Jan Zaanen

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Von Neumann entropy: entanglement and fields.

Bipartite vN entropy: measures entanglement = quantum information of Bell pairs

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|Bell =1200 + 11( ) ⇒ SvN = 2

Prod =120 + 1( ) 0 + 1( ) ⇒ SvN = 0

Field theory: infinite number of “qubits”, not two ..

A B

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ρA = TrBρ, SvN ,A = Tr ρA lnρA[ ]Divide space in two:

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“Space bipartite” Von Neumann entropy: success !

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ρA = TrBρ, SA = Tr ρA lnρA[ ]Divide space in two:

1+1 D CFT’s (Calabrese-Cardy):

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SA =c3logL c = central charge

Topological insulators (Kitaev-Preskil, Levin-Wen):

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SA = αΣ − γ +, Σ = Ld −1, γ = logD D = total quantum dimension.

Entanglement spectrum (Haldane):

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SA = − pλλ

∑ ln pλ , pλ =e−ξλ

e−ξλλ

∑Measures the edge/surface states without a surface!

Conformal fields (Klebanov etc.):

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SA = αΣ − F

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“Space bipartite” Von Neumann entropy: compressible systems.

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ρA = TrBρ, SA = Tr ρA lnρA[ ]Divide space in two:

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SA ,bosons = αΣ+, Σ = Ld −1€

SA = − pλλ

∑ ln pλ , pλ =e−ξλ

e−ξλλ

∑

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SA ,FL ∝ ΣlogΣ+

Bosonic field theory: scales with surface area.

momentum space

Single point in momentum space with massless excitations.

Fermi-liquid: extra log, longer range!

d-1 dimensional surface of massles excitations in momentum space = Fermi surface.

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AdS/CFT: entanglement entropy and the minimal surface.

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ρA = TrB ρ[ ]

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SvN = Tr ρA lnρA[ ]

The spatial bipartite entanglement entropy in the boundary is dual to the area of the minimal surface in the bulk, bounded by the cut in the space of the boundary (Takayanagi-Ryu conjecture).

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Holographic strange metal entanglement entropy.

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ds2 =1r2

−dt 2

r2d (z−1)/(d −θ )+ r2ϑ /(d −θ )dr2 + dxi

2⎛

⎝ ⎜

⎞

⎠ ⎟

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xi →ζ xi, t →ζzt, ds→ζθ / d ds

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S ∝ T (d −θ ) / z

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SvN ∝ Σ ,θ < d −1SvN ∝ ΣlnΣ ,θ = d −1SvN ∝ Σθ /(d −1) ,θ > d −1

Geometry in Einstein-Maxwell-Dilaton bulk:

“Hyperscaling violation” in boundary theory:

Thermal entropy:

Entanglement entropy (Sachdev, Huijse, Swingle):

Looks much like a deconfined Fermi-liquid, hidden from the gauge singlet UV propagators!

But this is longer ranged !?

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The Reissner-Nordstrom AdS2 horizon: local quantum criticality.

Near-horizon geometry of the extremal RN black hole: - Space directions: flat, codes for simple Galilean invariance in the boundary.

- Time-radial(=scaling) direction: emergent AdS2, codes for emergent temporal scale invariance!

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A(k,ω )∝G"AdS2 k,ω( )∝ω 2ν k

ν k =16

k 2 +1ξ2

(Fermion) spectral functions: “Algebraic Pseudogap metal”

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Holographic very strange metal entanglement entropy

Ground state entropy:

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S(T = 0)∝ µN 2 Needs both finite density and large N.

RN metal holographic entanglement entropy:

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SvN ∝ ld

For infinite l this turns into the T=0 entropy:

(Sachdev)

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ST =0 = Tr ρ lnρ][ ]Counting excitations: Massless point in momentum space = bosonic CFT’s: Massless surface in momentum space = Fermi liquid: SvN = (# points on FS) (CFT2 SvN)

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SvN ∝ ld −1

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SvN ∝ ld −1( ) ln l( )Massless volume in momentum space = RN metal: SvN = (#points in k space)(CFT1 SvN)

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SvN ∝ ld

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θ = d 1− 1z

⎛

⎝ ⎜

⎞

⎠ ⎟ , z→∞

Volume scaling!

Fermions at a finite density: the sign problem.

Imaginary time first quantized path-integral formulation

Boltzmannons or Bosons:

  integrand non-negative

  probability of equivalent classical system: (crosslinked) ringpolymers

Fermions:

  negative Boltzmann weights

  non probablistic: NP-hard problem (Troyer, Wiese)!!!

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A universal phase diagram

Quantum critical

Heavy fermions High Tc superconductors

Iron superconductors (?)

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Fermions at a finite density: land of opportunity?

