Superconductivity and charge density wave physics near an - PowerPoint PPT Presentation

Superconductivity and charge density wave physics near an antiferromagnetic quantum critical point: insights from Quantum Monte Carlo studies Xiaoyu Wang James Frank Institute University of Chicago Phys. Rev. B 95, 174520 (2017) ICTP Talk, Thursday 11/16/2017 arXiv:1710.02158

Collaborators Rafael Fernandes Yoni Schattner Erez Berg Yuxuan Wang (U. Minnesota) (Stanford) (U. Chicago) (U. Illinois)

Contents • Quantum critical phenomena • Sign-problem free determinant QMC • Nearly antiferromagnetic metal • Spin-fermion model • Previous analytical works • What do we learn from numerics? • Superconductivity • Emergent symmetry

Quantum Phase Transition • T=0 phase transition driven by an external parameter p • Quantum critical point (QCP) • Divergent correlation time — quantum coherence • Quantum critical fan • QCPs in metals • Landau damping; non-Fermi liquid; emergent orders Armitage et al, RMP (2010) • Signature in unconventional superconductors ∆ < k B T ∆ > k B T ∆ > k B T Nandi et al, PRL (2010) Hertz, PRB 1976; Millis, PRB 1993 Sachdev, Quantum Phase Transitions

• QCPs not easily obtained from microscopic models • Basic ingredients for a low-energy model • Quantum critical order parameter fluctuations • Fermi surface • Minimal coupling — space-time local • What do we look for? • Phase diagram • Collective excitations • Scaling behavior • Comparison to experiments and other microscopic calculations Even effective models are hard to solve! Need numerics!

Determinant Quantum Monte Carlo • Partition function • Fermion sign problem: Z h i ¯ , ; ~ Z s.f = exp( − S F − S B − S λ ) � D • fermion determinant is calculated from a time-ordered product • QCP tuned by bare boson mass • in general complex; especially S B = 1 1 Z � ) 2 + ( r ~ � ) 2 + r 0 ~ � 2 + u ~ ( @ τ ~ � 4 v 2 2 severe at low-T r , τ s • Electronic action is Gaussian: with sign Z 0.6 D [ ~ � ] ⇢ { ~ without sign Z s . f . = � ( r , ⌧ ) } Pair susceptibility 0.5 P d ⇢ { ~ � ( r , ⌧ ) } ≡ det ~ � exp( − S B ) 0.4 “fermion determinant” 0.3 DQMC: 0.2 • Construct a thermal ensemble by sampling; 0.1 • Unlimited by various approx. schemes 0 0 1 0.5 • Small system sizes; Finite size scaling T Scalapino, arXiv:cond-mat/0610710 Blanckenbecler, Scalapino & Sugar, PRD (1981) W 1 → 2 W 2 → 3 W N − 1 → N

• Fermion sign problem is generic • Sign-free QMC due to Kramer’s symmetry: U 2 = − 1; and [ H, ˜ ˜ U ] = 0 • e.g., negative-U Hubbard model; positive-U Hubbard model at half-filling Congjun Wu and Shou-Cheng Zhang, PRB (2005) • Engineered models: • Remove sign-problematic sector of the action • Need to show they preserve the low-energy physics qualitatively Ising-nematic QCP: AFM QCP: Schattner, Lederer, Kivelson and Berg, PRX (2016) Berg, Metlitski & Sachdev, Science (2012) Lederer, Schattner, Kivelson and Berg, PRL (2017) Schattner, Gerlach, Trebst and Berg, PRL (2016) Gerlach, Schattner, Berg and Trebst, PRB (2017) Many others: XW, Schattner, Berg and Fernandes, PRB (2017) Li, Jiang and Yao, PRL (2016) XW, Wang, Schattner, Berg and Fernandes, arXiv Dumitrescu, Serbyn, Scalettar, Vishwanath, PRB (2017) Xu, Sun, Schattner, Berg and Meng, PRX (2017) …

