Fluctuations and magnetoresistance oscillations near a half ...Fluctuations and magnetoresistance oscillations near a half-filled Landau level Michael Mulligan UC Riverside Jiang,

Fluctuations and magnetoresistance oscillations near

a half-filled Landau level

Michael Mulligan UC Riverside

Jiang, Stormer, Tsui, Pfeiffer, & West (1989)

directly influenced the work with Plamadeala and Nayak finding a particular realization of a perfect metal in (1+1)d

beautiful paper

thanks to my collaborators

Alfred Cheung (Stanford), Amartya Mitra (UCR), Sri Raghu (Stanford)

arXiv:1611.08910

arXiv:1901.08070

Objective: To argue that emergent gauge field fluctuations improve the comparison between composite fermion theories of the half-filled lowest Landau level and geometric resonance experiments.

Jiang, Stormer, Tsui, Pfeiffer, & West (1989)

composite fermions

heuristic picture: electrons at half-filling = composite fermions in zero effective field

Girvin & MacDonald (1987); Read (1989); Jain(1989); Hansson, Kivelson, & Zhang (1989); Lopez & Fradkin (1991); Halperin, Lee, & Read (1993); Kalmeyer & Zhang (1992)

B

+

B

+

electron composite fermion

$

from D. Arovas’ Ph.D thesis

b = B � 4⇡ne<latexit sha1_base64="bOSVjOoFj6Gf8Pj5yn0S05s1vec=">AAAB+nicbVBNS8NAEN34WetXqkcvi0XwYkm0oBeh1IvHCvYD2hA220m7dLMJuxulxP4ULx4U8eov8ea/cdvmoK0PBh7vzTAzL0g4U9pxvq2V1bX1jc3CVnF7Z3dv3y4dtFScSgpNGvNYdgKigDMBTc00h04igUQBh3Ywupn67QeQisXiXo8T8CIyECxklGgj+XYpwNe4js9wFfcShoUPvl12Ks4MeJm4OSmjHA3f/ur1Y5pGIDTlRKmu6yTay4jUjHKYFHupgoTQERlA11BBIlBeNjt9gk+M0sdhLE0JjWfq74mMREqNo8B0RkQP1aI3Ff/zuqkOr7yMiSTVIOh8UZhyrGM8zQH3mQSq+dgQQiUzt2I6JJJQbdIqmhDcxZeXSeu84l5U3LtquVbP4yigI3SMTpGLLlEN3aIGaiKKHtEzekVv1pP1Yr1bH/PWFSufOUR/YH3+AMaKkcE=</latexit>

composite fermions

heuristic picture: electrons at half-filling = composite fermions in zero effective field

Girvin & MacDonald (1987); Read (1989); Jain(1989); Hansson, Kivelson, & Zhang (1989); Lopez & Fradkin (1991); Halperin, Lee, & Read (1993); Kalmeyer & Zhang (1992)

B

+

electron composite fermion

$

from D. Arovas’ Ph.D thesis

b = B � 4⇡ne<latexit sha1_base64="bOSVjOoFj6Gf8Pj5yn0S05s1vec=">AAAB+nicbVBNS8NAEN34WetXqkcvi0XwYkm0oBeh1IvHCvYD2hA220m7dLMJuxulxP4ULx4U8eov8ea/cdvmoK0PBh7vzTAzL0g4U9pxvq2V1bX1jc3CVnF7Z3dv3y4dtFScSgpNGvNYdgKigDMBTc00h04igUQBh3Ywupn67QeQisXiXo8T8CIyECxklGgj+XYpwNe4js9wFfcShoUPvl12Ks4MeJm4OSmjHA3f/ur1Y5pGIDTlRKmu6yTay4jUjHKYFHupgoTQERlA11BBIlBeNjt9gk+M0sdhLE0JjWfq74mMREqNo8B0RkQP1aI3Ff/zuqkOr7yMiSTVIOh8UZhyrGM8zQH3mQSq+dgQQiUzt2I6JJJQbdIqmhDcxZeXSeu84l5U3LtquVbP4yigI3SMTpGLLlEN3aIGaiKKHtEzekVv1pP1Yr1bH/PWFSufOUR/YH3+AMaKkcE=</latexit>

ne = hf†fi = �1

2

1

4⇡h@xay � @yaxi

h@xay � @yaxi = �⇣@xAy � @yAx

⌘= �B < 0

HLR composite Fermi liquidHalperin, Lee, & Read (1993); Kalmeyer & Zhang (1992)

composite fermions provide a more “amenable” mean-field vacuum: an emergent Fermi liquid-like state in zero effective field

Lelectron = †e

⇣i@t +At +

1

2me(@j � iAj)

2⌘ e + Lint[

†e e]

$LCFL = Lf + Lgauge + Lint[f

†f ],

Lf = f†⇣i@t + (at +At) +

1

2mf(@j � i(aj +Aj))

