(Non)-Commutative Field Theories for some Composite Fermi ... · between composite fermions is weak...

(Non)-Commutative Field Theories for some Composite Fermi Liquids

T. Senthil (MIT)

Zhihuan Dong, grad student@ MIT

https://arxiv.org/abs/2006.01282

2d electrons in the ``quantum Hall” regime

Filling factor ν = 1,2., 3, ...... (IQHE)

ν = 1/3, 1/5, ...... (FQHE)

ν = 1/2, 1/4,…: Metal with ρxx ≠0, ρxy ≠ 0, but ρxx≪ρxy. (Composite Fermi Liquids)

Composite fermi liquid theory (Halperin, Lee, Read (HLR) 1993)

Assume (Jain 89) each electron captures two flux quanta to form a new fermion(“Composite fermions” )

See reduced effective field B* = B - (2h/e)ρ

At ν = 1/2, B* = 0

=> form Fermi surface of composite fermions

kth, it generates a phase of magnitude 2p × 2p. By defi-nition, a closed loop around a unit vortex generates aphase of 2p. Thus the product in equation 4 attaches 2pvortices to each electron in the noninteracting Slater-determinant wavefunction.

So we see that the flux quanta of our “Bohr theory”represent the microscopic vortices of the many-particlewavefunction, and the composite fermion is actually thebound state of an electron and 2p quantum vortices. A fluxquantum is topologically similar to a vortex; it also pro-duces an Aharonov–Bohm phase of 2p for a closed patharound itself. Therefore it is often useful to model the vor-tices as flux quanta and envision the composite fermion asan electron carrying 2p flux quanta.

How do the vortices cancel part of the external Bfield? Consider a path in which one particle executes acounterclockwise loop enclosing an area A, with all theother particles held fixed. Equating the sum of theAharonov–Bohm phase 2pBA/v0 and the phase ⊗2p2prAcoming from the encircled vortices to an effectiveAharonov–Bohm phase 2pAB*/v0, we get the new field B*

of equation 2. Of course a magnetometer will still measuresimply B. But, as far as a composite fermion is concerned,B* is the real field, as we shall see.

The form of the wavefunction C provides an insightinto why the repulsive interaction between electronsmight force vortices on them. The wavefunction is veryeffective in keeping the electrons apart. The probabilitythat any two will come within a distance r of each othervanishes like r2(2p+1). Contrast that with the r2 vanishingfor a typical state satisfying the Pauli principle. Inessence, then, electrons transmute into composite fermi-ons by capturing 2p vortices because that is how they bestscreen the repulsive Coulomb interaction. The interactionbetween composite fermions is weak because most of theCoulomb interaction has been screened out—or used up—in making them.

Equations 2, 3, and 4 are the master equationsdescribing the quantum fluid of composite fermions. Sincethe first two are the same and can be derived from the third,everything ultimately stems from a single equation. Thequantum numbers of the composite fermion follow straight-forwardly from the observation that each one is produced by

a single electron. It has the samecharge and spin as the electron, and itis also a fermion.

Seeing composite fermionsThe crucial, non-perturbative respectin which composite fermions distin-

guish themselves from electrons is that they experiencean effective magnetic field, B*, that is drastically differentfrom the external magnetic field. The effective magneticfield is so central, direct, and dramatic a consequence ofthe formation of composite fermions that its observation istantamount to an observation of the composite fermionitself.

At filling factors n less than 1, the experiments clear-ly show us composite fermions subject to the magneticfield B* rather than electrons subject to B. The most com-pelling experimental evidence for the composite fermioncomes simply from plotting the high-field magneto-resistance as a function of 1/n*, which is proportional to B*.

APRIL 2000 PHYSICS TODAY 41

2 4 6 8 10 12 14MAGNETIC FIELD (tesla)B

MAGNETIC FIELD (tesla)B

LO

NG

ITU

DIN

AL

RE

SIST

IVIT

Y(k

)r

Wxx

0.5

0

1

0

125

3

43

HA

LL

RE

SISTA

NC

E(

/)

rxy

he

2

T=35 mK

2 n=1

8

6

4

2

0

10

10 12 14

25

37

49

511

13n=

LO

NG

ITU

DIN

AL

RE

SIST

IVIT

Y(k

)r

Wxx

FIGURE 2. CAPTURING TWO FLUX

quanta transforms each electron in theplane into a composite fermion thatexperiences, in effect, a reduced residualmagnetic field.

