dmitry abanin (harvard) eugene demler (harvard) measuring entanglement entropy of a generic...
TRANSCRIPT
![Page 1: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/1.jpg)
Dmitry Abanin (Harvard)
Eugene Demler (Harvard)
Measuring entanglement entropy of a generic many-body system
MESO-2012, Chernogolovka
June 18, 2012
![Page 2: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/2.jpg)
-Many-body system in a pure state
-Divide into two parts,
-Reduced density matrix for left part
(effectively mixed state)
-Entanglement entropy:
-Characterizes the degree of entanglement in
Entanglement Entropy: Definition
![Page 3: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/3.jpg)
-Many-body quantum systems: scaling laws, a universal
way to characterize quantum phases
-Guide for numerical simulations of 1D quantum systems
(e.g., spin chains)
-Topological entanglement entropy: measure of
topological order
-Black hole entropy, Quantum field theories
Entanglement entropy across different fields
![Page 4: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/4.jpg)
-1D system, ?
-Gapped systems:
-1D Fermi gas
-Any critical system (conformal field theory):
IMPLICATIONS:
-Measure of the phase transition location and central charge
-Independent of the nature of the order parameter
Scaling law for entanglement entropy
c -- central charge
Wilczek et al’94Vidal et al’ 03Cardy, Calabrese’04
![Page 5: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/5.jpg)
Topological order
-no symmetry breaking or order parameter
-degeneracy of the ground state on a torus
-anyonic excitations
-gapless edge states (in some cases)
Physical realizations:
-Fractional quantum Hall states
-Z2 spin liquids (simulations)
-Kitaev model and its variations
DIFFICULT TO DETECT
Topological entanglement entropy
![Page 6: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/6.jpg)
Topological entanglement entropy
-Three finite regions, A, B, C
-Define topological entanglement entropy:
-In a topologically non-trivial phase,
-A unique way to detect top. order
-Proved useful in numerical studies
invariant
characterizing
the kind of top. order
(Kitaev, Preskill ’06; Levin, Wen ’06)
Isakov, Melko, Hasting’11Grover, Vishwanath’11…
![Page 7: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/7.jpg)
-Free fermions in 1D (e.g., quantum point contact)
-Relate entanglement entropy to particle number
fluctuations in left region in the ground state
(Physical reason: particle number fluctuations in a Fermi gas
grow as log(l))
-Limited to the case of free particles
-Breaks down when interactions are introduced
(e.g., for a Luttinger liquid)
Existing proposals to measure entanglement entropy experimentally
Klich, Levitov’06Song, Rachel, Le Hur et al ’10, ‘12
Hsu, Grosfield, Fradkin ’09Song, Rachel, Le Hur ‘10
![Page 8: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/8.jpg)
Is it possible to measure entanglement in a generic interacting many-body system?
(such that the measurement complexity would not grow exponentially with system size)
Challenging – nonlocal quantity, requires knowledge of exponentially many degrees of freedom..
