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Condensed Matter Theory
Physics, Applied Physics, Geology and Geophysics, Mechanical Engineering
Condensed matter systems
• Studies of many-body physics: number of particles is large, interactions among constituents can be strong. • A variety of phases and the transitions between them: superfluidity, superconductivity, Bose-Einstein condensate, ferromagnetic and anti-ferromagnetic phases, fractional quantum Hall states, topological phases (cannot be understood by symmetry breaking), … • Cold atom physics: interactions between particles can be tuned, BEC to BCS crossover, simulations of quantum condensed matter systems
Condensed matter physics is broadly divided into two subfields:
(I) Classical: hydrodynamics, granular material, foams, colloids, ice,…
(ii) Quantum: quantum phase transitions, superconductivity and superfluidity, quantum magnetism, quantum computing, cold atoms, quantum optics,…
Leonid Glazman Yoram Alhassid
Nicholas Read Steven Girvin R. Shankar
CMT at Yale: Physics
Meng Cheng
Vivuds Ozolins Sohrab Ismail-Beigi A. Douglas Stone
Applied Physics
John Wettlaufer
Geology & Geophysics Mechanical Engineering
Corey O’Hern
+ Topological insulators
Condensed matter theory Atomic Physics and Quantum Optics Quantum Computation and Quantum Optics of Electrical Circuits Cavity/Circuit QED
-coupling single microwave photons to superconducting qubits -entanglement of qubits, quantum error correction protocols -quantum measurements/amplification/noise -optomechanics: micromechanical cantilevers in optical cavities
Steven Girvin
Many body theory, quantum field theory, statistical mechanics Fractional quantum Hall systems and topological phases -non-Abelian statistics, new states of matter, connection with conformal field theory -Majorana zero modes -entanglement spectrum -Hall viscosity: a non-dissipative transport coefficient -application to topological quantum computation Two-dimensional critical phenomena and conformal field theory -related to integer quantum Hall effect and transport; algebraic techniques Combinatorial optimization and complexity -minimum spanning trees and relation to percolation -spin glasses -other computer science problems and quantum computation/information
Nicholas Read
One-dimensional quantum fluids (electrons, spinons, cold atoms): -Conduction of nano-wires; -Dynamic structure factors of spin excitations; -Quantum liquids in optical lattices
Leonid Glazman Quantum Many-Body Physics in Low Dimensions
Electron transport in topological insulators (TI): - Electron scattering mechanisms - Quantum effects in conduction properties - Theory is closely related to ongoing experiments
Low-dimensional and mesoscopic superconductors: -Quantum phase slips in mesoscopic superconductors; -Physics of superconducting qubits (close collaboration with experimental groups of Michel Devoret and Rob Scholekopf)
Fig. 1: Electron tunneling spectra explained by nonlinear Luttinger liquid theory
Fig. 2: Spectrum of a fluxonium qubit, experiment and theory
Fig. 3: Tunneling between helical edge states and electron puddles in 2D TI
Meng Cheng Quantum condensed matter theory
Topological order -- A new kind of order, invisible from local order parameters.
• Symmetries in topological order • Topological phenomena in 3+1 dimensions • Topological quantum computation
One-dimensional electrons – Quantum wires, 2D edge states, etc.
• Superconducting proximity effect in quantum Hall edge • Gapless phases in one dimension • Transport phenomena in quantum wires
Quantum spin liquid (QSL) -- Classification of gapped and gapless QSL; experimental signature of fractionalization
Quantum dots: sub-micron-scale artificial devices containing up to several thousand electrons
Yoram Alhassid Mesoscopic physics, nanoscience and cold atoms
Quantum gases (cold atoms): fermions in a harmonic trap with a tunable contact interaction • BCS to BEC crossover. • Unitary limit: pseudogap phase ? • Spin-orbit coupling new Rashbon-BEC phase
Nanoparticles: nano-scale metallic grains. • Their properties are different from bulk superconductors (conventional theory of superconductivity breaks down). • Competition between superconductivity and ferromagnetism.
• Statistical regime: interplay between one-body chaos and interactions. • “Universal” exchange interaction, spin effects.