FIGURE 3. COMPARING INTEGRAL AND FRACTIONAL

quantum Hall regimes, top and bottom panels, respectively.16, 3

Blue curve (shown only in the top panel) is the Hall resistancewith quantum plateaus. Red curves are ordinary longitudinal

resistivity with a dip, labeled by its electron filling factor n, foreach plateau. The filling factor n = n/(2n + 1) corresponds to

a composite-fermion filling factor n* = n. Thus the two redcurves, despite their very different electron filling factors, are

remarkably similar.

E↵ective theory:

L = CF

i@t � a0 � iAext

0 +(~r� i(~a+ ~A))2

2m

! CF +

1

8⇡aµ✏µ⌫�@⌫a� (1)

(Short-hand for better version with properly quantized Chern-Simons terms)

Other composite fermi liquids

1. Electrons at ν = 1/4,…..

2. Bosons at filling ν = 1 (numerics: ground state is a version gapped by pairing but the metallic state is still worth studying)

In these cases too, the HLR construction admits a composite fermi liquid ground state

Effective theory: Fermi surface + U(1) gauge field with Chern-Simons coupling

The many successes of HLR

Explanation/succesfull prediction of many aspects of the observed metallic state

Parent state of nearby Jain sequence of prominent quantum Hall states

Parent state of paired non-abelian quantum Hall state at same filling

Successful backing from numerics

Despite these successes there were some vexing open questions that were extensively thought about in the 1990s without resolution.

Unsatisfactory aspects of the HLR theory-I

Theory should make sense within the Lowest Landau Level (LLL) but HLR not suited to projecting to LLL.

Mean field effective mass = bare electron mass in HLR

LLL limit: take m to zero; what happens??

~!c

Vint ⌧ ~!c: project to LLL

~!c

Vint ⌧ ~!c: project to LLL

Much discussed in the late 90s (Shankar, Murthy; Read; Halperin, Stern, Simon, van Oppen; D.-H. Lee, Pasquier, Haldane,.......) but dust never settled.

Experience of learning to work within LLL will be useful in other contexts.

Unsatisfactory aspects of the HLR theory-II

At ν = 1/2 LLL theory has an extra particle-hole symmetry that HLR is blind to.

Issue identified in the 90s (Grotov, Gan, Lee, Kivelson,

96; Lee 98) but a possible resolution only in last few years

``Dirac composite fermions”

Son + many others (2015 - present) +μ

Dirac fermions at finite density + U(1) gauge fields

This progress sidestepped issue of LLL projection; I will not discuss p/h symmetry today.

The basic problem Particles in lowest Landau level with 2-body repulsive interactions.

|mi: Some basis for single particle states in LLL

Many body states spanned by |m1,m2,m3, .....,mN i (anti)symmetrized for

(fermions) bosons.

Hamiltonian has only interaction energy

H =1

2

Zd2q

(2⇡)2U(q)⇢L(q)⇢L(�q)

Projected “density” operators ⇢L(q) satisfy the “GMP algebra” (a.k.a W1algebra)

[⇢L(q), ⇢L(q0)] = 2i sin

✓(q⇥ q0

)l2B2

◆⇢L(q+ q0

)

U(q) = e�q2l2B

2 U0(q) with U0 the Fourier transform of the microscopic 2-

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Composite Fermi liquid of bosons at ν = 1: Important progress in the 1990s

Pasquier-Haldane (1998): Represent density operator satisfying GMP algebra in terms of fermionic partons (= LLL version of composite fermions)

Highly redundant description; impose constraints to recover physical Hilbert space. Constraint operators themselves satisfy GMP algebra (with opposite sign).

Read(1998): ``W∞ gauge structure” broken to U(1) by mean field composite fermi liquid solution.

Low energy theory: Fermi surface + U(1) gauge field

Fluctuation effects treated diagrammatically in a conserving approximation;

Physically sensible final results.

Not much further development since…..

Open questions about Pasquier-Haldane-Read

1. What is the low energy effective field theory?

2. How related to HLR?

3. How to treat competition with paired bosonic Pfaffian at same filling? Jain states at proximate filling?

4. Can these methods be generalized to other problems?

Open questions about Pasquier-Haldane-Read

1. What is the low energy effective field theory?

2. How related to HLR?

3. How to treat competition with paired bosons :Pfaffian at same filling? Jain states at proximate filling?

4. Can these methods be generalized to other problems?

A suggested effective field theory

36

(651440, TS). Part of this work was performed during a visit of TS at the Aspen Center for Physics,

which is supported by National Science Foundation grant PHY-1607611.

