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