Phase transitions and critical behavior in 2D Dirac materials Laura - PowerPoint PPT Presentation

Phase transitions and critical behavior in 2D Dirac materials Laura Classen Heidelberg, March 2017 1 / 26

Outline • 2D Dirac materials • From hopping electrons to Dirac fermions • Ordered phases/Chiral symmetry breaking • Multicriticality between density waves with Michael Scherer, Lukas T Janssen and Igor Herbut quantum critical • Kekul´ e order and fermion-induced quantum criticality ordered with Michael Scherer and Igor g g c Herbut 2 / 26

Dirac Materials • Graphene, Silicene and Germanene • 3D Dirac materials, artificial graphenes, ... A Z.K.Liu et al Science 343 (2014) • Universal properties as consequence of Dirac spectrum Energy Metal Dirac material Insulator particle � hole excitations empty � = 0 occpuied T.O.Wehling et al Adv. Phys. 76 (2014) Crystal momentum • Semimetal - stable against weak interactions 3 / 26

Graphene • In 2004 K. S. Novoselov and A. K. Geim fabricated free-standing graphene • 2D material • 1 layer of graphite • Hexagonal lattice of carbon atoms 4 / 26

Free electrons in graphene • Hopping of free electrons: b k 1 y A B � c † H = − t i , A , s c j , B , s + h.c. δ K δ 1 3 Γ a M δ k � i , j � , s 1 2 x K’ a 2 • ab initio t ≈ 2 . 8 eV b 2 • Energy bands show semimetallic behavior • Half filling: at E = 0 bands touch at Dirac points K , K ′ • Linear and isotropic energy spectrum at K , K ′ Castro Neto et al, Rev. Mod. Phys. 81 (2009) 5 / 26

From hopping to Dirac electrons • Approximation at low energies: Retain only modes near K , K ′ • Low energy effective action � 1 / T d τ d D − 1 x ¯ S F = Ψ γ µ ∂ µ Ψ 0 • with 8-component spinor Ψ = (Ψ ↑ , Ψ ↓ ) T and ¯ Ψ = Ψ † γ 0 � Λ Ψ † c † A , s ( K + q , ω n ) , c † B , s ( K + q , ω n ) , c † A , s ( K ′ + q , ω n ) , c † B , s ( K ′ + q , ω n ) q e i ω n τ + iq · x � � s ( x , τ ) = • and γ matrices γ 0 = ✶ 2 ⊗ σ z , γ 1 = σ z ⊗ σ y , γ 2 = ✶ 2 ⊗ σ x 6 / 26

Interactions and phase transitions • Repulsive Coulomb interactions ( n i , s = c † i , A / B , s c i , A / B , s ) � � H int = U n i , ↑ n i , ↓ + V n i , s n j , s ′ 2 i � i , j � , s , s ′ � + V 2 n i , s n j , s ′ + . . . �� i , j �� , s , s ′ • Long-ranged tail unscreened, but marginally irrelevant • Short-ranged interactions can induce quantum phase transition, but critical strength needed • Different orders depending on interaction profile 7 / 26

Quantum phase diagrams ED: Garc´ ıa-Mart´ ınez et al PRB 88 (2013) FRG: Pe˜ na et al arXiv:1606.01124 2.5 T= 0.046 SDW pha se SM pha se 2 • Semimetallic phase (SM) for CDW pha se 1.5 small interactions V/  1 • Spin Density Wave for large U 0.5 • Charge Density Wave for large V 0 3 4 5 6 7 U/  • Often Kekul´ e order for V ∼ V 2 V / HMC: Buividovich et al PoS (LATTICE2016) 244 8 / 26

