no DQMC sign problem. BEC-BCS crossover with interaction strength. Temperatures near state-of-the-art for experiment Hubbard model Eur. Phys. J. B. 2, 30 (1998)
44 ARPES with a QGM f Hubbard system rf probe Radiofrequency photon transfers to non-interacting state but preserves momentum Band mapping transforms quasimomentum to real momentum T 4 expansion in harmonic trap maps momentum space to real space (similar to time-of-flight measurement) Freeze atoms in deep lattice and detect
45 ARPES with a QGM f Radiofrequency photon transfers to non-interacting state but preserves momentum Band mapping transforms quasimomentum to real momentum T 4 expansion in harmonic trap maps momentum space to real space (similar to time-of-flight measurement) Freeze atoms in deep lattice and detect
46 ARPES with a QGM f Radiofrequency photon transfers to non-interacting state but preserves momentum Band mapping transforms quasimomentum to real momentum T 4 expansion in harmonic trap maps momentum space to real space (similar to time-of-flight measurement) Freeze atoms in deep lattice and detect
47 ARPES with a QGM f Radiofrequency photon transfers to non-interacting state but preserves momentum Band mapping transforms quasimomentum to real momentum T 4 expansion in harmonic trap maps momentum space to real space (similar to time-of-flight measurement) Freeze atoms in deep lattice and detect
48 ARPES data: increasing interaction strength Eur. Phys. J. B. 2, 30 (1998)
49 ARPES data: increasing interaction strength Expt U t = 4 T t = 0.5 ε k Determine U t, T t, and μ t from fitting correlators to equilibrium DQMC Spectral weight shifts to lower energy (U < 0) Spectral peak shifts away from μ at stronger interaction DQMC
50 ARPES data: increasing interaction strength Expt U t = 6 T t = 0.5 ε k Determine U t, T t, and μ t from fitting correlators to equilibrium DQMC Spectral weight shifts to lower energy (U < 0) Spectral peak shifts away from μ at stronger interaction DQMC
51 ARPES data: increasing interaction strength Expt U t = 8 T t = 0.5 ε k Determine U t, T t, and μ t from fitting correlators to equilibrium DQMC Spectral weight shifts to lower energy (U < 0) Spectral peak shifts away from μ at stronger interaction DQMC
52 ARPES data: increasing interaction strength Eur. Phys. J. B. 2, 30 (1998)
53 ARPES data: increasing temperature Expt U t = 8 T t = 0.44 ε k Determine U t, T t, and μ t from fitting correlators to equilibrium DQMC Second branch emerges with increasing temperature Lower branch: doublons Upper branch: singles DQMC
54 ARPES data: increasing temperature Expt U t = 8 T t = 1.0 ε k Determine U t, T t, and μ t from fitting correlators to equilibrium DQMC Second branch emerges with increasing temperature Lower branch: doublons Upper branch: singles DQMC
55 ARPES data: increasing temperature Expt U t = 8 T t = 2.8 ε k Determine U t, T t, and μ t from fitting correlators to equilibrium DQMC Second branch emerges with increasing temperature Lower branch: doublons Upper branch: singles DQMC
56 ARPES data: increasing temperature Expt U t = 8 T t = 5.0 ε k Determine U t, T t, and μ t from fitting correlators to equilibrium DQMC Second branch emerges with increasing temperature Lower branch: doublons Upper branch: singles DQMC
57 Challenges and opportunities ARPES of repulsive model: further cooling is a key challenge. Entropy redistribution. Immersion in bosonic baths. Floquet engineering of t-j models. Dynamical observables More challenging for theory Test approximations Toolkit small compared with materials
58 PI Waseem Bakr Graduate students Debayan Mitra Peter Brown Elmer Guardado-Sanchez Lysander Christakis Jason Rosenberg Benjamin Spar Post-docs Stanimir Kondov Peter Schauss Thank you! Undergraduates Joseph Scherrer Adam Bowman Mark Stone Joe Zhang Maria-Claudia Popescu Nathan Agmon Arka Adhikari Theory support David Huse (Princeton) Andre-Marie Tremblay (Sherbrooke) Jure Kokalj (U. of Ljubljana) Edwin Huang (Stanford) Tom Devereaux (Stanford) Funding
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