Quantum criticality of Fermi surfaces

Quantum criticality of Fermi surfaces Subir Sachdev Physics 268br, Spring 2018 HARVARD

Quantum criticality of Ising-nematic ordering in a metal y Occupied states x Empty states A metal with a Fermi surface with full square lattice symmetry

Quantum criticality of Ising-nematic ordering in a metal y x Spontaneous elongation along y direction: Ising order parameter < 0.

Quantum criticality of Ising-nematic ordering in a metal y x Spontaneous elongation along x direction: Ising order parameter > 0.

Ising-nematic order parameter Z d 2 k (cos k x cos k y ) c k c k Measures spontaneous breaking of square lattice point-group symmetry of underlying Hamiltonian

Quantum criticality of Ising-nematic ordering in a metal y x Spontaneous elongation along x direction: Ising order parameter > 0.

Quantum criticality of Ising-nematic ordering in a metal y x Spontaneous elongation along y direction: Ising order parameter < 0.

Quantum criticality of Ising-nematic ordering in a metal or =0 =0 c Pomeranchuk instability as a function of coupling

Quantum criticality of Ising-nematic ordering in a metal T Quantum critical TI-n =0 =0 c Phase diagram as a function of T and

Quantum criticality of Ising-nematic ordering in a metal T Quantum critical TI-n Classical d=2 Ising criticality =0 =0 c Phase diagram as a function of T and

Quantum criticality of Ising-nematic ordering in a metal T Quantum critical TI-n =0 D=2+1 rising criticality c? =0 Phase diagram as a function of T and

Quantum criticality of Ising-nematic ordering in a metal T Quantum critical TI-n =0 Strongly-coupled non-fermi liquid metal rwith no c quasiparticles =0 Phase diagram as a function of T and

Quantum criticality of Ising-nematic ordering in a metal T Quantum critical TI-n Fermi liquid =0 Strongly-coupled non-fermi liquid metal rwith no c quasiparticles =0 Fermi liquid Phase diagram as a function of T and

Quantum criticality of Ising-nematic ordering in a metal T Strange Metal TI-n Fermi liquid =0 Strongly-coupled non-fermi liquid metal rwith no c quasiparticles =0 Fermi liquid Phase diagram as a function of T and

The Fermi liquid Occupied states @ L = f r 2 2m µ + uf f f f f k F! Empty states

The Fermi liquid: RG L = f @ r 2 2m µ f + uf f f f Expand fermion kinetic energy at wavevectors about ~ k 0,by writing f ( ~ k 0 + ~q) = (~q)

The Fermi liquid: RG L = f @ r 2 2m µ f + uf f f f Expand fermion kinetic energy at wavevectors about ~ k 0,by writing f ( ~ k 0 + ~q) = (~q) L[ ]= @ i@ x @ 2 y + u

The Fermi liquid: RG S[ ]= Z d d 1 ydxd h @ i@ x @ 2 y + u i

The Fermi liquid: RG S[ ]= Z d d 1 ydxd h @ i@ x @ 2 y + u i The kinetic energy is invariant under the rescaling x! x/s, y! y/s 1/2, and! /s z,providedz = 1 and! s (d+1)/4. Then we find u! us (1 d)/2, and so we have the RG flow du d` = (1 d) 2 u Interactions are irrelevant in d =2!

The Fermi liquid: RG S[ ]= Z d d 1 ydxd h @ i@ x @ 2 y + u i The fermion Green s function to order u 2 has the form (upto logs) G(~q,!) = A! q x qy 2 + ic! 2 So the quasiparticle pole is sharp. And fermion momentum distribution function n( ~ D k)= f ( ~ k)f ( ~ E k) had the following form:

L = f r 2 @ 2m µ +4Fermiterms The Fermi liquid f Occupied states k F! Empty states Fermi wavevector obeys the Luttinger relation kf d fermion density Q, the Sharp particle and hole of excitations near the Fermi surface with energy! q z, with dynamic exponent z =1. The phase space density of fermions is e ectively one-dimensional, so the entropy density S T. It is useful to write this is as S T (d )/z, with violation of hyperscaling exponent = d 1.

