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Holographic methods forcondensed matter physics
Sean Hartnoll
Harvard University
Jan. 09 – CERN
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 1 / 29
Lecture I
Part I: Why AdS/CMT?
1 Motivations
2 Quantum criticality
3 Nonconventional superconductors
Part II: Geometric duals for scale invariant theories
1 The AdS/CFT logic
2 Scale invariance and z
3 Scale invariant geometries
4 What is z in the real world?
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 2 / 29
Future lectures II, III, IV
Lecture II
1 Away from scale invariance
2 Away from equilibrium – transport
Lecture III
1 The physics of spectral functions
2 Examples from experiments and from AdS/CFT
Lecture IV
1 Holographic superconductivity
2 Landscape of superconducting membranes
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 3 / 29
Part I: Why AdS/CMT?
1 Motivations
2 Quantum criticality
3 Nonconventional superconductors
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 4 / 29
Why condensed matter?
The logic of these lectures• Traditional condensed matter physics considers weakly interacting
‘quasiparticles’. Extremely successful:• e.g. Landau liquid theory + BCS theory of superconductivity.
• Doesn’t work for materials with strongly correlated electrons.• e.g. High temperature superconductors.• e.g. Near quantum critical points.
• AdS/CFT allows computations at strong coupling, including at finitetemperature, and a novel conceptual framework.
• Quantum critical points are scale invariant→ ideal kinematics for AdS/CFT.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 5 / 29
Why condensed matter?
Some ‘big picture’ comments
• There is a unique Lagrangian for particle physics in our universe.
• There are many Lagrangians in condensed matter systems, and anincreasing range can be engineered.
• e.g. Atoms in optical lattices.
• One day: SUSY in a lab? Experimental AdS/CFT?
• AdS/CFT suggests that there is not a unique ‘fundamental’formulation of physics, but different duality frames.
• e.g. Superconductivity ↔ new types of hairy black hole.• A unified approach to physics.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 6 / 29
Quantum criticality
• Quantum critical points: continuous phase transitions at T = 0.
• Lead to a strongly coupled quantum critical region of phase diagram:
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 7 / 29
Quantum criticality
Comments on phase transitions and dimensionality• Coleman-Mermin-Wagner-Hohenberg theorem: no second order phase
transitions in 2 dimensions. I.e.• 2+1 at finite temperature.• 1+1 at zero temperature.
• Massless goldstone bosons have an IR divergence below 2 dimensions:‘Fluctuations destroy order’.
• Exception I: At infinite N, no fluctuations. Pin system to extremum.
• ‘Exception’ II: Berezinsky-Kosterlitz-Thouless transition in 2+1dimensions at finite temperature (infinite order transition,condensation of vortices, algebraically quasi-ordered phase).
• We will often be working in 2+1 dimensions.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 8 / 29
Example I: Wilson-Fisher fixed point
• Let Φ be an N dimensional vector:
S [Φ] =
∫d3x
((∂Φ)2 + rΦ2 + u
(Φ2)2)
.
• Flows to strongly coupled fixed point when r → rc (quantum critical).
• This, relativistic, theory emerges at a critical point in an insulatingquantum magnet
HAF =∑〈ij〉
JijSi · Sj .
• Si : spin on a 2D square lattice.
• Antiferromagnetic: Jij > 0.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 9 / 29
• Take couplings J or J/g as shown below right (dashed = J/g).
• Tune g from 1 to ∞. Ground state:
→• g = 1 ground state is Neel ordered: 〈
∑i (−1)iSi 〉 = Φ 6= 0 .
• g =∞ ground state is decoupled, spin singlet, dimers. Spin rotationpreserved.
• Therefore: ∃ Quantum phase transition! Numerically: gc ≈ 1.91.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 10 / 29
• Excitations about Neel state: spin density waves. Φ4 theory withN = 3 in symmetry broken phase (Φ is the order parameter).
• Excitations about dimer state: triplons
• Three polarisations of triplon: Φ4 theory with N = 3 in symmetricphase.
• Low energy dynamics of entire phase diagram described by Φ4 theory,including the Wilson-Fisher fixed point.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 11 / 29
Example II: spinons and emergent photons
• Let A be an (emergent 2+1 dimensional) photon, z a complex spinor
S [z ,A] =
∫d3x
(|(∂ − iA) z |2 + r |z |2 + u
(|z |2)2
+1
2e20
F 2
).
• Flows to fixed point at r → rc . For r < rc can map to previousexample via
Φ = zασαβzβ .
