Gravitational response of the Quantum Hall effect

Transcription

1 Gravitational response of the Quantum Hall effect Semyon Klevtsov University of Cologne arxiv: , JHEP01(2014)133 () Gravitational response of QHE Bad Honnef / 18

2 Gravitational response of the Quantum Hall effect Semyon Klevtsov University of Cologne arxiv: , JHEP01(2014)133 (+ discussions with P. Wiegmann, A. Abanov, R. Bondesan, J. Dubail...) () Gravitational response of QHE Bad Honnef / 18

3 Quantum Hall effect Observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields. Hall conductance is quantized σ H = I/V H = ν, where ν is integer for integer QHE, or a fraction for fractional QHE. Involves many (N 10 6 ) electrons on lowest Landau level, described by a collective (Laughlin) state. Ψ({z i }) = N (z i z j ) β e B 2 i<j i z i 2, β = 1/ν Z + () Gravitational response of QHE Bad Honnef / 18

4 otivation In physics one can sometimes obtain important information about the system by putting it on a curved manifold. Prototypical example: a CFT partition function on a compact Riemann surface (, g 0 ) has the following behavior under the transformation of the reference metric g 0 to a new metric g = e 2σ g 0 log Z CFT (g) Z CFT (g 0 ) = c 12π S L(g 0, σ), where c R is a central charge of the CFT, and the Liouville action functional is S L (g 0, σ) = ( σ σ + R 0 σ)d 2 z, and R 0 is the scalar curvature of g 0. () Gravitational response of QHE Bad Honnef / 18

5 otivation Our main goals: 1) Consider the partition function of the integer Quantum Hall effect on a compact Riemann surface (, g 0 ). How does it transform under the change of metric g 0 g? Can we derive an analog of CFT formula for QHE (β = 1): log Z QHE (g) Z QHE (g 0 ) =? 2) Same question for the fractional Quantum Hall effect (β 1): log Z FQHE (g) Z FQHE (g 0 ) =? As a byproduct: The problem naturally generalizes to Kähler (, g) of dim C = n, equipped with holomorphic line bundle L. Relation to modern Kähler geometry. () Gravitational response of QHE Bad Honnef / 18

6 otivation Our main goals: 1) Consider the partition function of the integer Quantum Hall effect on a compact Riemann surface (, g 0 ). How does it transform under the change of metric g 0 g? Can we derive an analog of CFT formula for QHE (β = 1): log Z QHE (g) Z QHE = (main result) (g 0 ) 2) Same question for the fractional Quantum Hall effect (β 1): log Z FQHE (g) Z FQHE = (status report) (g 0 ) As a byproduct: The problem naturally generalizes to Kähler (, g) of dim C = n, equipped with holomorphic line bundle L. Relation to modern Kähler geometry. () Gravitational response of QHE Bad Honnef / 18

7 ain result Let k be the flux of the magnetic field. Then we have large k expansion log Z QHE (g) Z QHE (g 0 ) = k 2 2π S AY (g 0, φ) + k 4π S (g 0, φ) + 1 6π S L(g 0, φ) + remainder terms O(1/k) Instead of conformal form the metric g = e 2σ g 0, it is natural to use Kähler form g = g 0 + φ (g := g z z = det g ij ). New action functionals (unknown in physics - coming from Kähler geometry): (1 S AY (g 0, φ) = 2 φ φ ) + φg 0 d 2 z Aubin Yau ( S (g 0, φ) = φr0 + g log g ) d 2 z abuchi g 0 () Gravitational response of QHE Bad Honnef / 18

8 Lowest Landau level on Riemann surface On the plane LLL wavefunctions are ψ i = z i e k 2 z 2. What is the analog of LLL on (, g 0 )? Constant magnetic field: F = da = kg 0. Shrödinger equation for the lowest energy level reduces to ( + A z )ψ = 0, where A z = k log h 0 with many solutions ψ i (z, z) = s i (z)h0 k (z, z). athematically, magnetic field is described by the holomorphic line bundle L k, s i (z) is the basis of holomorphic sections (i = 1,.., N k ) and constant magnetic field F = kg 0 means choice of "polarization", implying positivity of the line bundle (F > 0 since the metric shall be positive definite everywhere on ). Examples: S 2, s i (z) = z i 1, i = 1,.., k + 1. T 2, s i (z) = θ i,0(kz, kτ), i = 1,.., k k on of genus h there are N k = k h + 1 sections. () Gravitational response of QHE Bad Honnef / 18