⇒
A.DouglasStoneComplexandnon-linearsystems• Lasertheoryandcomplexmicro/nanolasers• Quantum/wavechaos,randommatrixtheory• Classical/Quantumop?csincomplexmedia• Controloflightpropaga?oninrandommedia• Quantummeasurementandcontrol
Can light propagate through strong scattering opaque media? Yes, by inputting special states
Focusing and defocusing light through white paint
SLM
Nature Physics, 2017
A. Douglas StoneComplex and non-linear systems• Laser theory and complex micro/nano lasers• Quantum/wave chaos, random matrix theory• Classical/Quantum optics in complex media• Control of light propagation in random media• Quantum measurement and control
Can light propagate through
strong scattering opaque
media? Yes, by inputting
special states
Focusing and defocusing
light through white paint
Nature Physics, 2017
SLM
Theory: “Filtered” random matrix theory predicts
focusing enhancement
Shaped wavefront
Sohrab Ismail-Beigi Electronic structure & materials theory from first principles
Interfaces between two materials: How is the interfacial region different from either side? What do electrons do in such an asymmetric environment?
Si
Metal oxide
Σ = +
+ ...+
+
+ ...Electron correlation especially in transition metal oxides (many-body Green functions; slave bosons)
Nanoscale/low-dimensional materials • 2D materials: synthesis, bonding,
electronic & optical response, topology • Ferroelctric surface chemistry
Vidvuds Ozolins Theory of electronic structure & energy materials
Theory of real electronic materials Applications in energy storage, generation and conversion
Machine learning for quantum mechanics: • Solving the Schrödinger equation • Localized basis for electron correlation
- Wannier functions • Deep Boltzmann machines and
Convolutional Neural Nets for Quantum Monte Carlo
• Electron-phonon interactions in solids • Transport of heat and electrical current • Thermoelectric effect • Exotic magnetism – spin liquids
0
20
40
60
80
100
120
Γ X W K Γ L U W L K
Freq
uenc
y (c
m-1
)
Wave number PDOS
AuBaBi
Theoretical and Computational Studies of Soft Matter and Biological Materials
O’Hern Research Group
Current Projects:
Crowding in cell cytoplasm
Tissue formationin embryos Design of protein-
protein interactionsJamming in granular media
Glass-formabilityin metal alloys
Condensed Matter Theory at Yale
Broad research: From bulk systems and cold atoms to nanoscale physics and quantum information From mathematical foundations to numerical studies
Other departments: applied physics, mechanical engineering,…
Collaborations across departments: physics and applied physics (theory and experiment)
Interdisciplinary research
Yale Particle Theory• Faculty
• David Poland (Conformal Bootstrap): Soner Albayrak, Rajeev Erramilli
• Walter Goldberger (Double Copy, EFTs for gravity+BH’s): Jingping Li
• Witek Skiba (CFT Correlators, BSM): Alex Sandomirsky
• Tom Appelquist (Lattice, EFTs for Light Scalars)
• Senior Research Scientist
• George Fleming (Lattice): Kimmy Cushman, Daniel Berkowitz
• Postdocs
• Zhijin Li (Conformal Bootstrap)
• Yuan Xin (Hamiltonian Truncation, 2d CFT)
Close ties to CMT (Cheng, Read), NT (Alhassid), Astro (Moncrief, Nagai), Math (Neitzke), …
Nuclear Theory
Francesco Iachello
Yoram Alhassid
• Condensed matter theory: mesoscopic physics and nanoscience • Cold atoms
• Nuclear many-body theory
• Double beta decay
• Clustering in nuclei • Graphene
Group website: http://alhassidgroup.yale.edu
Yoram Alhassid
Graduate students: Scott Jensen Paul Fanto (DOE NNSA graduate fellowship) Sohan Vartak Postdoctoral associate: Wouter Ryssens
• Combination of statistical and many-body methods.
• Conventional methods have to be modified in a finite-size system.
• Combination of analytical and computational methods.
-- nuclei, quantum dots, nanoparticles, and cold atoms.
We study correlated quantum systems in which the number of particles is large but still small enough for finite-size effects to be important:
Frederick Phineas Rose Professor of Physics
Development of non-perturbative methods
e−βH = D σ⎡⎣ ⎤⎦∫ GσUσ
• Small-amplitude quantal fluctuations of can be calculated analytically
Gibbs ensemble ( ) can be written as a superposition of ensembles of non-interacting particles in external time-dependent fields
e−βH
Uσ( )σ τ
σ
auxiliary fields non-interacting propagator
σ
β = 1/ T
Large-amplitude fluctuations of can be evaluated by stochastic methods: auxiliary-field quantum Monte Carlo
Static path plus random-phase approximation (SPA+RPA) ⇒
Recent review: Y. Alhassid, in Emergent Phenomena in Atomic Nuclei from Large-Scale Modeling, ed. K.D. Launey (World Scientific, 2017)
Mesoscopic systems are intermediate between microscopic systems and macroscopic bulk matter.