L = f r 2 @ 2m µ +4Fermiterms The Fermi liquid f Occupied states q k F! Empty states Fermi wavevector obeys the Luttinger relation kf d fermion density Q, the Sharp particle and hole of excitations near the Fermi surface with energy! q z, with dynamic exponent z =1. The phase space density of fermions is e ectively one-dimensional, so the entropy density S T. It is useful to write this is as S T (d )/z, with violation of hyperscaling exponent = d 1.

Quantum criticality of Ising-nematic ordering in a metal or =0 =0 c Pomeranchuk instability as a function of coupling

Quantum criticality of Ising-nematic ordering in a metal E ective action for Ising order parameter S = d 2 rd ( ) 2 + c 2 ( ) 2 +( c) 2 + u 4

Quantum criticality of Ising-nematic ordering in a metal E ective action for Ising order parameter S = d 2 rd ( ) 2 + c 2 ( ) 2 +( c) 2 + u 4 E ective action for electrons: S c = d N f c i c i t ij c i c i =1 i i<j N f d c k ( + k ) c k =1 k

Quantum criticality of Ising-nematic ordering in a metal Coupling between Ising order and electrons S c = g Z d N f X =1 X k,q q (cos k x cos k y )c k+q/2, c k q/2, for spatially dependent > 0 < 0

Quantum criticality of Ising-nematic ordering in a metal S = d 2 rd ( ) 2 + c 2 ( ) 2 +( c) 2 + u 4 S c = N f d c k ( + k ) c k S c = =1 k Z g d N f X =1 X k,q q (cos k x cos k y )c k+q/2, c k q/2,

Quantum criticality of Ising-nematic ordering in a metal fluctuation at wavevector ~q couples most e ciently to fermions near ± ~ k 0. Expand fermion kinetic energy at wavevectors about ± ~ k 0 and boson ( ) kinetic energy about ~q = 0.

Quantum criticality of Ising-nematic ordering in a metal L[ ±, ]= + @ i@ x @ y 2 + + @ + i@ x @ y 2 + + + + 1 2g 2 (@ y ) 2 M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 (2010)

Quantum criticality of Ising-nematic ordering in a metal L = + @ i@ x @ y 2 + + @ + i@ x @ y 2 + + + + 1 2g 2 (@ y ) 2 One loop self-energy with N f fermion flavors: Z d 2 k d (~q,!) = N f 4 2 2 = N f 4! q y 1 [ i( +!)+k x + q x +(k y + q y ) 2 ] i k x + ky 2 Landau-damping

Quantum criticality of Ising-nematic ordering in a metal L = + @ i@ x @ y 2 + + @ + i@ x @ y 2 + + + + 1 2g 2 (@ y ) 2 Electron self-energy at order 1/N f : ( ~ k, ) = 1 N f Z d 2 q d! 4 2 2 [ i(! + )+k x + q x +(k y + q y ) 2 ] 1 " # q 2 y g 2 +! q y = i 2 g 2 2/3 p sgn( ) 2/3 3Nf 4

Quantum criticality of Ising-nematic ordering in a metal L = + @ i@ x @ y 2 + + @ + i@ x @ y 2 + + + + 1 2g 2 (@ y ) 2 Schematic form of and fermion Green s functions in d dimensions D(~q,!) = 1/N f q 2? +! q?, G f (~q,!) = q x + q 2? 1 isgn(!)! d/3 /N f In the boson case, q 2?!1/z b with z b =3/2. In the fermion case, q x q 2?!1/z f with z f =3/d. Note z f < z b for d > 2 ) Fermions have higher energy than bosons, and perturbation theory in g is OK. Strongly-coupled theory in d = 2.