Chiral symmetry breaking d D x ¯ � • Effective low-energy theory S F = Ψ γ µ ∂ µ Ψ • Describe interaction-induced phase transitions with chiral symmetry breaking • Gross-Neveu-Yukawa theory S = S F + S B + S Y • S B : Order parameter fields • Fermion and boson coupling � d τ d D − 1 xg i ϕ i ¯ S Y = Ψ M i Ψ M CDW = ✶ M SDW = � σ M Kekule = γ 3 , γ 5 • E.g. CDW: ¯ k , s c † A , k , s c A , k , s − c † ΨΨ ∼ � B , k , s c B , k , s → Difference of sublattice densities • Generalize number of Dirac points N f (graphene N f = 2) 9 / 26

FRG and truncation • Full theory in effective action Γ( g 1 , g 2 , . . . ) Γ k =Λ • Integrate out dof’s successively - Systematic O 3 implementation by (additive) regulator R k • Flow equation Wetterich PLB 301 (1993) {O i } ∂ t Γ = 1 Γ k =0 2STr(Γ (2) O 2 + R k ) − 1 ∂ t R k k O 1 with full propagator (Γ (2) + R k ) − 1 L. Fister k • Truncation � Ψ γ µ ∂ µ Ψ − 1 � � d D x Z Ψ , k ¯ 2 Z ϕ i , k ϕ i ∂ 2 g i , k ϕ i ¯ Γ k = µ ϕ i + ¯ Ψ M i Ψ + U k ( ϕ i ) • Differential equations for couplings ( β functions) encode scale evolution • Non-perturbative regime D = 2 + 1, N f = 2 directly accessible 10 / 26

Fixed points and critical behavior • Fixed points ( ∂ k g i = β g i = 0) → scale-free points → 2nd order phase transition • Scaling properties given by critical exponents P. Kopietz, Springer Verlag 2010 • Extract critical exponents from linearized β functions at FP � ∂β g i � ( g j − g ∗ � β g i ( { g n } ) = j ) � ∂ g j � g n = g ∗ j n • Sign of negative eigenvalues ( ± ) determines (ir)relevant directions • Relevant directions determine stability: Is FP approachable ? • Number of tuning parameters = number of relevant directions for stable FP • No such FP → 1st order phase transition 11 / 26

SDW and CDW: competition and multicriticality 12 / 26

Phase diagram with U and V V • Experiment: graphene is SM V c • But close to phase transition: CDW • Compare critical interactions with e.g. cRPA 1 Wehling et al PRL 106 (2011) • Mild increase of interaction SDW SM leads to phase transition U semimetal Ulybyshev et al PRL 111 (2013) Smith, Smekal PRB 89 (2014) U c 1 • Sizable charge-density and spin-current correlations Gra phe ne Gra phite Golor, Wessel PRB 92 (2015) Ba re cRPA Ba re cRPA • Isotropic strain of ∼ 15% can U A or B (e V) 17 .0 9 .3 17 .5, 17 .7 8 .0, 8 .1 00 induce transition U 01 (e V) 8 .5 5 .5 8 .6 3 .9 U A or B (e V) 5 .4 4 .1 5 .4, 5 .4 2 .4, 2 .4 02 H.-K Tang et al PRL 115 (2015) U 03 (e V) 4 .7 3 .6 4 .7 1.9 • Separate transitions: chiral Ising/Heisenberg universality class Janssen, Herbut PRB 89 (2014), Vacca, Zambelli PRD 91 (2015), Parisen et al PRB 91 (2015), Otsuka et al, PRX 6 (2016), Knorr PRB 94 (2016),... 13 / 26

Multicritical behavior in graphene V CDW mixe d V c SM SDW CDW ? χ � = 0 CDW 1 SM SDW SDW SM CDW � U φ � = 0 U c 1 SM SDW • Graphene parameters close to multicritical point • Competition of order parameters • Structure at MCP? Critical exponents? 14 / 26

