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Dynamics of composite fermions

Dam Thanh Son (University of Chicago)UQM Kickoff meeting, 09/12/2019

Plan

• Fractional quantum Hall effect

• Dirac composite fermion

• Hydrodynamics with “dynamical metric”

Lowest Landau level limit

H =X

a

(pa + eAa)2

2m+

X

ha,bi

e2

|xa � xb|

n=0

n=1B

m


H =X

a

(pa + eAa)2

2m+

X

ha,bi

e2

|xa � xb|

n=0

n=1B

m

m ! 0

! 1


H =X

a

(pa + eAa)2

2m+

X

ha,bi

e2

|xa � xb|

H = PLLL

X

a,b

e2

|xa � xb|

Projection tolowest Landau leveln=0

n=1B

m

m ! 0

! 1

Composite fermion

• A new quasiparticle: “composite fermion” = electron + 2 flux quanta

Flux attachment

per

ee

e

e

Jain, Lopez Fradkin, Ovchinnikov, Halperin Lee Read ~ 1990

⌫ =1

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per

CF CF

CF

Flux attachmentJain, Lopez Fradkin, Ovchinnikov, Halperin Lee Read ~ 1990

CF

full LL: gapped state

(CF = composite fermion)

HLR field theory

mean field:

L = i †(@0 � iA0 + ia0) � 1

2m|(@i � iAi + iai) |2 +

1

2

1

4⇡✏µ⌫�aµ@⌫a�

b = r⇥ a = 2⇥ 2⇡ †

Be↵ = B � b = B � 4⇡n

“flux attachment”

⌫ =1

2Be↵ = 0

low-energy dof: ψ excitations near Fermi surface

Problems of HLR theory

• The problem of energy scale (CF effective mass remains finite when me->0)

• The lack of particle-hole symmetry

• solved by Dirac CF theory

Particle-hole symmetry

PH symmetry

exact symmetry the Hamiltonian on the LLL, when mixing of higher LLs negligible

⌫ ! 1� ⌫

⇥|emptyi = |fulli

⇥c†k⇥�1 = ck

⇥ i⇥�1 = �i

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maps to itself

Fermionic particle-vortex duality

L = i e�µ(@µ � iAµ) e

Theory 1:

L = i �µ(@µ � iaµ) � 1

4⇡✏µ⌫�Aµ@⌫a�Theory 2:

DTS; Metlitski, Vishwanath; Wang, Senthil 2015

Conjecture: free fermion = “QED” in 2+1 D

emergent U(1) gauge field

Theory 1 𝜓e Theory 2 𝜓

magnetic field density

density magnetic field

physical EM field

Particle-vortex duality

S =

Zd

3x

i �

µ(@µ � iaµ) � 1

4⇡✏

µ⌫�Aµ@⌫a�

�

⇢ =�S

�A0= � b

4⇡

�S

�a0= 0 �! h �0 i = B

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e�0 e = †

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Zd

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Theory 1 in magnetic fieldzero charge density

Theory 2 at finite densityand zero magnetic field

Half-filled Landau level Fermi liquid

𝜓e = electrons 𝜓 = “composite fermion”

0

LLL projection

• The energy scale problem left unsolved by Dirac CF theory

• origin of the Fermi velocity

• Lack of LLL projection: is it a problem with low-energy effective theory?

• not necessarily, unless the LLL has consequences for low-energy physics

• one of such consequences: density-density correlator ~ q^4, not q^2

@⇢

@t+r · j = 0

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@

@t(mj) +r ·T = j⇥B

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j ⇠ qT

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⇢ ⇠ qj

!⇠ q2T

!B<latexit sha1_base64="hRBhESSNqHdX0qwOcUypr77ydwM=">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</latexit><latexit sha1_base64="sQYIa8R+ZnqUz6NXwPnUPYz6z6c=">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</latexit><latexit sha1_base64="sQYIa8R+ZnqUz6NXwPnUPYz6z6c=">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</latexit><latexit sha1_base64="AzRR2Io4K1ynbaLNo0QLPEWI6fQ=">AAACL3icdZDNSgMxFIUz/tb/qks3wSK4KpmC1GWpG5cV2lro1JJJM21sMpkmGaEM8yg+hM/gVtfiRnTpW5ixI7SiFwKH79zLzT1+xJk2CL06S8srq2vrhY3Nre2d3b3i/kFby1gR2iKSS9XxsaachbRlmOG0EymKhc/ptT++yPzrO6o0k2HTTCPaE3gYsoARbCzqF6ueGknoaSagFyhMkgm8TT0p6BDPUZhMbiqwmSa5U0/7xRIqn6GsICqjH5ETNyclkFejX/zwBpLEgoaGcKx110WR6SVYGUY4TTe9WNMIkzEe0q6VIRZU95LvA1N4YskABlLZFxr4TecnEiy0ngrfdgpsRvq3l8G/vG5sgvNewsIoNjQks0VBzKGRMEsLDpiixPCpFZgoZv8KyQjbRIzNdGGLL7JMfg6H/4t2peyisnuFSrV6nk4BHIFjcApcUAU1cAkaoAUIuAeP4Ak8Ow/Oi/PmvM9al5x85hAslPP5BdZKqXA=</latexit>

• Needed: a composite fermion theory that satisfies all constraints placed by LLL projection

• One requirement: CF has electric dipole moment with respect to the physical EM field

• It is not clear what is the place of this requirement in the field-theory duality

A hydrodynamic theory of the composite fermions

• Ref: arXiv:1907.07187

• related work: A.Gromov & DTS “bimetric theory”, Haldane’s dynamical gravity

Bosonic excitations

Low-energy, long-wavelength excitations: fluctuations of the shape of the Fermi surface

pF (t,x, ✓) = p0F +1X

n=�1un(t,x) e

�in✓.

2

θ

p

p

x

y

pF(θ)

FIG. 1. A deformed Fermi surface.

state, the composite fermions live in a magnetic fieldb = B/(2N + 1), e↵ectively forming an integer quantumHall state. We are interested in the regime of frequencyand momentum of the order of N�1 compared to theFermi energy and momentum. We now propose that alllow-energy excitations can be viewed as deformations ofthe Fermi surface from the circular shape, which we pa-rameterize by a function p

F

(t,x, ✓) that depends on timeand space and also on the direction in momentum space✓ (p

y

/px

= tan ✓) (see Fig. 1). Furthermore, we decom-pose the perturbation into di↵erent angular momentumchannels:

pF

(t,x, ✓) = p0F

+ u(t,x, ✓) = p0F

+1X

n=�1un

(t,x) e�in✓.

(1)In the language of Landau’s Fermi liquid theory, the stateparameterized by p

F

(t,x, ✓) corresponds to a distributionfunction np(t,x) which is one inside the Fermi line andzero outside the line.

We now derive the commutation relation between theun

s with the following prescription. If we define an op-erator F (and similarly G) as

F =

Zdx dp

(2⇡)2F (x,p)np(x), (2)

where np(x) is the quasiparticle distribution function,then we need to impose the condition on the commuta-tion relation so that

[F, G] = �i

Zdx dp

(2⇡)2{F, G}(x,p)np(x), (3)

where the {F, G} is the classical Poisson bracket betweenF and G,

{F, G} =@F

@pi

@G

@xi

� @G

@pi

@F

@xi

� b✏ij@F

@pi

@G

@pj

, (4)

where we have allowed the composite fermions to be inan external magnetic field b. For Jain’s sequences b =±B/(2N + 1). Restricting np to be of the form of thestep function (1 inside the Fermi line and 0 outside), F ,

G, and the right-hand side of Eq. (3) become functionalsof the shape of the Fermi surface, and one can easilyderive the commutator of the small perturbations u:

[u(x, ✓), u(x0, ✓0)] =i(2⇡)2

pF

✓�n

i

(✓)@

@xi

+b

pF

@

@✓

◆

[�(x� x

0)�(✓ � ✓0)] +O(u), (5)

where n(✓) = (cos ✓, sin ✓). In terms of un

, the formulareads

[um

(q), un

(q0)] =⇡

pF

2bm

pF

�m+n,0 + �

m+n,1q+

+ �m+n,�1q�

�(2⇡)2�(q+ q

0) +O(u), (6)

where q± = qx

±iqy

. This commutation relation has beenpreviously derived in Ref. [18] by extending Tomonaga’sbosonization method to higher dimensions. Note thatthe algebra depends only on the size of the Fermi surfacepF

but not on any dynamic properties (Fermi velocity,Landau’s parameters etc.).Gauging the Fermi surface.—The composite fermion

is coupled to a dynamical gauge field. A Fermi surfacecoupled to a gauge field is a long-standing theoreticalproblem, and the bosonized language allows us to partlyaddress it.In the bosonic description, the temporal component

of the gauge field a0 is coupled to u0, and the spatialcomponents are coupled to u±1. In the Dirac compositefermion theory, the leading term in action for a

µ

is theMaxwell term. If the dynamical gauge field is at infinitelystrong coupling, then the constraints u0 = u±1 = 0 ariseas the result of the equations of motion �S/�a

µ

= 0. Theassumption of strong gauge coupling should become bet-ter and better in the limit N ! 1. This is due to tworeasons. First, the coupling of the composite fermionsto the gauge field is set at the Fermi energy ✏

F

and mo-mentum p

F

, while the scales of interest for our problemare ✏

F

/N and pF

/N . This gauge coupling is relevantfor contact and marginal for Coulomb interactions. Sec-ond, at these low energies the Fermi surface is e↵ectivelyO(N) fermionic species (corresponding to O(N) patcheson the Fermi surface in the renormalization group treat-ment [22, 23]), boosting the ’t Hooft coupling by an ad-ditional factor of N . (The argument is more complicatedin the case of the HLR theory with a Chern-Simons termin the action for a

µ

, but the conclusion is the same).Hamiltonian and equation of motion.—Assuming the

composite fermions form a Fermi liquid with Landau’sparameters F

n

, the Hamiltonian of the system is

H =vF

pF

4⇡

Zdx

1X

n=�1(1 + F

n

)un

(x)u�n

(x), (7)

where Fn

are the Landau parameters. In the case ofa marginal Fermi liquid, we may understand by F

n

the

One scalar field per spin

At low momenta we can limit ourselves to a few lowest modes

q`B ⌧ 1/N ⌫ =N

2N + 1<latexit sha1_base64="BoT5e946w4u68L1QzEArC/KTlqI=">AAACIXicbZDLSgMxFIbP1Futt1GXboJFUIQ640ZBBKkbV6WCvUCnlEyaaUMzmWmSEcpQ8A18CJ/Bra5dKe7Ehfgmpq2Itx8CP/85h3Py+TFnSjvOi5WZmp6ZncvO5xYWl5ZX7NW1qooSSWiFRDySdR8rypmgFc00p/VYUhz6nNb83umoXrukUrFIXOhBTJsh7ggWMIK1iVr2Tt+jnLeKyOMcuXsl5PX7CW4jTyTHyAskJqiU7pd23WHLzjsFZyz0ZdzfJn9y9PZ4BQDllv3utSOShFRowrFSDdeJdTPFUjPC6TDnJYrGmPRwhzaMFTikqpmOvzREWyZpoyCS5gmNxun3iRSHSg1C33SGWHfV79oo/K/WSHRw2EyZiBNNBZksChKOdIRGfFCbSUo0HxiDiWTmVkS62HDQhuKPLX44YvKHwF9T3S+4TsE9N3CKMFEWNmATtsGFAziBMyhDBQhcwy3cwb11Yz1YT9bzpDVjfc6sww9Zrx/vV6Tc</latexit><latexit sha1_base64="mV7Msj64+rr3NoE9p2MkNRWC4Vc=">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</latexit><latexit sha1_base64="mV7Msj64+rr3NoE9p2MkNRWC4Vc=">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</latexit><latexit sha1_base64="6f8+l0/ZlNZA70YwySuoNASM+pU=">AAACIXicbZDLSsNAFIYn9VbrLerSzWARFKEm3ehGKHXjqlSwF2hCmEwn7dDJJJ2ZCCX0GXwIn8Gtrt2JO3Hlmzhpi9jWHwZ+/nMO58znx4xKZVmfRm5ldW19I79Z2Nre2d0z9w+aMkoEJg0csUi0fSQJo5w0FFWMtGNBUOgz0vIHN1m99UCEpBG/V6OYuCHqcRpQjJSOPPNs6BDGvCp0GIP2RQ06w2GCutDhyTV0AoEwrKXl2rk99syiVbImgr/GXjRFMFPdM7+dboSTkHCFGZKyY1uxclMkFMWMjAtOIkmM8AD1SEdbjkIi3XTypTE80UkXBpHQjys4Sf9OpCiUchT6ujNEqi8Xa1n4X62TqODKTSmPE0U4ni4KEgZVBDM+sEsFwYqNtEFYUH0rxH2kOShNcW6LH2ZMlggsm2a5ZFsl+84qVqozOnlwBI7BKbDBJaiAW1AHDYDBI3gGL+DVeDLejHfjY9qaM2Yzh2BOxtcPL+2iMw==</latexit>

