Post on 05-Jun-2018
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
⎛
⎝⎜⎞
⎠⎟
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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.)