![Page 9: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/9.jpg)
Proposed solution: entangle (a specially designed) composite many-body system with a qubit
Will show that Entanglement Entropy can be measured by studying just the dynamics of the qubit
![Page 10: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/10.jpg)
-Many-body system in a pure state
-Reduced density matrix
-n-th Renyi entropy:
PROPERTIES:
-Universal scaling laws
-Analytic continuation n1 gives von Neumann entropy
-Knowing all Renyi entropies reconstruct full
entanglement spectrum (of )
-As useful as the von Neumann entropy
Renyi Entanglement Entropy
![Page 11: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/11.jpg)
System of interest
-Finite many-body system
-short-range interactions and hopping (e.g., Hubbard model)
-Ground state separated from excited states by a gap
Gapped phase:
Correlation length
Gapless phase
Fermi velocity
![Page 12: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/12.jpg)
Useful fact: relation of entanglement and overlap of a composite many-body system-Consider two identical copies of the many-body system2 Different ways of connecting 4 sub-systems:
Way 1: Way 2:
-Overlap gives second Renyi entropy:
Ground state
Ground state
![Page 13: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/13.jpg)
DerivationSchmidt decomposition of a ground state for a single system
Orthogonal sets of vectors in L and R sub-systems
![Page 14: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/14.jpg)
DerivationSchmidt decomposition of a ground state for a single system
Orthogonal sets of vectors in L and R sub-systems
Represent ground states of the composite system using
Schmidt decomposition:
![Page 15: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/15.jpg)
DerivationSchmidt decomposition of a ground state for a single system
Zanadri, Zolka, Faoro ‘00, Horodecki, Ekert ’02; Cardy’11, others
Orthogonal sets of vectors in L and R sub-systems
Represent ground states of the composite system using
Schmidt decomposition:
![Page 16: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/16.jpg)
Main idea of the present proposal-Quantum switch coupled to composite system
(a two-level system)
-Controls connection of 4 sub-systems depending on its
state
Ground state
Ground state
Abanin, Demler, arXiv:1204.2819, Phys. Rev. Lett., in press
![Page 17: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/17.jpg)
Spectrum of the composite system
Energy
eigenfunction
Switch has no own dynamics (for now);
Two decoupled sectors
Eigenstates of a single system
![Page 18: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/18.jpg)
Introduce switch dynamics-Turn on
-Require:
(not too restrictive: gap is finite)
-For our composite many-body system,
such a term couples two ground states
-Effective low-energy Hamiltonian
Renormalized tunneling:
![Page 19: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/19.jpg)
Rabi oscillations: a way to measure the Renyi entanglement entropy Slowdown of the Rabi oscillations
due to the coupling to many-body
system
Bare Rabi frequency (switch uncoupled
from many-body system)
Rabi frequency is renormalized:
Gives the second Renyi entropy
Abanin, Demler, arXiv:1204.2819, Phys. Rev. Lett., in press
![Page 20: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/20.jpg)
Generalization for n>2 Renyi entropies-n copies of the many-body system
-Two ways to connect them
Ground state
Ground state
Overlap gives n-th Renyi entropy
![Page 21: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/21.jpg)
Proposed setup for measuring n>2 Renyi entropies
-Quantum switch controls the way in which 2n sub-system
are connected
-Renormalization of the Rabi frequency overlap
n-th Renyi entropy
![Page 22: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/22.jpg)
A possible design of the quantum switch in cold atomic systems
-quantum well
-polar molecule:
*forbids tunneling of blue particles -particle that constitutes many-body
system
tunneling
![Page 23: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/23.jpg)
A possible design of the quantum switch in cold atomic systems
-Doubly degenerate ground state that controls connection
of the composite many-body system
-Q-switch dynamics can be induced by tuning the
barriers between four wells
-Study Rabi oscillations by monitoring the population of the
wells
![Page 24: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/24.jpg)
Generalization to the 2D case
-2 copies of the system, engineer “double” connections across the boundary
AS/A
![Page 25: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/25.jpg)
Generalization to the 2D case
-Couple to an “extended” qubit living along the boundary
-Depending on the qubit state, tunneling either within or
between layers is blocked
-Measure n=2 Renyi entropy, and detect top. order
![Page 26: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/26.jpg)
Summary
-A method to measure entanglement entropy in a generic many-body systems
-Difficulty of measurement does not grow with the system size
APPLICATIONS
-Test scaling laws; detect location of critical points without
measuring order parameter
-Extensions to 2D – detect topological order?
MESSAGE: ENTANGLEMENT ENTROPY IS MEASURABLE
Details: Abanin, Demler, arXiv:1204.2819, Phys. Rev. Lett., in press(see also: Daley, Pichler, Zoller, arXiv:1205.1521)
![Page 27: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/27.jpg)
In collaboration with:
Michael Knap (Graz)
Yusuke Nishida (Los Alamos)
Adilet Imambekov (Rice)
Eugene Demler (Harvard)
PART 2: Time-dependent impurity in cold Fermi gas: orthogonality catastrophe and
beyond
![Page 28: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/28.jpg)
-Fermi-Fermi and Fermi-Bose mixtures realized
Strongly imbalanced mixtures of cold atoms
-Minority (impurity) atoms can
be localized by strong optical
lattice
-A controlled setting to study
impurity dynamics
Many groups: Salomon, Sengstock, Esslinger, Inguscio, I. Bloch,
Ketterle, Zwierlein, Hulet..