(length >> CF semiclassical orbit)

Nematic hydrodynamics• Degrees of freedom:

• density

• momentum density

• effective metric Z

dp

(2⇡)2pipjf(x,p) =

⇡i⇡j

n+ ⇡n(x)Gij(x)

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⇡i(x) =

Zdp

(2⇡)2pif(x,p)

<latexit sha1_base64="+lz80W3+cFDmonD+jSqvZfAkks4=">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</latexit><latexit sha1_base64="nnTheIdZOTV2AxC/yRaJJVIowIQ=">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</latexit><latexit sha1_base64="nnTheIdZOTV2AxC/yRaJJVIowIQ=">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</latexit><latexit sha1_base64="yAX+Ql+J3sKuREaqBj6l6s7LeZk=">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</latexit>

n(x) =

Zdp

(2⇡)2f(x,p)

<latexit sha1_base64="Wh4m05ICdup+y4Y3hXQ+zV8t7W4=">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</latexit><latexit sha1_base64="Rqw8lRFctjpWMaFsYG/uONEFL5E=">AAACZXicbVHNSyMxHM2Mn+v6UXcXLytrVhFaKGWmF/eyUujFYxesCs3YzaSZGkwyQ5IRS8hfuac9elj2b/BmOlXR6oPA473fR/KSFpxpE0V/g3BhcWl5ZfXD2sf1jc2t2vanM52XitA+yXmuLlKsKWeS9g0znF4UimKRcnqeXnen/vkNVZrl8tRMCpoIPJYsYwQbLw1rwqJqyECN08TGrahCM5onDsq6RWkGb10Dup8QMWnQd5QpTOwIVk7hnK23UcEal22HmjB7amg+2g04rB08DYTPJJ4nB53j3e6/9rffvWHtDo1yUgoqDeFY60EcFSaxWBlGOHVrqNS0wOQaj+nAU4kF1YmtXuTgoVdGMMuVP9LASn3ZYbHQeiJSXymwudLz3lR8zxuUJvuRWCaL0lBJZouykkOTw2nGcMQUJYZPPMFEMX9XSK6wj8r4n3i1JRXOZ/ImgbfkrN2Ko1b8y4fTATOsgq9gH9RBDI5AB5yAHugDAv6A+yAMFoL/4Ub4JdyZlYbBY89n8Arh3gPxDbXG</latexit><latexit sha1_base64="Rqw8lRFctjpWMaFsYG/uONEFL5E=">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</latexit><latexit sha1_base64="h8D8M5NY+wMx/Rl/KJNkBST1bzU=">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</latexit>

pdetG = n

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px

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py<latexit sha1_base64="gbnvSnBNBFpcsXkVV912jIrWbHc=">AAAB+XicbVC7SgNBFL0bXzG+opY2Q4JgFXZttAzYWEY0D0iWMDuZTYbMY5mZFZYlna2t1unE1q+x9E+cPAqTeODC4Zx7ufeeKOHMWN//9gpb2zu7e8X90sHh0fFJ+fSsZVSqCW0SxZXuRNhQziRtWmY57SSaYhFx2o7GdzO//Uy1YUo+2SyhocBDyWJGsHXSY9LP+uWqX/PnQJskWJJqvTJ9AYdGv/zTGyiSCiot4diYbuAnNsyxtoxwOin1UkMTTMZ4SLuOSiyoCfP5qRN06ZQBipV2JS2aq38nciyMyUTkOgW2I7PuzcT/vG5q49swZzJJLZVksShOObIKzf5GA6YpsTxzBBPN3K2IjLDGxLp0VrZEYuIyCdYT2CSt61rg14IHF04dFijCBVTgCgK4gTrcQwOaQGAIr/AG717uTb0P73PRWvCWM+ewAu/rF0O/lfc=</latexit><latexit sha1_base64="smah5DMnJLpJGUzS5dQdG4F+fNI=">AAAB+XicbVA9SwNBEJ2LXzF+RS0FWRIEq3Bno2XAxjKi+YDkCHt7e8mS3b1jd084jlTWtlqnE1t/jaU/w87NR2ESHww83pthZl6QcKaN6345hY3Nre2d4m5pb//g8Kh8fNLScaoIbZKYx6oTYE05k7RpmOG0kyiKRcBpOxjdTv32E1WaxfLRZAn1BR5IFjGCjZUekn7WL1fdmjsDWifeglTrlcnzz3kYN/rl714Yk1RQaQjHWnc9NzF+jpVhhNNxqZdqmmAywgPatVRiQbWfz04dowurhCiKlS1p0Ez9O5FjoXUmAtspsBnqVW8q/ud1UxPd+DmTSWqoJPNFUcqRidH0bxQyRYnhmSWYKGZvRWSIFSbGprO0JRBjm4m3msA6aV3VPLfm3dtw6jBHEc6gApfgwTXU4Q4a0AQCA3iBV3hzcmfivDsf89aCs5g5hSU4n7/vaJfg</latexit><latexit sha1_base64="smah5DMnJLpJGUzS5dQdG4F+fNI=">AAAB+XicbVA9SwNBEJ2LXzF+RS0FWRIEq3Bno2XAxjKi+YDkCHt7e8mS3b1jd084jlTWtlqnE1t/jaU/w87NR2ESHww83pthZl6QcKaN6345hY3Nre2d4m5pb//g8Kh8fNLScaoIbZKYx6oTYE05k7RpmOG0kyiKRcBpOxjdTv32E1WaxfLRZAn1BR5IFjGCjZUekn7WL1fdmjsDWifeglTrlcnzz3kYN/rl714Yk1RQaQjHWnc9NzF+jpVhhNNxqZdqmmAywgPatVRiQbWfz04dowurhCiKlS1p0Ez9O5FjoXUmAtspsBnqVW8q/ud1UxPd+DmTSWqoJPNFUcqRidH0bxQyRYnhmSWYKGZvRWSIFSbGprO0JRBjm4m3msA6aV3VPLfm3dtw6jBHEc6gApfgwTXU4Q4a0AQCA3iBV3hzcmfivDsf89aCs5g5hSU4n7/vaJfg</latexit><latexit sha1_base64="1ifyhcurWQ+xWR3CHbY3pRLWUSA=">AAAB+XicbVA9TwJBEJ3DL8Qv1NJmIzGxInc2WpLYWGIUMIEL2VsG2LC7d9ndM7lc+Am2WtsZW3+Npf/EBa4Q8CWTvLw3k5l5USK4sb7/7ZU2Nre2d8q7lb39g8Oj6vFJ28SpZthisYj1U0QNCq6wZbkV+JRopDIS2IkmtzO/84za8Fg92izBUNKR4kPOqHXSQ9LP+tWaX/fnIOskKEgNCjT71Z/eIGapRGWZoMZ0Az+xYU615UzgtNJLDSaUTegIu44qKtGE+fzUKblwyoAMY+1KWTJX/07kVBqTych1SmrHZtWbif953dQOb8KcqyS1qNhi0TAVxMZk9jcZcI3MiswRyjR3txI2ppoy69JZ2hLJqcskWE1gnbSv6oFfD+79WqNRpFOGMziHSwjgGhpwB01oAYMRvMArvHm59+59eJ+L1pJXzJzCEryvXxwklHA=</latexit>

Hydrodynamics

• Hydrodynamics can be formulated as a dynamical system with the Poisson brackets

• and Hamiltonian

Landau 1941

n = {H, n}<latexit sha1_base64="sNYnTTy+/OLrQVk15oK71v87WW0=">AAACDXicdVDLSgMxFL1TX7W+RsWVm9AiCJYyI4huhKKbLivYB3SGkknTNjSTGZKMUIZ+g7+gW127E7d+g0v/xExbwYoeCBzOua+cIOZMacf5sHJLyyura/n1wsbm1vaOvbvXVFEiCW2QiEeyHWBFORO0oZnmtB1LisOA01Ywus781h2VikXiVo9j6od4IFifEayN1LUPvF6kkUCXyEtRrYy8MhLepGuXnMqZkwE5FeebzBV3rpSqRe/kAQDqXfvTzCFJSIUmHCvVcZ1Y+ymWmhFOJwUvUTTGZIQHtGOowCFVfjo9f4KOjNJD/UiaJzSaqj87UhwqNQ4DUxliPVS/vUz8y+skun/hp0zEiaaCzBb1E450hLIsUI9JSjQfG4KJZOZWRIZYYqJNYgtbgjDL5Pvj6H/SPK24TsW9MeFcwQx5OIQiHIML51CFGtShAQRSeIQneLburRfr1Xqbleasec8+LMB6/wJVVZvK</latexit><latexit sha1_base64="ry3HnfmurkI4Ded2FBVxvIGhxbs=">AAACDXicdVDLSsNAFJ3UV62vqLhyM7QIgiUkguhGKLrpsoJ9QBPKZDpph04mYWYihNBvcOMPuNW1O3HrN3TpnzjpA6zogYHDOfc1x48Zlcq2J0ZhZXVtfaO4Wdra3tndM/cPWjJKBCZNHLFIdHwkCaOcNBVVjHRiQVDoM9L2R7e5334gQtKI36s0Jl6IBpwGFCOlpZ555PYjBTm8hm4G61XoViF3xz2zYlsXdg5oW/aCzBVnrlRqZffsaVJLGz3zS8/BSUi4wgxJ2XXsWHkZEopiRsYlN5EkRniEBqSrKUchkV42PX8MT7TSh0Ek9OMKTtWfHRkKpUxDX1eGSA3lby8X//K6iQquvIzyOFGE49miIGFQRTDPAvapIFixVBOEBdW3QjxEAmGlE1va4od5JouPw/9J69xybMu50+HcgBmK4BiUwSlwwCWogTpogCbAIAPP4AW8Go/Gm/FufMxKC8a85xAswfj8BnWZnVA=</latexit><latexit sha1_base64="ry3HnfmurkI4Ded2FBVxvIGhxbs=">AAACDXicdVDLSsNAFJ3UV62vqLhyM7QIgiUkguhGKLrpsoJ9QBPKZDpph04mYWYihNBvcOMPuNW1O3HrN3TpnzjpA6zogYHDOfc1x48Zlcq2J0ZhZXVtfaO4Wdra3tndM/cPWjJKBCZNHLFIdHwkCaOcNBVVjHRiQVDoM9L2R7e5334gQtKI36s0Jl6IBpwGFCOlpZ555PYjBTm8hm4G61XoViF3xz2zYlsXdg5oW/aCzBVnrlRqZffsaVJLGz3zS8/BSUi4wgxJ2XXsWHkZEopiRsYlN5EkRniEBqSrKUchkV42PX8MT7TSh0Ek9OMKTtWfHRkKpUxDX1eGSA3lby8X//K6iQquvIzyOFGE49miIGFQRTDPAvapIFixVBOEBdW3QjxEAmGlE1va4od5JouPw/9J69xybMu50+HcgBmK4BiUwSlwwCWogTpogCbAIAPP4AW8Go/Gm/FufMxKC8a85xAswfj8BnWZnVA=</latexit><latexit sha1_base64="bHXCt6jPrOHtX8S04cB4bBaBfP8=">AAACDXicdVBNS8MwGE7n15xfVfHkJTgED6OkguhFGHrZcYL7gLWMNEu3sDQtSSqMst/gb/CqZ2/i1d/g0X9iunXgRB8IPDzP+5UnSDhTGqFPq7Syura+Ud6sbG3v7O7Z+wdtFaeS0BaJeSy7AVaUM0FbmmlOu4mkOAo47QTj29zvPFCpWCzu9SShfoSHgoWMYG2kvn3kDWINBbyGXgYbNejVoPCmfbuKnAuUAyIHLUihuIVSBQWaffvLzCFpRIUmHCvVc1Gi/QxLzQin04qXKppgMsZD2jNU4IgqP5udP4WnRhnAMJbmCQ1n6s+ODEdKTaLAVEZYj9RvLxf/8nqpDq/8jIkk1VSQ+aIw5VDHMM8CDpikRPOJIZhIZm6FZIQlJtoktrQliPJMFh+H/5P2ueMix71D1fpNkU4ZHIMTcAZccAnqoAGaoAUIyMATeAYv1qP1ar1Z7/PSklX0HIIlWB/fK8+aQQ==</latexit>