Appendix A: Field theories for the bosonic composite fermi liquid

In this Appendix we present the field theory suggested in Ref. 23 and further discussed in Refs.

27, 28. In this theory, the composite fermion field may be given an interpretation as a (“fermion-

ized”) vortex of the physical bosons. Thus we will refer to this as the “Vortex Composite Fermi

Liquid” (VCFL), and will denote the corresponding composite fermion field v. The Lagrangian

takes the form

Lvcfl = v(@⌧ + ia0) v +1

2m⇤ | (@i + iai) v|2�

i

2⇡✏µ⌫�Aµ@⌫a� �

i

4⇡✏µ⌫�Aµ@⌫A� (A1)

Here a is the dynamical U(1) gauge field and Aµ is the background U(1) gauge field.

Though this form of the action was not explicitly written down in Ref. 23, the comments in Sec

II.D of that paper suggested that this e↵ective theory might describe the microscopic results in the

bulk of the paper. This e↵ective Lagrangian should be contrasted with that for the HLR theory:

LHLR = (@⌧ + i(a0 + A0)) v +1

2m⇤ | (@i + i(ai + Ai)) v|2�

i

4⇡✏µ⌫�aµ@⌫a� (A2)

In the microscopic derivation of HLR m⇤ is just the bare boson mass but if this Lagrangian emerges

in a LLL theory we should regard m⇤ as a renormalized e↵ective mass.

Both LHLR and Lvcfl describe (possibly distinct) composite fermi liquid phases of bosons at

⌫ = 1. In both theories all local operators are bosonic; in particular the operator with charge-1

under the background UA(1) gauge transformation is bosonic. The physical properties (deduced

within, for instance, the Random Phase Approximation) of both theories are similar and describe

metallic compressible phases. Nevertheless the two Lagrangians are di↵erent and it is not clear

whether they describe the same IR fixed point or not. Furthermore it has also not been clear which,

if any, of these two arises within a microscopic LLL treatment.

If we dispense with the LLL requirement, we can understand how to obtain either of these two

theories. The HLR Lagrangian can of course be obtained by a flux attachment transformation of the

original boson to composite fermion variables. The VCFL theory can be obtained as follows[27].

First perform a standard charge-vortex duality transformation of the boson system to pass to

a theory in terms of (bosonic) vortices coupled to a dynamical U(1) gauge field. At boson filling

Composite fermion as a fermionic vortex

Suggested in Read’s original 1998 paper but not explicitly derived.

No Chern-Simons term in action; contrast with usual HLR (with renormalized parameters).

36

(651440, TS). Part of this work was performed during a visit of TS at the Aspen Center for Physics,

which is supported by National Science Foundation grant PHY-1607611.

Appendix A: Field theories for the bosonic composite fermi liquid

In this Appendix we present the field theory suggested in Ref. 23 and further discussed in Refs.

27, 28. In this theory, the composite fermion field may be given an interpretation as a (“fermion-

ized”) vortex of the physical bosons. Thus we will refer to this as the “Vortex Composite Fermi

Liquid” (VCFL), and will denote the corresponding composite fermion field v. The Lagrangian

takes the form

Lvcfl = v(@⌧ + ia0) v +1

2m⇤ | (@i + iai) v|2�

i

2⇡✏µ⌫�Aµ@⌫a� �

i

4⇡✏µ⌫�Aµ@⌫A� (A1)

Here a is the dynamical U(1) gauge field and Aµ is the background U(1) gauge field.

Though this form of the action was not explicitly written down in Ref. 23, the comments in Sec

II.D of that paper suggested that this e↵ective theory might describe the microscopic results in the

bulk of the paper. This e↵ective Lagrangian should be contrasted with that for the HLR theory:

LHLR = (@⌧ + i(a0 + A0)) v +1

2m⇤ | (@i + i(ai + Ai)) v|2�

i

4⇡✏µ⌫�aµ@⌫a� (A2)

In the microscopic derivation of HLR m⇤ is just the bare boson mass but if this Lagrangian emerges

in a LLL theory we should regard m⇤ as a renormalized e↵ective mass.

Both LHLR and Lvcfl describe (possibly distinct) composite fermi liquid phases of bosons at

⌫ = 1. In both theories all local operators are bosonic; in particular the operator with charge-1

under the background UA(1) gauge transformation is bosonic. The physical properties (deduced

within, for instance, the Random Phase Approximation) of both theories are similar and describe

metallic compressible phases. Nevertheless the two Lagrangians are di↵erent and it is not clear

whether they describe the same IR fixed point or not. Furthermore it has also not been clear which,

if any, of these two arises within a microscopic LLL treatment.