Quantum criticality of Ising-nematic ordering in a metal L = + @ i@ x @ y 2 + + @ + i@ x @ y 2 + + + + 1 2g 2 (@ y ) 2 Schematic form of and fermion Green s functions in d = 2 D(~q,!) = 1/N f q 2 y +! q y, G f (~q,!) = q x + q 2 y 1 isgn(!)! 2/3 /N f In both cases q x qy 2! 1/z,withz =3/2. Note that the bare term! in G 1 f is irrelevant. Strongly-coupled theory without quasiparticles.

Quantum criticality of Ising-nematic ordering in a metal L = + @ i@ x @ y 2 + + @ + i@ x @ y 2 + + + + 1 2g 2 (@ y ) 2 Simple scaling argument for z =3/2.

Quantum criticality of Ising-nematic ordering in a metal L = + @ X i@ x @ y 2 + + X@ + i@ x @ y 2 + + + + 1 2g 2 (@ y ) 2 Simple scaling argument for z =3/2.

Quantum criticality of Ising-nematic ordering in a metal L = + @ X i@ x @ y 2 + + X@ + i@ x @ y 2 + + + + 1 2g 2 (@ y ) 2 Simple scaling argument for z =3/2. Under the rescaling x! x/s, y! y/s 1/2, and! /s z,we find invariance provided! s! s (2z+1)/4 g! gs (3 2z)/4 So the action is invariant provided z =3/2.

FL Fermi liquid q k F! k d F Q, the fermion density Sharp fermionic excitations near Fermi surface with q z, and z =1. Entropy density S T (d )/z with violation of hyperscaling exponent = d 1. Entanglement entropy S E k d 1 F P ln P.

FL Fermi liquid q kf! kfd Q, the fermion density NFL Nematic QCP kf! Fermi surface with kfd Q. Sharp fermionic excitations near Fermi surface with q z, and z = 1. Di use fermionic excitations with z = 3/2 n(k) to three loops. Entropy density S T (d )/z with violation of hyperscaling exponent = d 1. S T (d )/z with = d 1. Entanglement entropy SE kfd 1 P ln P. kf SE d 1 kf P ln P. k

FL Fermi liquid q NFL Nematic k F! k F! QCP q k d F Q, the fermion density Fermi surface with k d F Q. Sharp fermionic excitations near Fermi surface with q z, and z =1. Entropy density S T (d )/z with violation of hyperscaling exponent = d 1. Entanglement entropy S E k d 1 F P ln P. Di use fermionic excitations with z =3/2 to three loops. S T (d )/z with = d 1. S E k d 1 F P ln P. M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 (2010)

FL Fermi liquid q k F! NFL Nematic QCP q k F! k d F Q, the fermion density Fermi surface with k d F Q. Sharp fermionic excitations near Fermi surface with q z, and z =1. Entropy density S T (d )/z with violation of hyperscaling exponent = d 1. Entanglement entropy S E k d 1 F P ln P. Di use fermionic excitations with z =3/2 to three loops. S T (d )/z with = d 1. S E k d 1 F P ln P.

The Hubbard Model H = X i<j t ij c i c j + U X i 1 1 n i" n i# 2 2 µ X i c i c i t ij! hopping. U! local repulsion, µ! chemical potential Spin index =", # n i = c i c i c i c j + c j c i = ij c i c j + c j c i =0 Will study on the square lattice

Fermi surfaces in electron- and hole-doped cuprates Hole states occupied Electron states occupied E ective Hamiltonian for quasiparticles: H 0 = X i<j t ij c i c j X k " k c k c k with t ij non-zero for first, second and third neighbor, leads to satisfactory agreement with experiments. The area of the occupied electron states, A e, from Luttinger s theory is A e = 2 2 (1 x) for hole-doping x 2 2 (1 + p) for electron-doping p The area of the occupied hole states, A h, which form a closed Fermi surface and so appear in quantum oscillation experiments is A h =4 2 A e.

Fermi surface+antiferromagnetism Hole states occupied Electron states occupied + The electron spin polarization obeys S(r, ) = (r, )e ik r where K is the ordering wavevector.