⇡i = {H, ⇡i}<latexit sha1_base64="Qlzc+wq4bigmThj7MmjBL/PHLAE=">AAACFXicdZC7SgNBFIbPxluMt1VLEYYEQTCEXUG0EYI2KSOYC2SXMDuZJENmL8zMCmFJ5UPY2tpqbSe21pa+ibPZBIzogYGf79zm/F7EmVSW9WnklpZXVtfy64WNza3tHXN3rynDWBDaICEPRdvDknIW0IZiitN2JCj2PU5b3ug6zbfuqJAsDG7VOKKujwcB6zOClUZd89DphQo5EesydImcBNXKyClnwJl0zZJVObPSQFbFmosZsWekVC06J48AUO+aX3oeiX0aKMKxlB3bipSbYKEY4XRScGJJI0xGeEA7WgbYp9JNpmdM0JEmPdQPhX6BQlP6syPBvpRj39OVPlZD+TuXwr9ynVj1L9yEBVGsaECyRf2YIxWi1BPUY4ISxcdaYCKY/isiQywwUdq5hS2en3oyPxz9L5qnFduq2DfanCvIIg8HUIRjsOEcqlCDOjSAwD08wTO8GA/Gq/FmvGelOWPWsw8LYXx8A6senzg=</latexit><latexit sha1_base64="0kvTSk4giU/cdazJSTIU+693GlE=">AAACFXicdZDLSsNAFIYn9VbrLepShKFFECwhEUQ3QtFNlxXsBZoQJtNJO3RyYWYihNCVryD4DG517U7cuu7SN3HSVLCiBwZ+vnOb83sxo0Ka5lQrLS2vrK6V1ysbm1vbO/ruXkdECcekjSMW8Z6HBGE0JG1JJSO9mBMUeIx0vfF1nu/eES5oFN7KNCZOgIYh9SlGUiFXP7QHkYR2TF0KL6GdwWYd2vUC2BNXr5nGmZkHNA3zW8yJNSe1RtU+eZg20parf6p5OAlIKDFDQvQtM5ZOhrikmJFJxU4EiREeoyHpKxmigAgnm50xgUeKDKAfcfVCCWf0Z0eGAiHSwFOVAZIj8TuXw79y/UT6F05GwziRJMTFIj9hUEYw9wQOKCdYslQJhDlVf4V4hDjCUjm3sMULck++D4f/i86pYZmGdaPMuQJFlMEBqIJjYIFz0ABN0AJtgME9eALP4EV71F61N+29KC1p8559sBDaxxfLYqC+</latexit><latexit sha1_base64="0kvTSk4giU/cdazJSTIU+693GlE=">AAACFXicdZDLSsNAFIYn9VbrLepShKFFECwhEUQ3QtFNlxXsBZoQJtNJO3RyYWYihNCVryD4DG517U7cuu7SN3HSVLCiBwZ+vnOb83sxo0Ka5lQrLS2vrK6V1ysbm1vbO/ruXkdECcekjSMW8Z6HBGE0JG1JJSO9mBMUeIx0vfF1nu/eES5oFN7KNCZOgIYh9SlGUiFXP7QHkYR2TF0KL6GdwWYd2vUC2BNXr5nGmZkHNA3zW8yJNSe1RtU+eZg20parf6p5OAlIKDFDQvQtM5ZOhrikmJFJxU4EiREeoyHpKxmigAgnm50xgUeKDKAfcfVCCWf0Z0eGAiHSwFOVAZIj8TuXw79y/UT6F05GwziRJMTFIj9hUEYw9wQOKCdYslQJhDlVf4V4hDjCUjm3sMULck++D4f/i86pYZmGdaPMuQJFlMEBqIJjYIFz0ABN0AJtgME9eALP4EV71F61N+29KC1p8559sBDaxxfLYqC+</latexit><latexit sha1_base64="ph+qOB0ee+nFNV0wakdVZ640K2Q=">AAACFXicdZBNS8MwGMdTX+d8q3oUITgED2OkguhFGHrZcYJ7gbWUNEu3sDQtSSqMspMfws/gVc/exKtnj34T060DJ/pA4M/vecvzDxLOlEbo01paXlldWy9tlDe3tnd27b39topTSWiLxDyW3QArypmgLc00p91EUhwFnHaC0U2e79xTqVgs7vQ4oV6EB4KFjGBtkG8fuf1YQzdhPoNX0M1gowrd6gy4E9+uoNo5ygOiGpqLgjgFqYAimr79ZeaRNKJCE46V6jko0V6GpWaE00nZTRVNMBnhAe0ZKXBElZdNz5jAE0P6MIyleULDKf3ZkeFIqXEUmMoI66H6ncvhX7leqsNLL2MiSTUVZLYoTDnUMcw9gX0mKdF8bAQmkpm/QjLEEhNtnFvYEkS5J/PD4f+ifVZzUM25RZX6deFOCRyCY3AKHHAB6qABmqAFCHgAT+AZvFiP1qv1Zr3PSpesoucALIT18Q2BmJ2v</latexit>

{⇡i(x), ⇡j(y)} = [⇡j(x)@i + ⇡i(y)@j ]�(x� y)<latexit sha1_base64="dIa6tVHKDAwFA41PZ0T1jl906u0=">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</latexit><latexit sha1_base64="Ks1QSu1irphcL221i9JxEru1+mY=">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</latexit><latexit sha1_base64="Ks1QSu1irphcL221i9JxEru1+mY=">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</latexit><latexit sha1_base64="HBEPuF/degGW6tNa4iaY5BesH6Q=">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</latexit>

{⇡i(x), n(y)} = n(x)@i�(x� y)<latexit sha1_base64="O2s5qPss1D4McXyzZ/89uOvr7UA=">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</latexit><latexit sha1_base64="wJW8Qv6VrjOEl4Tea9a9/GaP0Ls=">AAACWXicbVBNS8NAFNzGj9b6Ve3Ry2IRWqglEUQvQsGLRwVbC00JL9tNXdxswu5GLCG/z98gnryIF696ddNKaasPHgwzb5jH+DFnStv2S8FaWV1bL5Y2yptb2zu7lb39rooSSWiHRDySPR8U5UzQjmaa014sKYQ+p3f+w2Wu3z1SqVgkbvU4poMQRoIFjIA2lFcBN8VuzDxWd0PQ936QPmWNJnabWMyYcdZwM3wxx5gb4wKpGXCPYXdIuYY59XjO6lVqdsueDJ4BZxnU2vjm471UPL32Km/uMCJJSIUmHJTqO3asB2meRjjNym6iaAzkAUa0b6CAkKpBOqkiw0eGGeIgkmaFxhN23pFCqNQ49M1l/qJa1nLyP62f6OB8kDIRJ5oKMg0KEo51hPNe8ZBJSjQfGwBEMvMrJvcggWjT/kKKH2amkz8N/AXdk5Zjt5wbU04bTaeEDtAhqiMHnaE2ukLXqIMIekaf6At9F16tglWyytNTq/DrqaKFsao/ILG5Rw==</latexit><latexit sha1_base64="wJW8Qv6VrjOEl4Tea9a9/GaP0Ls=">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</latexit><latexit sha1_base64="T/wYRTJLP7SlSdGyq8WjlxMV4Ik=">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</latexit>

H =

Zdx

1

2m

~⇡2(x)

n(x)+ ✏(n(x))

�

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Chiral metric hydro

• (det G)1/2 = n

{Gij(x), Gkl(y)} = �1

s("ikGjk + "ilGjk + "jkGil + "jlGik)�(x� y)

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{Gij(x), ⇡k(y)} = (Gik(x)@j +Gjk(x)@i + @kGij)�(x� y)<latexit sha1_base64="JXrtWZ0XbdU7g1SlDVtJ/Wt+kfs=">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</latexit><latexit sha1_base64="1DB1HGIguW9VyOhKkTQnU7NV69E=">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</latexit><latexit sha1_base64="1DB1HGIguW9VyOhKkTQnU7NV69E=">AAACiHicbVHLSgMxFM2Mr1pfVZeKBEulRS0zbqoLoehClxVsFTplyGQymk7mQZIRyzArf8Bf8LNc+hOCOzNtrY96SeDknHNzwo0TMyqkYbxq+szs3PxCYbG4tLyyulZa3+iIKOGYtHHEIn7rIEEYDUlbUsnIbcwJChxGbhz/PNdvHggXNAqv5SAmvQDdhdSjGElF2aVnK4UXdkr7WTW1HA8+ZrUDaKkVU9sfUYOsBq0MnsJqbvQzOHEqF+KSImb34X5+Td//vmaiUaVNDv44rVa0XMIk+rIffkXZpbJRN4YFJ8D8C8rN3Zenjx03atmlN8uNcBKQUGKGhOiaRix7aZ6HGcmKViJIjLCP7khXwRAFRPTS4eQyWFGMC72Iqx1KOGR/dqQoEGIQOMoZIHkv/mo5+Z/WTaR33EtpGCeShHgU5CUMygjm3wBdygmWbKAAwpyqt0J8jzjCUn3WrxQnyNRMpiYwDTpHddOom1dqOE0wqgLYArugCkzQAE1wCVqgDTB417a1iranF3VDb+gnI6uujXs2wa/Szz4BPkHFpA==</latexit><latexit sha1_base64="rXYIuoUwAgUTV3K8UVkZ7qYILGs=">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</latexit>

s related from the Hall viscosity

= average “orbital spin” of composite fermion

n = 1<latexit sha1_base64="4VSwMTN7+MsGND3ssJUOsD//QuY=">AAAB+XicbVC7SgNBFL0bXzG+Vi1tBoNgFXZttBEDNpYRzQOSJcxOZpMhM7PLzKwQl3yCrdZ2Yusv+BOW+RNnkxQm8cCFwzn3cu89YcKZNp734xTW1jc2t4rbpZ3dvf0D9/CooeNUEVonMY9VK8SaciZp3TDDaStRFIuQ02Y4vM395hNVmsXy0YwSGgjclyxiBBsrPchrv+uWvYo3BVol/pyUb76fJ1UAqHXdSacXk1RQaQjHWrd9LzFBhpVhhNNxqZNqmmAyxH3atlRiQXWQTU8dozOr9FAUK1vSoKn6dyLDQuuRCG2nwGagl71c/M9rpya6CjImk9RQSWaLopQjE6P8b9RjihLDR5Zgopi9FZEBVpgYm87CllCMbSb+cgKrpHFR8b2Kf++Vq3kyOYpwAqdwDj5cQhXuoAZ1INCHF3iFNydz3p0P53PWWnDmM8ewAOfrFyLWlqI=</latexit><latexit sha1_base64="aI0GS5XWk6X3Eikp8SFCp1J/Gl0=">AAAB+XicbVC7SgNBFL0bXzG+4qOzGQyCVdi1iY0YsNAyonlAsoTZyWwyZGZ2mZkV4pJPsNXaTixs/AV/wjJ/4uRRmMQDFw7n3Mu99wQxZ9q47o+TWVldW9/Ibua2tnd29/L7BzUdJYrQKol4pBoB1pQzSauGGU4bsaJYBJzWg/712K8/UqVZJB/MIKa+wF3JQkawsdK9vPTa+YJbdCdAy8SbkcLV99Po5vMorbTzo1YnIomg0hCOtW56bmz8FCvDCKfDXCvRNMakj7u0aanEgmo/nZw6RKdW6aAwUrakQRP170SKhdYDEdhOgU1PL3pj8T+vmZjwwk+ZjBNDJZkuChOOTITGf6MOU5QYPrAEE8XsrYj0sMLE2HTmtgRiaDPxFhNYJrXzoucWvTu3UC7DFFk4hhM4Aw9KUIZbqEAVCHThGV7g1UmdN+fd+Zi2ZpzZzCHMwfn6Beb6l+Y=</latexit><latexit sha1_base64="aI0GS5XWk6X3Eikp8SFCp1J/Gl0=">AAAB+XicbVC7SgNBFL0bXzG+4qOzGQyCVdi1iY0YsNAyonlAsoTZyWwyZGZ2mZkV4pJPsNXaTixs/AV/wjJ/4uRRmMQDFw7n3Mu99wQxZ9q47o+TWVldW9/Ibua2tnd29/L7BzUdJYrQKol4pBoB1pQzSauGGU4bsaJYBJzWg/712K8/UqVZJB/MIKa+wF3JQkawsdK9vPTa+YJbdCdAy8SbkcLV99Po5vMorbTzo1YnIomg0hCOtW56bmz8FCvDCKfDXCvRNMakj7u0aanEgmo/nZw6RKdW6aAwUrakQRP170SKhdYDEdhOgU1PL3pj8T+vmZjwwk+ZjBNDJZkuChOOTITGf6MOU5QYPrAEE8XsrYj0sMLE2HTmtgRiaDPxFhNYJrXzoucWvTu3UC7DFFk4hhM4Aw9KUIZbqEAVCHThGV7g1UmdN+fd+Zi2ZpzZzCHMwfn6Beb6l+Y=</latexit><latexit sha1_base64="/SaRW9cyr0R6ynmld6yV6gMAgMw=">AAAB+XicbVA9SwNBEJ2LXzF+RS1tFoNgFe5stBECNpYRzQckR9jbbJIlu3vH7pwQjvwEW63txNZfY+k/cZNcYRIfDDzem2FmXpRIYdH3v73CxubW9k5xt7S3f3B4VD4+ado4NYw3WCxj046o5VJo3kCBkrcTw6mKJG9F47uZ33rmxopYP+Ek4aGiQy0GglF00qO+DXrlil/15yDrJMhJBXLUe+Wfbj9mqeIamaTWdgI/wTCjBgWTfFrqppYnlI3pkHcc1VRxG2bzU6fkwil9MoiNK41krv6dyKiydqIi16kojuyqNxP/8zopDm7CTOgkRa7ZYtEglQRjMvub9IXhDOXEEcqMcLcSNqKGMnTpLG2J1NRlEqwmsE6aV9XArwYPfqVWy9MpwhmcwyUEcA01uIc6NIDBEF7gFd68zHv3PrzPRWvBy2dOYQne1y9x65QE</latexit>