9 Laughlin wave function on Riemann surface The Laughlin wave function of N k non-interacting fermions (integer QHE) is given by Slater determinant Ψ(z 1,..., z Nk ) = 1 Nk! det ψ i(z j ) For the metric g 0, the (normalized) wavefunctions are ψ 0 i (z) = s i (z)h k 0. Then for the new metric g = g 0 + φ the wavefunctions are ψ i (z) = ψ 0 i (z)e kφ The Laughlin wave function for the fractional QHE is Ψ β (z 1,..., z Nk ) = 1 ( det ψi (z j ) ) β Nk! () Gravitational response of QHE Bad Honnef / 18

10 Partition function for integer QHE Partition function (generating functional) N k Z QHE (g 0, g) = Ψ(z 1,..., z Nk ) 2 g(z i )d 2 z i = N k = 1 N k! N k i=1 det s i (z j ) 2 e k i φ(z i ) h0 k (z i)g(z i )d 2 z i. Varying Z QHE (g 0, g) with respect to δφ(z) allows to define density correlation functions ρ(z) = 1 k i δ(z z i): ρ(x)ρ(y)... ρ(z). Example of S 2, or C compactified, Wiegmann-Zabrodin (2006): Z FQHE,S2 (W ) = 1 (z) 2β e k N k i W (z i ) d 2 z i, N k! C N k They derived large k expansion, using loop equation, for arbitrary W. () Gravitational response of QHE Bad Honnef / 18 N k i=1 i=1

11 Derivation The integer QHE partition function enjoys determinantal representation Z QHE (g 0, g) = det s i s j h0 k ij e kφ gd 2 z, studied by Donaldson (2004). Variation of the free energy wrt δφ(z) δ log Z QHE ( ) (g 0, g) = kρk + ρ k δφ gd 2 z. (1) is controlled by the density of states function ρ k (z) = N k i=1 N k ψ i (z)ψ i (z) = s i (z) 2 h0 k e kφ. In math this is known as Bergman kernel on the diagonal. Number of states N k = ρ k(z)gd 2 z. i=1 () Gravitational response of QHE Bad Honnef / 18

12 Density of states (Bergman kernel) On Kähler (, g) of dim C = n with L k there is asymptotic expansion N k ρ k (z) = s i (z) 2 h0 k e kφ = k n + k n R + k n 2 1 R i=1 Tian-Yau-Zelditch expansion (Zelditch 1998). Check: number of states is topological, for n = 1, N k = ρ k(z)gd 2 z = k (2 2h). In physics can be derived from path integral representation of the density of LLL states (Douglas - SK 2008) ρ k (z) = lim T z=x(t ) z=x(0) e 1 T 0 (g ij ẋ i ẋ j +A i ẋ i )dt Dx(t). (2) All-order closed formula (Hao Xu 2011) valid in normal coordinate system ρ k (z) = k n s c s.... g (3) s=0 c s are known - full control over the expansion. () Gravitational response of QHE Bad Honnef / 18

13 Derivation, cont d Using the expansion of ρ k we can integrate out the free energy order by order in k (in principle to all orders) δ log Z QHE ( ) (g 0, g) = kρk + ρ k δφ gd 2 z log Z QHE (g 0, g) = k 2 2π S AY (g 0, φ) + k 4π S (g 0, φ) + 1 6π S L(g 0, φ) 5 ( ) R 2 gd 2 z R0 2 96πk g 0d 2 z + O(1/k 2 ) The action functionals here satisfy one-cocycle condition on the space of metrics: S(g 0, g 2 ) = S(g 0, g 1 ) + S(g 1, g 2 ). Starting from order 1/k this becomes easy, since S(g 0, g) = S(g) S(g 0 ) (exact one-cocycle). Conjecture: remainder term is exact one-cocycle. () Gravitational response of QHE Bad Honnef / 18