Fermi surface+antiferromagnetism We use the operator equation (valid on each site i): 1 1 U n " n # = 2U S 2 2 3 ~ 2 + U 4 (1) Then we decouple the interaction via exp 2U X Z! Z d S 3 ~ i 2 = D J ~ i ( )exp i X i Z d apple 3 8U ~ J 2 i ~J i ~ Si! (2) We now integrate out the fermions, and look for the saddle point of the resulting e ective action for ~ J i. At the saddle-point we find that the lowest energy is achieved when the vector has opposite orientations on the A and B sublattices. Anticipating this, we look for a continuum limit in terms of afield~' i where ~J i = ~' i e ik r i (3)

Fermi surface+antiferromagnetism In this manner, we obtain the spin-fermion model Z Z = Dc D~' exp ( S) Z X S = d k Z c k d X i @ " k c k @ c i ~' i ~ c i e ik r i + Z d d 2 r apple 1 2 (r r ~' ) 2 + 1 2 (@ ~' ) 2 + s 2 ~' 2 + u 4 ~' 4

Fermi surface+antiferromagnetism In the Hamiltonian form (ignoring, for now, the time dependence of ~' ), the coupling between ~' and the electrons takes the form X H sdw = ~' q c k+q, ~ c k+k, k,q,, where ~ are the Pauli matrices, the boson momentum q is small, while the fermion momenum k extends over the entire Brillouin zone. In the antiferromagnetically ordered state, we may take ~' / (0, 0, 1), and the electron dispersions obtained by diagonalizing H 0 + H sdw are E k± = " k + " k+k 2 ± s "k " k+k 2 2 + 2 ~' 2 This leads to the Fermi surfaces shown in the following slides as a function of increasing ~'.

Fermi surface+antiferromagnetism Metal with large Fermi surface

Fermi surface+antiferromagnetism Fermi surfaces translated by K =(, ).

Fermi surface+antiferromagnetism Hot spots

Fermi surface+antiferromagnetism Electron and hole pockets in antiferromagnetic phase with h~' i6=0

Square lattice Hubbard model with hole doping Increasing SDW order Hole pockets Electron pockets S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

Square lattice Hubbard model with hole doping Increasing SDW order Hole pockets Electron pockets Hot spots where " k = " k+k S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

Square lattice Hubbard model with hole doping Increasing SDW order Hole pockets Hole pockets Electron pockets Hot spots where " k = " k+k Fermi surface breaks up at hot spots into electron and hole pockets S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

Square lattice Hubbard model with hole doping Increasing SDW order Hole pockets Electron pockets Hot spots where " k = " k+k Fermi surface breaks up at hot spots into electron and hole pockets S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

Square lattice Hubbard model with hole doping h~' i6=0 and large h~' i6=0 and small h~' i =0 In ncreasing SDW Metal with hole pockets Metal with electron and hole pockets Metal with large Fermi surface s

Square lattice Hubbard model with electron doping h~' i6=0 and large h~' i6=0 and small h~' i =0 Metal with electron pockets Metal with electron and hole pockets Metal with large Fermi surface s

Square lattice Hubbard model with no doping h~' i6=0 and large h~' i6=0 and small h~' i =0 In ncreasing SDW Insulator Metal with electron and hole pockets Metal with large Fermi surface s

Fermi surface+antiferromagnetism Fermi surfaces translated by K =(, ).

Hot spots

Low energy theory for critical point near hot spots

Theory has fermions 1,2 (with Fermi velocities v 1,2 ) and boson order parameter ~', interacting with coupling v 1 v 2 k y k x 1 fermions occupied 2 fermions occupied Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).

L f = 1 (@ iv 1 r r ) 1 + 2 (@ iv 2 r r ) 2 v 1 v 2 k y k x 1 fermions occupied 2 fermions occupied Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).