11

then one can identify

s = ±1

2

✓N +

1

2

◆, (53)

where the upper and lower signs correspond to ⌫ = N/(2N + 1) and ⌫ = (N + 1)/(2N +

1), respectively. The replacement can also be justified by the following argument: up to

terms containing second derivatives of Gij, the prefactor is ⌦/n, which in the chiral metric

hydrodynamics is conserved and determined completely by the initial condition. Provided

that at infinite past the system is in the ground state, ⌦/n is then equal to hbi/n. Note

that our semiclassical prescription for computing the Poisson bracket, Eqs. (48) and (49),

cannot in principle yield terms with second derivatives; one may hope a more sophisticated

treatment will reveal their existence. In any case, for the calculation of linear response, the

replacement (52) can always be made.

One may be concerned that the parameter s in Eq (53), which has been previously

identified with the orbital spin per particle, is neither integer nor half-integer. This is not a

problem, as s is the average orbital angular momentum per particle. This suggests another

way to determine s. Recall that in the composite fermion picture, the ⌫ = N/(2N +1) state

has the composite fermions filling up N + 12 Landau levels: half of the zero-energy Landau

level (n = 0) and the positive-energy levels with quantum numbers n = 1, 2, . . . , N . Now

the orbital spin of a particle on the n-th Landau level is n, so the average orbital spin is

s =1

N + 12

✓1

2· 0 + 1 + 2 + · · ·+N

◆=

N(N + 1)

2N + 1. (54)

For the ⌫ = (N + 1)/(2N + 1) state, s flips sign. The value in Eq. (54) does not coincides

with that in Eq. (53), but the relative di↵erence is of order 1/N2 at large N . This discrep-

ancy is should be attributed to the imperfectness of the semiclassical procedure (48). We

thus conjecture that (54) represents the exact value of the parameter s and use it in later

calculations. Note that even for N = 1 the values of the s in Eqs. (53) and (54) di↵er from

each other only by a factor of 9/8.

From Eq. (30) and (54) one can find the odd viscosity of the Hall fluid,

⌘H = ±N(N + 1)

2N + 1

B

8⇡. (55)

This should be compared with the expectation: ⌘H = 14⇢e(S � 1) where S is the shift, and

S � 1 is the part of the shift associated with the guiding centers. Equation (55) exactly

matches this expectation, given that S = N + 2 for ⌫ = N/(2N + 1) and S = �N + 1 for

⌫ = (N + 1)/(2N + 1).

C. Equations of motion of chiral metric hydrodynamics of fractional quantum Hall

states

We now derive the relevant formulas for the metric hydrodynamics of the composite

fermions. Due to the dipolar interaction of the composite fermion with the external electric

Hydrodynamic equation

H = H0[n,⇡i, Gij ] +

Zdx

✓�a0n+

"ijEj

B⇡i

◆

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A = {H, A}<latexit sha1_base64="E1QnR64vWWAwcK94wltLWKKUuKg=">AAACC3icbVC7SgNBFL3rM8bXakqbwSBYhLBro42QYJMyAfOA7BJmJ7PJkNkHM7PCsuQT/AZbre3E1o+w087PcDYRMYkHLhzOuS+OF3MmlWW9G2vrG5tb24Wd4u7e/sGheXTckVEiCG2TiEei52FJOQtpWzHFaS8WFAcep11vcpP73TsqJIvCW5XG1A3wKGQ+I1hpaWCWnGGkUB1dIydrVJwKqjvTgVm2qtYM6JfYy6RcQ62vDwBoDsxPvYUkAQ0V4VjKvm3Fys2wUIxwOi06iaQxJhM8on1NQxxQ6Waz56foTCtD5EdCV6jQTP07keFAyjTwdGeA1Vgue7n4n9dPlH/lZiyME0VDMj/kJxypCOVJoCETlCieaoKJYPpXRMZYYKJ0XgtXvCDPZCWBVdK5qNpW1W7pcGowRwFO4BTOwYZLqEEDmtAGAik8wCM8GffGs/FivM5b14yfmRIswHj7Bk9knAE=</latexit><latexit sha1_base64="2b2GlzMBt8MH/O1ZoZEEY/L9l8M=">AAACC3icbVDLSgMxFM34aq2v0S7dBIvgopQZQXQjtLjpsgX7gM5QMmnahiaZIckIw9BP8Bvc6tqduPUjXLoRP8NMK2JbD1w4nHNfnCBiVGnHebfW1jc2t3L57cLO7t7+gX141FZhLDFp4ZCFshsgRRgVpKWpZqQbSYJ4wEgnmNxkfueOSEVDcauTiPgcjQQdUoy0kfp20RuEGtbgNfTSetkrw5o37dslp+LMAH+Ju0xKVdj8+sznLhp9+8NswTEnQmOGlOq5TqT9FElNMSPTghcrEiE8QSPSM1QgTpSfzp6fwlOjDOAwlKaEhjP170SKuFIJD0wnR3qslr1M/M/rxXp45adURLEmAs8PDWMGdQizJOCASoI1SwxBWFLzK8RjJBHWJq+FKwHPMllJYJW0zyuuU3GbJpwqmCMPjsEJOAMuuARVUAcN0AIYJOABPIIn6956tl6s13nrmvUzUwQLsN6+AbZ1nEs=</latexit><latexit sha1_base64="2b2GlzMBt8MH/O1ZoZEEY/L9l8M=">AAACC3icbVDLSgMxFM34aq2v0S7dBIvgopQZQXQjtLjpsgX7gM5QMmnahiaZIckIw9BP8Bvc6tqduPUjXLoRP8NMK2JbD1w4nHNfnCBiVGnHebfW1jc2t3L57cLO7t7+gX141FZhLDFp4ZCFshsgRRgVpKWpZqQbSYJ4wEgnmNxkfueOSEVDcauTiPgcjQQdUoy0kfp20RuEGtbgNfTSetkrw5o37dslp+LMAH+Ju0xKVdj8+sznLhp9+8NswTEnQmOGlOq5TqT9FElNMSPTghcrEiE8QSPSM1QgTpSfzp6fwlOjDOAwlKaEhjP170SKuFIJD0wnR3qslr1M/M/rxXp45adURLEmAs8PDWMGdQizJOCASoI1SwxBWFLzK8RjJBHWJq+FKwHPMllJYJW0zyuuU3GbJpwqmCMPjsEJOAMuuARVUAcN0AIYJOABPIIn6956tl6s13nrmvUzUwQLsN6+AbZ1nEs=</latexit><latexit sha1_base64="TWOgC/o0z7rlQkAXrFe5izLxVns=">AAACC3icbVDLSsNAFJ34rPUV7dLNYBFclJK40Y3Q4qbLCvYBTSiT6bQdOpmEmRshhH6C3+BW1+7ErR/h0j9x0haxrQcuHM65L04QC67Bcb6sjc2t7Z3dwl5x/+Dw6Ng+OW3rKFGUtWgkItUNiGaCS9YCDoJ1Y8VIGAjWCSZ3ud95ZErzSD5AGjM/JCPJh5wSMFLfLnmDCHAd32Iva1S8Cq57075ddqrODPiXuKukjBZo9u1vs4UmIZNABdG65zox+BlRwKlg06KXaBYTOiEj1jNUkpBpP5s9P8UXRhngYaRMScAz9e9ERkKt0zAwnSGBsV71cvE/r5fA8MbPuIwTYJLODw0TgSHCeRJ4wBWjIFJDCFXc/IrpmChCweS1dCUI80zWElgn7auq61Tde6dcqy3SKaAzdI4ukYuuUQ01UBO1EEUpekYv6NV6st6sd+tj3rphLWZKaAnW5w/Uj5mI</latexit>

CF dipole moment

Electron density

⇢ =B � b

4⇡� ✏ij@i

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<latexit sha1_base64="ZAu3cmRf4PaWPJQl8XKtFML8Mfk=">AAACRnicbVBNaxRBFHyzRo3xazVHL02CEA9ZZkQwFyGsF08hgpsEttehp/fNbic93U33G2EZ5l/lR+ToVS8S4tmbeLV3V0I+LGgoqt6jXlfhtAqUpt+Szp2Vu/furz5Ye/jo8ZOn3WfPD4KtvcSBtNr6o0IE1MrggBRpPHIeRVVoPCxO3s/9wy/og7LmE80cjioxMapUUlCU8u4e91PL3jFeeiFZ098u2uYNd6pl24yjC0pb87lRxy3jTnhSQueKcY0lbS1WmjibH7d97tVkSq/y7mbaSxdglyS7STZ3d358ZQCwn3fP+djKukJDUosQhlnqaNTMo6TGdo3XAZ2QJ2KCw0iNqDCMmsW/W/YyKmNWWh+fIbZQr240ogphVhVxshI0DTe9ufg/b1hTuTNqlHE1oZHLoLLWjCybl8jGyqMkPYtESK/irUxORayDYtXXUoqqjZ3cauA2OXjdy9Je9jGW04clVuEFbMAWZPAWduED7MMAJJzCd7iAn8lZ8iv5nfxZjnaSfzvrcA0d+Av/kbQ1</latexit><latexit sha1_base64="30jWrc4XOcPCZWLRvVVAE7+bk4A=">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</latexit><latexit sha1_base64="30jWrc4XOcPCZWLRvVVAE7+bk4A=">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</latexit><latexit sha1_base64="0eITepax+x6oibxuDH2Fd+vaGIw=">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</latexit>

= n� b+ !