If we dispense with the LLL requirement, we can understand how to obtain either of these two

theories. The HLR Lagrangian can of course be obtained by a flux attachment transformation of the

original boson to composite fermion variables. The VCFL theory can be obtained as follows[27].

First perform a standard charge-vortex duality transformation of the boson system to pass to

a theory in terms of (bosonic) vortices coupled to a dynamical U(1) gauge field. At boson filling

Do they describe the same IR physics, or are they distinct theories?

Which of these is actually obtained in the Pasquier-Haldane-Read microscopic theory?

See also: Alicea, Hermele, Motrunich, Fisher(2005) - flux attachment to fermionize vortices (not in LLL)Wang, TS (2016) - standard HLR + Fermi surface Berry phase

Our answers

1. This bosonic CFL is described by a non-commutative field theory of fermions + U(1) gauge field with no Chern-Simons term

2. Approximate map(*) to commutative field theory for long wavelength, low amplitude gauge fluctuations: HLR (+ subheading corrections) with parameters determined by interactions

(*) Using Seiberg, Witten (1999)

Remarks on non-commutative field theory

Field theories defined in a space-time where spatial coordinates do not commute.

Long history in physics/math literature; much studied in high energy physics 1998-2003

Of course the best example of such a space is the Landau Level - so perhaps not surprising that LLL effective theories are non-commutative.

Incompressible states: Proposals for a non-commutative Chern-Simons description (Susskind 2001, Polychronakos 2002,…….)

Connection to microscopics? Added value over standard commutative TQFT description?

Here I will focus on the metallic state of bosons at ν = 1.

Pasquier-Haldane-Read parton construction

Represent states in many body boson Hilbert space by

|m1, ......,mN i = ✏n1n2.....nN c†n1m1c†n2m2

......c†nNmN|0i

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Symmetric Anti-symmetricTensor

Physical states: singlets under SU(N) rotations on the right:

cmn ! cmn0URn0n

Generators ⇢Rnn0 = c†nmcmn0 of these SU(N)R satisfy constraint:

⇢Rnn0 | physi = �nn0 | physi<latexit sha1_base64="el6AXAP2bbV195jqKvcmuF/9x2Y=">AAADU3icfVNNbxMxEHUSPspCaQpHLiOSquUS7YaoCZWQKnGAEwqFtJW66crxzm6seu2V7S2Klv2PCIkDf4QLB/AmQWqpykiWxm9mPG+exrNccGN9/0ej2bpz9979jQfew0ebj7fa20+OjSo0wwlTQunTGTUouMSJ5Vbgaa6RZjOBJ7OLN3X85BK14Up+soscpxlNJU84o9ZB0XaDj+cL464CjKUWzQEYLlOB1kAhY9TQ/TjZe/+iC1rZZY0BJcHOETRP5/YAvHCGKZdlzE0u6CKjdl55LCozWUG4zKFaq8+whHYrmJwfRaXclZUXooyvl3k74aXJKcPS7wUsq7y3KFFTq7SBbqjnallbv/Ia2HkY0zRF7ZCsWr/eBZXU5AyueUdHXTCOt0kWwBx5qymXt7B24NUWX8Lc8KjMnT71JLRWxfUNYxSW/i/n5mDgeR5E7Y7fezXa7w/2we/5/jDoB7XTHw5eDiBwSG0dsrZx1P4WxooVGUrLBDXmLPBzOy2ptpwJdPoVBp1YFzTFM+dKmqGZlsudqGDHITEkSrsjLSzRqxUlzYxZZDOXWVM0/8ZWvG/GzgqbjKYll3lhUbJVo6QQYBXUCwYx18isWDiHMs0dV2Bzqimzbg09J8LfSeF257jfC9wGfOh3DkdrOTbIM/Kc7JGADMkheUfGZEJY42vjZ+N3kzS/N3+13C9ZpTYb65qn5Jq1Nv8AkkAQMg==</latexit><latexit sha1_base64="el6AXAP2bbV195jqKvcmuF/9x2Y=">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</latexit><latexit sha1_base64="el6AXAP2bbV195jqKvcmuF/9x2Y=">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</latexit><latexit sha1_base64="el6AXAP2bbV195jqKvcmuF/9x2Y=">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</latexit>

cmn: usual fermion anticommutation relations; destroy fermionic partons.<latexit sha1_base64="yEu/ea6cdOEUBnUiWfs++yizys4=">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</latexit><latexit sha1_base64="yEu/ea6cdOEUBnUiWfs++yizys4=">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</latexit><latexit sha1_base64="yEu/ea6cdOEUBnUiWfs++yizys4=">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</latexit><latexit sha1_base64="yEu/ea6cdOEUBnUiWfs++yizys4=">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</latexit>

‘Left’ SU(N) rotations generated by ⇢Lmm0 = c†nm0cmn are physical operators.