2⌘f

Lgauge =1

2⇡✏µ⌫⇢aµ@⌫ a⇢ �

2

4⇡✏µ⌫⇢aµ@⌫ a⇢

“flux attachment”

l<latexit sha1_base64="HUHAyqJrVMO8w9K/Aw4Vadein3s=">AAAB83icbZDLSgMxFIbPeK31VnXpJlgEV2VGBF0W3bisYC/QGUomk2lDM8mQi6WUvoYbF4q49WXc+Tam7Sy09YfAx3/O4Zz8cc6ZNr7/7a2tb2xubZd2yrt7+weHlaPjlpZWEdokkkvVibGmnAnaNMxw2skVxVnMaTse3s3q7SeqNJPi0YxzGmW4L1jKCDbOCkObJ3IksFJy1KtU/Zo/F1qFoIAqFGr0Kl9hIonNqDCEY627gZ+baIKVYYTTaTm0muaYDHGfdh0KnFEdTeY3T9G5cxKUSuWeMGju/p6Y4EzrcRa7zgybgV6uzcz/al1r0ptowkRuDRVksSi1HBmJZgGghClKDB87wEQxdysiA6wwMS6msgshWP7yKrQua4Hjh6tq/baIowSncAYXEMA11OEeGtAEAjk8wyu8edZ78d69j0XrmlfMnMAfeZ8/rWiSFg==</latexit><latexit sha1_base64="HUHAyqJrVMO8w9K/Aw4Vadein3s=">AAAB83icbZDLSgMxFIbPeK31VnXpJlgEV2VGBF0W3bisYC/QGUomk2lDM8mQi6WUvoYbF4q49WXc+Tam7Sy09YfAx3/O4Zz8cc6ZNr7/7a2tb2xubZd2yrt7+weHlaPjlpZWEdokkkvVibGmnAnaNMxw2skVxVnMaTse3s3q7SeqNJPi0YxzGmW4L1jKCDbOCkObJ3IksFJy1KtU/Zo/F1qFoIAqFGr0Kl9hIonNqDCEY627gZ+baIKVYYTTaTm0muaYDHGfdh0KnFEdTeY3T9G5cxKUSuWeMGju/p6Y4EzrcRa7zgybgV6uzcz/al1r0ptowkRuDRVksSi1HBmJZgGghClKDB87wEQxdysiA6wwMS6msgshWP7yKrQua4Hjh6tq/baIowSncAYXEMA11OEeGtAEAjk8wyu8edZ78d69j0XrmlfMnMAfeZ8/rWiSFg==</latexit><latexit sha1_base64="HUHAyqJrVMO8w9K/Aw4Vadein3s=">AAAB83icbZDLSgMxFIbPeK31VnXpJlgEV2VGBF0W3bisYC/QGUomk2lDM8mQi6WUvoYbF4q49WXc+Tam7Sy09YfAx3/O4Zz8cc6ZNr7/7a2tb2xubZd2yrt7+weHlaPjlpZWEdokkkvVibGmnAnaNMxw2skVxVnMaTse3s3q7SeqNJPi0YxzGmW4L1jKCDbOCkObJ3IksFJy1KtU/Zo/F1qFoIAqFGr0Kl9hIonNqDCEY627gZ+baIKVYYTTaTm0muaYDHGfdh0KnFEdTeY3T9G5cxKUSuWeMGju/p6Y4EzrcRa7zgybgV6uzcz/al1r0ptowkRuDRVksSi1HBmJZgGghClKDB87wEQxdysiA6wwMS6msgshWP7yKrQua4Hjh6tq/baIowSncAYXEMA11OEeGtAEAjk8wyu8edZ78d69j0XrmlfMnMAfeZ8/rWiSFg==</latexit><latexit sha1_base64="HUHAyqJrVMO8w9K/Aw4Vadein3s=">AAAB83icbZDLSgMxFIbPeK31VnXpJlgEV2VGBF0W3bisYC/QGUomk2lDM8mQi6WUvoYbF4q49WXc+Tam7Sy09YfAx3/O4Zz8cc6ZNr7/7a2tb2xubZd2yrt7+weHlaPjlpZWEdokkkvVibGmnAnaNMxw2skVxVnMaTse3s3q7SeqNJPi0YxzGmW4L1jKCDbOCkObJ3IksFJy1KtU/Zo/F1qFoIAqFGr0Kl9hIonNqDCEY627gZ+baIKVYYTTaTm0muaYDHGfdh0KnFEdTeY3T9G5cxKUSuWeMGju/p6Y4EzrcRa7zgybgV6uzcz/al1r0ptowkRuDRVksSi1HBmJZgGghClKDB87wEQxdysiA6wwMS6msgshWP7yKrQua4Hjh6tq/baIowSncAYXEMA11OEeGtAEAjk8wyu8edZ78d69j0XrmlfMnMAfeZ8/rWiSFg==</latexit>