L f = 1 (@ iv 1 r r ) 1 + 2 (@ iv 2 r r ) 2 v 1 v 2 Hot spot k y k x Cold Fermi surfaces Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).

L f = 1 (@ iv 1 r r ) 1 + 2 (@ iv 2 r r ) 2 Order parameter: L ' = 1 2 (r r ~' ) 2 + 1 2 (@ ~' ) 2 + s 2 ~' 2 + u 4 ~' 4 Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).

L f = 1 (@ iv 1 r r ) 1 + 2 (@ iv 2 r r ) 2 Order parameter: L ' = 1 2 (r r ~' ) 2 + 1 2 (@ ~' ) 2 + s 2 ~' 2 + u 4 ~' 4 Yukawa coupling: L c = ~' 1 ~ 2 + 2 ~ 1 Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004).

L f = 1 (@ iv 1 r r ) 1 + 2 (@ iv 2 r r ) 2 v 1 v 2 Metal with large Fermi surface h~' i =0 " k1 < 0 " k2 < 0 Fermion dispersions: " k1 = v 1 k and " k2 = v 2 k

L f = 1 (@ iv 1 r r ) 1 + 2 (@ iv 2 r r ) 2 ~' 1 ~ 2 + 2 ~ 1 E k± > 0 Metal with E k+ > 0 E k < 0 hole and electron pockets h~' i6=0 E k± < 0 Fermion dispersions: s "k1 E k± = " k1 + " k2 2 ± 2 " k2 2 + 2 ~' 2

Hertz action. Upon integrating the fermions out, the leading term in the ~' e ective action is (q,! n ) ~' (q,! n ) 2, where (q,! n ) is the fermion polarizability. This is given by a simple fermion loop diagram (q,! n )= Z d d k (2 ) d Z d n 2 1 [ i( n +! n )+v 1 (k + q)][ i n + v 2 k]. We define oblique co-ordinates p 1 = v 1 k and p 2 = v 2 k. It is then clear that the integrand is independent of the (d 2) transverse momenta, whose integral yields an overall factor d 2 (in d =2this factor is precisely 1).

Also, by shifting the integral over k 1 we note that the integral is independent of q. So we have (q,! n )= d 2 v 1 v 2 Z dp1 dp 2 d n 8 3 1 [ i( n +! n )+p 1 ][ i n + p 2 ]. Next, we evaluate the frequency integral to obtain (q,! n ) = = d 2 v 1 v 2! n d 2 4 v 1 v 2. Z dp1 dp 2 4 2 [sgn(p 2 ) sgn(p 1 )] i! n + p 1 p 2 In the last step, we have dropped a frequency-independent, cuto dependent constant which can absorbed into a redefinition of r. Inserting this fermion polarizability in the e ective action for ~', we obtain the Hertz action for the SDW transition: S H = Z d d k (2 ) d T X 1 k 2 +! n + s ~' (k,! n ) 2 2! n + u Z d d xd ~' 2 (x, ) 2. 4

Fate of the fermions. Let us, for now, assume the validity of the Hertz Gaussian action, and compute the leading correction to the electronic Green s function. This is given by the following Feynman graph for the electron self energy,. At zero momentum for the 1 fermion we have 1 (0,! n )= 2 Z d d q (2 ) d Z d n 2 1 [q 2 + n ][ i( n +! n )+v 2 q ]. We first perform the integral over the q direction parallel to v 2, while ignoring the subdominant dependence on this momentum in the boson propagator. Then we have 2 Z d d 1 Z q d n sgn( n +! n ) 1 (0,! n )=i v 2 (2 ) d 1 2 q 2 + n 2 Z d d 1 q q 2 = i sgn(! n ) v 2 (2 ) d 1 ln +! n q 2.

