Hall viscosity of quantum fluids

Nicholas Read

Yale University

ESI, Vienna, August 20, 2014 NSF-DMR

Outline: Definitions for viscosity and Hall viscosity 1) Adiabatic transport BCS paired states Magnetic field Fractional quantum Hall states Quantization arguments 2) Kubo formulas --- stress-stress response 3) Relation with conductivity 4) Effective action approach

Hall viscosity in quantum systems: Avron, Seiler, and Zograf, 1995 -- general set-up, filled lowest LL Levay, 1995 -- single ptcle, any LL Tokatly and Vignale, 2007 -- “any LLL state” Read, 2008/09 -- paired states and frac qu Hall states, relation with orbital spin and shift Tokatly and Vignale, 2009 -- rederivation for Laughlin state Haldane, 2009 (unpublished) -- general discussion and rederivation Read and Rezayi, 2010/11 -- general explication, numerics, and quantization arguments Hughes, Leigh, and Fradkin, 2011 -- Dirac fermions/topological insulator Dam Son and coworkers, 2011 -- effective field theory approaches, conductivity relation Bradlyn, Goldstein, and NR, 2012 -- Kubo formulas, general conductivity relation

Viscosity is a fourth rank tensor Momentum density obeys continuity: ---defines stress tensor . In a solid, we have for expectation of stress where the local strain is , is displacement field, are elastic coefficients (moduli), and is the viscosity tensor. and are both symmetric tensors if rotational invariance holds. In a fluid with local velocity , elastic part becomes (pressure), we replace and also add (momentum flux) to stress tensor.

Landau and Lifshitz, “Elasticity”

@ga=@t + @b¿ba = 0

¿ab = ¡¸abefuef ¡ ´abef@uef=@t + : : : ;

uab = 12

(@bua + @aub)

@uab=@t = 12

(@bva + @avb)

¿ab(x; t)

¿ab

Rate of loss of mechanical energy, or rate of entropy production, is Symmetric and antisymmetric parts: ---at zero frequency, only symmetric part gives dissipation; if rotation invariant, it reduces to usual bulk and shear viscosities, and ---antisymmetric part vanishes if time reversal is a symmetry; if rotation invariant, it reduces to one number in two dimensions (odd under reflections), i.e. ; none in higher dimensions

Avron, Seiler, and Zograf (1995)

Hall viscosity is analog of Hall conductivity

´H

´abef = ´H(±be²af ¡ ±af ²eb)

kBT

µ@s

@t+r ¢ js

¶= ´abef

@uab@t

@uef@t

¸ 0

³ ´sh

Avron et al.: framework of calculating in a quantum fluid by adiabatic transport (Berry phase), varying aspect ratio of periodic boundary conditions. N.R. (2009): extended to gapped paired superfluids and fractional quantum Hall fluids, discovered general form where is number density, and is (minus) mean “orbital spin” per particle, e.g. for p-ip superfluid (and also for filled LLL in Avron et al.). Further, if the ground state for particles on a sphere requires quanta of magnetic flux, and then the “shift” is Quantized if translation and rotation invariance are preserved ---then characteristic of a topological phase.

Wen and Zee (1992)

´H = 12¹n¹s¹h;

N.R. and Rezayi (2011)

Ways to calculate viscosity in quantum fluids ---adiabatic transport: antisymmetric part in gapped systems only ---first part ---Kubo formula for stress-stress response: any kind of system or frequency ---second part ---momentum-momentum response: relation with conductivity (Gal inv case) ---third part ---effective action approach ---fourth part

! = 0

Strain is geometric Relate . Then metric in two systems is or usual In terms of , aspect ratio and boundary conditions are fixed. Volume is . or describe strain: Group of all invertible matrices is .

Avron, Seiler, and Zograf (1995)

ds2 = gabdxadxb

¤ GL(d;R)

pgLd

Stress is variation with respect to metric In coordinates, stress (analog of electric current ) Time-varying is “gravitational field” ( is electric field) So and we examine adiabatic response to slowly-varying spatially-uniform (analog of slowly-varying uniform , or boundary condition). Use square in space. Avron, Seiler, and Zograf (1995)

(w. periodic boundary conditions)

¿ab = ¡2±H

±gab

¿ab = ¡ 12´abef

@gef@t

Adiabatic response and Berry phase Suppose Hamiltonian depends on parameters and that is some “current” operator. Also for each value of (ignore “persistent currents”) and is gapped. Then as , using quantum adiabatic theorem, where and are Berry or adiabatic “connection” and “curvature”. Or is a Berry phase. For us, and as matrices, then (Analog of “Chern number” approach to quantized Hall conductivity.)

