Dmitry Abanin (Harvard)
Eugene Demler (Harvard)
Measuring entanglement entropy of a generic many-body system
MESO-2012, Chernogolovka
June 18, 2012
-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
-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
-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
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
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…
-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
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..
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
-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
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
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
DerivationSchmidt decomposition of a ground state for a single system
Orthogonal sets of vectors in L and R sub-systems
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:
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:
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
Spectrum of the composite system
Energy
eigenfunction
Switch has no own dynamics (for now);
Two decoupled sectors
Eigenstates of a single system
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:
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
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
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
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
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
Generalization to the 2D case
-2 copies of the system, engineer “double” connections across the boundary
AS/A
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
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)
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
-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..
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)
-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
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,…)
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,…)
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,…)
-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
-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
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
-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
-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…..
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
-No impurity bound state
-Leading power-law decay
-Sub-leading oscillating
contribution due to van Hove
singularity at band bottom
Results: overlap, a<0
-Impurity potential does not
create a bound state
-Single threshold
Universal RF spectra for a<0
-Single threshold
-New non-analytic
Feature at
Universal RF spectra for a<0
-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
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
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
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)
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
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
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
no bound state
-Power-law decay
-Weak oscillations from van
Hove singularity at band
bottom
Results: overlap
-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
-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
-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
-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
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
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
-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
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
-So far, concentrated on measuring impurity properties
-Measurable property of the Fermi gas which reveals
OC physics?
Seeing OC in the state of fermions
-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
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
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
-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
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
Spectrum of the composite system
entangled
entangled
Energy
eigenfunction
Switch has no own dynamics;
Two decoupled sectors
Eigenstates of a single system
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