�<latexit sha1_base64="YhZiJh5LzN0XNhuJAbFnN31OsR4=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBiyVRQY9FLx6r2A9oQ9lsJ+3SzSbsboQS+g+8eFDEq//Im//GTduDtj4YeLw3w8y8IBFcG9f9dgorq2vrG8XN0tb2zu5eef+gqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLRbe63nlBpHstHM07Qj+hA8pAzaqz0cFbqlStu1Z2CLBNvTiowR71X/ur2Y5ZGKA0TVOuO5ybGz6gynAmclLqpxoSyER1gx1JJI9R+Nr10Qk6s0idhrGxJQ6bq74mMRlqPo8B2RtQM9aKXi/95ndSE137GZZIalGy2KEwFMTHJ3yZ9rpAZMbaEMsXtrYQNqaLM2HDyELzFl5dJ87zqXVS9+8tK7WYeRxGO4BhOwYMrqMEd1KEBDEJ4hld4c0bOi/PufMxaC8585hD+wPn8AanCjMg=</latexit>

geometric resonances

quantum oscillations that occur because of the presence of a, e.g. 1D, periodic scalar/vector ( ) potential of wavelength a� = ⌥1

4

SdH oscillations

geometric resonances (m = 3, 2, 1)

c~e|Bm| =

a

2kF(m+ �), m = 1, 2, 3

Weiss, v. Klitzing, Ploog, & Weimann (1989)

m = 1 m = 2

m = 3

“SdH oscillations”

geometric resonances (m = 2, 1)

Smet, Jobst, v. Klitzing, Weiss, Wegscheider, & Umansky (1999)

m = 1 m = 2

m = 3

(i) composite fermions respond to an applied periodic electric modulation as a magnetic modulation

(composite fermions are electrically-neutral dipoles)

(ii) composite fermion Fermi momentum kCFF =

p4⇡ne

Read (1995); Haldane & Pasquier (1997); Lee (1998); Stern, Halperin, von Oppen & Simon (1999); Murthy & Shankar (2003)

geometric resonances of composite fermions (pre 2014)

Kamburov, Liu, Mueed, Shayegan, Pfeiffer, West, & Baldwin (2014)

e e

e

Be↵

geometric resonances of composite fermions (post 2014)



e e

e

Be↵



e e

e

Be↵



kF (⌫ < 1/2) =p4⇡ne

kF (⌫ > 1/2) =p4⇡(B/2⇡ � ne) =

p4⇡nh

inferred from solid line

inferred from dotted line

e e

e

Be↵


minority hypothesisKamburov, Liu, Mueed, Shayegan, Pfeiffer, West, & Baldwin (2014)

Barkeshli, MM, and Fisher (2015)composite fermion theories of holes and electrons …?

In this talk, I will argue:

(i) for a different interpretation. In mean-field theory, where fluctuations of the emergent gauge field are ignored, HLR composite fermions experience BOTH periodic scalar and vector potentials.

Cheung, Raghu, & MM (2017)

see also: Wang, Cooper, Halperin, & Stern (2017); Levin & Son (2017)

-0.04 -0.02 0.02 0.04

0.1

0.2

0.3

0.4

Δρxx

(Arb. u

nits

� �0.00

��(B�−�B1/2)/B1/2

)

In this talk, I will argue:

(ii) In the Dirac composite fermion theory (to be described), emergent gauge fluctuations improve the comparison between theory and experiment (we expect similar agreement in the HLR theory).

Mitra & MM (2019)

see also: Wang, Cooper, Halperin, & Stern (2017); Levin & Son (2017)

ne ⌘�Le

�At= †

e e +B

4⇡

half-filled lowest Landau “interpreted” as a half-filled zeroth Landau level of a Dirac fermion (similar to graphene)

⌫e =1

2() †

e e = 0

B

me! 1singular LLL is “regularized” by a Dirac electron Lagrangian

Son (2015)

Dirac electrons

Le = e�µ(i@µ +Aµ) e +

1

8⇡✏µ⌫⇢Aµ@⌫A⇢ + . . .

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t 7! �t

e 7! ��t ⇤e

(At, Ax, Ay) 7! (�At, Ax, Ay)

+

Le 7! Le +1

4⇡✏µ⌫⇢Aµ@⌫A⇢

particle� hole = CT+ T

CT =

T =


1

8⇡✏µ⌫⇢Aµ@⌫A⇢ + . . .

<latexit sha1_base64="1auKK0j4/9YRuv7L7fL2Qhe0uck=">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</latexit>

Dirac electrons

Witten (2003)


1

8⇡✏µ⌫⇢Aµ@⌫A⇢ + . . .

l

L = �µ(i@µ + aµ) � 1

4⇡✏µ⌫⇢aµ@⌫A⇢ +

1

8⇡✏µ⌫⇢Aµ@⌫A⇢ + . . .

<latexit sha1_base64="QQwek/IlVfRXLLuBZ6l1nR8rg2o=">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</latexit> Son (2015)

Dirac composite fermions

Son (2015)


L = �µ(i@µ + aµ) � 1

4⇡✏µ⌫⇢aµ@⌫A⇢ +

1

8⇡✏µ⌫⇢Aµ@⌫A⇢ + . . .

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Son (2015)


L = �µ(i@µ + aµ) � 1

4⇡✏µ⌫⇢aµ@⌫A⇢ +

1

8⇡✏µ⌫⇢Aµ@⌫A⇢ + . . .