<latexit sha1_base64="6/ebs0dragzu74nn1jbwccuft4m=">aaaimhicjvvbj9tefhylbiq5begrlym2lfk1dxba7valuggx7qmprxs7lda70dg+dka1z5yzculqop+if34kviaeqrzykzh2nmt2iryrkeyzc/2+b874wck1czzfb9x86+13dlq33rxfe/+ddz/avf3xcy1zfebjibopxvpmy8ifnhhuenyzkwspn+cp/+qbav90jepzkz6baybnkysfj3jadc0nbu8cez7gxbqbcooqtasvyspgz4/gmbkegyeziksounlqzc/2uiqr8+/qqk67eekfqk66wltou6x9xizhitftzyynqmxremjyrtktwikqbpiqiqqyzxkzceuqhvzedicvwtogjkx3mkfvjqup4ilcff9oihjrm/tg+zbrof/mxyjany7klkkijxqnjnequsoak5yn7arczyhaceaaalu87ulb+4hhkwv34csdr1fjsgvkdecplb3bxqb5wg0vgyejwpcn0fps+ca/ardhey13as8id1zoof3wkgrmbuiahgeq7qkfsos+efwy8clfgik8cgfufh1azpjbrvhe2gqpqyqeebrluzuuyc/23jkixmicjsjghfbilqymzmuhmuf5fy+h5nxhebc4dnb7ldoqvdtqviiylwpkeotsll5h5tpi2qnpsjgnpiq7asbhpndhxhxkhwlwgjwsl1xpaxkqwjotaaw2klhjqjjcfxa3odoq8quqmkymcsavitqrmtpdmax5qkfiivxfduwvcz+ukremaasxoxpqbghxcqqksqkui9rcl7pkogtgymakqhsfwm54xtmpmfjswtr7+7cvta4vmdurk1rlyjye9gexgbwowrtuuulxvxatzq3l4kw+/lhqssi3ckd5r7ntiijxpahjffb35/p5tvopmdq9njhky+lxgf9fg4mawngdsbvjk3u2ewdnc2wjtynnqw0ozv0pbcnrtzscfqpbzqd2ql2wmq2hcqjpnzkrjxw2hvvp83sjtouiqw0dfnkvn4u6lhgsesmnbm/ttcptctkgmv1nq3b4acffvahxthbt1lcvzuqplckavrysvhttak4q6m7nvnswqptvhurb7p7tcw6dowcool37947cw0n6ce4/co89blfn1m+e1tzpbru/ekem8uombgldgmeuk5nzgindgwqpfk4xy8erfumzvqqwoj4v6guvhm9pjaxncsqrmkvvty+cpvppu58su2ag+s29avg/9s5yez08l7iolimrlbjfet1uqtuzmutjlf5yodjvcsgqzljal6g2cyrlp7d95uw/5zo99/v+3uovgzhuwz9an1kdy7wormfwu+uzdwifoz/t/lrzx86frz9bv7x+av29ml15o/h5xlr0tp75f9bjzoi=</latexit> <latexit sha1_base64="6/ebs0dragzu74nn1jbwccuft4m=">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</latexit> <latexit sha1_base64="6/ebs0dragzu74nn1jbwccuft4m=">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</latexit> <latexit sha1_base64="hp+6lruf2d3tzaldqaqqvekmxyw=">aaab2xicbzdnsgmxfixv1l86vq1rn8eiucozbnqpuhfzwbzco5rm5k4bmskmyr2hdh0bf25efc93vo3pz0jbdwq+zknivscullqubn9ebwd3b/+gfugfnfzjk9nmo2fz0gjsilzl5jnmfpxu2cvjcp8lgzylffbj6f0i77+gstlxtzqrmmr4wmtuck7o6oyaraadlmw2ivxdc9yanb+gss7kddujxa0dhefbucunsafw7g9liwuxuz7ggupnm7rrtrxzzi6dk7a0n+5oykv394ukz9bostjdzdhn7ga2mp/lbiwlt1eldvesarh6kc0vo5wtdmajnchizrxwyasblykjn1yqa8z3hysbg29d77odbu3wmya6nmmfxeein3ahd9cblghi4bxevyn35n2suqp569lo4i+8zx84xio4</latexit> <latexit sha1_base64="azuc7c136kqlz8jcnn+xksejcw8=">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</latexit> <latexit sha1_base64="eeckffjemaalhdmmlx7zl0xejcm=">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</latexit> <latexit sha1_base64="rmzatibmwmlwbilvjvjalzllges=">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</latexit> <latexit sha1_base64="6/ebs0dragzu74nn1jbwccuft4m=">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</latexit> <latexit sha1_base64="6/ebs0dragzu74nn1jbwccuft4m=">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</latexit> <latexit sha1_base64="6/ebs0dragzu74nn1jbwccuft4m=">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</latexit> <latexit sha1_base64="6/ebs0dragzu74nn1jbwccuft4m=">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</latexit> <latexit sha1_base64="6/ebs0dragzu74nn1jbwccuft4m=">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</latexit> <latexit sha1_base64="6/ebs0dragzu74nn1jbwccuft4m=">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</latexit> Evaluation of the q integral shows that (d 1)/2 1 (0,! n )! n The most important case is d = 2, where we have 1 (0,! n )=i 2 v 2 p sgn(! n ) p! n, d =2.