4⇡<latexit sha1_base64="xKdkZriih1p4mFZhwfuHo3v1rjI=">AAACE3icdVDLSgNBEOyNrxhfUY+5DAZBEMOuCOYiBL14jGAekA1hdjKbDJmZXWZmhbDk4Ef4DV717E28Cl49iX/iJFEwPgoaiqpuuruCmDNtXPfVyczNLywuZZdzK6tr6xv5za26jhJFaI1EPFLNAGvKmaQ1wwynzVhRLAJOG8HgbOw3rqjSLJKXZhjTtsA9yUJGsLFSJ184QRIdID9UmKTBvh8J2sOj9MiP2aiTL3oldwLk/iJfVrFSfntBAFDt5N/9bkQSQaUhHGvd8tzYtFOsDCOcjnJ+ommMyQD3aMtSiQXV7XTyxAjtWqWLwkjZkgZN1O8TKRZaD0VgOwU2ff3TG4t/ea3EhOV2ymScGCrJdFGYcGQiNE4EdZmixPChJZgoZm9FpI9tHsbmNrMlEDOZ/E/qhyXPLXkXNpxTmCILBdiBPfDgGCpwDlWoAYFruIU7uHdunAfn0Xmatmacz5ltmIHz/AGfAKAO</latexit><latexit sha1_base64="TP54jrqJFem816UH7C1DwFHaPew=">AAACE3icdVC7SgNBFJ2NrxhfqxYiaQaDIIhhVwTTCEEbywjmAdkQZiezyZCZ2WVmVgjLFn6E32CrtZ3YCrbaiJ2f4SRRMD4OXDiccy/33uNHjCrtOM9WZmp6ZnYuO59bWFxaXrFX12oqjCUmVRyyUDZ8pAijglQ11Yw0IkkQ9xmp+/2ToV+/IFLRUJzrQURaHHUFDShG2khtO38EBdyDXiARTvxdL+Ski9LkwIto2rYLbtEZATq/yJdVKJdenzZe3jcrbfvN64Q45kRozJBSTdeJdCtBUlPMSJrzYkUihPuoS5qGCsSJaiWjJ1K4bZQODEJpSmg4Ur9PJIgrNeC+6eRI99RPbyj+5TVjHZRaCRVRrInA40VBzKAO4TAR2KGSYM0GhiAsqbkV4h4yeWiT28QWn09k8j+p7Rddp+iemXCOwRhZkAdbYAe44BCUwSmogCrA4BJcgxtwa11Zd9a99TBuzVifM+tgAtbjBym2oeA=</latexit><latexit sha1_base64="TP54jrqJFem816UH7C1DwFHaPew=">AAACE3icdVC7SgNBFJ2NrxhfqxYiaQaDIIhhVwTTCEEbywjmAdkQZiezyZCZ2WVmVgjLFn6E32CrtZ3YCrbaiJ2f4SRRMD4OXDiccy/33uNHjCrtOM9WZmp6ZnYuO59bWFxaXrFX12oqjCUmVRyyUDZ8pAijglQ11Yw0IkkQ9xmp+/2ToV+/IFLRUJzrQURaHHUFDShG2khtO38EBdyDXiARTvxdL+Ski9LkwIto2rYLbtEZATq/yJdVKJdenzZe3jcrbfvN64Q45kRozJBSTdeJdCtBUlPMSJrzYkUihPuoS5qGCsSJaiWjJ1K4bZQODEJpSmg4Ur9PJIgrNeC+6eRI99RPbyj+5TVjHZRaCRVRrInA40VBzKAO4TAR2KGSYM0GhiAsqbkV4h4yeWiT28QWn09k8j+p7Rddp+iemXCOwRhZkAdbYAe44BCUwSmogCrA4BJcgxtwa11Zd9a99TBuzVifM+tgAtbjBym2oeA=</latexit><latexit sha1_base64="d3Yjq/QbjdF/Q3ydwbJuwTxjFUg=">AAACE3icdVDJSgNBEO2JW4zbqMdcGoMgiGFGBL0IQS8eI5gFMkPo6fQkTXoZunuEMMzBj/AbvOrZm3j1Azz6J3YWwbg8KHi8V0VVvShhVBvPe3cKC4tLyyvF1dLa+sbmlru909QyVZg0sGRStSOkCaOCNAw1jLQTRRCPGGlFw8ux37olSlMpbswoISFHfUFjipGxUtctn0MBj2AQK4Sz6DCQnPRRnp0ECc27bsWvehNA7xf5sipghnrX/Qh6EqecCIMZ0rrje4kJM6QMxYzkpSDVJEF4iPqkY6lAnOgwmzyRw32r9GAslS1h4ET9PpEhrvWIR7aTIzPQP72x+JfXSU18FmZUJKkhAk8XxSmDRsJxIrBHFcGGjSxBWFF7K8QDZPMwNre5LRGfy+R/0jyu+l7Vv/YqtYtZOkVQBnvgAPjgFNTAFaiDBsDgDjyAR/Dk3DvPzovzOm0tOLOZXTAH5+0TaQydxw==</latexit>

! = ~r⇥✓~⇡

n

◆

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vorticity

dipole contribution

Kelvin’s circulation theorem

engineering.stackexchange.com

1869

• In ideal hydrodynamics vorticity is carried with the flow

• Leads to an infinite number of conserved quantities (Casimirs of the Poisson algebra)

! = ~r⇥✓~⇡

n

◆

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vorticity

Kelvin’s circulation theorem

• In the presence of magnetic field and metric degree of freedom, Kelvin’s theorem is modified

• Ω = constant

b+ ! +s

2

pGR[G]

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⌦ =<latexit sha1_base64="TgwgaK/Z+VXjswP0cTo2iRbbEN0=">AAAB/nicbVDLSgNBEOz1GeMr6tHLYBA8hV0vehEDXrwZwTwgCWF20psMmdldZmaFuAT8Bq969iZe/QB/wmP+xNlExCQWNBRV3XR3+bHg2rjul7O0vLK6tp7byG9ube/sFvb2azpKFMMqi0SkGj7VKHiIVcONwEaskEpfYN0fXGV+/R6V5lF4Z4YxtiXthTzgjBorNVo3EnuUXHQKRbfkTkB+iTdPipefD+MyAFQ6hXGrG7FEYmiYoFo3PTc27ZQqw5nAUb6VaIwpG9AeNi0NqUTdTif3jsixVbokiJSt0JCJ+ncipVLrofRtp6Smr+e9TPzPayYmOG+nPIwTgyGbLgoSQUxEsudJlytkRgwtoUxxeythfaooMzaimS2+HNlMFhJYJLXTkueWvFu3WM6SyZCDQziCE/DgDMpwDRWoAgMBT/AML86j8+q8Oe/T1iXnZ+YAZuB8fAOxKpib</latexit><latexit sha1_base64="XXW573kv7nB/apQn18aOUydHENo=">AAAB/nicbVDLSgNBEJyNrxhf8XHzMhgET2HXi17EgAe9GcE8IFnC7KQ3GTIzu8zMCnEJ+A1e9exNvfoB/oTH/ImTRMQkFjQUVd10dwUxZ9q47peTWVhcWl7JrubW1jc2t/LbO1UdJYpChUY8UvWAaOBMQsUww6EeKyAi4FALehcjv3YHSrNI3pp+DL4gHclCRomxUr15LaBD8FkrX3CL7hj4l3izpHD+eT+8fNtLy638sNmOaCJAGsqJ1g3PjY2fEmUY5TDINRMNMaE90oGGpZII0H46vneAD63SxmGkbEmDx+rfiZQIrfsisJ2CmK6e9Ubif14jMeGpnzIZJwYknSwKE45NhEfP4zZTQA3vW0KoYvZWTLtEEWpsRFNbAjGwmcwlME+qx0XPLXo3bqFUQhNk0T46QEfIQyeohK5QGVUQRRw9oif07Dw4L86r8z5pzTg/M7toCs7HN3Vdmd8=</latexit><latexit sha1_base64="XXW573kv7nB/apQn18aOUydHENo=">AAAB/nicbVDLSgNBEJyNrxhf8XHzMhgET2HXi17EgAe9GcE8IFnC7KQ3GTIzu8zMCnEJ+A1e9exNvfoB/oTH/ImTRMQkFjQUVd10dwUxZ9q47peTWVhcWl7JrubW1jc2t/LbO1UdJYpChUY8UvWAaOBMQsUww6EeKyAi4FALehcjv3YHSrNI3pp+DL4gHclCRomxUr15LaBD8FkrX3CL7hj4l3izpHD+eT+8fNtLy638sNmOaCJAGsqJ1g3PjY2fEmUY5TDINRMNMaE90oGGpZII0H46vneAD63SxmGkbEmDx+rfiZQIrfsisJ2CmK6e9Ubif14jMeGpnzIZJwYknSwKE45NhEfP4zZTQA3vW0KoYvZWTLtEEWpsRFNbAjGwmcwlME+qx0XPLXo3bqFUQhNk0T46QEfIQyeohK5QGVUQRRw9oif07Dw4L86r8z5pzTg/M7toCs7HN3Vdmd8=</latexit><latexit sha1_base64="1uLliY/yXim8eYRFddDlUN5T8lQ=">AAAB/nicbVDLSgNBEOyNrxhfUY9eBoPgKex60YsQ8OLNCOYByRJmJ51kyMzsMjMrhCXgN3jVszfx6q949E+cPBCTWNBQVHXT3RUlghvr+19ebm19Y3Mrv13Y2d3bPygeHtVNnGqGNRaLWDcjalBwhTXLrcBmopHKSGAjGt5M/MYjasNj9WBHCYaS9hXvcUatk5rtO4l9Sq47xZJf9qcgvyRYJiWYo9opfre7MUslKssENaYV+IkNM6otZwLHhXZqMKFsSPvYclRRiSbMpveOyZlTuqQXa1fKkqn6dyKj0piRjFynpHZglr2J+J/XSm3vKsy4SlKLis0W9VJBbEwmz5Mu18isGDlCmebuVsIGVFNmXUQLWyI5dpmsJLBK6hflwC8H936pUpmnk4cTOIVzCOASKnALVagBAwHP8AKv3pP35r17H7PWnDefOYYFeJ8/AE6V/Q==</latexit> ⌦+ ~r · (⌦~v) = 0

<latexit sha1_base64="ytTvPZArLSedHs0u3ivfubeKwcQ=">AAACKnicbVDLSgNBEOz1/Tbq0cugCIoQdnPRixDw4k0Fo0I2hN5JJw7OzC4zs4EQ8h9+hN/gVc/exIuCBz/D2UTEV8FAUdVN9VSSSWFdGD4FY+MTk1PTM7Nz8wuLS8ulldVzm+aGU42nMjWXCVqSQlPNCSfpMjOEKpF0kVwfFv5Fl4wVqT5zvYwaCjtatAVH56VmqRK3UsfiY0UdZLss7hKPNSYSWcwLZ/vTKgzW3WEHLGyWNsNyOAT7ItFvslllp+8vAHDSLL36EJ4r0o5LtLYehZlr9NE4wSUN5uLcUob8GjtU91SjItvoD/82YFteabF2avzTjg3V7xt9VNb2VOInFbor+9srxP+8eu7a+42+0FnuSPNRUDuXzKWsKIq1hCHuZM8T5Eb4Wxm/QoPc+Tp/pCRq4Dv508Bfcl4pR2E5OvXlVGGEGViHDdiGCPagCkdwAjXgcAN3cA8PwW3wGDwFz6PRseBzZw1+IHj7ABtXp+k=</latexit><latexit sha1_base64="nazzU3CDHwGL/adu78oKihb+B/4=">AAACKnicbVDLSgMxFM3UV31XXboJilARykxBdCMU3LizBatCp5Q76W0bmmSGJFMopf/hR/gNbnXtTlwJIn6GmVbE14HA4Zx7OTcnSgQ31vefvNzM7Nz8Qn5xaXlldW29sLF5aeJUM6yzWMT6OgKDgiusW24FXicaQUYCr6L+aeZfDVAbHqsLO0ywKaGreIczsE5qFcphO7Y0PJfYBXpAwwGyUEEkgIYsc4qfVmbQwT49oX6rsOuX/AnoFwl+k90Krb2/5RcOq63CiwthqURlmQBjGoGf2OYItOVM4HgpTA0mwPrQxYajCiSa5mjytzHdc0qbdmLtnrJ0on7fGIE0ZigjNynB9sxvLxP/8xqp7Rw3R1wlqUXFpkGdVFAb06wo2uYamRVDR4Bp7m6lrAcamHV1/kiJ5Nh18qeBv+SyXAr8UlBz5VTIFHmyTXZIkQTkiFTIGamSOmHkhtyRe/Lg3XqP3pP3PB3NeZ87W+QHvNcPgmioMw==</latexit><latexit sha1_base64="nazzU3CDHwGL/adu78oKihb+B/4=">AAACKnicbVDLSgMxFM3UV31XXboJilARykxBdCMU3LizBatCp5Q76W0bmmSGJFMopf/hR/gNbnXtTlwJIn6GmVbE14HA4Zx7OTcnSgQ31vefvNzM7Nz8Qn5xaXlldW29sLF5aeJUM6yzWMT6OgKDgiusW24FXicaQUYCr6L+aeZfDVAbHqsLO0ywKaGreIczsE5qFcphO7Y0PJfYBXpAwwGyUEEkgIYsc4qfVmbQwT49oX6rsOuX/AnoFwl+k90Krb2/5RcOq63CiwthqURlmQBjGoGf2OYItOVM4HgpTA0mwPrQxYajCiSa5mjytzHdc0qbdmLtnrJ0on7fGIE0ZigjNynB9sxvLxP/8xqp7Rw3R1wlqUXFpkGdVFAb06wo2uYamRVDR4Bp7m6lrAcamHV1/kiJ5Nh18qeBv+SyXAr8UlBz5VTIFHmyTXZIkQTkiFTIGamSOmHkhtyRe/Lg3XqP3pP3PB3NeZ87W+QHvNcPgmioMw==</latexit><latexit sha1_base64="Be90cnFjHbzZc+qPa2cW9Ou7Rig=">AAACKnicbVDLSgMxFM3Ud31VXboJFkERyowb3QgFN+6sYKvQKeVOetuGJpkhyRRK6X/4EX6DW127K64EP8RMW8RaDwQO59zLuTlRIrixvj/2ckvLK6tr6xv5za3tnd3C3n7NxKlmWGWxiPVjBAYFV1i13Ap8TDSCjAQ+RL3rzH/oozY8Vvd2kGBDQkfxNmdgndQsnIet2NLwVmIH6BkN+8hCBZEAGrLMOZlZmUH7p/SK+s1C0S/5E9AfEvwlRTJDpVn4dCEslagsE2BMPfAT2xiCtpwJHOXD1GACrAcdrDuqQKJpDCd/G9Fjp7RoO9buKUsn6u+NIUhjBjJykxJs1/z1MvE/r57a9mVjyFWSWlRsGtROBbUxzYqiLa6RWTFwBJjm7lbKuqCBWVfnXEokR66ThQYWSe28FPil4M4vlsuzdtbJITkiJyQgF6RMbkiFVAkjT+SFvJI379l798bex3Q05812DsgcvK9voIKlcA==</latexit>