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Son (2015)


particle� hole = T+ T

⌫e =1

2() b = 0

�L�at

= 0 () † =B

4⇡<latexit sha1_base64="UBtN+yS/+6+CFP32uv63ZKTvyiA=">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</latexit>

ne =1

4⇡(�b+B)

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unbroken particle-hole symmetry forbids a Dirac mass and Chern-Simons term for the emergent gauge field

�ij =1

4⇡

⇣✏ij �

1

2✏ik(�

)�1kl ✏lj

⌘

<latexit sha1_base64="Fe10jazYFbBDQ5DdQDw78NVa144=">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</latexit>

(aside: particle-hole symmetric conductivity if the composite fermions exhibit zero Hall response)

�~a =⇣0,W sin(Kx1)

⌘,

<latexit sha1_base64="j5g7hjl4+nEz/IfMAMpqk3Oqvco=">AAACFnicbVBNS8NAEN34WetX1aOXxSK0UEuigl6EUi+Clwr2A5oQNttpu3SzCbubYgn9FV78K148KOJVvPlvTNoetPXBwOO9GWbmeSFnSpvmt7G0vLK6tp7ZyG5ube/s5vb2GyqIJIU6DXggWx5RwJmAumaaQyuUQHyPQ9MbXKd+cwhSsUDc61EIjk96gnUZJTqR3NyJ3QGuCbaHQGMyxlfYrrJewSzhJrYVE4Vb/OBaxVQslrJuLm+WzQnwIrFmJI9mqLm5L7sT0MgHoSknSrUtM9ROTKRmlMM4a0cKQkIHpAfthArig3LiyVtjfJwoHdwNZFJC44n6eyImvlIj30s6faL7at5Lxf+8dqS7l07MRBhpEHS6qBtxrAOcZoQ7TALVfJQQQiVLbsW0TyShOkkyDcGaf3mRNE7L1lnZujvPV6qzODLoEB2hArLQBaqgG1RDdUTRI3pGr+jNeDJejHfjY9q6ZMxmDtAfGJ8/G8CcJg==</latexit>

periodic scalar potential gives rise to a periodic vector potential

�⇢ii / |✏ij |�� jj<latexit sha1_base64="SwlhYnuewJTzWFlIsGp63cKOH4U=">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</latexit>

geometric resonances appear as corrections to resistivity due to periodic vector potential; largest corrections in direction x1

<latexit sha1_base64="wZ7aekLp5jkt/r6OHqfsptkK9pA=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi8eK9gPaUDbbTbt0swm7E7GE/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvSKQw6LrfTmFldW19o7hZ2tre2d0r7x80TZxqxhsslrFuB9RwKRRvoEDJ24nmNAokbwWjm6nfeuTaiFg94DjhfkQHSoSCUbTS/VPP65UrbtWdgSwTLycVyFHvlb+6/ZilEVfIJDWm47kJ+hnVKJjkk1I3NTyhbEQHvGOpohE3fjY7dUJOrNInYaxtKSQz9fdERiNjxlFgOyOKQ7PoTcX/vE6K4ZWfCZWkyBWbLwpTSTAm079JX2jOUI4toUwLeythQ6opQ5tOyYbgLb68TJpnVe+86t1dVGrXeRxFOIJjOAUPLqEGt1CHBjAYwDO8wpsjnRfn3fmYtxacfOYQ/sD5/AEMrI2j</latexit>

setup: electrical conductivity dictionary

Kivelson, Lee, Krotov, & Gan (1997)

�� ij =1

L1L2⌃M

⇣@EM fD(EM )

⌘⌧(EM )vMi vMj ,

<latexit sha1_base64="OlAUkLGTafoVwsU3mtVSEZbdYdg=">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</latexit>

we assume:

and we calculate the spectrum of the quadratic Hamiltonian

H = ~� ·⇣

@

@~x+ ~a

⌘+m�3

<latexit sha1_base64="x+gz4NWcmEJHZZeLlQp0vkdM4g8=">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</latexit>

in mean-field theory:

m = 0

µ =pB

<latexit sha1_base64="/9Ce53PioQ73/eM27Bwc4ws9pmY=">AAACBXicbVC7TsMwFHV4lvAKMMJgUYGYqgSQYEGqysJYJPqQmqhyXKe1ajvBdpCqqAsLv8LCAEKs/AMbf4ObZoCWI13p+Jx75XtPmDCqtOt+WwuLS8srq6U1e31jc2vb2dltqjiVmDRwzGLZDpEijArS0FQz0k4kQTxkpBUOryd+64FIRWNxp0cJCTjqCxpRjLSRus4Bh8fwCrrQx9L2eZq/fHUvdVYbd52yW3FzwHniFaQMCtS7zpffi3HKidCYIaU6npvoIENSU8zI2PZTRRKEh6hPOoYKxIkKsvyKMTwySg9GsTQlNMzV3xMZ4kqNeGg6OdIDNetNxP+8TqqjyyCjIkk1EXj6UZQyqGM4iQT2qCRYs5EhCEtqdoV4gCTC2gRnmxC82ZPnSfO04p1VvNvzcrVWxFEC++AQnAAPXIAquAF10AAYPIJn8ArerCfrxXq3PqatC1Yxswf+wPr8ATuLleI=</latexit>

