Nematic quantum criticality

Walter Metzner

1. Introduction: Nematic quantum criticality and non-Fermi liquid

π repulsive

2. Nematic lattice model 0 attractive

0.16 1st order transition −π 2nd order transition −π 0 π 0.14 1st order (MFT) 2nd order (MFT) 0.12

0.1

3. Turning a first order transition continuous T 0.08 0.06

0.04

0.02

0 -0.85 -0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.5 -0.45 µ (0,π) T=0.15

1.5 ky T=0.20 T=0.30

(0,0) kx (π,0) T=0.40

,ω) 1 Tc=0.14 k

4. Fermi surface truncation from thermal fluctuations Α( 0.5

0 q -1.5 -1 -0.5 0 0.5 1 1.5 1 ω k−p k−p 1 2 q q N 2

5. Validity of Hertz action at QCP? k−p N k−p3

q 3 k−p 4

Coworkers: Luca Dell’Anna (Padua) Hiroyuki Yamase (Tsukuba) Pawel Jakubczyk (Warsaw) Stephan Thier (Mainz) 1. Introduction

Quantum phase transitions: Phase transition at T = 0 driven by control parameter δ (pressure, density, ...)

T If order parameter vanishes

quantum continuously at transition: critical quantum critical point.

Quantum fluctuations lead to ordered quantum disordered unusual physical properties.

QCP δ

Quantum phase transitions in metals: non-Fermi liquid behavior in quantum critical regime d-wave Pomeranchuk instability – nematic transition

π π Spontaneous breaking of tetragonal symmetry free free generated by d-wave 0 deformed 0 deformed forward scattering

−π −π Halboth, wm ’00 −π π−π π 0 0 Yamase, Kohno ’00 (a) (b) P Order parameter nd = k dkhnki where dk = cos kx − cos ky

Realization of ”nematic” electron liquid (→ Kivelson et al. ’98)

Experimental evidence for nematic phase in

Sr3Ru2O7 (Mackenzie group) and YBCO (Keimer group, Taillefer group) Incipient nematic instability in cuprates:

Sizable in-plane anisotropy observed (neutron scattering) for magnetic excitations in YBCO (Hinkov et al. ’04-’08)

Natural explanation: Enhancement of bare anisotropy from structural orthorhombicity by nematic correlations (Yamase + wm ’06) or fluctuating stripes (Vojta et al. ’06) Non-Fermi liquid behavior near d-wave Pomeranchuk instability:

Scattering at Fermi surface fluctuations leads to large decay rates

• At quantum critical point (T = 0, ξ = ∞):

ImΣ(k , ω) ∝ d2 |ω|2/3 for ω → 0 wm, Rohe, Andergassen ’03 F kF

∗ large anisotropic decay rate of single-particle excitations ∗ maximal near van Hove points, minimal near diagonal in Brillouin zone: ”cold spots” ⇒ no quasi-particles away from Brillouin zone diagonal

• Decay rate for DC transport in quantum critical regime:

γtr ∝ d2 T 4/3 Dell’Anna, wm ’07,’09 kF kF 2. Nematic lattice model: wm, Rohe, Andergassen ’03 X 1 k k + q H = H + f 0(q) n (q) n 0(−q) kin 2L kk k k k,k0,q

P † where nk(q) = σ ck+q,σ ck,σ k’ k’− q and only small momentum transfers q contribute (forward scattering)

0 Hkin tight-binding kinetic energy (from hopping t, t on square lattice)

π repulsive Interaction with d-wave attraction: fkk0(q) = −g(q) dk dk0 0 attractive with dk = cos kx − cos ky and g(q) > 0 yields Pomeranchuk instability −π −π 0 π Mean-field phase diagram:

0.14 1st order transition 2nd order transition Transition typically 0.12 first order 0.1 0.08 at low temperatures T 0.06

Khavkine et al. ’04 0.04 nematic Yamase et al. ’05 0.02

0 -0.85 -0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.5 -0.45 µ

t = 1, t0 = −1/6, g = 0.8

Could order parameter fluctuations lead to a second order (continuous) transition at low temperatures? Grand canonical potential (mean-field theory)

Ω φ2 2T X
X = − ln 1 + e−(k−φdk−µ)/T where φ = g d hn i L 2g L k k k k

X 2n Ω(φ) Coefficients of expansion Ω(φ) = a2nφ n≥0 all negative for n ≥ 2 at low temperatures !