Take N ! 1, and |mi, |ni as single particle basis states for a Landau level.<latexit sha1_base64="RpXcxlOlN5KjqeFgx16ywc5b4VY=">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</latexit><latexit sha1_base64="RpXcxlOlN5KjqeFgx16ywc5b4VY=">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</latexit><latexit sha1_base64="RpXcxlOlN5KjqeFgx16ywc5b4VY=">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</latexit><latexit sha1_base64="RpXcxlOlN5KjqeFgx16ywc5b4VY=">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</latexit>

Momentum space formulation

Fourier transform cmn using magnetic translation operator ⌧k = eik·R:

cmn =

Zd2k

(2⇡)2hm|⌧k|nick

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Guiding center coordinate

Usual fermion operator

GMP algebra

anti-GMP algebra

Fourier-transformed density operators (= fermion bilinears) satisfy:

[⇢L(q), ⇢L(q0)] = 2i sin

✓(q⇥ q0)l2B

2

◆⇢L(q+ q0)

[⇢R(q), ⇢R(q0)] = �2i sin

✓(q⇥ q0)l2B

2

◆⇢R(q+ q0)

[⇢L(q), ⇢R(q0)] = 0<latexit sha1_base64="0DXC00YsNjjFHBj+XuvXDtqLa9U=">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</latexit><latexit sha1_base64="0DXC00YsNjjFHBj+XuvXDtqLa9U=">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</latexit><latexit sha1_base64="0DXC00YsNjjFHBj+XuvXDtqLa9U=">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</latexit><latexit sha1_base64="0DXC00YsNjjFHBj+XuvXDtqLa9U=">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</latexit>

The Hamiltonian H =12

R d2q(2⇡)2U(q)⇢L(q)⇢L(�q)

<latexit sha1_base64="ae6XoOqrk6NApE+buc+DfYF2AQU=">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</latexit><latexit sha1_base64="ae6XoOqrk6NApE+buc+DfYF2AQU=">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</latexit><latexit sha1_base64="ae6XoOqrk6NApE+buc+DfYF2AQU=">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</latexit><latexit sha1_base64="ae6XoOqrk6NApE+buc+DfYF2AQU=">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</latexit>

Constraint ⇢R(q)|physi = 0 for all non-zero q.<latexit sha1_base64="E3/kOjlFPrV1h0d++KB9vwscQnA=">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</latexit><latexit sha1_base64="E3/kOjlFPrV1h0d++KB9vwscQnA=">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</latexit><latexit sha1_base64="E3/kOjlFPrV1h0d++KB9vwscQnA=">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</latexit><latexit sha1_base64="E3/kOjlFPrV1h0d++KB9vwscQnA=">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</latexit>

Hartree-Fock solution

H is quartic in fermions - first study within Hartree-Fock.

A simple (global) symmetry preserving state:

hc†kck0i = nk�(2)

(k� k0)