Avron and Seiler (1985)

Avron, Seiler, and Zograf (1995)

´abcd = ¡ 1

LdgbegdfFae;cf

I

C

A¹d¸¹

Geometry of shear: Two independent area-preserving shears in two dimensions: or These moves don’t commute: if undo in reverse order, we get a net rotation:

The transformations are described by transformations of coordinates, Pure shears are symmetric matrices, e.g.: and is a small rotation. If the state has angular momentum (“spin”), then it picks up a phase related to the spin. This is the Berry phase we will calculate.

Single-particle toy model Consider Hamiltonian in infinite two dimensional space and non-degenerate eigenstate . Rotational invariance of (in ) implies that . Again view in terms of , states . Now , where in coord rep are generators of GL(2,R). Adiabatic curvature: (at any , in terms of coords). ---commutation of two pure shears gives rotation. Similar to Berry phase when dragging orientation of a quantum spin—group SU(2)

N.R. and Rezayi (2011)

← Angular momentum!

BCS paired states Think of as relative coordinate in a pair of particles, bound by , zero center of mass momentum. Generalize further to many such pairs (BCS wavefunction, -wave pairs, spinless). Use periodic boundary condition: unfortunately breaks the invariance properties. But if pairs small compared with system size, this effect drops out. Find as , with fixed, , where . Alternative derivation: , depends on through discrete in finite size. Calculate adiabatic curvature in thermo limit, result is same. Note we see the full angular momentum of every pair, even at weak coupling. C.f. resolution of “intrinsic angular momentum” controversy.

N.R. (2009) N.R. and Rezayi (2011)

(Independent of shape, as should be for a fluid.)

Magnetic field case --- non-interacting particles Several ways to do it. Hamiltonian, one particle in infinite plane: Define Then for , Each Landau level has “extensive” degeneracy; must transport the subspace (see below), not single states. Find by similar calculation ---angular momentum (“spin”) is that of the cyclotron motion only

Avron, Seiler, and Zograf (1995) Levay (1995) N.R. and Rezayi (2011)

(times identity matrix in degeneracy space)

(B = 1)

← Angular momentum of cyclotron motion!

Non-interacting gas: --- e.g. for filled LLL At high T, we recover classical plasma result

Lifshitz and Pitaevskii, “Physical Kinetics”

´H = 12 ¹n(N + 1=2)¹h =

hHi2!cL2

´H = nkBT

2!c

Fractional quantum Hall states A droplet of QH fluid in plane for “special” (soluble) Hamiltonians has infinite multiplicity of degenerate edge states. For transport of a degenerate subspace, the curvature evaluated in ground state is (the non-Abelian/Yang-Mills curvature) where is projector onto degenerate subspace. (See below for J’s.)

First term is related to total angular momentum . With some effort, for “nice” trial wavefunctions, the last two terms give . Remaining angular momentum is per particle, and Hall viscosity is therefore as claimed.

N.R. and Rezayi (2011)

Fab;cd(0) = ih'I j[Jab; Jcd]j'Ii¡ ih'I jJabP0Jcdj'Ii+ ih'I jJcdP0Jabj'Ii

P0

12(º¡1N2 ¡NS)

¡12º¡1N2 +O(N1=2)

´H = 12ns¹h

¡s = ¡ 12S

The “shift”: For QH states and for paired states on a sphere, in ground state need magnetic flux quanta, In “conformal block” states, , and holds even more generally (e.g. under particle-hole conjugation). E.g. for Moore-Read state If and have no common factors, then is an integer. Assuming , it follows that is integer, which was not obvious initially. ( for BCS paired states, in which ; is an integer.)

s = 12º¡1 + 1=2

p-ip paired state!

Quantization/robustness arguments No “Chern number” arguments here. Suppose we have a rotationally-invariant perturbation of a rotationally-invariant Hamiltonian. For sufficiently small pert, gap is still non-zero. Work at fixed . Return to for pure shear, coefficients of perturbations. Consider adiabatic transport in all these. for is the (traceless) stress, and vanishes by rotational invariance as system size , even for slowly varying . Then so for . But so , which says that is unchanged by the perturbation. Translation and rotation invariance (no disorder) appears to be essential here.

N.R. and Rezayi (2011)

Ns

Kubo formulas --- B=0 In variables (but write as ), a time-dependent strain yields modified Hamiltonian and again is the strain generator . Moreover, from continuity for momentum, Fourier transform on and expand in to first order, find Now for viscosity, we want response of stress to applied rate of strain , as frequency goes to zero. We should extract stress response to static strain, which gives elastic moduli (not a pole in bulk viscosity at zero frequency).