AFM QCP and Spin-fermion model

Abanov, Chubukov & Schmalian, Adv. in Phys. (2003) Spin-fermion model Metlitski & Sachdev, PRB (2010) … • Electrons near the Fermi surface coupled to quantum critical antiferromagnetic fluctuations Z Z 0 ( q , i Ω ) ~ � q · ~ � − 1 ¯ X S B = � − q S F = ψ k α ( ∂ τ + ε k − µ ) ψ k α q ,i Ω τ k α • Spin fluctuation peaked at Q • Fermi surface - π π � π π 0 ( q , i Ω ) = r 0 + ( q − Q ) 2 + Ω 2 χ − 1 v 2 s � � r 0 < 0 : r 0 > 0 : k y k x - π - π Q = ( π , π ) - π � π Spin-fermion coupling: Z ~ � · ¯ S λ = � α ~ � αβ β N´ eel order x , τ

• Hot spots: Points on the Fermi surface that couple strongly to spin fluctuations • Low-energy physics governed by linearized hot spot approximation: ε i, k ≈ v ( i ) F · ( k − k ( i ) hs ); i = 1 , 2 θ hs k y v (2) ~ 2 F v (1) ~ k x F Q = π , π ( ) Q = ( π , π ) 1 Abanov, Chubukov & Schmalian, Adv. in Phys. (2003) Metlitski & Sachdev, PRB (2010) …

• Emergent SU(2) symmetry at each pair of hot spots ! ψ † ✓ ψ i, k ↑ ◆ i, − k ↓ ; i = 1 , 2 → − ψ † ψ i, k ↓ i, − k ↑ • Enlarged order parameter O(4): complex SC and CDW • Relevant to hole-doped cuprates? ∆ 1 , SC = h ψ 1 , ↑ ψ 1 0 , ↓ � ψ 1 , ↓ ψ 1 0 , ↑ i 1’ ∆ 1 , CDW = h ψ 1 , ↑ ψ † 1 0 , ↑ + ψ 1 , ↓ ψ † 1 0 , ↓ i ∆ 1 , SC ∆ 1 , CDW 1 1 Metlitski & Sachdev, PRB (2010) Wang, Agterberg & Chubukov, PRB (2015)

• Low frequency spin fluctuations are strongly renormalized due to the hot spots — Landau damping 1 λ 2 1 χ ( q , i Ω n ) = r 0 + ( q − Q ) 2 + Ω 2 γ ∝ n /v 2 s + | Ω n | / γ v 2 f sin( θ hs ) - π π � π π Polarization bubble: e, k + q φ q φ − q � � e, k - π - π - π � π Abanov, Chubukov & Schmalian, Adv. in Phys. (2003) Metlitski & Sachdev, PRB (2010) Mross et al, PRB (2010) …

• How to study SC and non-FL due to quantum critical spin fluctuations? —Hot-spot Eliashberg approximation damped spin fluct. θ hs v (2) ~ F v (1) ~ F ( ) Q = π , π Regular part of the self-energy Anomalous part of the self-energy ✓ λ 2 ◆ 2 Σ ( ω ) ∼ √ ω γ ∼ λ 2 sin( θ hs ) T c ∝ v F • How to understand the angle dependence of T c ? • Spin fluct. strongly damped; insufficient to mediate pairing θ hs → 0 : Abanov, Chubukov & Schmalian, Adv. in Phys. (2003) Metlitski & Sachdev, PRB (2010) …

How to achieve sign-free QMC? • How to avoid the fermion sign problem? • Kramer’s symmetry: • Two electron bands ˜ U = i σ 2 ⊗ τ 3 C • Spin fluct. couple inter-band • Hot spots dominate low-energy physics 2 2 Q Q 1 1 Berg, Metlitski & Sachdev, Science (2012)