<latexit sha1_base64="fzKTUyetvvqsJ3vgPJ9kYbnArOo=">AAADAHicdVJNb9NAEF2br2I+moLEhcuIFJEcGtk50BQJqQgJ5Vgk0laK02i9HjurrNdmdx0UWb7wV7hwACGu/Axu/BvWSYAUwUiWnue92Z15s1EhuDa+/8Nxr1y9dv3Gzk3v1u07d3dbe/dOdV4qhiOWi1ydR1Sj4BJHhhuB54VCmkUCz6L5y4Y/W6DSPJdvzLLASUZTyRPOqLGp6Z7zAParkFEBw3ofuIa3JVWGM+ASElSZFWk4gIQrbUCbMl7CO25mlh1anUI8eJWzeQ888MKFLijDyu8FLKu9F6B5VgiETiryiIou6GWWoVFLsB1qVAsuU3skNfgMwJZHmHJZxVwXgi4zama1FwoqU3sEuwhjmqaoplXYMFEC85pt/TypIVRr7XOQ2yoIYxSGXlSdfrfu/M7bmf4Ud70QZXzp6mmr7fd83w+CABoQHD71LTg6GvSDAQQNZaNNNnEybX0P45yVGUrDBNV6HPiFmVQrMwXaUUqN1p45TXFsoaQZ6km1WmANj20mhiRX9pMGVtntiopm2roXWWXTnf6bW7X8D25cmmQwqbgsSoOSrS9KSgEmh+Y1QMwVMiOWFlCmeLN4NqOKMmPfjGdN+DUp/B+c9nuB3fnrfvt4sLFjhzwkj0iHBOSQHJMhOSEjwpza+eB8cj67792P7hf361rqOpua++RSuN9+AgeQ8IQ=</latexit><latexit sha1_base64="fzKTUyetvvqsJ3vgPJ9kYbnArOo=">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</latexit><latexit sha1_base64="fzKTUyetvvqsJ3vgPJ9kYbnArOo=">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</latexit><latexit sha1_base64="fzKTUyetvvqsJ3vgPJ9kYbnArOo=">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</latexit>

Fermions acquire a dispersion (*) and form a Fermi sea with area set by their density.

Mean field composite fermi liquid

(*) Approximate as parabolic - what matters is near Femi surface anyway.

Mean field action:

SHF =

Zd⌧

d2k

(2⇡)2ck

dckd⌧

�✓

k2

2m⇤

◆c†kck

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Beyond mean field

Mean field breaks infinite gauge symmetry generated by ⇢R - the breakingis weak as q ! 0.

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Read (1998): Diagrammatic conserving approximation but here we seek an effective Lagrangian.

=> Include small-q gauge fluctuations on top of mean field

Di�culty: In k-space, mean field state is simple but ⇢R transformations mixstates at di↵erent momenta.

In m,n (Landau orbital) basis, gauge transformations look simple but meanfield state is complicated.

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Convenient formulation: Fields in non-commutative space and time

Define abstract fields as a function of the non-commuting guiding center coordinate

The Fourier transform definition helps deduce many of their properties.

For instance, products of such fields are given by the ``star product" which is associative but not commutative:

c(R, ⌧) =

Zd2k

(2⇡)12

eik·Rck,⌧<latexit sha1_base64="j1RaWDSn/LzOM0ODYLyEhlelaFA=">AAACj3icdVHbitswEJXd29bdttn2sS9DQyELJVh+aLL0QqAv7du2NLsLcRJkeZwVkWUjyYUg/Dv9oL71byrnsnRLOyA4zDmjOTOT1VIYG8e/gvDO3Xv3Hxw9jB4dP37ytHfy7MJUjeY45ZWs9FXGDEqhcGqFlXhVa2RlJvEyW3/s+MvvqI2o1De7qXFespUSheDM+tSy9wMigDTDlVAuF6aWbFMye91GfJB2ICvg62tILWtOAd6D1wplIS004y5fJAfNunWDJK3F6WJH0dYlbQu4cOJGAinPK197+LYFvnQ37K5JG6Wo8ttOIm9x2evHwziOKaXQATp6E3twdjZO6BhoR/nok32cL3s/07ziTYnKcsmMmdG4tnPHtBVcom/TGKwZX7MVzjxUrEQzd9t9tvDKZ3IoKu2fn3ab/bPCsdKYTZl5ZWfR/M1tff+DmzW2GM+dUHVjUfFdo6KRYCvojgO50Mit3HjAuBbeK/Br5hdq/Qkjv4TDpPB/cJEMaTykX5L+ZLxfxxF5QV6SAaFkRCbkEzknU8KD4yAJ3gbvwpNwFH4IJztpGOxrnpNbEX7+DVqbw8I=</latexit><latexit sha1_base64="j1RaWDSn/LzOM0ODYLyEhlelaFA=">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</latexit><latexit sha1_base64="j1RaWDSn/LzOM0ODYLyEhlelaFA=">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</latexit><latexit sha1_base64="j1RaWDSn/LzOM0ODYLyEhlelaFA=">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</latexit>

f(R) ⇤ g(R) = ei✓ij2 @Ri@R0

j f(R)g(R0)��R0=R

<latexit sha1_base64="1iXM4TEjE1UrR2KsOHuV+6zgSWQ=">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</latexit><latexit sha1_base64="1iXM4TEjE1UrR2KsOHuV+6zgSWQ=">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</latexit><latexit sha1_base64="1iXM4TEjE1UrR2KsOHuV+6zgSWQ=">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</latexit><latexit sha1_base64="1iXM4TEjE1UrR2KsOHuV+6zgSWQ=">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</latexit>

Can define derivatives, and integrals

Here [Ri, Rj ] = i✓ij = i⇥✏ij .