• Taking again an insulating quantum magnet (J,Q > 0)
HVBS = J∑〈ij〉
Si · Sj − Q∑〈ijkl〉
(Si · Sj −
1
4
)(Sk · Sl −
1
4
),
• Preserves lattice rotations (Z4), unlike previous example.
• Second term favours 4 spins on a plaquette forming 2 singlet pairs.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 12 / 29
• For J/Q � 1, back to Neel state. Already discussed.• Order parameter: Φ = 〈
∑i (−1)iSi 〉 6= 0.
• Breaks spin rotation but preserves lattice rotation.
• For J/Q � 1 get valence bond solid (VBS).• Preserves spin rotation but breaks lattice rotation.• Order parameter: Ψ = (−1)jx Sj · Sj+x + i(−1)jy Sj · Sj+y 6= 0.• Has a spin half excitation: spinon.
→
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 13 / 29
• Emergent photon has a beautiful description in the VBS phase.
• In 2+1 dimensions we can dualise the photon to a scalar
?3F = dζ .
Revealing a ‘dual’ symmetry ζ → ζ + δζ.
• The VBS order parameter turns out to be a condensate of monopoles
Ψ ∼ e i2πζ/e20 .
• The spontaneously broken symmetry: ζ → ζ + δζ is in fact therotational symmetry of the lattice.
• The dual photon ζ is precisely the goldstone boson for spatialrotations.
• The dual photon is massive for r > rc because the lattice breaksU(1)→ Z4, but becomes massless at the quantum critical point.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 14 / 29
Nonconventional superconductors
• Conventional superconductivity is described by BCS theory (1957).
• BCS theory: Lattice vibrations (phonons) mediate an attractive forcebetween (dressed) electrons. At low temperatures an electron bilinearcondenses: 〈ΨΨ〉 6= 0.
• Many superconductors not describable by BCS theory (eg. high-Tc).
• Non-BCS superconductivity might mean:• weak: ‘pairing mechanism’ does not involve phonons but a different
‘glue’ such as paramagnons (mediate spin-spin forces).• strong: inherently strongly coupled. No charged quasiparticles to ‘pair’
in the first place.
• Candidates for the stronger case if superconductivity close to aquantum critical point.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 15 / 29
Two phase diagrams:
• Left: ‘heavy fermion’ compound. Clear connection to quantumcritical point.
• Right: high-Tc cuprate. Experimental evidence for nearby VBS orderand possible quantum critical point.
• Quantum critical points similar to those we discussed(antiferromagnetism).
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 16 / 29
Part II: Geometric duals for scale invariant theories
1 The AdS/CFT logic
2 Scale invariance and z
3 Scale invariant geometries
4 What is z in the real world?
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 17 / 29
The AdS/CFT logic
• The large N limit is a classical limit.
• For gauge theories: what are the classical saddles?
• AdS/CFT: for some theories, at least, the classical saddles aresolutions to a theory of gravity in one (or more) higher dimensions!
• AdS/CFT geometrises the energy scale. (cf. renormalsation groupequations suggest ‘locality’ in energy).
• Minimal AdS/CFT structure:
Large N gauge theoryd spacetime dimensions
!Classical gravitational theoryd + 1 spacetime dimensions.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 18 / 29
Scale invariance and z
• Field theories can be defined with a cutoff or at a UV fixed point.
• At a fixed point, theory is invariant under space and time scaling
t → λz t , ~x → λ~x .
• z is the dynamical critical exponent. There is no reason for z = 1.
• Minimal algebra has {Mij ,Pk ,H,D}. Dilatations act
[D,Mij ] = 0 , [D,Pi ] = iPi , [D,H] = izH .
• Sometimes called the Lifshitz algebra.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 19 / 29
Scale invariant geometries
• AdS/CFT asks: can we realise the Lifshitz algebra geometrically?
• Kachru et al. reply (2008):
ds2 = L2
(−dt2
r2z+
dx idx i
r2+
dr2
r2
).
• Killing vectors generating algebra
Mij = −i(x i∂j − x j∂i ) , Pi = −i∂i , H = −i∂t ,
D = −i(z t ∂t + x i∂i + r ∂r ) .
• z = 1 is AdSd+1, ehancement to Lorentzian conformal algebra.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 20 / 29
Comments on the Lifshitz geometry
• If z > 1 then near the boundary gtt diverges faster than gxixi .Lightcones flatten. Effective speed of light diverges near the boundary
→ nonrelativistic causality of field theory dual.
• If z 6= 1 the spacetimes are singular at the ‘horizon’ r =∞.(null ‘pp’ singularity – implies no global extension. Physics?).
• The case z = 1 is a solution to Einstein gravity
S =1
2κ2
∫dd+1x
√−g
(R +
d(d − 1)
L2
).