Minimum of Ω(φ) not captured by Taylor expansion; Ω(φ) cannot be truncated at any finite order! φ

How to compute fluctuation effects in this situation? Standard techniques require expansion in φ! 3. Turning a first order transition continuous Jakubczyk, Yamase, wm ’09

Introduce order parameter fluctuations φq, integrate out fermions ⇒ Hertz action Z   T X d2q |ω | S[φ] = φ A n + Z q2 φ + U[φ] 2 (2π)2 q,ωn 0|q|z−2 0 −q,−ωn ωn

• dynamical exponent z = 3 for nematic transition • potential U[φ] usually truncated at quartic order;

• momentum cutoff |q| ≤ Λ0

To treat fluctuation effects requires integration of e−S[φ] over all field configurations (functional integral) Functional renormalization group Review article: Metzner et al., Rev. Mod. Phys. (in press)

Exact functional flow equation for effective action ΓΛ[φ] : Wetterich ’93

1 ∂ RΛ δ2ΓΛ ∂ ΓΛ[φ] = tr Λ where Γ(2)[φ] = Λ 2 Γ(2)[φ] + RΛ δφ δφ

acts as Regulator function: RΛ(q) = ZΛ(Λ2 − q2)Θ(Λ2 − q2) momentum cutoff ZΛ = wave function renormalization

Λ initial condition: Γ 0[φ] = S[φ] , Λ0 ultraviolet cutoff (largest |q|)

final effective action: Γ[φ] = lim ΓΛ[φ] Λ→0 yields thermodynamics (phase diagram etc.) and correlation functions Effective action with full local potential:

Ansatz for effective action: Z   T X d2q |ω | ΓΛ[φ] = φ AΛ n + ZΛq2 φ + U Λ[φ] 2 (2π)2 q,ωn |q|z−2 −q,−ωn ωn where Z Z 1/T with arbitrary function U Λ(φ) U Λ[φ] = dτ d2r U Λ(φ(r, τ)) 0 no expansion in powers of φ

Flow equation for U Λ(φ) partial differential equation involving derivatives with respect to Λ and φ

Initial condition for potential: U Λ0(φ) = L−1Ω(φ) (mean-field pot.) Flow of potential U Λ(φ):

-0.8932 Λ=Λ0=0.37 Λ=0.22 Λ=0.11 Λ=0.05 Parameters: -0.8936 Λ=0.00

0 t = 1, t = −1/6 ) φ -0.894 g = 0.8 U(

Z0 = 10 -0.8944 A0 = 1

-0.8948 0 0.05 0.1 0.15 0.2 φ

For Λ = 0 flat for φ ∈ [0, φ0] as required by convexity. Phase diagram including fluctuations:

0.16 1st order transition 2nd order transition Critical temperature T (µ) 0.14 1st order (MFT) c 2nd order (MFT) for 0.12 −1 0.1 Λ0 = e = 0.37

T 0.08 and 0.06 Λ0 = 1 0.04 compared to 0.02 mean-field result 0 -0.85 -0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.5 -0.45 µ

Fluctuations suppress Tc (as expected). Continuous transition down to T = 0 can be realized! ⇒ quantum criticality Critical temperature and Ginzburg region

• µ-dependence of Tc near QCP µ0: µ − µ0 Tc ∝ Jakubczyk, Strack, Katanin, wm ’08 log(µ − µ0)

• µ-dependence of Ginzburg temperature TG near QCP: µ − µ0 TG ∝ Millis ’93 log(µ − µ0)

• difference between Tc and TG:

µ − µ0 Tc TG − Tc ∝ 2 ∝ Bauer, Jakubczyk, wm ’11 log (µ − µ0) | log(µ − µ0)|

⇒ Large Ginzburg region with strong thermal fluctuations −1 and large correlation length ξ ∝ (T − Tc) 4. Fermi surface truncation from thermal fluctuations Yamase + wm ’11

How do thermal nematic fluctuations in (large) Ginzburg region affect spectral function? Pseudogap?