![Page 29: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/29.jpg)
Probing impurity physics: cold atomic vs. solid state systems
Cold atoms:
-Wide tunability via Feshbach
resonance: strong interactions
regime
-Fast control: quench-type
experiments possible
-Rich atomic physics toolbox:
direct, time-domain
measurements
Solid state systems
-Limited tunability
-Many-body time scales too
fast; dynamics beyond linear
response out of reach
-No time-domain experiments
Energy-domain only (X-ray
absorption)
![Page 30: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/30.jpg)
-Relevant overlap:
-- scattering phase shift at Fermi energy
-Manifestation: a power-law edge singularity in the X-ray
absorption spectrum
Orthogonality catastrophe and X-ray absorption spectra in solids
Without impurity
With impurity
Nozieres, DeDominicis; Anderson ‘69
-Response of Fermi gas to a suddenly introduced impurity
![Page 31: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/31.jpg)
Previously: (very long times)
Preview: Universal OC in cold atoms
(very small energies)
-No universality at short times/large energies (band
structure,scattering parameters unknown,…)
![Page 32: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/32.jpg)
Previously: (very long times)
Preview: Universal OC in cold atoms
(very small energies)
-This work: exact solution for (all times and energies);
-No universality at short times/large energies (band
structure,scattering parameters unknown,…)
![Page 33: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/33.jpg)
Previously: (very long times)
Preview: Universal OC in cold atoms
(very small energies)
-This work: exact solution for (all times and energies);
-Universal, determined only by impurity scattering length
-Time domain: new important oscillating contribution
to overlap
-Energy domain: cusp singularities in with a new exponent at
energy above absorption threshold
-No universality at short times/large energies (band
structure,scattering parameters unknown,…)
![Page 34: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/34.jpg)
-Fermi gas+single localized impurity
-Two pseudospin states of impurity, and
- -state scatters fermions
-state does not
-Scattering length
Setup
-Pseudospin can be manipulated optically
*flip
*create coherent superpositions, e.g.,
-Study orthogonality catastrophe in frequency and time domain
![Page 35: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/35.jpg)
-Entangle impurity pseudospin and Fermi gas;
-Utilize optical control over pseudospin study Fermi gas
dynamics
-Ramsey protocol
1) pi/2 pulse
2) Evolution
3) pi/2 pulse, measure
Ramsey interferometry –probe of OC in the time domain
![Page 36: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/36.jpg)
Free atom
RF spectroscopy of impurity atom: OC in the energy domain
Atom in a Fermi sea – OC completely changes absorption function
New cusp
singularity
![Page 37: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/37.jpg)
-Certain sets of excited states are important
-Edge singularity (standard): multiple low-energy e-h pairs
-Singularity at : extra electron -- band bottom to Fermi surface +
multiple low-energy e-h pairs
Origin of singularities in the RF spectra:an intuitive picture
Singularity at EfThreshold singularity
![Page 38: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/38.jpg)
-Solution in the long-time limit is known (Nozieres-
DeDominicis’69); based on solving singular integral equation
OUR GOAL: full solution at all times
-Approach 1: write down an integral equation with exact
Greens functions; solve numerically (possible, but difficult)
-Approach 2: reduce to calculating functional determinants
(easy)
Functional determinant approach to orthogonality catastrophe
Combescout, Nozieres ‘71; Klich’03, Muzykantskii’03; Abanin, Levitov’04; Ivanov’09; Gutman, Mirlin’09-12…..