Other cases need additional matter.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 21 / 29
More symmetry! – adding Galilean boosts
• In classical mechanics Galilean boosts: {xi → xi + vi t, t → t} satisfy
[Mij ,Kk ] = i(δikKj − δjkKi ) , [Pj ,Ki ] = 0 , [H,Ki ] = −iPi .
• Quantum mechanical representations require a central extension
[Pj ,Ki ] = 0 [Pj ,Ki ] = −iδijN .
N can be interpreted as the particle number/total mass.
• Galilean algebra is consistent with scale invariance if
[D,Ki ] = i(1− z)Ki , [D,N] = i(2− z)N .
• Algebra {Mij ,Pi ,H,D,Ki ,N} sometimes called Schrodinger algebra.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 22 / 29
Comments on the Schrodinger algebra
• When z = 2, symmetry can be further extended by a ‘special’conformal generator.
• Dilatation only commutes with the particle number (also: the totalmass) if z = 2. This is because mass is dimensionless if z = 2.
• Implication: cannot have a discrete spectrum for the particle numberin a scale invariant theory if z 6= 2.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 23 / 29
Galilean scale-invariant geometry
• The Schrodinger geometry (Son, McGreevy + Balasubramanian 2008)
ds2 = L2
(−dt2
r2z− 2dtdξ
r2+
dx idx i
r2+
dr2
r2
).
• Killing vectors generating the new symmetries
Ki = −i(−t∂i + x i∂ξ) , N = −i∂ξ ,
D = −i(zt∂t + x i∂i + (2− z)ξ∂ξ + r∂r ) .
• Extra dimension ξ for particle number symmetry. To get discreteparticle spectrum
ξ ∼ ξ + 2πLξ .
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 24 / 29
Comments on Schrodinger geometries
• When z 6= 2, dilatation is not a symmetry if ξ identified.
• ξ is a null direction. Identification is dangerous (e.g. wound strings).
• The sign of gtt can be changed leaving the solution Lorentzian.However, with the opposite sign lightcones no longer collapse near theboundary. Furthermore, the spacetime appears to be unstable in thiscase.
• For z 6= 1, need matter fields to solve equations of motion.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 25 / 29
A physical consequence of z
S =
∫dd−1kdω
(2π)d
(r + k2 + ω2/z
)|Φ(ω, k)|2 +
∫dd−1xdt u (Φ2)2 .
• The scaling dimension of the coupling u can be worked out to be
[u] = (5− d − z) .
• Therefore the coupling u is irrelevant if
d > dc = 5− z .
• For z > 1 critical dimension is lowered!• E.g. in d = 3 = 2 + 1:
• z = 1: u relevant.• z = 2: u marginal.• z = 3: u irrelevant.
• Higher z more amenable to perturbative treatment in d = 2 + 1.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 26 / 29
Examples of z in the real world
• z = 1:• Insulating quantum antiferromagnets (relevant for high-Tc).• Bose Hubbard-like models at p/q filling (e.g. optical lattices).
• z = 2: Itinerant anitferromagnetism (heavy fermions).
SiA.F . =
∫dtdd−1x
[−γΦ∂tΦ + (∂xΦ)2 + rΦ2 + u
(Φ2)2]
.
• z = 3: Itinerant ferromagnetism (heavy fermions).
SiF . = −∫
dt dd−1k
(2π)d−1
1
v |k |Φ∂tΦ+
∫dtdd−1x
[(∂xΦ)2 + rΦ2 + u
(Φ2)2]
.
• z nonuniversal: ‘local quantum criticality’ (heavy fermions).E.g. z ≈ 2.6 in CeCu6−xAux at critical doping.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 27 / 29
A choice of z
For the remainder of these lectures I will focus on z = 1 for the followingreasons:
• Plenty of interesting quantum critical points in nature with z = 1.
• In 2+1 dimensions and within certain classes of models, z = 1theories are the most likely to strongly coupled.
• The AdS/CFT dictionary is not yet fully developed for z 6= 1.
• The z = 1 background solves the Einstein equations without extramatter. This may make predictions more robust.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 28 / 29
Summary so far
• Quantum critical phases have an underlying scale invariant and oftenstrongly coupled field theory.
• May be important for nonconventional superconductivity.
• Aim to study these theories using AdS/CFT.
• Discussed geometric realisations of scale invariance for different valuesof dynamical critical exponent z .
• Options for z 6= 1: With and without Galilean symmetry
• Galilean symmetry most natural if z = 2.
Sean Hartnoll (Harvard) AdS/CMT Jan. 09 – CERN 29 / 29