Thermal fluctuation propagator

gd˜ kdk0 −1 D 0(q) = − correlation length ξ ∝ (T − T ) kk ξ−2 + q2 c

Spectral function for single-electron excitations 1 1 1 A(k, ω) = − ImG(k, ω) = − Im π π ω − (k − µ) − Σ(k, ω) (0,π) T=0.15

1.5 ky T=0.20 Perturbative calculation T=0.30

(0,0) kx (π,0) T=0.40

,ω) 1 Tc=0.14 D k Α(

Σ = 0.5 G0 0 -1.5 -1 -0.5 0 0.5 1 1.5 ω

Splitting of quasi-particle peak for T → Tc, reminiscent of pseudogap

Artefact of perturbation theory? Small momentum transfers q ∝ ξ−1 dominate Exact calculation ⇒ Current vertex related to density vertex

Λ(k, ω; q) = vkΛ(k, ω; q) D Ward identity ⇒ Σ = G−1(k−q/2, ω) − G−1(k+q/2, ω) G Λ(k, ω; q) = Λ vk · q ⇒ Asymptotically exact linear integral equation for G

1.2 T=0.15 T=0.20 T=0.30 0.8 Quasiparticle peak broadened, T=0.40 ,ω) no splitting k Tc=0.14 Α( √ 2 0.4 width ∝ dk log ξ

0 -2 -1.5 -1 -0.5 0ω 0.5 1 1.5 2 2 Prefactor dk ⇒ momentum dependent smearing of Fermi surface:

(0,π) T=0.20 8 6 4 ky 2 Spectral function at 0 zero frequency A(k, 0)

(0,0) kx (π,0)

Reminiscent of Fermi arcs 5. Validity of Hertz action at QCP?

Integrating out fermions may lead to singular interactions between order parameter fluctuations; approximation by a local φ4-interaction questionable. What is the true behavior of the effective order parameter interaction in the relevant scaling limit?

q 1 Effective N-point interaction k−p k−p 1 2 q q between order parameter fluctuations N 2 as obtained from Hubbard-Stratonovich k−p N k−p3 transformation given by q fermionic N-point loop ΠN (q1, . . . , qN ) 3 k−p4 k-integral of product of N propagators G0(k − pi) Which scaling limit?

Low energy limit dominated by small momentum and energy transfers, with |ωi|  |qi|, and |qi| increasingly collinear (Metlitski & Sachdev ’10)

3 2 ⇒ Collinear scaling limit ωi 7→ λ ωi, qix 7→ λ qix, qiy 7→ λqiy

Naive expectation:

Same as static limit (first ωi → 0, then qi → 0), which is finite; order parameter interactions would then be irrelevant.

True behavior: Thier, wm ’11

2(3−N) ΠN 7→ λ ΠN divergent for N ≥ 4 ! ⇒ all N-point interactions are non-local and marginal ⇒ no simple truncation! π repulsive

Conclusion: 0 attractive

−π −π 0 π

The nematic quantum phase transition is an interesting new paradigm in the theory of quantum criticality in two-dimensional metals, with new theoretical aspects and some relevance for correlated electron materials.

0.16 1st order transition 2nd order transition 0.14 1st order (MFT) q (0,π) T=0.15 1 2nd order (MFT) 1.5 =0.20 0.12 ky T k−p k−p T=0.30 1 2 0.1 =0.40 q q (0,0) kx (π,0) T N 2 ,ω) 1 Tc=0.14 k

T 0.08 Α( k−p k−p 0.06 N 3 0.5 0.04 q 3 0.02 0 -1.5 -1 -0.5 0 0.5 1 1.5 k−p ω 4 0 -0.85 -0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.5 -0.45 µ