Geometric responses of Quantum Hall systems Alexander Abanov July 2, 2015 With: Andrey Gromov & Kristan Jensen Amsterdam Summer Workshop Low-D Quantum Condensed Matter

Fractional Quantum Hall state – exotic fluid I Two-dimensional electron gas in magnetic field forms a new type of quantum fluid I It can be understood as quantum condensation of electrons coupled to vortices/fluxes I Quasiparticles are gapped, have fractional charge and statistics I The fluid is ideal – no dissipation! I Density is proportional to vorticity! I Transverse transport: Hall conductivity and Hall viscosity, thermal Hall e ff ect I Protected chiral dynamics at the boundary of the system

Transverse transport Signature of FQH states – quantization and robustness of Hall conductance � H � H = ⌫ e 2 j i = � H ✏ ij E j , - Hall conductivity. h Are there other “universal” transverse transport coe ffi cients? Hall viscosity: transverse momentum transport Thermal Hall conductivity: transverse energy/heat transport What are the values of the corresponding kinetic coe ffi cients for various FQH states? Are there corresponding “protected” boundary modes?

Acknowledgments and References Andrey Gromov – future postdoc in Chicago at the Kadano ff Center for Theoretical Physics I A. G. Abanov and A. Gromov, Phys. Rev. B 90 , 014435 (2014). Electromagnetic and gravitational responses of two-dimensional noninteracting electrons in a background magnetic field. I A. Gromov and A. G. Abanov, Phys. Rev. Lett. 113 , 266802 (2014). Density-Curvature Response and Gravitational Anomaly. I A. Gromov and A. G. Abanov, Phys. Rev. Lett. 114 , 016802 (2015). Thermal Hall E ff ect and Geometry with Torsion. I A. Gromov, G. Cho, Y. You, A. G. Abanov, and E. Fradkin, Phys. Rev. Lett. 114 , 016805 (2015). Framing Anomaly in the E ff ective Theory of the Fractional Quantum Hall E ff ect. I A. Gromov, K. Jensen, and A.G. Abanov, arXiv:1506.07171 (2015). Boundary e ff ective action for quantum Hall states.

Essential points of the talk I Induced Action encodes linear responses of the system I Coe ffi cients of geometric terms of the induced action – universal transverse responses. I Hall conductivity, Hall viscosity, thermal Hall conductivity. I These coe ffi cients are computed for various FQH states. I Framing anomaly is crucial in obtaining the correct gravitational Chern-Simons term! I Non-vanishing Hall viscosity does not lead to protected gapless edge modes.

Induced action Partition function of fermions in external e/m field A µ is given by: Z D D † e iS [ ψ , ψ † ; A µ ] = e iS ind [ A µ ] Z = with i ~ @ t + eA 0 � 1 � i ~ r � e ⌘ 2 � Z ⇣ d 2 x dt † S [ , † ; A µ ] = c A 2 m + interactions Induced action encodes current-current correlation functions h j µ j ν i = � 2 S ind h j µ i = � S ind , , . . . � A µ � A µ � A ν + various limits m ! 0, e 2 /l B ! 1 , ...

Induced action [phenomenological] Use general principles: gap+symmetries to find the form of S ind I Locality ! expansion in gradients of A µ I Gauge invariance ! written in terms of E and B I Other symmetries: rotational, translational, . . . h ✏ 2 E 2 � 1 S ind = ⌫ Z Z 2 µB 2 + � B r E + . . . i d 2 x dt AdA + 4 ⇡ Find responses in terms of phenomenological parameters ⌫ , ✏ , µ, � ,. . . Compute these parameters from the underlying theory. For non-interacting particles in B with ν = N see Abanov, Gromov 2014. Any functional of E and B is gauge invariant, but . . .

Chern-Simons action Z S CS = ν AdA 4 π

Linear responses from the Chern-Simons action In components ⌫ Z AdA ⌘ ⌫ Z d 2 x dt ✏ µ νλ A µ @ ν A λ S CS = 4 ⇡ 4 ⇡ ⌫ Z h i d 2 x dt = A 0 ( @ 1 A 2 � @ 2 A 1 ) + . . . 4 ⇡ Varying over A µ ⇢ = � S CS j 1 = � S CS = ⌫ = � ⌫ 2 ⇡ B , 2 ⇡ E 2 , � A 0 � A 1 2 π and � H = ∂ρ ν We have: � H = ∂ B – Streda formula.

Properties of the Chern-Simons term I Gauge invariant in the absence of the boundary (allowed in the induced action) I Not invariant in the presence of the boundary I Leads to protected gapless edge modes I First order in derivatives (more relevant than F µ ν F µ ν , B 2 or E 2 at large distances) I Relativistically invariant (accidentally) I Does not depend on metric g µ ν (topological, does not contribute to the stress-energy tensor)

Elastic responses: Strain and Metric I Deformation of solid or fluid r ! r + u ( r ) I u ( r ) - displacement vector I u ik = 1 2 ( @ k u i + @ i u k ) - strain tensor I u ik plays a role of the deformation metric I deformation metric g ik ⇡ � ik + 2 u ik with ds 2 = g ik dx i dx k I stress tensor T ij - response to the deformation metric g ij

Stress tensor and induced action Studying responses I Microscopic model S = S [ ] I Introduce gauge field and metric background S [ , A, g ] I Integrate out matter degrees of freedom and obtain and S ind [ A, g ] I Obtain E/M, elastic, and mixed responses from ✓ ◆ Z dx dt p g j µ � A µ + 1 2 T ij � g ij � S ind = I Elastic responses = gravitational responses Important: stress is present even in flat space!