Dealing with fermions at finite density we just know two serious (mathematical) operations:

1. Declare them to be a Fermi-gas that you might want to hit with uncontrolled perturbation theory.

2. Declare that they run into a BCS/Hartree-Fock “bosonic” instability.

But behind the “sign brick wall” there might be a vast landscape of new states of truly extreme quantum matter waiting to be discovered!

Is this what the holographic strange metals try to tell to us?

But first some other glimpses behind the sign brick wall …

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Von Neumann entropy: entanglement versus quantum statistics.

“Entanglement” entropy = quantum information of Bell pairs.

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|Bell =1200 + 11( ) ⇒ SvN = 2

Is the Fermi gas a “better” quantum information resource??

Von Neumann entropy can speak with two tongues: it scrambles quantum-statistical non-locality and quantum information.

Fermi-Dirac is classical non-local information: “Mottness in momentum space”.

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SvN ,FL ∝ ΣlogΣ+Fermi gas vN entropy is “longer” ranged:

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Quantum statistics and first quantized path integrals

Fermions: infinite cycles set in at TF, but cycles with length w and w+1 cancel each other approximately. Free energy pushed to EF!

Cycle decomposition

Bose condensation: Partition sum dominated by infinitely long cycles

Feynman

The nodal hypersurface

Antisymmetry of the wave function

Nodal hypersurface

Pauli hypersurface Free Fermions

Test particle

d=2

Constrained path integrals Formally we can solve the sign problem!!

Self-consistency problem:

Path restrictions depend on !

Ceperley, J. Stat. Phys. (1991)

Ceperley path integral: Fermi gas in momentum space

Sergei Mukhin

Single particle propagator:

single particle momentum conserved N particle density matrix:

‘harmonic potential’

Fermi gas = cold atom Mott insulator in harmonic trap!

Mukhin, JZ, ..., Iranian J. Phys. (2008)

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ρF (K,K";τ) = 2πδ ki − ki"( )

k1≠k2≠≠kN

∏ e−|ki |

2 τ

2m

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Mottness: traffic jam non-locality

This car is stuck

Because of this ..

By von Neumann cut it might look “quantum” entangled …

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The origin of the long range Fermi-gas “entanglement” entropy

Extreme non-locality in real space. Fermi gas: Mottness in k space.

Fourier transformation

Change one particle coordinate and the nodal surface changes everywhere.

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SA ,FL ∝ ΣlogΣ+

is a consequence of Fermi-Dirac statistics which is in turn a carrier of purely classical non-local information.

The constrained path integral leaves no doubt:

Mott insulating Cold atoms in real space

Extreme non-locality in k space

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Cuprates start as doped Mott-insulators Anderson

Mott insulator Doped Mott insulator

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H = t ciσ+

ij∑ c jσ +U ni↑

i∑ ni↓

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Ht−J = t c iσ+

ij∑ c jσ + J

S i

ij∑ •

S j

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Mott insulator: the vanishing of Fermi-Dirac statistics

Mott-insulator: the electrons become distinguishable, stay at home principle!

Spins live in tensor-product space. “Spin signs” are like hard core bosons in a magnetic field, can be gauged away on a bipartite lattice (“Marshall signs”)

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τ c = +1

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Doped Mott-insulator: Weng statistics Zheng-Yu Weng

t-J model: spin up is background, spin down’s (‘spinons’) and holes (‘holons’) are hard core bosons.

Exact Partition sum:

The sign of any term is set by:

The (fermionic) number of holon exchanges

The number of spinon-holon ‘collisions’

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Nhh c[ ]

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Nh↓ c[ ]

arXiv: 0802.0273

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Anderson Localization without quenched disorder!

Zhu et al., subm. Science

“Numerically exact” DMRG of one hole t-J model on n-leg ladders: remnant signs cause Anderson localization of the hole on the perfect lattice!

Constrained path integrals Formally we can solve the sign problem!!

Self-consistency problem:

Path restrictions depend on !

Ceperley, J. Stat. Phys. (1991)

Reading the worldline picture Fermi-energy: confinement energy imposed by local geometry

Average node to node spacing

Fermi surface encoded globally:

Change in coordinate of one particle changes the nodes everywhere

Finite T:

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λnl = vFτ inel = vFEF

kBT⎛

⎝ ⎜

⎞

⎠ ⎟ kBT⎛

⎝ ⎜

⎞

⎠ ⎟

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ρF = Det eiki rj( ) = 0

Non-locality length:

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ρF = (4πλβ)−dN / 2Det exp −(ri − rj0)

2

4λτ

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

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λ = 2 /(2M)

Key to fermionic quantum criticality

At the QCP scale invariance, no EF Nodal surface has to become fractal !!!

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Vacuum structure

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ρF R,R(τ );τ →∞( ) = Ψ* R( )Ψ R(∞)( )

Long time, zero temperature:

IR fermionic information encoded in the ground state wavefunction.

Need the answer: the wave function!