13

where ! is the vorticity defined in Eq. (32). One can also compute the electron current

density from jie = �S/�Ai, but the formulas are somewhat cumbersome and will not be

presented here.

Two remarks on the electron density are in order. First, one can easily check that

the Poisson bracket between the electron density (63) at two di↵erent points can be again

expressed in terms of the electron density:

{⇢e(x), ⇢e(y)} = � 1

8⇡2n✏ij@i⇢e(x)@j�(x� y). (64)

This is the long-wavelength version of the Girvin-MacDonald-Platzman algebra [6]. The

dipole contribution to the electron density encodes a nontrivial aspect of the physics of the

lowest Landau level [27, 28].

Second, using Eqs. (58) and (38), one can rewrite the electron density as

⇢e =B � ⌦

4⇡+

s

8⇡

pGR[G]. (65)

If we consider a problem in which a fractional quantum Hall ground state is perturbed by

an external field, then ⌦ can be determined by its value at t = �1, which is hbi. In this

case the perturbation of the electron density from the average value (B � hbi)/(4⇡) is givenby the curvature of the dynamical metric,

�⇢e =s

8⇡

pGR[G]. (66)

Since the Poisson bracket of ⌦ with any field is either zero or proportional to ⌦ itself, the

Gaussian curvature density also satisfies the long-wavelength Girvin-MacDonald-Platzman

algebra [7]. The identification of the Gaussian curvature density with the electron density

was part of Haldane’s proposal [8] and the bimetric theory [7]. Equation (66) will play an

important role in the later calculation of the static structure factor.

D. Illustrative calculation of the density response

We demonstrate the working of the procedure on an example of a model, where the

Hamiltonian has the quadratic form

H =

Zdx

⇡2

2m⇤n0+

�

2(�n)2 +

µ

4Q2

ij

�, (67)

which, for the simplicity, is assumed to be independent of b. In this case the constraints (58)

become

n =B

4⇡, vi = 0, (68)

and the linearized hydrodynamic equations

µ@jQij = nei + n✏ijV

jb, (69)

@tQij + @iVj + @jVi � �ij@kVk �µ

sn(✏ikQkj + ✏jkQki) = 0. (70)

⇢e =B

4⇡� b+ !

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An immediate consequence

�⇢e =s

8⇡

pGR[G]

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⇠ @i@jGij<latexit sha1_base64="23TDEQf29IDReSFYCY60Jw3vEfM=">AAACGXicbZC7TsMwFIZPyq2UW4GRxbRCYqoSFhgrMcBYJHqR2ihyXKd1ayeR7SBVUTckHoKJB2CFmQ2xMjHyJjgtAtryS5Y+/eccH/v3Y86Utu0PK7e0vLK6ll8vbGxube8Ud/caKkokoXUS8Ui2fKwoZyGta6Y5bcWSYuFz2vSH51m9eUOlYlF4rUcxdQXuhSxgBGtjecXDjmICdWIsNcPcY784QBdeygZjr1i2K/ZE6AeceShXSw+3YFTzip+dbkQSQUNNOFaq7dixdtPsWsLpuNBJFI0xGeIebRsMsaDKTSdfGaMj43RREElzQo0m7t+JFAulRsI3nQLrvpqvZeZ/tXaigzM3ZWGcaBqS6aIg4UhHKMsFdZmkRPORAUwkM29FpI8lJtqkN7PFF1kmCwksQuOk4tgV58qEU4Wp8nAAJTgGB06hCpdQgzoQuINHeIJn6956sV6tt2lrzvqe2YcZWe9fsjeipg==</latexit><latexit sha1_base64="XNQJIsv0P3YexSLg4wHpSnJcnpI=">AAACGXicbZC7TsMwFIadcivlFmBEQqYVElOVsMBYiQHGItGL1ESR4zitWzuJbAepijox8BBMPAArzGyIlYmRx2DDaRHQliNZ+vT/5/jYv58wKpVlvRuFhcWl5ZXiamltfWNzy9zeaco4FZg0cMxi0faRJIxGpKGoYqSdCIK4z0jLH5zlfuuaCEnj6EoNE+Jy1I1oSDFSWvLMA0dSDp0ECUUR8+gv9uG5l9H+yDMrVtUaF/wBexYqtfL9zed+ENc988MJYpxyEinMkJQd20qUm+XXYkZGJSeVJEF4gLqkozFCnEg3G39lBA+1EsAwFvpECo7VvxMZ4lIOua87OVI9Oevl4n9eJ1XhqZvRKEkVifBkUZgyqGKY5wIDKghWbKgBYUH1WyHuIYGw0ulNbfF5nslcAvPQPK7aVtW+1OHUwKSKYA+UwRGwwQmogQtQBw2AwS14AI/gybgzno0X43XSWjC+Z3bBVBlvX13vpI8=</latexit><latexit sha1_base64="XNQJIsv0P3YexSLg4wHpSnJcnpI=">AAACGXicbZC7TsMwFIadcivlFmBEQqYVElOVsMBYiQHGItGL1ESR4zitWzuJbAepijox8BBMPAArzGyIlYmRx2DDaRHQliNZ+vT/5/jYv58wKpVlvRuFhcWl5ZXiamltfWNzy9zeaco4FZg0cMxi0faRJIxGpKGoYqSdCIK4z0jLH5zlfuuaCEnj6EoNE+Jy1I1oSDFSWvLMA0dSDp0ECUUR8+gv9uG5l9H+yDMrVtUaF/wBexYqtfL9zed+ENc988MJYpxyEinMkJQd20qUm+XXYkZGJSeVJEF4gLqkozFCnEg3G39lBA+1EsAwFvpECo7VvxMZ4lIOua87OVI9Oevl4n9eJ1XhqZvRKEkVifBkUZgyqGKY5wIDKghWbKgBYUH1WyHuIYGw0ulNbfF5nslcAvPQPK7aVtW+1OHUwKSKYA+UwRGwwQmogQtQBw2AwS14AI/gybgzno0X43XSWjC+Z3bBVBlvX13vpI8=</latexit><latexit sha1_base64="r9jvtUgBWLv8jpDQzJt2Hcvp4us=">AAACGXicbZC7TsMwFIZPyq2UW4CRxVAhMVUJC4yVGGAsEr1ITRU5rtO6tZ3IdpCqqDMPwTOwwsyGWJkYeROStgLaciRLn/7/HB/7D2LOtHGcT6uwsrq2vlHcLG1t7+zu2fsHDR0litA6iXikWgHWlDNJ64YZTluxolgEnDaD4VXuN++p0iySd2YU047APclCRrDJJN8+9jQTyIuxMgxzn/3iAF37KRuMfbvsVJxJoR9wF6EMs6r59pfXjUgiqDSEY63brhObTppfSzgdl7xE0xiTIe7RdoYSC6o76eQrY3SaKV0URio70qCJ+ncixULrkQiyToFNXy96ufif105MeNlJmYwTQyWZLgoTjkyE8lxQlylKDB9lgIli2VsR6WOFicnSm9sSiDyTpQSWoXFecZ2Ke+uUq9VZOkU4ghM4AxcuoAo3UIM6EHiAJ3iGF+vRerXerPdpa8GazRzCXFkf34qcoR8=</latexit>

! h�⇢e�⇢ei!,q ⇠ q4<latexit sha1_base64="yYMOCxt3Y7wUGlA7CaYg0Vi3IzA=">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</latexit><latexit sha1_base64="75TKa2LlkNCY6uf14r6VGym/HYc=">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</latexit><latexit sha1_base64="75TKa2LlkNCY6uf14r6VGym/HYc=">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</latexit><latexit sha1_base64="tfV3v/LMDoKEoT2c92tY9oGLWAI=">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</latexit>

Property of the lowest Landau level

• Since R ~ ∂∂G: density-density correlation functions ~ q4

• The static structure factor is independent of H

Gij = �ij + hij<latexit sha1_base64="W3X3yJ9ZuyXT1eS+93wNvYupE6g=">AAACFXicdVDLSgMxFL1TX7W+qi5FiBZBEIYZQXSjFFzosoJ9QFtKJs20scnMkGSUMhQEP8JvcKtrd+LWtUv/xEwfYEUPBM49515u7vEizpR2nE8rMzM7N7+QXcwtLa+sruXXNyoqjCWhZRLyUNY8rChnAS1rpjmtRZJi4XFa9XrnqV+9pVKxMLjW/Yg2Be4EzGcEayO18tsXrYTdDNAparQp13hUHaDukLTyBcc+clIgx3YmZKy4Y6VQ3Lnj9wBQauW/Gu2QxIIGmnCsVN11It1MsNSMcDrINWJFI0x6uEPrhgZYUNVMhmcM0J5R2sgPpXmBRkP150SChVJ94ZlOgXVX/fZS8S+vHmv/pJmwIIo1DchokR9zpEOUZoLaTFKied8QTCQzf0WkiyUm2iQ3tcUTaSaTw9H/pHJou47tXplwzmCELGzBLuyDC8dQhEsoQRkIPMATPMOL9Wi9Wm/W+6g1Y41nNmEK1sc38YigpA==</latexit><latexit sha1_base64="SCaLZNqp5bkv+NPLx88mPTHpe7Y=">AAACFXicdVDLSsNAFJ34rPUVdSnCaBEEISSC6EYpdKHLCvYBbQiTyaQdO3kwM1FK6EJcuha/wa2u3Ylb1y79EydNC1b0wMC559zLnXvcmFEhTfNTm5qemZ2bLywUF5eWV1b1tfW6iBKOSQ1HLOJNFwnCaEhqkkpGmjEnKHAZabi9SuY3rgkXNAovZT8mdoA6IfUpRlJJjr515qT0agBPYNsjTKK82ofdIXH0kmkcmhmgaZhjMlKskVIqb9+w+8rDbdXRv9pehJOAhBIzJETLMmNpp4hLihkZFNuJIDHCPdQhLUVDFBBhp8MzBnBXKR70I65eKOFQ/TmRokCIfuCqzgDJrvjtZeJfXiuR/rGd0jBOJAlxvshPGJQRzDKBHuUES9ZXBGFO1V8h7iKOsFTJTWxxgyyT8eHwf1I/MCzTsC5UOKcgRwFsgh2wByxwBMrgHFRBDWBwB57AM3jRHrVX7U17z1untNHMBpiA9vEN4fKiCA==</latexit><latexit sha1_base64="SCaLZNqp5bkv+NPLx88mPTHpe7Y=">AAACFXicdVDLSsNAFJ34rPUVdSnCaBEEISSC6EYpdKHLCvYBbQiTyaQdO3kwM1FK6EJcuha/wa2u3Ylb1y79EydNC1b0wMC559zLnXvcmFEhTfNTm5qemZ2bLywUF5eWV1b1tfW6iBKOSQ1HLOJNFwnCaEhqkkpGmjEnKHAZabi9SuY3rgkXNAovZT8mdoA6IfUpRlJJjr515qT0agBPYNsjTKK82ofdIXH0kmkcmhmgaZhjMlKskVIqb9+w+8rDbdXRv9pehJOAhBIzJETLMmNpp4hLihkZFNuJIDHCPdQhLUVDFBBhp8MzBnBXKR70I65eKOFQ/TmRokCIfuCqzgDJrvjtZeJfXiuR/rGd0jBOJAlxvshPGJQRzDKBHuUES9ZXBGFO1V8h7iKOsFTJTWxxgyyT8eHwf1I/MCzTsC5UOKcgRwFsgh2wByxwBMrgHFRBDWBwB57AM3jRHrVX7U17z1untNHMBpiA9vEN4fKiCA==</latexit><latexit sha1_base64="5l4ADk9SDz27StwRdybmWX+6GnQ=">AAACFXicdVDLSgMxFM34rPU16lKEYBEEoWQE0Y1ScKHLCvYB7TBkMpk2NskMSUYoQ1d+hN/gVtfuxK1rl/6JM+0UrOiBwLnn3MvNPX7MmTYIfVpz8wuLS8ullfLq2vrGpr213dRRoghtkIhHqu1jTTmTtGGY4bQdK4qFz2nLH1zmfuueKs0ieWuGMXUF7kkWMoJNJnn23pWXsrsRPIfdgHKDJ9UR7I+JZ1dQ9QTlgKiKpqRQnEKpgAJ1z/7qBhFJBJWGcKx1x0GxcVOsDCOcjsrdRNMYkwHu0U5GJRZUu+n4jBE8yJQAhpHKnjRwrP6cSLHQeij8rFNg09e/vVz8y+skJjxzUybjxFBJJovChEMTwTwTGDBFieHDjGCiWPZXSPpYYWKy5Ga2+CLPZHo4/J80j6sOqjo3qFK7KNIpgV2wDw6BA05BDVyDOmgAAh7AE3gGL9aj9Wq9We+T1jmrmNkBM7A+vgF00Z7g</latexit>