Critical point theory is strongly coupled in d =2 Results are independent of coupling v 1 v 2 k y G fermion p i! 1 v.k k x A. J. Millis, Phys. Rev. B 45, 13047 (1992) Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, 255702 (2004)

M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010) Critical point theory is strongly coupled in d =2 Results are independent of coupling k k k? k y k x G fermion = Z(k k )! v F (k k )k?, Z(k k ) v F (k k ) k k

Strong coupling physics in d =2 The theory so far has the boson propagator 1 q 2 +! which scales with dynamic exponent z b = 2, and now a fermion propagator 1 i! + c 1! (d 1)/2 + v q. First note that for d<3, the bare i! term is less important than the contribution from the self energy at low frequencies. Ignoring the i! term, we see that the fermion propagator scales with dynamic exponent z f =2/(d 1). For d>2, z f <z b, and so at small momenta the boson fluctuations have lower energy than the fermion fluctuations. Thus it seems reasonable to assume that the fermion fluctuations are not as singular, and we can focus on an e ective theory of the SDW order parameter ~' alone. In other words, the Hertz assumptions appear valid for d>2.

<latexit sha1_base64="uq7oav7n8mv/qi0adincouv8snk=">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</latexit> <latexit sha1_base64="uq7oav7n8mv/qi0adincouv8snk=">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</latexit> <latexit sha1_base64="uq7oav7n8mv/qi0adincouv8snk=">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</latexit> <latexit sha1_base64="hp+6lruf2d3tzaldqaqqvekmxyw=">aaab2xicbzdnsgmxfixv1l86vq1rn8eiucozbnqpuhfzwbzco5rm5k4bmskmyr2hdh0bf25efc93vo3pz0jbdwq+zknivscullqubn9ebwd3b/+gfugfnfzjk9nmo2fz0gjsilzl5jnmfpxu2cvjcp8lgzylffbj6f0i77+gstlxtzqrmmr4wmtuck7o6oyaraadlmw2ivxdc9yanb+gss7kddujxa0dhefbucunsafw7g9liwuxuz7ggupnm7rrtrxzzi6dk7a0n+5oykv394ukz9bostjdzdhn7ga2mp/lbiwlt1eldvesarh6kc0vo5wtdmajnchizrxwyasblykjn1yqa8z3hysbg29d77odbu3wmya6nmmfxeein3ahd9cblghi4bxevyn35n2suqp569lo4i+8zx84xio4</latexit> <latexit sha1_base64="/gaw60x2ignjqzsafnz3ddbxlly=">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</latexit> <latexit sha1_base64="0fcjx1vto3mmy7nraa792wq5afy=">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</latexit> <latexit sha1_base64="n5jzmv+88k0i5jutxuqwps8xbb0=">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</latexit> <latexit sha1_base64="uq7oav7n8mv/qi0adincouv8snk=">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</latexit> <latexit