Hydrodynamic backflow

Feynman-Cohen: mass enhancement in 4He Classical fluid: incompressible flow

Wave function ansatz for „foreign“ atom moving through He superfluid with velocity small compared to sound velocity:

Backflow wavefunctions in Fermi systems

Widely used for node fixing in QMC

→  Significant improvement of variational GS energies

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Frank’s fractal nodes … Feynman‘s fermionic backflow wavefunction:

Frank Krüger

Fractal dimension of the nodal surface

Calculate the correlation integral on random d=2 dimensional cuts

Backflow turns nodal surface into a fractal !!!

Fermionic quantum phase transitions in the heavy fermion metals

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m* =1EF

EF → 0⇒ m* →∞

QP effective mass

Turning on the backflow Nodal surface has to become fractal !!!

Try backflow wave functions

Collective (hydrodynamic) regime:

MC calculation of n(k)

Divergence of effective mass as a→ac

mm* ∝ 1− a

ac

⎛

⎝⎜⎞

⎠⎟

3

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Fractal nodes versus conformal fermion matter

Is the fractal geometry of the nodal surface necessary condition for the finite density conformal fermion liquid?

Credible but no proof …

Is the hydrodynamical back flow system representative?

Suspicious: hidden free system …

Bare fermion propagators: quasi-local quantum criticality?

Computations that can be done (hard work in progress):

vN entropy: the fractal nodes encode more strongly non-local “statistical” information than “Fermi-gas Mottness” !?

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The full density matrix: a trace too far.

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Ψ X ( ) =ψ

x 1( )ψ x 2( )ψ

x N −1( )ψ x N( )

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ρ X , X ' τ( );τ( ) ≡ Ψ

X ,0( )Ψ

X ',τ( ) , τ →∞

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ρA = TrB ρ[ ]

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SvN = Tr ρA lnρA[ ]

Field theory: the density matrix is the n-point propagator.

The von Neumann entropy departs from this information.

But by taking the traces it scrambles fermion statistics information and quantum information.

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The nodal surface and the dictionary.

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Ψ X ( ) =ψ

x 1( )ψ x 2( )ψ

x N −1( )ψ x N( )

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ρ X , X ' τ( );τ( ) ≡ Ψ

X ,0( )Ψ

X ',τ( ) = 0, τ →∞

Diagnosing generalized (non Fermi-Dirac) fermion statistics:

One needs an answer first, the nodal surface is diagnostics:

Are the holographic strange metals characterized by unconventional (non Fermi-gas) nodal structure?

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The nodal surface and the dictionary (I).

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Ψ X ( ) =ψ

x 1( )ψ x 2( )ψ

x N −1( )ψ x N( )

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ρ X , X ' τ( );τ( ) ≡ Ψ

X ,0( )Ψ

X ',τ( ) = 0, τ →∞

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g x , x ';τ( ) = ψ x ,0( )ψ x ',τ( )

Computing the n-point propagator holographically:

A first guess: identify with the O(1/N) gauge singlet fermions.

Liu, McGreevy, … : in this order AdS/CFT turns into an effective weak-weak duality with the ramification that n-point propagators are described by weak coupling (bulk) diagrams, inserting fully dressed (strong coupling) single fermion propagators

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ψ x ( )

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The nodal surface and the dictionary (II).

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Ψ X ( ) =ψ

x 1( )ψ x 2( )ψ

x N −1( )ψ x N( )

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ρ X , X ' τ( );τ( ) ≡ Ψ

X ,0( )Ψ

X ',τ( ) = 0, τ →∞

Computing the n-point propagator holographically:

But we would like to know how this acts out for the “deconfined stuff behind the horizon” of the large N limit, look for color ….

The vN entropy (full trace) shows that the information is there.

Conundrum: what is the dictionary entry yielding nodal surface information?

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Summary …

Fermions at finite density: the fermion signs are wrecking established mathematical machinery, but it leaves room for BIG surprises.

Is AdS/CFT revealing such surprises? The AdS2 strange metals, the emergent Fermi liquids, the holographic superconductors.

General measure of non FD fermion statistics: Ceperley’s nodal surface. When signs are the cause, the nodal surfaces of the temporally conformal AdS/CFT strange metals have to be fractals!

What is the dictionary entry for nodal surface information?

The entanglement entropy knows about “fermion signs” but is ambiguous: in the Fermi gas it reveals Fermi-Dirac = classical non-local information.

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References

Fermions and conformal invariance: the fractal nodal surface.

Fermions, information and the constrained path integral. J. Zaanen, F. Kruger, J.-H. She, D. Sadri, S.I. Mukhin, arXiv: 0802.2455 (Iranian J. Phys).

F. Kruger and J. Zaanen, arXiv:0804.2161 (Phys. Rev. B 78, 035104)

Mottness: fermion statistics which is not Fermi-Dirac. J. Zaanen, B. Overbosch, arXiv:0911.4070 (Phil. Trans. Roy. Soc.)