15

In particular, one notices that µ cancels out after substituting Eq. (26). This fact has a

deeper reason, going beyond the simple model (67): the long-wavelength behavior of the

equal-time density-density correlator is determined by the Poisson algebra, but not by the

detail form of the Hamiltonian.

To show that, we note that in the quantum theory, to quadratic order the “holomorphic”

component Q ⌘ Qzz = 14(Qxx � 2iQxy � Qyy) and the “antiholomorphic” component Q ⌘

Qzz =14(Qxx + 2iQxy �Qyy) of the metric tensor have the commutator

[Q(x), Q(y)] =1

sn�(x� y), (78)

and hence can be consider as creation and annihilation operators. If s > 0, then Q is the

creation and Q annihilation operator, and the roles are reverse for s < 0. To quadratic

order the Hamiltonian contains terms of the form QQ, and the problem reduces itself in the

quadratic level to the problem of a harmonic oscillator. Terms that does not contain one Q

and one Q appears in a very high order in the derivative expansion, (@2Q)2 or (@2Q)2 and

can be neglected. The vacuum is then annihilated by the annihilation operator.

Q(x)|0i = 0, if s > 0,

Q(x)|0i = 0, if s < 0.(79)

It is now straightforward to compute the equal-time correlator of the electron density

operator. As shown before, the fluctuating part of the electron density is proportional to

the Gaussian curvature density of the dynamical metric, i.e.,

�⇢e =s

8⇡

pGR =

s

2⇡(@2Q+ @2Q), (80)

which implies

h�⇢e�⇢eiq =Zdx e�iq·xh�⇢e(x)�⇢e(0)i =

s2q4

64⇡2

Zdx e�iq·x ⇥h0|Q(x)Q(0) + Q(x)Q(0)|0i

⇤.

(81)

Now by using Eqs. (78) and (79) we find

h�⇢e�⇢eiq =|s|

64⇡2nq4. (82)

For ⌫ = N/(2N + 1) and ⌫ = (N + 1)/(2N + 1) states, using the value of s from Eq. (54),

h�⇢e�⇢eiq =N(N + 1)

2N + 1

q4

16⇡B. (83)

By convention, the static structure factor is defined as

s(q) =h�⇢e�⇢eiq

h⇢ei=

(18(N + 1)(q`B)4, ⌫ = N

2N+1 ,18N(q`B)4, ⌫ = N+1

2N+1 .(84)




[hzz(x, hzz(y)] =1

s�(x� y)

<latexit sha1_base64="OrZGByqGGzCu1cqypHkZoyLtF30=">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</latexit><latexit sha1_base64="ES8c84DUthnSIxFiOEsOexQMGmA=">AAACSHicbVDLSgMxFM3UV62vUZeKBItQoZaZLtSNUOjGZQX7gM4wZtJMG5p5kGTEdpjf8iP8A8GNi7p2J+5MX2CrFxJOzjmXe3PciFEhDeNVy6ysrq1vZDdzW9s7u3v6/kFDhDHHpI5DFvKWiwRhNCB1SSUjrYgT5LuMNN1+daw3HwkXNAzu5SAito+6AfUoRlJRjl5r95zEchGHw+mdFtTTg09p0SpCpQ3nzCA9t+ENtDyOMDQFtDqESTR3X8w9jp43Ssak4F9gzkC+cnlcHZVPHmqO/m51Qhz7JJCYISHaphFJO0FcUsxImrNiQSKE+6hL2goGyCfCTiY/T+GZYjrQC7k6gYQT9ndHgnwhBr6rnD6SPbGsjcn/tHYsvWs7oUEUSxLg6SAvZlCGcBwj7FBOsGQDBRDmVO0KcQ+paKQKe2GK66cqE3M5gb+gUS6ZRsm8U+FUwLSy4AicggIwwRWogFtQA3WAwTN4AyPwob1on9qX9j21ZrRZzyFYqEzmB2uZs6I=</latexit><latexit sha1_base64="ES8c84DUthnSIxFiOEsOexQMGmA=">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</latexit><latexit sha1_base64="wbpRdow5o5oLRA3i//bm4AxHpU0=">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</latexit>

15








[Q(x), Q(y)] =1

sn�(x� y), (78)







Q(x)|0i = 0, if s > 0,

Q(x)|0i = 0, if s < 0.(79)




�⇢e =s

8⇡

pGR =

s

2⇡(@2Q+ @2Q), (80)

which implies


s2q4

64⇡2


⇤.

(81)



64⇡2nq4. (82)


h�⇢e�⇢eiq =N(N + 1)

2N + 1

q4

16⇡B. (83)



h⇢ei=

(18(N + 1)(q`B)4, ⌫ = N

2N+1 ,18N(q`B)4, ⌫ = N+1

2N+1 .(84)



a<latexit sha1_base64="oboyzemSzDfF81rLRzfvOR6yipM=">AAAB93icdZA9SwNBEIbn4leMX1FLm8UgWIU7G23EoI1lAuYDkiPsbeaSJXt7x+6ecISAva3WdmLrr/A3WPpP3CQKSdAXFh7ed4aZnSARXBvX/XRyK6tr6xv5zcLW9s7uXnH/oKHjVDGss1jEqhVQjYJLrBtuBLYShTQKBDaD4c0kb96j0jyWdyZL0I9oX/KQM2qsVaPdYskru1MR9xe8ZShdfRQuHwCg2i1+dXoxSyOUhgmqddtzE+OPqDKcCRwXOqnGhLIh7WPboqQRan80XXRMTqzTI2Gs7JOGTN35jhGNtM6iwFZG1Az0cjYx/8raqQkv/BGXSWpQstmgMBXExGTya9LjCpkRmQXKFLe7EjagijJjb7MwJYjG8zf5HxpnZc8tezW3VLmGmfJwBMdwCh6cQwVuoQp1YIDwCE/w7GTOi/PqvM1Kc85PzyEsyHn/BvMblUY=</latexit><latexit sha1_base64="XNwveYBfVS5hGh0ygcly7yfUafE=">AAAB93icdZDLSgMxFIYzXut4q7p0EyyCqzLjRjdi0Y3LFuwF2qFk0jNtaJIZkowwDH0Ct7p2IYhbn8JHEJe+iWmr0Bb9IfDx/+dwTk6YcKaN5306S8srq2vrhQ13c2t7Z7e4t9/Qcaoo1GnMY9UKiQbOJNQNMxxaiQIiQg7NcHg9zpt3oDSL5a3JEggE6UsWMUqMtWqkWyz5ZW8i7P2Cvwily3f3Inn+cKvd4lenF9NUgDSUE63bvpeYICfKMMph5HZSDQmhQ9KHtkVJBOggnyw6wsfW6eEoVvZJgyfubEdOhNaZCG2lIGagF7Ox+VfWTk10HuRMJqkBSaeDopRjE+Pxr3GPKaCGZxYIVczuiumAKEKNvc3clFCMZm/yPzROy75X9mteqXKFpiqgQ3SETpCPzlAF3aAqqiOKAN2jB/ToZM6T8+K8TkuXnJ+eAzQn5+0b+naWug==</latexit><latexit sha1_base64="XNwveYBfVS5hGh0ygcly7yfUafE=">AAAB93icdZDLSgMxFIYzXut4q7p0EyyCqzLjRjdi0Y3LFuwF2qFk0jNtaJIZkowwDH0Ct7p2IYhbn8JHEJe+iWmr0Bb9IfDx/+dwTk6YcKaN5306S8srq2vrhQ13c2t7Z7e4t9/Qcaoo1GnMY9UKiQbOJNQNMxxaiQIiQg7NcHg9zpt3oDSL5a3JEggE6UsWMUqMtWqkWyz5ZW8i7P2Cvwily3f3Inn+cKvd4lenF9NUgDSUE63bvpeYICfKMMph5HZSDQmhQ9KHtkVJBOggnyw6wsfW6eEoVvZJgyfubEdOhNaZCG2lIGagF7Ox+VfWTk10HuRMJqkBSaeDopRjE+Pxr3GPKaCGZxYIVczuiumAKEKNvc3clFCMZm/yPzROy75X9mteqXKFpiqgQ3SETpCPzlAF3aAqqiOKAN2jB/ToZM6T8+K8TkuXnJ+eAzQn5+0b+naWug==</latexit><latexit sha1_base64="8dxkaSjUzicdi53Fb/K5ulyuNRk=">AAAB93icdVDLSgNBEOyNrxhfUY9eBoPgKex60WPQi8cEzAOSJcxOOsmQmdllZlZYlnyBVz17E69+jkf/xMlDMEELGoqqbrq7okRwY33/0ytsbG5t7xR3S3v7B4dH5eOTlolTzbDJYhHrTkQNCq6wabkV2Ek0UhkJbEeTu5nffkRteKwebJZgKOlI8SFn1DqpQfvlSlD15yD+DwnWSQWWqPfLX71BzFKJyjJBjekGfmLDnGrLmcBpqZcaTCib0BF2HVVUognz+aFTcuGUARnG2pWyZK7+nsipNCaTkeuU1I7NujcT//K6qR3ehDlXSWpRscWiYSqIjcnsazLgGpkVmSOUae5uJWxMNWXWZbOyJZLT35n8T1pX1cCvBg2/UrtdplOEMziHSwjgGmpwD3VoAgOEJ3iGFy/zXr03733RWvCWM6ewAu/jG2jgk3k=</latexit>