⌧(EM ) = ⌧

vMi =@EM

@pi<latexit sha1_base64="9phDWA+U9hYcUx3vN2rJqfFVmco=">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</latexit>

setup: Kubo formula

Charbonneau, van Vliet & Vasilopoulos (1982); Peeters & Vasilopoulos (1993)

Cheung, Raghu, & MM (2017)

(ii) IR equivalence? in the regime of experimental parameters: HLR and the Dirac composite fermion theories have the same mean-field predictions

(i) systematic 2% or less “inward shift” from the mean-field result

mean-field theory result

H = ~� ·⇣

@

@~x+ ~a

⌘+m�3


strategy: (i) determine fluctuation corrections to quadratic Hamiltonian

by determining the fluctuation corrections to the (inverse of the) Dirac composite fermion propagator G(x,y) that approximately (RPA) solves the Schwinger-Dyson equations,

(ii) recompute the contribution to the resistivity from the periodic potential using the corrected quadratic Hamiltonian

iG�1(x, y)� iG�10 (x, y) = �↵G(x, y)��⇧�1

↵�(x� y),

i⇧↵�(x� y) = trh�↵G(x, y)��G(y, x)

i

<latexit sha1_base64="lBfc+YnS51RBrltJgR2D5BBUfxc=">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</latexit>

going beyond mean-field theory

away from half-filling, particle-hole symmetry is broken (since there’s an effective magnetic field) and so we expect fluctuations to generate all terms consistent with symmetry: this means a mass is no longer forbidden in the composite fermion Hamiltonian

(we use the bare vertices to leading order; consistent with our later expansion of the fermion self-energy)

G0(x, y) = ei�(x,y)

Zd3p

(2⇡)3eip↵(x�y)↵G0(p)

<latexit sha1_base64="bgBOaBxrsGqVBCUlvDKCwxepF5k=">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</latexit>

�iG0(p) ⌘(p↵ + µ0�↵,0)�↵ +m0I

(p0 + µ0 + i✏p0)2 � p2i �m2

0

+ b(p0 + µ0)I+m0�0

⇣(p0 + µ0 + i✏p0)

2 � p2i �m20

⌘2 +O(b2)

<latexit sha1_base64="FosBY9/GGuSnBiDgZo0ulJERJG0=">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</latexit>

and pseudo-momentum tree-level fermion propagator

with Schwinger phase

�(x, y) = � b

2(x2 � y2)(x1 + y1)

<latexit sha1_base64="g0joUXltdppSrJkArFTFlm21z2E=">AAACG3icbZDLSsNAFIYn9VbrLerSzWARWtSSREE3QtGNywr2Ak0Ik+m0HTq5MDORhtD3cOOruHGhiCvBhW/jpM1CW38Y+PjPOZw5vxcxKqRhfGuFpeWV1bXiemljc2t7R9/da4kw5pg0cchC3vGQIIwGpCmpZKQTcYJ8j5G2N7rJ6u0HwgUNg3uZRMTx0SCgfYqRVJarW3ZjSCvjk6QKr+ApTD1oh6ofWhNYGbuWshLXqmZswmPFZrXk6mWjZkwFF8HMoQxyNVz90+6FOPZJIDFDQnRNI5JOirikmJFJyY4FiRAeoQHpKgyQT4STTm+bwCPl9GA/5OoFEk7d3xMp8oVIfE91+kgOxXwtM/+rdWPZv3RSGkSxJAGeLerHDMoQZkHBHuUES5YoQJhT9VeIh4gjLFWcWQjm/MmL0LJq5lnNvDsv16/zOIrgAByCCjDBBaiDW9AATYDBI3gGr+BNe9JetHftY9Za0PKZffBH2tcPn7+cIw==</latexit>

assuming the Schwinger phase is uncorrected, we may work in pseudo-momentum space with standard methods

tree-level fermion propagator

Dirac fermions in a magnetic field

Schwinger (1951); Ritus (1978)

Watson & Reinhardt (2014)

i⌃↵(q)�↵ + i⌃m(q)I =

Zd3p

(2⇡)3�↵G(p+ q)��⇧�1

↵�(p),

i⇧↵�(�q) = N

Zd3p

(2⇡)3trh�↵G(p)��G(p+ �q)