• Sufficiently small to be governed by the laws of quantum mechanics.
• Finite-size effects are important.
Quantum dots: sub-micron-scale devices containing up to several thousands electrons (“artificial atom”).
Statistical regime: single-particle dynamics are chaotic. • Interplay between one-body chaos and electron-electron interactions.
I. Mesoscopic Physics and Nanoscience
Flake of graphene: SYK Hamiltonian ?
A. Chen, PRL 2018
“Quantum holography in a graphene flake with an irregular boundary”
What are the signatures of SYK in the transport properties of the device?
Quantum holography in a graphene flake with an irregular boundary
An↵any Chen,1, 2 R. Ilan,3 F. de Juan,4 D.I. Pikulin,5 and M. Franz1, 2
1Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada V6T 1Z12Quantum Matter Institute, University of British Columbia, Vancouver BC, Canada V6T 1Z4
3Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel4Rudolf Peierls Centre for Theoretical Physics, Oxford, 1 Keble Road, OX1 3NP, United Kingdom
5Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA(Dated: July 23, 2018)
Electrons in clean macroscopic samples of graphene exhibit an astonishing variety of quantumphases when strong perpendicular magnetic field is applied. These include integer and fractionalquantum Hall states as well as symmetry broken phases and quantum Hall ferromagnetism. Here weshow that mesoscopic graphene flakes in the regime of strong disorder and magnetic field can exhibitanother remarkable quantum phase described by holographic duality to an extremal black hole intwo dimensional anti-de Sitter space. This phase of matter can be characterized as a maximallychaotic non-Fermi liquid since it is described by a complex fermion version of the Sachdev-Ye-Kitaevmodel known to possess these remarkable properties.
Tensions between the laws of quantum mechanics andclassical gravity that are emblematic of the extreme en-vironments occurring in the early universe and near hori-zons of black holes constitute the most enigmatic mys-teries in modern physics. A promising avenue to resolvesome of the paradoxes encountered in these studies, suchas the black hole information paradox, is the holographicprinciple [1]. In holographic duality, quantum gravitydegrees of freedom in a (d + 1)-dimensional spacetime“bulk” are represented by a many-body system definedon its d-dimensional boundary.
Important new insights into these fundamental ques-tions have been gained recently through the study of theSachdev-Ye-Kitaev (SYK) model [2, 3] which describesa system of N fermions in (0+1) dimensions subject torandom all-to-all four-fermion interactions and is dual todilaton gravity in (1+1) dimensional anti-de Sitter spaceAdS
2
[4, 5]. Despite being maximally strongly interact-ing this model is, remarkably, exactly solvable in the limitof large N . It has been shown to exhibit physical proper-ties characteristic of the black hole, including the exten-sive ground state entropy S
0
⇠ N , emergent conformalsymmetry at low energy and fast scrambling of quan-tum information that saturates the fundamental boundon the relevant Lyapunov chaos exponent �
T
. Exten-sions of this model also show interesting behaviors, in-cluding unusual spectral properties [6–8], supersymme-try [9], quantum phase transitions of an unusual type[10–12], quantum chaos propagation [13–15], patterns ofentanglement [16, 17] and strange metal behavior [18].
In this letter we propose a simple experimental real-ization of the SYK model with complex fermions in amesoscopic graphene flake with an irregular boundaryand subject to a strong applied magnetic field. Unlikethe earlier proposals in solid state systems [19, 20], whichtargeted the Majorana fermion version of the model, ourproposed device does not require superconductivity oradvanced fabrication techniques and should therefore be
B
A B
δ1
δ2δ3
FIG. 1. Schematic depiction of the proposed device. Ir-regular shaped graphene flake in applied magnetic field Bforms the (0+1) dimensional many-body system equivalentto a black hole in (1+1) anti-de Sitter space. Inset: latticestructure of graphene with A and B sublattices marked andnearest neighbor vectors denoted by �j .
