Abanov, Artem

Abanov, Artem. (2018). Non-BCS superconductivity near a quantum-critical point. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/197533

University of Minnesota Digital Conservancy

Non-BCS superconductivity near aquantum-critical pointArtem AbanovTexas A&M UniversityMay, 2018Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.Where do incoherent fermions come from and how do they pair?Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.Yuxuan Wang Boris L. Altshuler Emil A. Yuzbashyan Andrey V.ChubukovPRL 117, 157001 (2016)Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.What is BCS?Pairing boson is gapped – ωD .Coupling is weak λ 1. (λ ∼ g2/ω2D)Tc ∼ ωDe−1/λ  ωD .Only virtual boson’s transitions.Integrate out bosons.Hubbard-Stratonovich. . .etc.Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.What is BCS?We can do better. Let’s increase the coupling constant.To find Tc we need to linearize these equations.=Σ(ω < ωD) = 0.Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.What is BCS?Tc ∼ ωDe− 1+λλ  ωD .All fermions are coherent.Only virtual bosonic transitions.Still BCS.Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.What is BCS?Lessons:Σ suppresses Tc .The linearized equation∆m =λ2T∑n∆n|ωn|ω2D(ωn − Ωm)2 + ω2Das Tc  ωD can be written1 = λ∫ ωDTcdω|ω| .All electronic Matsubara frequencies up to ωD participate.Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.Towards larger λ.What to keep from BCS?Migdal theorem – neglect vertex corrections – large N.Momentum factorization.What will changeBoson propagator.Electron self energy.Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.Limit λ→∞.We want to consider the limit λ→∞.In order to keep Tc finite we need to take ωD → 0.Different for different models.For phonon Tc ∼ ωD√λ. (P. B. Allen and R. C. Dynes, PRB 12, 905 (1975).)For Spin-Fermion model Tc ∼ ωDλ2.etc.Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.γ-Model.Cannot integrate out bosons.Low energy particle-hole excitations.Landau damping for bosons.Landau damping + momentum factorization:∫dqD(ωn, q) ∼ 1|ωn|γcolor superconductivity γ = 0+, (D. T. Son, PRD 59, 094019 (1999); A. V.Chubukov and J. Schmalian, PRB 72, 174520 (2005).),spin- and charge-mediated pairing in D = 3−  dimensionγ = O() 1, D. F. Mross, J. McGreevy, H. Liu, and T. Senthil, PRB 82, 045121 (2010); M.A. Metlitski, D. F. Mross, S. Sachdev, and T. Senthil, PRB 91, 115111 (2015); S. Raghu, G. Torroba, andH. Wang, PRB 92,205104 (2015).2D pairing model with interaction peaked at 2kF : γ = 1/4 (N.E. Bonesteel, I. A. McDonald, and C. Nayak, PRL 77, 3009 (1996).); S. Lederer, Y. Schattner, E. Berg,and S. A. Kivelson, PRL 114, 097001 (2015); Z. Wang, W. Mao, and K. Bedell, PRL 87, 257001 (2001);R. Roussev and A. J. Millis, PRB 63, 140504 (2001); A. V. Chubukov, A. M. Finkelstein, R. Haslinger, andD. K. Morr, PRL 90, 077002 (2003).etc., etc., etc.,Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.InstabilityEquation for TcΦ(ωm) = gγpiT∑n 6=mΦ(ωn)|ωn + Σ(ωn)|1|ωm − ωn|γΣ(ωm) = gγpiT∑n 6=msgn(ωn)|ωm − ωn|γIn order to check the competition between Σ and Φ we use large N(S. Raghu, G. Torroba, and H. Wang, PRB 92, 205104 (2015).)Φ(ωm) =gγNpiT∑n 6=mΦ(ωn)|ωn + Σ(ωn)|1|ωm − ωn|γΣ(ωm) = gγpiT∑n 6=msgn(ωn)|ωm − ωn|γArtem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.Linear equation at T = 0.At T = 0Σ(ω) =gγ1− γ |ω|1−γsgn(ω)Φ(Ω) =1N1− γ2∫Φ(ω)|ω − Ω|γ |ω|1−γdω1 + (1− γ)|ω/g |γNot BCS!No log at small ω.Upper cutoff is provided by bare fermionic ω.Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.Exact spectrum.The linear operator defined asLˆΦ =1− γ2∫Φ(ω)|ω − Ω|γ |ω|1−γdω1 + (1− γ)|ω/g |γhas a continuous spectrum0 < E < Ec = (1− γ)Γ2(γ/2)Γ(γ)cos2(piγ/4)cos(piγ/2)For 0 < γ < 1, the largest E = Ec > 1, but finite.If N > Nc = Ec – no solution.Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.Similar analysis for BCS.Similar analysis of the linear operator at T = 0 for BCS gives0 < E < Ec = O(λ)At small λ BCS has no solution.But the kernel of BCS has a log singularity, which must be cutoff!Comparing to the strong couplingNo log singularity.No solution for N > Nc .Large Σ kills superconductivity.Not so!Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.Return to finite T : ΣSelf-energy (ωn = piT (2n + 1))Σ(ωm) = gγpiT∑n 6=msgn(ωn)|ωm − ωn|γNoticeΣ(ωm=0) = Σ(ωm=−1) = 0andΣ(ωm 6=0,−1) = gγT 1−γΣ¯mSelf-energy is not large at m = 0,−1.Σ¯m does not depend on T and g .Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.Return to finite T : ΦAt large N at T ∼ Tc we can neglect bare ωn in comparisonwith Σ(ωn) in the equation for Φ.We separate m = 0,−1 terms (Φ0 = Φ−1)Φ0 =gγN(2piT )γΦ0 +1N∑m>0Φmχm,0Φm>0 =gγN(2piT )γΦ0[1mγ+1(m + 1)γ]+1N∑n>0Φnχn,mSet of numbers χn,m does not depend on N, g , and T .Consider what happens at T ∼ g2pi 1N1/γ and large N.Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.Large NUp to the first 1/N correctionTc ≈ g2pi1N1/γ(1 +δγNγ),where δγ = O(1).No Nc . Tc is always finite.The corrections go as 1/(Nγ).At large N or γ the transition comes from only two fermionicMatsubara frequencies and one bosonic.Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.Comparison to numerics.100 101 10210-1510-5105Tc/gγ = 0.1ActualEq. (5)100 101 102N0.010.1Tc/gγ = 2 ActualEq. (5)Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.Comparison to numerics.N = 110-1 100 101 102γ1101001000Tc/gTc = g/2πActualFirst MatsubaraAt small γ the result fails. We cannot neglect the bare ω term.Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.ConclusionsAt strong coupling and large N or γ the instability comes fromthe first two fermionic Matsubara frequencies.Very different from BCS, where all frequencies up toωD  Tc are involved.The upper cutoff is provided by the fermionic bare ω.No Nc .Unclear role of the “real” bosons.Artem Abanov Correlated Electron Systems – Novel Developments. FTPI, Minnesota.

Non-BCS superconductivity near a quantum-critical point

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Non-BCS superconductivity near a quantum-critical point

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