Numerical characterization of low-energy properties (c) r = 0 . 7 1 . 0 0 . 9 quasiparticle weight Z 0 . 8 0 . 7 Q 0 . 6 0 . 5 0 . 4 hs 0 . 3 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 momentum k x / π Electrons lose coherence near hot spots | | − − (b) 0 . 30 2 . 5 SDW susceptibility χ − 1 excluding superconducting dome (c) λ = 2 0 . 25 0 . 029 ≤ T ≤ 0 . 1 2 . 0 3 . 15 ≤ r ≤ 4 . 70 gap ∆ k x 0 . 20 8 ≤ L ≤ 14 0 . 00 ≤ | ω n | ≤ 1 . 08 * 1 . 5 0 . 15 0 . 00 ≤ | q − Q | ≤ 1 . 05 0 . 10 1 . 0 0 . 05 hs 0 . 5 0 . 00 28,638 data points, χ 2 dof = 6.5 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 0 . 0 momentum k x / π 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 0 . 71 ( q − Q ) 2 + 0 . 94 | ω n | + 0 . 43 [ r − 3 . 03 ] SC gap function k -independent Damping dynamics of spin fluct. Schattner et al, PRL (2016); Gerlach et al, PRB (2017)

Superconductivity near QCP • Are the SC properties governed by the hot spots? • Is Eliashberg approximation valid? • Phys. Rev. B 95, 174520 (2017)

Band structure • Study a series of band structures with different δ /t • Different low-energy properties, while maintaining same bandwidth 8t 0 0 - π π - π π π π π π t − δ t + δ 0 δ / t = 0.2 0 0 δ / t = 0.3 0 µ - π - π - π - π 0 0 - π π - π π 0 0 - π π - π π π π π π t + δ θ hs t − δ 0 δ / t = 0.5 0 0 δ / t = 0.8 0 − µ - π - π - π - π 0 0 - π π - π π e 1 e 2 Blue band shifted by Q ; pair of hot spots overlap

• For each band parameter : δ /t • QMC procedure: λ 2 = 8 t Spin-fermion interaction: • Locate AF QCP by varying System sizes: L = 8 , 10 , 12 , 14 bare mass r 0 of spin fluct. Temperatures: T ≥ 0 . 04 t • Obtain T c via BKT criterion ρ s ( T c ) = 2 T c t ∼ 100 meV ⇒ T ∼ 40 K π 6 β t = 2 L = 8 0 . 12 ( β L 2 ) − 1 χ φ ( Q , i Ω n = 0) 5 β t = 4 L = 10 β t = 6 L = 12 4 β t = 8 L = 14 0 . 08 β t = 10 3 ρ s 2 0 . 04 AF QCP 1 T c 0 . 00 0 0 2 4 6 8 10 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 r 0 T/t

• QMC procedure: • For each band parameter : δ /t • Locate AF QCP by varying λ 2 = 8 t Spin-fermion interaction: bare mass r 0 of spin fluct. System sizes: L = 8 , 10 , 12 , 14 • Obtain T c via BKT criterion ρ s ( T c ) = 2 T c Temperatures: T ≥ 0 . 04 t π thermodynamic limit (estimate) lower bound value

• T c is not correlated with density of states at the Fermi energy 0 - π π π π π π π π 0 δ / t = 0.2 0 δ - π - π π π 0 - π π π π - π π π π π π π π θ δ δ π π π π π π π π π π π π π π π π δ δ π π π π π π - π π 0 π π - π π π π π π θ hs 0 δ / t = 0.8 0 δ π - π - π - π 0 π π - π π

π π π π π π π π δ δ π π π π π π - π π 0 π π - π π π π π π θ hs 0 δ / t = 0.8 0 δ π - π - π - π • T c is strongly correlated with the relative angle between Fermi 0 π π - π π velocities at a pair of hot spots T Eliash = (0 . 14 sin θ hs ) t c T QMC = (0 . 13 sin θ hs ) t c

• Static pair susceptibility: Z h ˆ Γ ( r , τ )ˆ Γ † (0 , 0) i ˆ χ pair = Γ ( r , τ ) ∼ ψ ↑ ( r , τ ) ψ ↓ ( r , τ ) r , τ