⇥ = �l2B = “non-commutativity” parameter.<latexit sha1_base64="/U9Lyfi0l4F90skYWm8dZY2HbIE=">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</latexit><latexit sha1_base64="/U9Lyfi0l4F90skYWm8dZY2HbIE=">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</latexit><latexit sha1_base64="/U9Lyfi0l4F90skYWm8dZY2HbIE=">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</latexit><latexit sha1_base64="/U9Lyfi0l4F90skYWm8dZY2HbIE=">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</latexit>

How does this help?

Mean field action also looks simple:

SHF =

Zd⌧d2R

✓c(R, ⌧) ⇤ dcR

d⌧+

1

2mrc(R, ⌧) ⇤ rc(R, ⌧)

◆

<latexit sha1_base64="pdJuzL/V4k/TsKkv8iGRM/3s2oc=">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</latexit><latexit sha1_base64="pdJuzL/V4k/TsKkv8iGRM/3s2oc=">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</latexit><latexit sha1_base64="pdJuzL/V4k/TsKkv8iGRM/3s2oc=">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</latexit><latexit sha1_base64="pdJuzL/V4k/TsKkv8iGRM/3s2oc=">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</latexit>

Simple action of transformations generated by ⇢L,R:

c(R, ⌧) ! UL(R, ⌧) ⇤ c(R, ⌧) ⇤ UR

(R, ⌧)

Here take UL,R= ei✓L,R(R,⌧)

with exponential defined through power series

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Now require that action be invariant under long wavelength `right’ gauge transformations: Replace all derivatives by covariant derivatives

Non-commutative action for composite fermi liquid

S =

Zd2Rd⌧ c ⇤D0c+ ia0⇢+

1

2m⇤DicDic

The covariant derivatives

Dµc = @µc� ic ⇤ aµ � iAµ ⇤ c

aµ: dynamical non-commutative U(1) gauge field coupling to ‘right’ currents.

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Gauge invariance:

c ! c+ ic ⇤ ✓R + i✓L ⇤ caµ ! aµ + @µ✓R + i(aµ ⇤ ✓R � ✓R ⇤ aµ)Aµ ! Aµ + @µ✓L + i(✓L ⇤Aµ �Aµ ⇤ ✓L)

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Fermions at finite density, form a Fermi surface; gauge fields vary slowly on scale lB.

Comments

1. Theory has Fermi surface + dynamical U(1) gauge field without any Chern-Simons term but is formulated in non-commutative space-time.

2. Hartree-Fock theory gives estimate of bare effective mass m* in terms of interaction strength.

How to compare with previous proposals?

Previous proposed field theories (HLR, fermionic vortex liquid) are all ordinary commutative field theories.

However, the non-commutativity occurs at scale of magnetic length, we can hope to approximate long wavelength gauge fluctuations in terms of a `coarge-grained’ commutative theory.

Key tool: The Seiberg-Witten map

Expansion is local (coefficients only involve fields and derivatives at same space-time point).

Gauge transformation parameters are mapped as functions of the gauge field configurations themselves.

Here we will need a small generalization that includes the fermion fields.

Seiberg, Witten (1999): Map pure non-commutative gauge theory withgauge fields aµ to a commutative gauge theory with gauge fields aµ in a sys-tematic expansion in powers of ⇥.

<latexit sha1_base64="GVQ73D1eJXgvDCPlWRD3WuWk+5E=">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</latexit><latexit sha1_base64="GVQ73D1eJXgvDCPlWRD3WuWk+5E=">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</latexit><latexit sha1_base64="GVQ73D1eJXgvDCPlWRD3WuWk+5E=">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</latexit><latexit sha1_base64="GVQ73D1eJXgvDCPlWRD3WuWk+5E=">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</latexit>

Pure U(1) gauge theory: Exact an non-perturbative Seiberg-Witten map

exists (Liu, Michelson, Okawa, Ooguri 01, Mukhi, Suryanarayana 01).