i

<latexit sha1_base64="Ej8CLGP2Xw2NdE3Co3/Nx6HqHCY=">AAADGnicfVJNixNBEO0Zv9b4ldWjl8IgJKyGmY2gl4VlFdSLRNbsLqSTUNPpTJrtnmm7e4QwzO/w4l/x4kERb+LFf2PnQ91kxYKGx6tXr6q6O9FSWBdFP4PwwsVLl69sXa1du37j5q369u0jmxeG8R7LZW5OErRcioz3nHCSn2jDUSWSHyenT+f543fcWJFnb9xM84HCNBMTwdB5arQdRALooUgVjihKPcXm2xbQFJXC4ZKAHfgjUYusQjdNkvJlBXtAReagHA87oIHmvhM0d4Fq0Rp2qt8+5dKogudNvXPGPuEOgXbFsHwYV6OVChZ01dStB0BpjUpMuCwPn6VYpLyqiWXBupaOufRW3noPXv1/pJIaBc5U9ECk/fU9/XQbs3nGb//XfV40GNUbUTtaBJwH8Qo0yCq6o/p3Os5ZoXjmmERr+3Gk3aBE4wSTfiVaWK6RnWLK+x5mqLgdlIunreC+Z8YwyY0/fq0Fe7aiRGXtTCVeOX8Wu5mbk//K9Qs3eTIoRaYLxzO2bDQpJLgc5v8ExsJw5uTMA2RG+FmBTdEgc/431fwlxJsrnwdHu+24045fP2rsH6yuY4vcJfdIk8TkMdknL0iX9AgL3gcfg8/Bl/BD+Cn8Gn5bSsNgVXOHrEX44xcZavVm</latexit>

in pseudo-momentum space

N=1 is the number of fermion flavors; 1/N is an artificial small parameter used to organize the calculation

⌃↵ = ⌃(1)↵ + ⌃(2)

↵ + . . . = 0 + ⌃(2)↵ + . . . ,

⌃m = ⌃(1)m + ⌃(2)

m + . . . = ⌃(1)m + ⌃(2)

m + . . . .<latexit sha1_base64="4qAPY+nzd98Ceek9H574OFc45Io=">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</latexit>

as a result, (i) the chemical potential is un-corrected and (ii) the fermion obtains the dynamically generated mass (at leading order)

m = ⌃(1)m

<latexit sha1_base64="urDg2PGijMmp5Sys0S6jvDm4xuw=">AAAB+3icbVBNS8NAEJ34WetXrEcvi0Wol5KooBeh6MVjRfsBbQyb7bZdupuE3Y1YQv6KFw+KePWPePPfuG1z0NYHA4/3ZpiZF8ScKe0439bS8srq2npho7i5tb2za++VmipKJKENEvFItgOsKGchbWimOW3HkmIRcNoKRtcTv/VIpWJReK/HMfUEHoSszwjWRvLtkkCXqHvHBgL74iGtuMeZb5edqjMFWiRuTsqQo+7bX91eRBJBQ004VqrjOrH2Uiw1I5xmxW6iaIzJCA9ox9AQC6q8dHp7ho6M0kP9SJoKNZqqvydSLJQai8B0CqyHat6biP95nUT3L7yUhXGiaUhmi/oJRzpCkyBQj0lKNB8bgolk5lZEhlhiok1cRROCO//yImmeVN3Tqnt7Vq5d5XEU4AAOoQIunEMNbqAODSDwBM/wCm9WZr1Y79bHrHXJymf24Q+szx8urZM9</latexit>

Schwinger-Dyson equations

we ignore the particle-hole even component (Landau damping etc)

logic: (i) finite effective field leads to generation of a Chern-Simons term for the gauge field, which then induces a fermion mass, consistent with the broken particle-hole symmetry

i⇧↵�even(�q)� ✏↵��q�⇧odd(�q) = N

Zd3p

(2⇡)3trh�↵G(0)(p)��G(0)(p+ �q)

i

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⇧odd(0) =N

4⇡

⇣⇥(|⌃(1)

m |� µ0)⌃(1)

m

|⌃(1)m |

+⇥(µ0 � |⌃(1)m |)⌃

(1)m

µ0

⌘

<latexit sha1_base64="aTniYNERYUdsq6EButDHl6GQltM=">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</latexit>

gauge field self-energy

(ii) the particle-hole odd component is enhanced at small, but finite b relative to the particle-hole even component

the Schwinger-Dyson equation implies the mass (self-energy) solves

dimensional analysis implies

in perturbation theory, we can determine the wave function renormalization and verify

⌃(1)m = �2µ0sign(⌃

(1)m )

N+

2

3

bµ20

N |⌃(1)m |3

<latexit sha1_base64="vOkZLLD//IflssFVG+HYe7bmACw=">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</latexit>

⌃(1)m =

µ0

Nf⇣bN3

µ20

⌘

<latexit sha1_base64="LEnltl4ndTojyaiw2ATnKOy3PPs=">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</latexit>

limN!1

⌃(2)↵ = 0

<latexit sha1_base64="7cE8n1l6iJDBpVRaV8obAbcXjQ8=">AAACHHicbVA9SwNBEN3z2/gVtbRZDII24U4FbQTRxkoUjQq5eMxt9pIlu3vH7pwSjvwQG/+KjYUiNhaC/8ZNTOHXg4HHezPMzIszKSz6/oc3Mjo2PjE5NV2amZ2bXygvLl3YNDeM11gqU3MVg+VSaF5DgZJfZYaDiiW/jDuHff/yhhsrUn2O3Yw3FLS0SAQDdFJU3gqlUFFxTEMjWm0EY9JbGgqdYLdHwzPRUnBdrG9u9KIQZNYGukf9qFzxq/4A9C8JhqRChjiJym9hM2W54hqZBGvrgZ9howCDgkneK4W55RmwDrR43VENittGMXiuR9ec0qRJalxppAP1+0QBytquil2nAmzb315f/M+r55jsNgqhsxy5Zl+LklxSTGk/KdoUhjOUXUeAGeFupawNBhi6PEsuhOD3y3/JxWY12KoGp9uV/YNhHFNkhaySdRKQHbJPjsgJqRFG7sgDeSLP3r336L14r1+tI95wZpn8gPf+Ca6YoQ4=</latexit>

(recall the 1/N expansion begins with ⌃(1)↵ = 0.)

