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Metals without quasiparticles A. Review of Fermi liquid theory B. A non-fermi liquid: the Ising-nematic quantum critical point C. Fermi surfaces and gauge fields

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.

Entanglement entropy B A P i ) Ground state of entire system, = ih A =Tr B = density matrix of region A Entanglement entropy S E = Tr ( A ln A )

Entanglement entropy i ) Ground state of entire system, = ih Take i = 1 p 2 ( "i A #i B #i A "i B ) Then A =Tr B = density matrix of region A = 1 2 ( "i A h" A + #i A h# A ) Entanglement entropy S E = Tr ( A ln A ) =ln2

D. Gioev and I. Klich, Physical Review Letters 96, 100503 (2006) B. Swingle, Physical Review Letters 105, 050502 (2010) Entanglement entropy of the Fermi liquid B A P Logarithmic violation of area law : S E = 1 12 (k F P )ln(k F P ) for a circular Fermi surface with Fermi momentum k F, where P is the perimeter of region A with an arbitrary smooth shape. The prefactor 1/12 is universal: it is independent of the shape of the entangling region, and of the strength of the interactions.

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.

Metals without quasiparticles A. Review of Fermi liquid theory B. A non-fermi liquid: the Ising-nematic quantum critical point C. Fermi surfaces and gauge fields

c T (K) S. Kasahara, H.J. Shi, K. Hashimoto, S. Tonegawa, b Y. Mizukami, T. Shibauchi, K. Sugimoto, T. Fukuda, T. Terashima, A.H. Nevidomskyy, and d Y. Matsuda, Nature 486, 382 (2012). 200 100 a M Paramagnetic (tetragonal) (010) T T * Electronic nematic H (100) Fe As/P T T > T* T < T* (a.u.) 2 (a.u.) 35 K 6 Antiferromagnetic (orthorhombic) Superconducting 0 0 0.2 0.4 0.6 x BaFe 2 (As 1 x P x ) 2 a Torque = 0 M H 0 a 4 (a.u.) c 0 180 Single 0crystal 18 (deg.) (d M H b

High temperature superconductors YBa 2 Cu 3 O 6+x

Pseudogap

Pseudogap Strange Metal

Visualization of the emergence of the pseudogap state and the evolution to superconductivity in a lightly hole-doped Mott insulator Y. Kohsaka, T. Hanaguri, M. Azuma, M. Takano, J. C. Davis, and H. Takagi Nature Physics, 8, 534 (2012). c 500 400 300 300 400 Cu 50 f R, Ca 1.88 Na 0.12 CuO 2 Cl 2 d 500 400 300 300 O x 400 O y 5 0.80 0.65 (mv) 2 nm 0.50 Evidence for nematic order (i.e. breaking of 90 rotation symmetry) in Ca 1.88 Na 0.12 CuO 2 Cl 2.

Visualization of the emergence of the pseudogap state and the evolution to superconductivity in a lightly hole-doped Mott insulator Y. Kohsaka, T. Hanaguri, M. Azuma, M. Takano, J. C. Davis, and H. Takagi T > T Nature Physics, 8, 534 (2012). c c Y X 2 nm 0.5 0.9 500 400 300 300 400 Cu 500 g 2 nm - 2 n Q xx (R), Ca 1.88 Na 0.12 CuO 2 Cl 2 + 0 h 500 400 300 300 O x 400 O y 500 Evidence for nematic order (i.e. breaking of 90 rotation symmetry) in Ca 1.88 Na 0.12 CuO 2 Cl 2. 2 nm + 0 - O ( O(2) O (

Broken rotational symmetry in the pseudogap phase of a high-tc superconductor R. Daou, J. Chang, David LeBoeuf, Olivier Cyr- Choiniere, Francis Laliberte, Nicolas Doiron- Leyraud, B. J. Ramshaw, Ruixing Liang, D. A. Bonn, W. N. Hardy, and Louis Taillefer Nature, 463, 519 (2010).