sha1_base64="uq7oav7n8mv/qi0adincouv8snk=">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</latexit> <latexit sha1_base64="uq7oav7n8mv/qi0adincouv8snk=">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</latexit> <latexit sha1_base64="uq7oav7n8mv/qi0adincouv8snk=">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</latexit> <latexit sha1_base64="uq7oav7n8mv/qi0adincouv8snk=">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</latexit> <latexit sha1_base64="uq7oav7n8mv/qi0adincouv8snk=">aaahh3iclvxbbtw2en2kiz2olzjty18g9raww0ew1lcuszgkrea8bhbrjzzgoqaxolaejvemqbu3m/wf9kf61r/pive+ns6k6ollcjhnzjkz268lawwu/xxn7mf37s/mpngyfp7fl189mnv89tujgs3fw64kpff7zihcvuktlbyq+7uwrowxyq9//is73xskbasqdu2ofoclg1qyk5xzbb09vv9h0hcdwy25qkzqk2cc9dpasvpva+kqaqys6nxkjdckk+qmz3rdszcikr2m2c0f2vwopskjkgoacmbcdvwvurxvwtvswkzsqzstlayu2khknsdbrpauzjrxctwzn7zv0y+udfhzmvqyqfim2mc24y2w01zynaxnhtb0km1o6ahipeqkzmpvart1pxz1axejwjvspu6jusz0kf8vpickayoe0phrfinr/xghfrivlvcmopdudseu8ftz6a+tyss8bforqy0fiot4y5yypuhs0xug9awxlah7lbrlolkkaqilqowyvtoy1ifgykdwvvkt+40tnci0kv0ubjgrqiqejahzchcqaxhsiiplyuj6paiudvq6y92b2zbo1d3qanqxpmxup8np6bn1lj1f3tb45av82enhz54s5gnfgw+qxpsskkgejmqyawbkiobbxlvwwftd4wor+qi2r0nwhel1+yht5g1is66a0hc6a08ynahzjdwmkw8p8kzgjwfewmuyokspmf4khgpqxkfapjgyiaprkwwcwzm87aiv+yd3ft0jpvoarplmpqdh1e2ggo+tidn1liddygwyc4pxfsmeadpfgjlaykto+2gktm5b1bvapw1zidpwqka2dilzjam2qa/afumyy72cnrlwfldvxom29lr1qz9xl2cd609yghmg0ojkglmeqc7m6smqnt20znbtkzegbtr/ucixvdxwyq26zcwtzauzywyipl0wjek1si2grmqhwqfouvy8ytvpj9gcnr73rqjxfac5b6bj8eofdvzjdfdb0kn1xfku+9ia7jbypsougqnorgjaskvdctytfqhihwwygcgwu5kaanotnr8iz98iw+cny+mf3anpdxjpuzfecr0vhz/fubqwxr3ntvry9smuz/8uhlkc0svpqnjstzyo2lilexu00iviahexdi7m5qmwwsd2rfg4urirr69jea2uxsubfier/+y77bd9npdnksknrof4abscxfftd8dmo6jcyk41rtsmh4ptayzrssicjv3/yis+x87usxnmcfnd6xfjvhozkvu46eay+eezh263nb00nts8hmuqbixmzzrr1hrofvenrknuaboik0o2zz94jjbh6bljsliolp598a4xxley/9abf/6ypenb59vod52fttzz6dzvbhw2o287fobeza8zkzorsw9nl2fxzzenv+/eawo+6dz4zl/8dupchhq=</latexit> However, in d = 2, we have z f = z b = 2. Thus fermionic and bosonic fluctuations are equally important, and it is not appropriate to integrate the fermions out at an initial stage. We have to return to the original theory of coupled bosons and fermions. This turns out to be strongly coupled, and exhibits complex critical behavior. For more details, see M. A. Metlitski and S. Sachdev, arxiv:1005.1288 (Physical Review B 82, 075127 (2010)).