Appealing physical interpretation: Relation to map between Lagrangian and

Eulerian descriptions of a fluid (Jackiw, Pi, Polychronakos, 02; also Susskind

01)<latexit sha1_base64="Pt6OT9i9+SV0xF9gQQPQkqd4kYw=">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</latexit><latexit sha1_base64="Pt6OT9i9+SV0xF9gQQPQkqd4kYw=">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</latexit><latexit sha1_base64="Pt6OT9i9+SV0xF9gQQPQkqd4kYw=">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</latexit><latexit sha1_base64="Pt6OT9i9+SV0xF9gQQPQkqd4kYw=">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</latexit>

Seiberg-Witten map to linear order

A(A) = A+�A(A) a(a) = a+�a(a)

✓L(✓L, A) = ✓L +�✓L(✓L, A) ✓R(✓R, a) = ✓R +�✓R(✓R, a) (1)

c( , A, a) = +� ( , A, a)

Here �A, �a, �✓R, �✓L, and �c are all of o(⇥).<latexit sha1_base64="AH2dIKfdLLKGiww+bvAKhb0hWxI=">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</latexit><latexit sha1_base64="AH2dIKfdLLKGiww+bvAKhb0hWxI=">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</latexit><latexit sha1_base64="AH2dIKfdLLKGiww+bvAKhb0hWxI=">AAAD43ichVPLbtNAFJ3YPEp4pbBkMyKhSkUVORUINkitYNFFFqVq2kqZKFxPbpJRx2NrZoyIrGzZsAAhtvwUOz6FHZPEdtNQ1CtZOnPOuQ9fj8NECmOD4HfF82/cvHV740717r37Dx7WNh+dmDjVHLs8lrE+C8GgFAq7VliJZ4lGiEKJp+H527l++hG1EbE6ttME+xGMlRgJDtZRg83KH8pCHAuVgRRjNasyCSHKLIIEPyWzKnWx32QTsNn+bHvrTY6es3coLaxIdIvC8gClD0rfhUSZilUahagZW1RndoIWBp2lY3maDTo7az1LoSh5XZ4bKLccXbYc7ayNWQp56evS8sF5kyVGFP1Wizq6nNLhq330YhNVhmpYfIAD1EgbxX4bOyWGFVyM+C/VcRSoYUnzBgVXEKSk8Yg24iY7nvu2G61BrR60XgbzoEErKEDOtHOmTvI4HNR+sWHM0wiV5RKM6bWDxPYz0FZwie7qpAYT4Ocwxp6DCiI0/WxxR2f0mWOGdBRr9yhLF+xqRgaRMdModM4I7MSsa3PyKq2X2tHrfiZUklpUfNlolEpqYzq/8HQoNHIrpw4A18LNSvkENHDrfouqW0LxpvT/4GS31Q5a7fe79b0X+To2yBPylDRJm7wie+SAHJIu4d4H77P31fvmo//F/+7/WFq9Sp7zmFwK/+dfvR079w==</latexit><latexit sha1_base64="AH2dIKfdLLKGiww+bvAKhb0hWxI=">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</latexit>

The map can be found explicitly by requiring that the hatted fields satisfy ordinary U(1) gauge invariance for both dynamical and background gauge fields.

Strategy: Plug in the map to get an effective commutative gauge theory

Emergence of HLR

E↵ective commutative Lagrangian

L = LHLR + Lcorr

LHLR = @0 � i(a0 + A0) + ia0⇢+1

2m⇤

��⇣@i � i(ai + Ai)

⌘

��2

� i1

4⇡✏↵�� a↵@� a�

Lcorr = �⇥

2✏↵�

✓(f0� � F0�)@↵( )� @↵(a� � A�)( D0 � 1

2m⇤ |Di |2)◆

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First term is just the standard HLR theory!(but with correct effective mass)

Second term is a correction that is small for long wavelength, low amplitude gauge fluctuations

Comments

1. The commutative theory has a Chern-Simons term (as expected in HLR) which is absent in the non-commutative theory.

This is traced to the non-trivial form factors in the density operator in the non-commutative theory (roughly correspond to a non-trivial Berry phase)

2. Can use the commutative effective theory to, say, study Jain states near ν = 1 (technically a bit hard within the microscopic Pasquier-Haldane-Read approach) The correction terms may need to be included for some purposes.

3. We do not find the fermionic vortex field theory.

Conclusions/outlook

Other composite fermi liquids?

Some simple generalizations easy (eg spinfull bosons at total filling 1)

Fermions at 1/2-filling?

In all cases it is natural that a LLL theory is non-commutative .

Other generalizations possible - eg mean field theory of multicomponent quantum-Hall like systems at total integer filling.

(Non)-Commutative Field Theories for some Composite Fermi ... · between composite fermions is weak...

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