14 Conjecture check Conjecture: remainder term is exact one-cocycle. Check to the order 1/k 2 : log Z QHE (g 0, g) = k 2 2π S AY (g 0, φ) + k 4π S (g 0, φ) + 1 6π S L(g 0, φ) 5 ( ) R 2 gd 2 z R0 2 96πk g 0d 2 z + ( πk 2 (29R 3 66R R) gd 2 z ) (29R0 3 66R 0 0 R 0 ) g 0 d 2 z + O(1/k 3 ) () Gravitational response of QHE Bad Honnef / 18

15 Conjecture check Conjecture: remainder term is exact one-cocycle. Check to the order 1/k 2 : log Z QHE (g 0, g) = k 2 2π S AY (g 0, φ) + k 4π S (g 0, φ) + 1 6π S L(g 0, φ) 5 ( ) R 2 gd 2 z R0 2 96πk g 0d 2 z + ( πk 2 (29R 3 66R R) gd 2 z ) (29R0 3 66R 0 0 R 0 ) g 0 d 2 z + O(1/k 3 ) () Gravitational response of QHE Bad Honnef / 18

16 Fractional QHE Result for FQHE on S 2 (Can-Laskin-Wiegmann, ) log Z FQHE,β (g 0, g) = = β k 2 2π S AY (g 0, φ) + β k 4π S (g 0, φ) + ( β 1 ) 1 2 2π S L(g 0, φ) +... using loop equation method. T 2 and higher genus (SK-Wiegmann et al, on the way). Nonperturbative relation: Z β (g) = Z FQHE,β (g 0, g) (Z QHE (g 0, g)) β Z β (g) is independent of g 0 (background independent) log Zβ (g) Z β (g 0 ) = β 1 12π S L(g 0, σ) exact relation at k =. Compare to the CFT partition function log Z CFT (g) Z CFT (g 0 ) = c 12π S L(g 0, σ), () Gravitational response of QHE Bad Honnef / 18

17 Physical interpretation There is a long-term effort to understant the effective theory of QHE systems. One interpretation is that electrons in QHE form an inconpressible "quantum Hall liquid" (Girvin-acDonald-Platzman 1986, see also recent papers by Hoyos, Son, Wiegmann, Abanov etc). The Bergman kernel is the density profile of the liquid. log Z FQHE = β k 2 2π S AY (g 0, φ)+β k 4π S (g 0, φ)+ ( 1 3 +β 1 ) 1 2 2π S L(g 0, φ) The first coefficient is inverse Hall conductance. The response to the curvature is interpreted as an anomalous Hall viscosity η = δρ k δr where (δs = R δs L = R) The Liouville term is related to heat transport (.Stone, J.Cardy). () Gravitational response of QHE Bad Honnef / 18

18 Beta-deformed Bergman kernel Begman kernel is ubiquitous in Kähler geometry. Analogs of Z QHE have been studied by Donaldson (2004) and R. Berman (2008-) on Kähler - derived leading order term in the large k expansion of Z QHE. Darboux-Cristoffel formula for the Bergman kernel ρ k (z 1 ) = 1 det s i (z j ) 2 e k (N k 1)! N k 1 We define "β-deformed" Bergman kernel ρ β k (z 1) = 1 det s i (z j ) 2β e βk (N k 1)! N k 1 This is a new object in Kähler geometry. N k i φ(z i ) h0 k (z i)g n (z i )d 2n z i i=2 N k i φ(z i ) i=2 h βk 0 (z i)g n (z i )d 2n z i () Gravitational response of QHE Bad Honnef / 18

19 FQHE and off-diagonal Bergman kernel Statement: β-deformed Bergman kernel has large k local asymptotic expansion (no rigorous proof available at the moment). Conjecture: there exists asymptotic expansion ρ β k = β (βk) n s P s (β)(r s +...), s=0 where P s (β) is a polynomial in β of degree s, and R s schematically denotes curvature invariants of degree s. Another representation for the FQHE partition function Z β (g) Z β (g) = 1 N k! N k Where B k (z i, z j ) = i N k (det B k (z i, z j )) 2β g n (z i )d 2n z i i=1 ψ(z i )ψ(z j ) - off-diagonal Bergman kernel. B k (z i, z j ) = e k z i z j 2 (k n +...) () Gravitational response of QHE Bad Honnef / 18

20 Thank you () Gravitational response of QHE Bad Honnef / 18