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

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,where~q is tangent to the Fermi surface, and so the initial and final fermion states, after scattering of, are nearly degenerate. 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

Computation of Landau damping We will only be interested in terms in that are singular in q and! n, and will drop regular contributions from regions of high momentum and frequency. In this case, it is permissible to reverse the conventional order of integrating over frequency first, and to first integrate over k x. It is a simple matter to perform the integration over k x in using the method residues to yield (working temporarily with d 1 spatial dimensions along the y direction) (~q,! n )= N f 2v F Z d d 1 k y (2 ) d 1 Z d n 2 sgn( n +! n ) sgn( n )! n + iv F q x + iappleq 2 y/2+iapple~q y ~k y = N f! n 2 v F Z d d 1 k y (2 ) d 1 1! n + iv F q x + iappleq 2 y/2+iapple~q y ~k y. We now integrate along the component of ~ k y parallel to the direction of ~q y to obtain (~q,! n ) = = N f! n 2 v F apple q y N f! n 2 v F apple q y d 2 Z d d 2 k y (2 ) d 2 Note that in d = 2 the last non-universal factor is not present, and the result for universal with d 2 = 1. Also, in our action v F = 1 and apple = 2. is

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 d/3 in dimension d.

Computation of fermion self energy The fermion self energy can be evaluated by methods similar to those used for the computation of Landau damping. Integrating over q x,withd 1 spatial dimensions along the y direction, we find (with =1/(2 v F apple)) (k, n )=i 1 v F = i 1 v F sgn( n ) Z d d 1 Z q y d!n sgn(! n + n ) q y (2 ) d 1 2 q y 3 +! n Z d d 1 q y (2 ) d 1 q qy 3 + n y ln q y 3. Evaluation of the q y integral yields sgn( ) d/3 near d = 2. In the physically important case of d = 2, the q y integral evaluates to (k, n )= 1 v F 1/3 p 3 sgn( n) n 2/3, 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 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 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.

Entanglement entropy of the non-fermi liquid B A P Logarithmic violation of area law : S E = C E k F P ln(k F P ) for a circular Fermi surface with Fermi momentum k F, where P is the perimeter of region A with an arbitrary smooth shape. The prefactor C E is expected to be universal but 6= 1/12: independent of the shape of the entangling region, and dependent only on IR features of the theory. B. Swingle, Physical Review Letters 105, 050502 (2010) Y. Zhang, T. Grover, and A. Vishwanath, Physical Review Letters 107, 067202 (2011)

Metals without quasiparticles A. Review of Fermi liquid theory B. A non-fermi liquid: the Ising-nematic quantum critical point C. Fermi surfaces and gauge fields

Fermi surfaces and gauge fields A Fermi surface of c k fermions coupled to a gauge field a µ = (a, a): S c = N f X X Z d c k (@ ia + " k a ) c k =1 k S a = Z d d rd apple 1 2g 2 ( µ @ a ) 2 The a component couples to the fermion density, and the density fluctuations screen the long-range Coulomb interactions. So we are only in interested in the transverse component of the gauge field, which couples to the fermion current. We work in the Coulomb gauge q a(q) = 0.

Fermi surfaces and gauge fields ~a fluctuation at wavevector ~q couples most e ciently to fermions near ± ~ k 0, where ~q is tangent to the Fermi surface, and so the initial and final fermion states, after scattering of, are nearly degenerate. In the Coulomb gauge, we can write ~a =(, 0), with x, y axes chosen as above. Expand fermion kinetic energy at wavevectors about ± ~ k 0 and boson ( ) kinetic energy about ~q = 0.

Fermi surfaces and gauge fields L[ ±, ]= + @ i@ x @ 2 y + + @ + i@ x @ 2 y + + + 1 2g 2 (@ y ) 2 M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 (2010)

Fermi surfaces and gauge fields Only difference from the nematic case is the change in this sign! All previous analysis of the breakdown of quasiparticles applies here too. L[ ±, ]= + @ i@ x @ 2 y + + @ + i@ x @ 2 y + + + 1 2g 2 (@ y ) 2 M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 (2010)