![Page 39: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/39.jpg)
Represent as a determinant in
single-particle space
Functional determinant approach to orthogonality catastrophe
Fermi distribution
function
Time-dep. scattering
operator
-Long-time behavior: analytical solution possible
Muzykantskii, Adamov’03, Abanin, Levitov’04,…
-Arbitrary times (this work): evaluate the determinant
numerically; certain features (prefactors, new cusp singularity)
found analytically
Desired response function
Many-body trace
![Page 40: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/40.jpg)
-No impurity bound state
-Leading power-law decay
-Sub-leading oscillating
contribution due to van Hove
singularity at band bottom
Results: overlap, a<0
![Page 41: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/41.jpg)
-Impurity potential does not
create a bound state
-Single threshold
Universal RF spectra for a<0
![Page 42: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/42.jpg)
-Single threshold
-New non-analytic
Feature at
Universal RF spectra for a<0
![Page 43: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/43.jpg)
-Origin: combined dynamics of hole at band bottom+e-h pairs
-Becomes more pronounced for strong scattering
-Smeared on the energy scale
-At the unitarity, evolves into true power-law
singularity with universal exponent ¼!
Cusp singularity at Fermi energy
Zoom
Knap, Nishida, Imambekov, DA, Demler, to be published
![Page 44: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/44.jpg)
Universal RF spectra for a>0
-Impurity potential creates a
bound state
-Double threshold (bound state
filled or empty)
-Non-analytic feature
at distance from first threshold
-Characteristic three-peak shape
![Page 45: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/45.jpg)
Summary
-New regimes and manifestation of orthogonality catastrophe in cold atoms
-Exact solutions for Fermi gas response and RF spectra obtained; New singularity found
-Spin-echo sequences probe more complicated dynamics of Fermi gas
-Extensions to multi-component cold atomic gases simulate quantum transport and more…
Knap, Nishida, Imambekov, Abanin, Demler, to be published
![Page 46: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/46.jpg)
a<0; no bound stateWeak oscillations from van
Hove singularity at band
bottom
Results: overlap
a>0; bound stateStrong oscillations
(bound state either filled or empty)
![Page 47: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/47.jpg)
Represent
Functional determinant approach to orthogonality catastrophe
w/o impurity with impurity
Density matrix
Trace is over the full many-body state; dimensionality
-number of single-particle states
![Page 48: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/48.jpg)
Consider quadratic many-body operators
A useful relation
Then
Trace over many-body space (dimensionality )
Determinant in the single-particle space (dimensionality )
-Holds for an arbitrary number of exponential operators
-Derivation:
step1 – prove for a single exponential (easy)
step2 – for two or more exponentials, use Baker-Hausdorf formula
reduce to step 1
![Page 49: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/49.jpg)
Rich many-body physics
Single impurity problems in condensed matter physics
-Edge singularities in the
X-ray absorption spectra(asympt. exact solution of non-
Equilibrium many-body problem)
-Kondo effect: entangled
state of impurity spin and
fermions
Influential area, both for methods (renormalization group) and for strongly correlated materials
![Page 50: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/50.jpg)
no bound state
-Power-law decay
-Weak oscillations from van
Hove singularity at band
bottom
Results: overlap
![Page 51: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/51.jpg)
-Many unknowns;
Simple models hard to test(complicated band structure, unknown
impurity parameters, coupling to phonons,
hole recoil)
-Limited probes(usually only absorption spectra)
-Dynamics beyond linear response
out of reach (relevant time scales GHz-THz,
experimentally difficult)
Probing impurity physics in solids is limited
X-ray absorption in Na
![Page 52: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/52.jpg)
-Parameters known fully universal
properties
-Tunable by the Feshbach resonance
(magnetic field controls scatt.