<latexit sha1_base64="mg3QrUWoqfpnk/Hfsrpor/Mja1A=">AAACAnicbVBNS8NAEN3Ur1q/op7Ey2IR2ktIVNCLUPTisaJthSaGyXbbLt18sLsRSihe/CtePCji1V/hzX/jts1BWx8MPN6bYWZekHAmlW1/G4WFxaXlleJqaW19Y3PL3N5pyjgVhDZIzGNxF4CknEW0oZji9C4RFMKA01YwuBz7rQcqJIujWzVMqBdCL2JdRkBpyTf33BvWC+E+qzjVke8CT/qAz7FtVX2zbFv2BHieODkpoxx13/xyOzFJQxopwkHKtmMnystAKEY4HZXcVNIEyAB6tK1pBCGVXjZ5YYQPtdLB3VjoihSeqL8nMgilHIaB7gxB9eWsNxb/89qp6p55GYuSVNGITBd1U45VjMd54A4TlCg+1ASIYPpWTPoggCidWkmH4My+PE+aR5ZzbDnXJ+XaRR5HEe2jA1RBDjpFNXSF6qiBCHpEz+gVvRlPxovxbnxMWwtGPrOL/sD4/AGKgJWS</latexit>

fermion self-energy

the result of this rather complicated and approximate analysis

H = ~� ·⇣

@

@~x+ ~a

⌘+m�3


with chemical potential

µ =pB

<latexit sha1_base64="nUMlkeTAPTa17U0qq7qW0EsoyJc=">AAAB9XicbVBNSwMxEM36WetX1aOXYBE8lV0V9CKUevFYwX5Ady3ZNNuGJtk1mVXK0v/hxYMiXv0v3vw3pu0etPXBwOO9GWbmhYngBlz321laXlldWy9sFDe3tnd2S3v7TROnmrIGjUWs2yExTHDFGsBBsHaiGZGhYK1weD3xW49MGx6rOxglLJCkr3jEKQEr3fsyxVfYNw8astq4Wyq7FXcKvEi8nJRRjnq39OX3YppKpoAKYkzHcxMIMqKBU8HGRT81LCF0SPqsY6kikpkgm149xsdW6eEo1rYU4Kn6eyIj0piRDG2nJDAw895E/M/rpBBdBhlXSQpM0dmiKBUYYjyJAPe4ZhTEyBJCNbe3YjogmlCwQRVtCN78y4ukeVrxzire7Xm5WsvjKKBDdIROkIcuUBXdoDpqIIo0ekav6M15cl6cd+dj1rrk5DMH6A+czx/t85Ik</latexit>

and dynamically-generated mass (given by continued to N=1) ⌃(1)m

<latexit sha1_base64="4w8pnQwLCA2pxGRBpScJ7ATCuCc=">AAAB9XicbVBNSwMxEJ2tX7V+VT16CRahXsquCnosevFY0X5Auy3ZNNuGJtklySpl6f/w4kERr/4Xb/4b03YP2vpg4PHeDDPzgpgzbVz328mtrK6tb+Q3C1vbO7t7xf2Dho4SRWidRDxSrQBrypmkdcMMp61YUSwCTpvB6GbqNx+p0iySD2YcU1/ggWQhI9hYqdu5ZwOBe6Kblr3TSa9YcivuDGiZeBkpQYZar/jV6UckEVQawrHWbc+NjZ9iZRjhdFLoJJrGmIzwgLYtlVhQ7aezqyfoxCp9FEbKljRopv6eSLHQeiwC2ymwGepFbyr+57UTE175KZNxYqgk80VhwpGJ0DQC1GeKEsPHlmCimL0VkSFWmBgbVMGG4C2+vEwaZxXvvOLdXZSq11kceTiCYyiDB5dQhVuoQR0IKHiGV3hznpwX5935mLfmnGzmEP7A+fwBsTKR+g==</latexit>