relatively straightforward to assemble using only the ex-isting technologies. The proposed design is illustrated inFig. 1. Magnetic field B applied to graphene is knownto produce a variety of interesting quantum phases [21–30]. At the noninteracting level the field simply reorga-nizes the single-particle electron states into Dirac Lan-dau levels with energies [31] E
n
' ±~vp2n(eB/~c) and
n = 0, 1, · · · . We argue that when the graphene flakeis su�ciently small and irregular the electrons in then = 0 Landau level (LL
0
) are generically described bythe SYK model. This remarkable property is rooted inthe celebrated Aharonov-Casher construction [32] whichimplies that, in the absence of interactions, LL
0
remainsperfectly sharp even in the presence of strong disorderthat respects the chiral symmetry of graphene. As weshall see a flake with a highly irregular boundary, il-lustrated in Fig. 1, is chirally symmetric. Electrons inLL
0
, therefore, remain nearly perfectly degenerate, de-spite the fact that their wavefunctions acquire randomspatial structure. When Coulomb repulsion is projectedonto these highly disordered states, random all-to-all in-teractions between the zero modes are generated, exactlyas required to define the SYK model.
arX
iv:1
802.
0080
2v2
[con
d-m
at.st
r-el
] 20
Jul 2
018
The residual two-body interaction is random and correspond to an SYK (Sachdev, Ye, Kitaev) model (describing non-Fermi liquids)
Nanoparticles
Nano-scale metallic grains
• “Superconducting” but properties are very different from bulk superconductors: conventional theory of superconductivity (BCS) breaks down.
Discrete energy levels extracted from non-linear conductance measurements
Levels of aluminum grain vs. a magnetic field
By tuning the interaction it is possible to go continuously from “paired” fermionic atoms (BCS) to a Bose-Einstein condensate (BEC) of dimers.
Many interesting experimental results for theorists.
II. Cold atoms (quantum gases)
Exotic behavior was conjectured at the unitary limit (of strongest interaction) At low temperatures, there is a superfluid phase with a large gap, but there are claims that a a gap exists even above the critical temperature.
⇒
• The existence and extend of a pseudogap regime is extensively debated • in the literature
pseudogap regime (non-Fermi liquid)
III. Nuclear many-body theory
The strong interactions make the problem difficult; conventional methods are intractable.
The challenge: microscopic derivations of statistical nuclear properties from the underlying effective interactions.
• Important for astrophysical processes: nucleosynthesis, supernovas, … • “Phase transitions” in finite systems
• Tests of fundamental symmetries
Methods: we developed quantum Monte Carlo methods to solve in much larger configuration spaces (~ 1040) than those that can be treated by conventional methods (~ 1011).
Quantum shape transition (at T=0) vs. neutron number N→
Thermal shape transition vs. temperature (or excitation energy)
↑T
Example: a unified approach to quantum and thermal shape phase transitions
β,γ
00.1 0.2
00.1 0.2
π/ 3
π/ 6
148Sm
00.1 0.2
00.1 0.2
π/ 3
π/ 6
150Sm
00.1 0.2
00.1 0.2
π/ 3
π/ 6
152Sm
0
γ
0.1 0.20
γ
0.1 0.2
π/ 3
π/ 6
154Sm
00.1 0.2
00.1 0.2
π/ 3
π/ 6
00.1 0.2
00.1 0.2
π/ 3
π/ 6
00.1 0.2
00.1 0.2
π/ 3
π/ 6
0
γ
0.1 0.20
γ
0.1 0.2
π/ 3
π/ 6
00.1 0.2
β
00.1 0.2
β
π/ 3
π/ 6
00.1 0.2
β
00.1 0.2
β
π/ 3
π/ 6
00.1 0.2
β
00.1 0.2
β
π/ 3
π/ 6
0
γ
0.1 0.2
β
0
γ
0.1 0.2
β
π/ 3
π/ 6
0
1
2
3
4
log 10P(T,β,γ)
T = 4 MeV
T = 0.8 MeV
T = 0.07 MeV
Conclusion
• Interdisciplinary work on the interface of mesoscopic physics/nanoscience, cold atoms and nuclear many-body theory has led to important progress in our understanding of finite-size correlated quantum systems.
• A major challenge in understanding the rich physical behavior exhibited by these systems is the inclusion of correlations beyond mean-field theory. • Novel quantum phases.