length) access new regimes
-Fast control of microscopic parameters
(compared to many-body scales)
-Rich toolbox for probing many-body states
Cold atoms: new opportunities for studying impurity physics
![Page 53: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/53.jpg)
-Overlap
- as system size , “orthogonality catastrophe”
-Infinitely many low-energy electron-hole pairs produced
Introduction to Anderson orthogonality catastrophe (OC)
Fundamental property of the Fermi gas
![Page 54: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/54.jpg)
-Relevant overlap:
-- scattering phase shift at Fermi energy
-Manifestation: a power-law singularity in the X-ray absorption spectrum
Orthogonality catastrophe and X-ray absorption spectra
Without impurity
With impurity
Nozieres, DeDominicis; Anderson
![Page 55: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/55.jpg)
Represent
Functional determinant approach to orthogonality catastrophe
w/o impurity with impurity
Density matrix
Trace is over the full many-body state; dimensionality
-number of single-particle states
![Page 56: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/56.jpg)
Consider quadratic many-body operators
A useful relation
Then
Trace over many-body space (dimensionality )
Determinant in the single-particle space (dimensionality )
-Holds for an arbitrary number of exponential operators
-Derivation:
step1 – prove for a single exponential (easy)
step2 – for two or more exponentials, use Baker-Hausdorf formula
reduce to step 1
![Page 57: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/57.jpg)
-Response of Fermi gas to process in which impurity
switches between different states several times
Spin echo: probing non-trivial dynamics of the Fermi gas
-Advantage: insensitive to slowly fluctuating magnetic fields
(unlike Ramsey)
-Such responses cannot be probed in solid state systems
![Page 58: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/58.jpg)
Spin echo response: features
-Power-law decay at long times with an enhanced exponent
-Unlike the usual situation, where
spin-echo decays slower than
Ramsey!
-Universal
-Generalize to n-spin-echo;
yet faster decay
![Page 59: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/59.jpg)
-So far, concentrated on measuring impurity properties
-Measurable property of the Fermi gas which reveals
OC physics?
Seeing OC in the state of fermions
![Page 60: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/60.jpg)
-Yes, distribution of energy fluctuations
following a quench
1) Flip pseudospin starting with interacting state
2) Measure distribution of total energy of fermions with new Hamiltonian
-Measurable in time-of-flight experiments
Seeing OC in the state of fermions
Overlap function
Also: Silva’09; Cardy’11
![Page 61: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/61.jpg)
Generalizations: non-equilibrium OC, non-commuting Riemann-Hilbert problem -Impurity coupled to several Fermi
seas at different chemical potentials
-Theoretical works in the context of quantum transport
-Mathematically, reduces to non-commuting Riemann-Hilbert problem (general solution not known)
-Experiments lacking
Muzykantskii et al’03Abanin, Levitov ‘05
![Page 62: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/62.jpg)
Multi-component Fermi gas: access to non-equilibrium OC and quantum transport in cold atomic system
DA, Knap, Nishida Demler, in preparation
-Fermions with two hyperfine states, u and d, +impurity
-Imbalance,
-pi/2 pulse on fermions
play the role of fermions in two leads
-Impurity scattering creates both “reflection” and “transmission”-”Simulator” of the non-equilibrium OC and quantum transport
![Page 63: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/63.jpg)
-OC for interacting fermions (e.g., Luttinger liquid)
-Dynamics: many-body effects in Rabi oscillations of
impurity spin
-Very different physics for an impurity inside BEC
Other directions
![Page 64: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/64.jpg)
Summary
-New manifestations of OC in atomic physics experiments and in energy counting statistics
-Exact solutions for Fermi gas response and RF spectra obtained; New singularities at Fermi energy
-Extensions to multi-component cold atomic gases simulate quantum transport and more
Knap, Nishida, DA, Demler, in preparation
![Page 65: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/65.jpg)
![Page 66: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/66.jpg)
![Page 67: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/67.jpg)
Spectrum of the composite system
entangled
entangled
Energy
eigenfunction
Switch has no own dynamics;
Two decoupled sectors
Eigenstates of a single system
![Page 68: Dmitry Abanin (Harvard) Eugene Demler (Harvard) Measuring entanglement entropy of a generic many-body system MESO-2012, Chernogolovka June 18, 2012 June](https://reader035.vdocuments.site/reader035/viewer/2022062313/56649d1f5503460f949f2df0/html5/thumbnails/68.jpg)
Multi-component Fermi gas: access to non-equilibrium OC and quantum transport in cold atomic system
DA, Knap, Demler, in preparation
-Imbalance different species
-Mix them by pi/2 pulse on
-Realization of non-equilibrium OC problem
-”Simulator” of quantum transport
and non-abelian Riemann-Hilbert problem
-Charge full counting statistics can be probed
Specie 1
Specie 2