Coupled order parameter fields � ¯ and SDW field � � • CDW field χ = ΨΨ φ = � Ψ � σ Ψ � • Symmetry of CDW and SDW fields is ❩ 2 and O (3) • Two Yukawa terms � � � ΨΨ + g φ � d D x g χ χ ¯ φ · ¯ S Y = Ψ � σ Ψ • Coupling between different oder parameter fields � � 1 χ ) χ + 1 � φ ) � d D x 2 χ ( − ∂ 2 µ + m 2 φ · ( − ∂ 2 µ + m 2 S B = φ 2 + λ χ 8 χ 4 + λ φ � 2 + λ χφ φ 2 + . . . � � � φ · � 4 χ 2 � φ 8 15 / 26

Fixed point structure • 2 tuning parameters, i.e. stable FP can have 2 relevant directions • Sign of third critical exponent determines stability 1.0 Θ 3 , B 0.5 Θ 3 0.0 � 0.5 Θ 3 , D � 1.0 2 3 4 5 10 20 50 100 N f • Two candidates for stable FP • Chiral Heisenberg + Ising for small N f • New universality from coupled FP for large N f • Mid-size N f : no stable FP → 1st order transition 16 / 26

Multicritical behavior at stable FP • Determine phase structure from effective potential • Positions of minima determined by ∆ = λ χ λ φ − λ 2 χφ ∆ > 0 ∆ = 0 ∆ < 0 ∆ > 0 ∆ < 0 V V T etracritical Bicritical 1 st order V c V c CDW coexistence CDW 1 1 SDW SDW SM SM U U 1 U c 1 U c 17 / 26

Phase diagram as function of N f • ∆ = λ χ λ φ − λ 2 χφ determines multiciritcal behavior B: chiral Heisenberg + Ising D: new coupled FP V V V graphene T etracritical Bicritical 1 st order V c V c V c coexistence CDW CDW CDW 1 st order MCP 1 1 1 SDW SDW SM SDW SM SM U U U U c 1 U c U c 1 1 18 / 26

Kekul´ e order and fermion-induced criticality 19 / 26

Kekul´ e Valence Bond Solid • Bond-dependent nearest-neighbor hopping ∆ t i ,δ c † � H K = − i , A , s c i + δ, B , s + h.c. i , s ,δ • Breaks lattice translation and rotation symmetry C 6 → C 3 • Order can be induced by sufficiently strong • electronic interactions V ∼ V 2 Hou et al (2007), Weeks/Franz (2010), Roy/Herbut (2010),... • Electron-phonon interaction Nomura et al (2009), Kharitonov (2012), Classen et al (2014),... • Observed in graphene on Copper substrate and artificial graphene t t 2 t 1 Gomes et al Nature 483 (2012) 20 / 26

Low energy model for Kekul´ e order C C B • Described by complex order A A parameter φ = φ 1 + i φ 2 with C Z 3 symmetry B B A C C • Dirac fermions L F = ¯ ψγ µ ∂ µ ψ • Coupling between fermions and order parameter fields L Y = ih ¯ ψ ( φ 1 γ 3 + φ 2 γ 5 ) ψ • Cubic terms in free energy allowed µ φ + m 2 | φ | 2 + g ( φ 3 + φ ∗ 3 ) + λ | φ | 4 + . . . L B = − φ ∗ ∂ 2 21 / 26

Landau Criterion and Fermion-Induced QCP F LG � Φ � � m 2 Φ 2 � g Φ 3 �ΛΦ 4 • First order transition due to cubic terms F LG � Φ � • Presence of fermion critical mode can change Landau picture → Fermion-induced quantum critical point Φ • RG picture: need stable FP for continuous transition • Here: 1 tuning parameter, i.e. stable FP would have 1 relevant direction µ φ + m 2 | φ | 2 + g ( φ 3 + φ ∗ 3 ) + λ | φ | 4 + . . . L B = − φ ∗ ∂ 2 • At Gaussian FP 2 relevant directions [ m 2 ] = 2 [ g ] = 3 − D / 2 [ λ ] = 4 − D • At interacting FP power counting modified 22 / 26