a†<latexit sha1_base64="+eS0ip8IpUikL6ITFd+pK10qctY=">AAAB/3icdZDLSgMxFIbP1Futt6pLN8EiuCozbnQjFt24rGAv0taSyWSmoUlmSDJCGQr6DG517U7c+gQ+g0vfxLRVaIv+EPj4/3M4J8dPONPGdT+d3MLi0vJKfrWwtr6xuVXc3qnrOFWE1kjMY9X0saacSVozzHDaTBTFwue04fcvRnnjjirNYnltBgntCBxJFjKCjbVu8G07wFFEVbdY8sruWMj9BW8eSmcfhdN7AKh2i1/tICapoNIQjrVueW5iOhlWhhFOh4V2qmmCSR9HtGVRYkF1JxsvPEQH1glQGCv7pEFjd7ojw0LrgfBtpcCmp+ezkflX1kpNeNLJmExSQyWZDApTjkyMRr9HAVOUGD6wgIlidldEelhhYuyNZqb4Yjh9k/+hflT23LJ35ZYq5zBRHvZgHw7Bg2OowCVUoQYEBDzCEzw7D86L8+q8TUpzzk/PLszIef8GGqKYug==</latexit><latexit sha1_base64="zfnglPDjbDpheh+xfCd32JcXvtY=">AAAB/3icdZC7SgNBFIZnvcb1FrW0WQyCVdi10UYM2lhGMBdJ1jA7e3YzZGZ2mZkVwpLCZ7DVysJObH0CH0EsfRMniUIS9IeBj/8/h3PmBCmjSrvupzU3v7C4tFxYsVfX1jc2i1vbdZVkkkCNJCyRzQArYFRATVPNoJlKwDxg0Ah658O8cQtS0URc6X4KPsexoBElWBvrGt+0QxzHIDvFkld2R3LcX/BmoXT6bp+kTx92tVP8aocJyTgITRhWquW5qfZzLDUlDAZ2O1OQYtLDMbQMCsxB+flo4YGzb5zQiRJpntDOyJ3syDFXqs8DU8mx7qrZbGj+lbUyHR37ORVppkGQ8aAoY45OnOHvnZBKIJr1DWAiqdnVIV0sMdHmRlNTAj6YvMn/UD8se27Zu3RLlTM0VgHtoj10gDx0hCroAlVRDRHE0T16QI/WnfVsvViv49I566dnB03JevsGIf2aLg==</latexit><latexit sha1_base64="zfnglPDjbDpheh+xfCd32JcXvtY=">AAAB/3icdZC7SgNBFIZnvcb1FrW0WQyCVdi10UYM2lhGMBdJ1jA7e3YzZGZ2mZkVwpLCZ7DVysJObH0CH0EsfRMniUIS9IeBj/8/h3PmBCmjSrvupzU3v7C4tFxYsVfX1jc2i1vbdZVkkkCNJCyRzQArYFRATVPNoJlKwDxg0Ah658O8cQtS0URc6X4KPsexoBElWBvrGt+0QxzHIDvFkld2R3LcX/BmoXT6bp+kTx92tVP8aocJyTgITRhWquW5qfZzLDUlDAZ2O1OQYtLDMbQMCsxB+flo4YGzb5zQiRJpntDOyJ3syDFXqs8DU8mx7qrZbGj+lbUyHR37ORVppkGQ8aAoY45OnOHvnZBKIJr1DWAiqdnVIV0sMdHmRlNTAj6YvMn/UD8se27Zu3RLlTM0VgHtoj10gDx0hCroAlVRDRHE0T16QI/WnfVsvViv49I566dnB03JevsGIf2aLg==</latexit><latexit sha1_base64="doKRrUedAh3mPBJp+3ooSWB4YpI=">AAAB/3icdVA9SwNBEJ3zM8avqKXNYRCswp2NlkEbywjmQ5IY5vb2Lkt2947dPSEcKfwNtlrbia0/xdJ/4uZDSII+GHi8N8PMvCDlTBvP+3JWVtfWNzYLW8Xtnd29/dLBYUMnmSK0ThKeqFaAmnImad0ww2krVRRFwGkzGFyP/eYjVZol8s4MU9oVGEsWMYLGSvf40Akxjqnqlcp+xZvA9X6Jv0zKMEOtV/ruhAnJBJWGcNS67Xup6eaoDCOcjoqdTNMUyQBj2rZUoqC6m08OHrmnVgndKFG2pHEn6vxEjkLroQhsp0DT18veWPzLa2cmuuzmTKaZoZJMF0UZd03ijr93Q6YoMXxoCRLF7K0u6aNCYmxGC1sCMZrP5H/SOK/4XsW/9crVq1k6BTiGEzgDHy6gCjdQgzoQEPAML/DqPDlvzrvzMW1dcWYzR7AA5/MHkFiW7Q==</latexit>


[hzz(x, hzz(y)] =1

s�(x� y)


15








[Q(x), Q(y)] =1

sn�(x� y), (78)







Q(x)|0i = 0, if s > 0,

Q(x)|0i = 0, if s < 0.(79)




�⇢e =s

8⇡

pGR =

s

2⇡(@2Q+ @2Q), (80)

which implies


s2q4

64⇡2


⇤.

(81)



64⇡2nq4. (82)


h�⇢e�⇢eiq =N(N + 1)

2N + 1

q4

16⇡B. (83)



h⇢ei=

(18(N + 1)(q`B)4, ⌫ = N

2N+1 ,18N(q`B)4, ⌫ = N+1

2N+1 .(84)





�⇢el ⇠ @2zhzz + @2

zhzz<latexit sha1_base64="cKYndSXML0r3x3Zuf/Al3T8wSXg=">AAACS3icbVDLShxBFL090fiOk7h0UygBQRi63ehS4salojMjTE+a2zV3nMKq7qaqOjDT9If5EfmAQMBkp2t34sLqafF9oIrDOfdRdeJMCmN9/4/X+DQz+3lufmFxaXnly2rz67eOSXPNqc1TmeqzGA1JkVDbCivpLNOEKpbUjS8OKr/7i7QRaXJqxxn1FZ4nYig4WidFzZNwQNJiqEdpVIRaMZIlC41QLMxQW4EymvzcYSNnxq jZpL5Ltv3sPzplXTaZlFFz02/5U7AnErwlm/sbW3//AcBR1PwfDlKeK0osl2hML/Az2y+q+VxSuRjmhjLkF3hOPUcTVGT6xfTzJfvulAEbptqdxLKp+rKjQGXMWMWuUqEdmbdeJX7k9XI73OsXIslySwmvFw1zyWzKqiTZQGjiVo4dQa6FeyvjI9TIrcv71ZZYVZm8S+A96ey0Ar8VHLtwfkCNeViHDdiCAHZhHw7hCNrA4RKu4BpuvN/erXfn3delDe+xZw1eoTH7ACpHtrY=</latexit><latexit sha1_base64="yLW4AlXB4/ufw8wFk6ss9v8hxnA=">AAACS3icbVDLSsQwFE3H93vUpZugCIIwtG7GpejGpaKjwnQst5k7TjBpS5IKM6Vf4pf4CS78AFEQd7p2Jy5Mp+L7QMLhnPtITpgIro3r3jqVoeGR0bHxicmp6ZnZuer8wpGOU8WwwWIRq5MQNAoeYcNwI/AkUQgyFHgcnu8U/vEFKs3j6ND0EmxJOIt4hzMwVgqqB34bhQFfdeMg85WkKHLqay6pn4AyHETQP92gXWuGoGi/vH O6/uV/OHlZ1u/nQXXFrbkD0E/i/SYrW8tr93f168u9oProt2OWSowME6B103MT08qK+UxgPumnGhNg53CGTUsjkKhb2eDzOV21Spt2YmVPZOhA/d6RgdS6J0NbKcF09W+vEP/zmqnpbLYyHiWpwYiVizqpoCamRZK0zRUyI3qWAFPcvpWyLihgxub9Y0soi0z+JPCXHG3UPLfm7dtwtkmJcbJElska8UidbJFdskcahJEr8kCeyLNz47w4r85bWVpxPnoWyQ9URt4BYCW4TA==</latexit><latexit sha1_base64="yLW4AlXB4/ufw8wFk6ss9v8hxnA=">AAACS3icbVDLSsQwFE3H93vUpZugCIIwtG7GpejGpaKjwnQst5k7TjBpS5IKM6Vf4pf4CS78AFEQd7p2Jy5Mp+L7QMLhnPtITpgIro3r3jqVoeGR0bHxicmp6ZnZuer8wpGOU8WwwWIRq5MQNAoeYcNwI/AkUQgyFHgcnu8U/vEFKs3j6ND0EmxJOIt4hzMwVgqqB34bhQFfdeMg85WkKHLqay6pn4AyHETQP92gXWuGoGi/vH O6/uV/OHlZ1u/nQXXFrbkD0E/i/SYrW8tr93f168u9oProt2OWSowME6B103MT08qK+UxgPumnGhNg53CGTUsjkKhb2eDzOV21Spt2YmVPZOhA/d6RgdS6J0NbKcF09W+vEP/zmqnpbLYyHiWpwYiVizqpoCamRZK0zRUyI3qWAFPcvpWyLihgxub9Y0soi0z+JPCXHG3UPLfm7dtwtkmJcbJElska8UidbJFdskcahJEr8kCeyLNz47w4r85bWVpxPnoWyQ9URt4BYCW4TA==</latexit><latexit sha1_base64="z80TYWx5a26lgvyLEYgVbEHFgDM=">AAACS3icbVDLSgMxFM3UV31XXboJFkEQyowbXRbduKxoVejU4U56a4PJzJBkhHaYD/Mj/ABX4k7X7sSFmbaorR5IOJxzH8kJE8G1cd0npzQzOze/UF5cWl5ZXVuvbGxe6jhVDJssFrG6DkGj4BE2DTcCrxOFIEOBV+HdSeFf3aPSPI4uTD/BtoTbiHc5A2OloHLud1AY8FUvDjJfSYoip77mkvoJKMNBBIObA9qzZgiKDkZ3Tv d//LGTj8oGgzyoVN2aOwT9Jt40qZIxGkHlxe/ELJUYGSZA65bnJqadFfOZwHzJTzUmwO7gFluWRiBRt7Ph53O6a5UO7cbKnsjQofq7IwOpdV+GtlKC6elprxD/81qp6R61Mx4lqcGIjRZ1U0FNTIskaYcrZEb0LQGmuH0rZT1QwIzNe2JLKItM/iTwl1we1Dy35p251frxOJ0y2SY7ZI945JDUySlpkCZh5IE8k1fy5jw6786H8zkqLTnjni0ygdLcF/dPtHA=</latexit>


[hzz(x, hzz(y)] =1

s�(x� y)


15








[Q(x), Q(y)] =1

sn�(x� y), (78)







Q(x)|0i = 0, if s > 0,

Q(x)|0i = 0, if s < 0.(79)




�⇢e =s

8⇡

pGR =

s

2⇡(@2Q+ @2Q), (80)

which implies


s2q4

64⇡2


⇤.

(81)



64⇡2nq4. (82)


h�⇢e�⇢eiq =N(N + 1)

2N + 1

q4

16⇡B. (83)



h⇢ei=

(18(N + 1)(q`B)4, ⌫ = N

2N+1 ,18N(q`B)4, ⌫ = N+1

2N+1 .(84)





�⇢el ⇠ @2zhzz + @2


should be good for large Nfor N=1 (nu=1/3, 2/3): reproduces exact value from Laughlin wf


[hzz(x, hzz(y)] =1

s�(x� y)


15








[Q(x), Q(y)] =1

sn�(x� y), (78)







Q(x)|0i = 0, if s > 0,

Q(x)|0i = 0, if s < 0.(79)




�⇢e =s

8⇡

pGR =

s

2⇡(@2Q+ @2Q), (80)

which implies


s2q4

64⇡2


⇤.

(81)



64⇡2nq4. (82)


h�⇢e�⇢eiq =N(N + 1)

2N + 1

q4

16⇡B. (83)



h⇢ei=

(18(N + 1)(q`B)4, ⌫ = N

2N+1 ,18N(q`B)4, ⌫ = N+1

2N+1 .(84)





�⇢el ⇠ @2zhzz + @2


should be good for large Nfor N=1 (nu=1/3, 2/3): reproduces exact value from Laughlin wf

(HLR: 1/2 or 2 times Laughlin’s value)


[hzz(x, hzz(y)] =1

s�(x� y)


15








[Q(x), Q(y)] =1

sn�(x� y), (78)







Q(x)|0i = 0, if s > 0,

Q(x)|0i = 0, if s < 0.(79)




�⇢e =s

8⇡

pGR =

s

2⇡(@2Q+ @2Q), (80)

which implies


s2q4

64⇡2


⇤.

(81)



64⇡2nq4. (82)


h�⇢e�⇢eiq =N(N + 1)

2N + 1

q4

16⇡B. (83)



h⇢ei=

(18(N + 1)(q`B)4, ⌫ = N

2N+1 ,18N(q`B)4, ⌫ = N+1

2N+1 .(84)

Conclusion

• Low-q regime of FQH liquid: described by a fluid with internal metric degree of freedom, coupled to a gauge field

• Electron density ~ curvature of dynamic metric

• Static structure factor: algebraic calculation

Thank you

dynamics of composite fermions - harvard university › files › ultra-qm › ... · where we have...

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