Quantum Hall in Geometric Background (by Gil Cho) e e e = electron e = magnetic field

Geometric background I For 2+1 dimensions and spatial metric g ij we introduce “spin connection” ! µ so that 1 p gR = @ 1 ! 2 � @ 2 ! 1 – gravi-magnetic field, 2 E i = ! i � @ i ! 0 ˙ – gravi-electric field, I For small deviations from flat space g ik = � ik + � g ik we have explicitly ! 0 = 1 ! i = � 1 2 ✏ jk � g ij ˙ 2 ✏ jk @ j � g ik g ik , I Close analogy with E/M fields A µ $ ! µ !

Geometric terms of the induced action Terms of the lowest order in derivatives S ind = ⌫ Z h s ! dA + � 0 ! d ! i AdA + 2¯ . 4 ⇡ Geometric terms: I AdA – Chern-Simons term ( ⌫ : Hall conductance, filling factor) I ! dA – Wen-Zee term (¯ s : orbital spin, Hall viscosity, Wen-Zee shift) I ! d ! – “gravitational CS term” ( � 0 : Hall viscosity - curvature, thermal Hall e ff ect, orbital spin variance) I In the presence of the boundary � 0 can be divided into chiral central charge c and s 2 (Bradlyn, Read, 2014). The latter does not correspond to an anomaly. (Gromov, Jensen, AGA, 2015)

The Wen-Zee term Responses from the Wen-Zee term S WZ = ⌫ ¯ s Z ! dA . 2 ⇡ Emergent spin (orbital spin) ¯ s ⌫ 4 ⇡ ( A + ¯ s ! ) d ( A + ¯ s ! ) Wen-Zee shift for sphere � N = ⌫ S ; S = 2¯ s ⌫ ¯ 2 ⇡ A 0 d ! ! �⇢ = ⌫ ¯ s 2 ⇡ d ! ! � N = ⌫ ¯ s s Z d 2 x p gR = ⌫ ¯ s � = ⌫ ¯ s (2 � 2 g ) 4 ⇡ Hall viscosity (per particle) ⌘ H = ¯ s 2 n e : 2 ⇡! dA ! B ⌫ ¯ ⌫ ¯ s s s ¯ 2 ✏ jk � g ij ˙ 2 ⇡ ! 0 = n e ¯ s ! 0 = n e g ik

Hall viscosity Gradient correction to the stress tensor T ik = ⌘ H ( ✏ in v nk + ✏ kn v ni ) , where v ik = 1 2( @ i v k + @ k v i ) = 1 2 ˙ g ik – strain rate (a) Shear viscosity (b) ⌘ H – Hall viscosity picture from Lapa, Hughes, 2013 Avron, Seiler, Zograf, 1995

The Wen-Zee construction for ν = 1 Integrate out fermions but leave currents j = � 1 2 π da (Wen, Zee, 1992) S [ a ; A, ! ] = � 1 Z ✓ A + 1 ◆ � ada + 2 da . 2 ! 4 ⇡ (Wen, Zee, 1992) + framing anomaly: A + 1 I minimize over a : a = � � � 2 ! I substitute back into the action and obtain I take into account framing anomaly (Gromov et.al., 2015) ✓ ◆ ✓ ◆ Z 1 A + 1 A + 1 � 1 S ind = 2 ! d 2 ! 48 ⇡! d ! 4 ⇡

Digression: the quantum Chern-Simons theory The partition function for Chern-Simons theory in the metric background (Witten, 1989) ⇢ � i k � ⇢ � i c ✓ Γ d Γ + 2 ◆� Z Z Z 3 Γ 3 Da exp ada = exp tr 4 ⇡ 96 ⇡ ⇢ � i c � Z = exp ! d ! , 48 ⇡ where c = sgn( k ) and the last equality is correct for our background. I We specialized Witten’s results to the Abelian CS theory I The result is obtained from the fluctuation determinant det( d ) I The dependence on metric comes from the gauge fixing dV φ D µ a µ R I Action does not depend on metric, path integral does: anomaly (framing anomaly)

Obtaining the e ff ective field theory for FQH states I Reduce problem to noninteracting fermions with ⌫ - integer interacting with statistical Abelian and non-Abelian gauge fields. Can be done, e.g., by flux attachment or parton construction (Zhang, Hansson, Kivelson, 1989; Wen, 1991; Cho, You, Fradkin, 2014) I Integrate out fermions and obtain the e ff ective action S [ a, A, g ] using the results for free fermions. (Gromov, AA, 2014) I Integrate out statistical gauge fields taking into account the framing anomaly. (Gromov et al., 2015) I Obtain the induced action S geom ind [ A, g ] and study the corresponding responses.

Example: Laughlin’s states 1 Flux attachment for Laughlin’s states ⌫ = 2 m +1 Z 2 m 4 ⇡ bdb + 1 � S 0 [ , A + a + m ! , g ] � 2 ⇡ adb Integrating out , a , b Z 1 1 ✓ A + 2 m + 1 ◆ ✓ A + 2 m + 1 ◆ � 1 S geom = d 48 ⇡! d ! ! ! ind 4 ⇡ 2 m + 1 2 2 Coe ffi cients 1 s = 2 m + 1 ⌫ = 2 m + 1 , ¯ , c = 1 . 2

Other states Geometric e ff ective actions have been obtained for: I Free fermions at ⌫ = N I Laughlin’s states I Jain series I Arbitrary Abelian QH states I Read-Rezayi non-Abelian states I The method can be applied to other FQH states A. Gromov et.al., PRL 114 , 016805 (2015). Framing Anomaly in the E ff ective Theory of the Fractional Quantum Hall E ff ect.

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