Bradlyn, Goldstein, NR (2012)

H1 = ¡@¸ab@t

Jab

Jab = ¡ 12

X

j

fxj;a; pj;bg

@gb=@t + @a¿ab = 0

Tab = ¡@Jab=@t = ¡i[H0; Jab]

µTab =

Zd2x ¿ab(x)

¶

Tab

For response: Stress-strain form: Stress-stress form: Strain-strain form: To get viscosity, we must extract the change in the expectation so finally

� “contact term” � as expected!

---reduces to adiabatic approach as freq -> 0

hTab(t)i ¡ hTab(t)i0 = ¡Z 1

¡1dt0Xabcd(t¡ t0)

@¸cd@t

(t0)

´abcd(!) = Xabcd(!)=Ld ¡i

!+(·¡1 ¡ P )±ab±cd

Xabcd(!) = ¡ lim"!0

i

Z 1

0

dt ei!+th[Tab(t); Jcd(0)]i0

Xabcd(!) = lim"!0

1

!+

µh[Tab(0); Jcd(0)]i0 +

Z 1

0

dt ei!+th[Tab(t); Tcd(0)]i0

¶

Xabcd(!) = lim"!0

µ¡ih[Jab(0); Jcd(0)]i0 + !+

Z 1

0

dt ei!+th[Jab(t); Jcd(0)]i0

¶

hTabi = PLd±ab[·¡1 = ¡Ld(@P=@Ld)N ]

Kubo formulas –- B > 0 (two dimensions) Now (Lorentz force) For (symmetric gauge), which obey gl(2,R) commutation relations, we recover and three forms of response function as before. Example: non-interacting electrons ---Hall and (at non-zero frequency) shear viscosity; no bulk viscosity

@gb=@t + @a¿ab =B

mp²bcgc

Tab = ¡@Jab=@t = ¡i[H0; Jab]

Tab =1

2mp

X

j

f¼j;a; ¼j;bg

´abcd(!) =hH0i

L2[(!+)2 ¡ (2!c)2][i!+(±ad±bc ¡ ²ad²bc)¡ 2!c(±bc²ad ¡ ±ad²cb)]

Relation with conductivity In Galilean-invariant systems, number current density is . Use to relate: For B=0, relation is well-known (e.g. Taylor and Randeria 2010) usually without Hall viscosity.

For B>0, we obtain ( is Hall conductivity) which generalizes the result of Hoyos and Son (2012). Maybe this can be measured?

j = g=mp

¡@gb@t

+B

mp²bcgc = @a¿ab

´¹º®¯(!) = 12m2

p (!±º¸ ¡ i!c²º¸)@2¾¸½(q; !)

@q¹@q®

¯̄¯̄q=0

(!±½¯ ¡ i!c²½¯)

¡ i·¡1int

2!+(±¹º±®¯ + ±¹¯±º®)

´¹º®¯(!) = 12[´¹º®¯(!) + ´®º¹¯(!)]

12B2 @2

@q2x

¾H(q; ! = 0)jq=0 = ´H(! = 0)¡ ·¡1int

!c+

2i

!clim!!0

!´sh(!)

¾H

Effective action approach for quantum Hall cases Integrate out matter in presence of U(1) gauge field and gravitational background to obtain effective (induced) action---Chern-Simons etc Shift is due to 1st Wen-Zee term which involves spin connection (rotations in space only): Varying with respect to gives stress response---Hall viscosity ---as above. Not the integral of a gauge-invariant local expression, so this response cannot be renormalized by a perturbation as long as trans & rotation invariance is maintained (“Chern-Simons-like”). Not from a torsion term---see Bradlyn and N.R. (2014).

¢S(WZ1) =ºS4¼

Zd3xb²¹º¸!¹@ºA¸

Wen and Zee (1992)

!¸ ´1

2² ba ! a

¸ b =1

2²abe¹aeºb@ºg¹¸ +

1

2²abe¹a@¸eb¹

g¹º = ´abea¹ebº

N.R. and Goldberger (2009), unpublished [see talk at KITP, February 23, 2009] Hoyos and Son (2012) Bradlyn and N.R. (2014)

Use as numerical diagnostic tool In QH numerics, usually determine for a state at given by looking for best ground state on sphere. Prey to “aliasing”. Much better to numerically measure a parameter in periodic b.c. geometry ---unbiased, no aliasing. Now being used widely in numerics on QH states performed by “matrix product state” methods. Zaletel, Mong, and others E.g. 2/5 state, , as expected. Also Hansson et al

N.R. and Rezayi (2011)

¹s = 2

Conclusion

• Hall viscosity: a non-dissipative transport coefficient:

• Orbital spin per particle, , is a true emergent property --- quantized

• Use as numerical diagnostic tool

• Experimental detection: quantum Hall systems, superfluids, cold atoms…?

´H = 12¹n¹s¹h

¹s