m ⇡ .69sign(b)|b|1/3B1/6<latexit sha1_base64="xpsYEuLJAiwt4q7AsysRqG6KqMU=">AAACF3icbVC7TsMwFHXKq5RXgJHFokIqS0goKrBVZWEsEn1ITakc122t2k5kO4gqzV+w8CssDCDECht/Q9J2gJYjXd2jc+6VfY8XMKq0bX8bmaXlldW17HpuY3Nre8fc3asrP5SY1LDPfNn0kCKMClLTVDPSDCRB3GOk4Q2vUr9xT6SivrjVo4C0OeoL2qMY6UTqmBaHLgoC6T9Aq3QJI1dyqGhfxAXvGI698V3knBRjWEl7Kc51zLxt2RPAReLMSB7MUO2YX27XxyEnQmOGlGo5dqDbEZKaYkbinBsqEiA8RH3SSqhAnKh2NLkrhkeJ0oU9XyYlNJyovzcixJUacS+Z5EgP1LyXiv95rVD3LtoRFUGoicDTh3ohg9qHaUiwSyXBmo0SgrCkyV8hHiCJsE6iTENw5k9eJPVTyylazs1ZvlyZxZEFB+AQFIADzkEZXIMqqAEMHsEzeAVvxpPxYrwbH9PRjDHb2Qd/YHz+AARZnUw=</latexit>

asymmetric at fixed density; symmetric at fixed external field B

summary of analysis

-0.04 -0.02 0.02 0.04

0.1

0.2

0.3

0.4

Δρxx

(Arb. u

nits

� �0.00

��(B�−�B1/2)/B1/2

)

commensurability oscillations

�⇢xx / 1� T/TD

sinh(T/TD)

h1� 2 sin2

⇣2⇡l2bpB �m2

d� ⇡

4

⌘i

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T�1D =

4⇡2l2bd

1q1� m2

B<latexit sha1_base64="OHRang4h2n4edphxeiIC3fQ7lqc=">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</latexit>

Mitra & MM (2019)

(blue curve with m = 0 is the mean-field result)

“resolving” the IR equivalence?

�⇢xx / T/TNR

sinh(T/TNR)<latexit sha1_base64="1xnkW8NRKIAQg0Ba9pp6eFa8FmE=">AAACInicbZDLSgMxFIYzXmu9jbp0EyxC3dQZFdRdUReupEpv0ClDJk3b0EwyJBlpGeZZ3Pgqblwo6krwYUwvgrb+EPj5zjlJzh9EjCrtOJ/W3PzC4tJyZiW7ura+sWlvbVeViCUmFSyYkPUAKcIoJxVNNSP1SBIUBozUgt7lsF67J1JRwct6EJFmiDqctilG2iDfPveuCNMIerIr/KTfT6EXSRFpAZPyYdlPbu4MEeYG6CnKu/kfeJD6ds4pOCPBWeNOTA5MVPLtd68lcBwSrjFDSjVcJ9LNBElNMSNp1osViRDuoQ5pGMtRSFQzGa2Ywn1DWrAtpDlcwxH9PZGgUKlBGJjOEOmumq4N4X+1RqzbZ82E8ijWhOPxQ+2YQZPAMC/YopJgzQbGICyp+SvEXSQR1ibVrAnBnV551lSPCu5xwb09yRUvJnFkwC7YA3ngglNQBNegBCoAgwfwBF7Aq/VoPVtv1se4dc6azOyAP7K+vgHBBaPF</latexit>

T�1NR =

4⇡2l2bd

m⇤p4⇡ne

<latexit sha1_base64="6uYiKgULzeZsDvtGpWiSfMsSikk=">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</latexit>

m⇤ /pB

<latexit sha1_base64="V9OO8gy8RUKvwz+GdBQAVsmYxtg=">AAACAHicbVC7TsMwFHV4lvIKMDCwWFRITFUCSDBWZWEsEn1ITagc12mtOnGwb5CqKAu/wsIAQqx8Bht/g9tmgJYjWTo6515dnxMkgmtwnG9raXlldW29tFHe3Nre2bX39ltapoqyJpVCqk5ANBM8Zk3gIFgnUYxEgWDtYHQ98duPTGku4zsYJ8yPyCDmIacEjNSzD6N7j2jAXqJkAhJ7+kFBVs97dsWpOlPgReIWpIIKNHr2l9eXNI1YDFQQrbuuk4CfEQWcCpaXvVSzhNARGbCuoTGJmPazaYAcnxilj0OpzIsBT9XfGxmJtB5HgZmMCAz1vDcR//O6KYRXfsbjJAUW09mhMBXYJJ20gftcMQpibAihipu/YjokilAwnZVNCe585EXSOqu651X39qJSqxd1lNAROkanyEWXqIZuUAM1EUU5ekav6M16sl6sd+tjNrpkFTsH6A+szx/07Jaj</latexit>

in HLR mean-field theory

where

If the HLR composite fermion effective mass

and if this result holds when fluctuations are included, then the finite-temperature behavior of commensurability oscillations could distinguish the two theories

Manoharan, Shayegan, & Klepper (1994)

open questions

a careful analysis on the effects of the particle-hole even component of the gauge field self-energy

fluctuation effects in the HLR composite fermion theory

can this “finite magnetic field technology” be used to study the renormalization group properties of non-Fermi liquids in a non-zero field?

Documents

Fluctuations and magnetoresistance oscillations near a half ...Fluctuations and magnetoresistance oscillations near a half-filled Landau level Michael Mulligan UC Riverside Jiang,