Quantum matter & black hole ringing

Transcription

1 Quantum matter & black hole ringing Sean Hartnoll Harvard University Work in collaboration with Chris Herzog and Gary Horowitz : , Frederik Denef : Frederik Denef and Subir Sachdev : 0907.????. AdS/CMT July 09 KITP Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 1 / 21

2 Plan of talk Motivation unconventional phases at finite density 1 Low temperature and finite density (bosons and fermions) 2 Two uses of magnetic fields 3 Experimental examples 4 Free fermions and bosons Strongly coupled theories with gravity duals 1 The normal state 2 Large N magnetic susceptibility 3 Black hole instabilities (superconducting instabilities) 4 1/N corrections to the free energy 5 Black hole ringing 6 Quantum oscillations Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 2 / 21

3 Motivation unconventional phases at finite density 1 Low temperature and finite density (bosons and fermions) 2 Two uses of magnetic fields 3 Quantum oscillations in High - T c superconductors 4 Quantum criticality under the dome in High - T c superconductors 5 Free fermions 6 Free bosons Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 3 / 21

4 Low temperature and finite density Effective field theories in condensed matter physics often have a finite charge density. Weak coupling intuition at low temperatures and finite density: Charged fermions: Fermi surface is built up. Charged bosons: condensation instabilities (e.g. superconductivity). Weakly interacting low energy excitations about a condensate or Fermi surface are very well characterised. There seem to be materials where these descriptions do not work. Perspective of this talk: AdS/CFT gives a tractable theory with an exotic finite density ground state. Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 4 / 21

5 Two uses of magnetic fields Magnetic fields useful for probing both fermions and bosons. de Haas - van Alphen effect: a Fermi surface leads to oscillations in the magnetic susceptibility as a function of 1/B. In a magnetic field [P x, P y ] ib P x dp y 2π(l )B. When the area of the orbit is a cross section of the Fermi surface there is a sharp response. I.e. at 1/B l/a F l/k 2 F l/µ2. Large magnetic field will suppress superconducting instabilities. Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 5 / 21

6 Quantum oscillations in High - T c superconductors Doiron-Leyraud et al (Nature), Vignolle et al (Nature). de Haas - van Alphen oscillations in underdoped and overdoped cuprates. In underdoped region, carrier density much lower than naïve expectation: small Fermi surface. Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 6 / 21

7 Criticality under the dome in High - T c superconductors Daou et al (Nature Physics) Resistivity in normal phase linear in temperature (anomalous). Applying a large magnetic field shows persistance down to T = 0 at critical doping. Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 7 / 21

8 Free fermions Free bosons or fermions in magnetic fields have Landau levels ε l = m qb (l ). Free energy for fermions (D=2+1) Ω = qb AT ( ) log 1 + e (ε l±qµ)/t. 2π Zero temperature limit lim Ω = qb A T 0 2π l ± (qµ ε l )θ(qµ ε l ). Magnetic susceptibility has oscillations χ 2 Ω B 2 = qb A q 2 (l )2 2π ε 2 δ(qµ ε l ) +, l l l Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 8 / 21

9 Free bosons Free energy for bosons unstable if ε 0 < qµ Ω = qb A ( ) log 1 e (ε l±qµ)/t + Ω 2π T =0. l ± Magnetic susceptibility at T=0 if stable (Hurwitz zeta function) m Χ A q Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 9 / 21 m

10 Strongly coupled theories with gravity duals 1 The normal state 2 Large N magnetic susceptibility 3 Black hole instabilities (superconducting instabilities) 4 1/N corrections to the free energy 5 Black hole ringing 6 Quantum oscillations Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 10 / 21

11 The normal state The minimal ingredient is Einstein-Maxwell theory S E [A, g] = d 4 x [ g 1 (R 2κ 2 + 6L ) ] 4g 2 F 2. The normal state is dual to a dyonic black hole ds 2 = (f L2 r 2 (r)dτ 2 + dr 2 ) f (r) + dx i dx i, [ A = iµ 1 r ] dτ + B x dy. r + Free energy is the action evaluated on shell Ω 0 = AL2 2κ 2 r 3 + (1 + r +µ 2 2 γ 2 3r +B 4 2 γ 2 ). Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 11 / 21

12 Large N magnetic susceptibility Easy to compute χ 2 Ω 0 B 2 Plot result: Κ 2 Μ Χ L 2 A Looks just like free bosons... (but massless!) Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 12 / 21

13 Black hole superconducting instabilities Add some matter to the bulk to make things more interesting... Charged bosons: S E [φ] = d 4 x [ g φ iqaφ 2 + m 2 φ 2]. Charged fermions: S E [ψ] = d 4 x g [ ( ψγ + 1 ) ] 4 ω abγ ab iqa ψ + m ψψ. Bosons: Criterion for homogeneous instability at T = 0 q 2 γ m 2, γ 2 = 2g 2 L 2 κ 2. Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 13 / 21

14 Γ q [Denef-SAH 09] Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 14 / 21

15 Landscape of superconducting membranes 1000 Solutions [Denef-SAH 09] Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 15 / 21

16 1/N corrections to the free energy Suppose no superconducting instability (or suppress with B) Nontrivial Landau-level structure subleading in 1/N? Quantum contribution from charged matter: [ Ω 1-loop = T tr log ˆ 2 + m 2] [ ] T tr log Γ ˆD + m +. It is difficult to compute determinants in black hole backgrounds and it is hardly ever done... Reformulate the problem using quasinormal modes. Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 16 / 21

17 Black hole ringing Late times: a perturbed black hole rings with characteristic frequencies. Quasinormal modes: poles of the retarded Green s function (bulk or boundary). Some typical quasinormal for charged AdS black holes at low temperature (not easy to make these plots!) Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 17 / 21

18 The free energy and quasinormal modes We derived (new to my knowledge) formulae for the determinant as a sum over quasinormal modes z (l) of the black hole Ω 1-loop, B = qb AT 2π log l z (l) ( z (l) 2πT ( ) ) Γ iz (l) 2. 2πT Ω 1-loop, F = qb AT 2π ( ( log Γ iz (l) 2πT + 1 ) ) 2. 2 l z (l) For the BTZ black hole we did the sum explicitly and checked agreement with the known result. Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 18 / 21

19 Quantum oscillations The power of these formulae is that if an individual quasinormal mode does something non-analytic, then this is directly identified. Faulkner-Liu-McGreevy-Vegh have shown that at zero temperature there is a fermion quasinormal mode whose trajectory in the complex frequency plane bounces off the real axis at k = k F. At a finite magnetic field, this gives a bounce when 2Bl = k 2 F. The contribution of this mode to the free energy is Ω = qba 2π 2 log µ T Im z. Taking two derivatives, delta functions in the magnetic susceptibility with period ( ) 1 = 2πq. B A F Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 19 / 21

20 Preliminary comments on bosons It looks possible that stable bosons can also give oscillations if a zero temperature bosonic quasinormal mode crosses a branch cut. Summing numerically over the bosonic modes does not converge very quickly but may indicate some structure d 2 F mu 4. db Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 20 / 21

21 Conclusions There exist systems with finite charge density that are described as neither conventional Fermi liquids or superfluids. AdS/CFT provides model exotic stable finite density systems. Magnetic fields are an essential experimental and theoretical tool for probing such systems. There is interesting structure at 1/N in AdS/CFT related to Landau levels for fermions and bosons. Found a method for computing determinants about black holes using quasinormal modes. Fermionic loops are shown to give de Haas - van Alphen oscillations. Sean Hartnoll (Harvard U) Quantum matter & black hole ringing July 09 KITP 21 / 21