diff --git "a/QtFRT4oBgHgl3EQf7Tgp/content/tmp_files/load_file.txt" "b/QtFRT4oBgHgl3EQf7Tgp/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/QtFRT4oBgHgl3EQf7Tgp/content/tmp_files/load_file.txt" @@ -0,0 +1,1099 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf,len=1098 +page_content='Density of states and spectral function of a superconductor out of a quantum-critical metal Shang-Shun Zhang1 and Andrey V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Chubukov1 1School of Physics and Astronomy and William I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA (Dated: February 1, 2023) We analyze the validity of a quasiparticle description of a superconducting state at a metallic quantum-critical point (QCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' A normal state at a QCP is a non-Fermi liquid with no coherent quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' A superconducting order gaps out low-energy excitations, except for a sliver of states for non-s-wave gap symmetry, and at a first glance, should restore a coherent quasiparticle behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We argue that this does not necessarily hold as in some cases the fermionic self-energy remains singular slightly above the gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This singularity gives rise to markedly non-BCS behavior of the density of states and to broadening and eventual vanishing of the quasiparticle peak in the spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We analyze the set of quantum-critical models with an effective dynamical 4-fermion interaction, mediated by a gapless boson at a QCP, V (Ω) ∝ 1/Ωγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We show that coherent quasiparticle behavior in a superconducting state holds for γ < 1/2, but breaks down for larger γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We discuss signatures of quasiparticle breakdown and compare our results with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Metals near a quantum critical point (QCP) display a number of non-Fermi liquid properties like linear-in-T resistivity, a broad peak in the spectral function near kF with linear-in-ω width, singular behavior of optical conductivity, etc [1–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' These properties are often thought to be caused by the coupling of fermions to near-gapless fluctuations of an order parameter, which condenses at a QCP [19– 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The same fermion-boson interaction gives rise to superconductivity near a QCP [30–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' A superconducting order gaps out low-energy excitations, leaving at most a tiny subset of gapless states for a non-s−wave order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' A general belief has been that this restores fermionic coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' A frequently cited experimental evidence is the observed re-emergence of a quasiparticle peak below Tc in near-optimally doped cuprates (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' From theory side, the argument is that the fermionic self-energy in a superconductor has a conventional Fermi- liquid form Σ(ω) ∼ ω at the lowest ω, in distinction from a non-Fermi-liquid Σ(ω) ∝ ωa with a < 1 in the normal state [44–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In this paper, we analyze theoretically whether fermions in a superconducting state at a QCP can be viewed as well-defined coherent quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We argue that this is not necessarily the case as fermionic self-energy can still be singular on a real frequency axis immediately above the gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This singularity gives rise to markedly non-BCS behavior of the density of states (DoS) and to broadening and eventual vanishing of the quasiparticle peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For superconductivity away from a QCP, mediated by a massive boson, numerous earlier studies have found that the spectral function A(k, ω) at T = 0 has a δ-functional peak at ω = (∆2 + (ξk/Z)2)1/2, where ξk = vF (k − kF ) is a fermionic dispersion (vF is a Fermi velocity), ∆ is a superconducting gap, and Z is an inverse quasiparticle residue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' A δ-functional peak FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Three possible forms of the electronic spectral function A(k, ω) at T = 0 in a quantum critical superconductor at a small but finite k − kF and in the absence of impurity broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (a): A(k, ω) vanishes at |ω| = ∆ and has a well-defined peak at ω > ∆, (b): A(k, ω) diverges at |ω| = ∆, but it non-monotonic at larger ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The peak in A(k, ω) at |ω| > ∆ broadens, but still exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (c): A(k, ω) diverges at |ω| = ∆, and monotonically decreases at larger ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In case (a) fermions can be viewed as well-defined quasiparticles, in case (c) the quasiparticle picture completely breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The case (b) is the intermediate one between (a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' holds for momenta near the Fermi surface, as long as ω < ∆ + ω0, where ω0 is a mass of a pairing boson in energy units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' At larger ω, fermionic damping kicks in, and the peak broadens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The same physics leads to peak-dip-hump behavior of A(k, ω) as a function of ω, observed most spectacularly in near-optimally doped cuprate Bi2Sr2CaCu2O8+δ (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='g, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [54, 55]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' At a QCP, the pairing boson becomes massless and ω0 vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This creates a singular behavior near the gap edge at ω = ∆, which holds even when ξk is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' A simple experimentation shows that there are three possible forms of A(k, ω), which we present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 1: it (i) either vanishes at ω = ∆ and has a well-defined peak at ω > ∆ whose width at small ξk is parametrically smaller than its energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' or (ii) diverges at ω = ∆, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='13679v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='supr-con] 31 Jan 2023 Three possible forms of A( k,w) in quantum-critical superconductor (a) Quasiparticle picture (b) Partial breakdown (c) Complete breakdown of quasiparticle of quasiparticle (k,w) A △ 0 △ 0 △ 0 3 3 32 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 2 Leading exponent 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 Subleading exponent 8 c 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='=7g 0 2 4 6 8 10 D(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=') .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=" 8 ' 1:18 gap edge 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 (" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' )8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 D(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=') !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 1 10 5 0 5 10 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='=7g 0 1 2 3 4 5 DoS 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='1 100 (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' ")=7g 100 101 9 1=x0:5 9 1=x0:59 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:35 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:8 (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (a) Exponents ν and c for the leading and the subleading terms in the expansion D(ω) ≃ 1+α(∆−ω)ν +β(∆−ω)ν+c, where D(ω) = ∆(ω)/ω and the gap edge ∆ is the solution of D(ω = ∆) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (b) Numerical result for D(ω) for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Inset shows the power-law behavior near the gap edge with ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='18, consistent with (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (c) Fermionic DoS at T = 0 for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='35 (thick green line) and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8 (thin pink line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In both cases, the DoS vanishes below the gap edge ∆ and has a power-law singularity above it N(ω) ∝ 1/(ω − ∆)ν/2, but the exponent ν is different in the two cases, as we show in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' but is non-monotonic at larger ω and displays a broad maximum at some ω > ∆, or (iii) diverges at ω = ∆ and monotonically decreases at larger ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In the first case, fermions in a quantum-critical superconductor can be viewed as well-defined quasiparticles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' in the last case the quasiparticle picture completely breaks down;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' the second case is the intermediate one between the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Our goal is to understand under what circumstances A(k, ω) of a quantum-critical superconductor has one of these forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For our study, we consider dispersion- full fermions, Yukawa-coupled to a massless boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We assume, like in earlier works (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [56]), that a boson is Landau overdamped, and its effective velocity is far smaller than vF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In this situation, the interaction that gives rise to non-Fermi liquid in the normal state and to superconductivity, is a purely dynamical V (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The fermionic self-energy and the pairing gap, tuned into a proper spatial pairing channel, are then determined by two coupled equations in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' At a QCP, V (Ω) is singular at vanishing Ω in spatial dimension D ≤ 3, and behaves as V (Ω) ∝ (¯g/Ω)γ, where ¯g is the effective fermion-boson coupling, and the exponent γ is determined by the underlying microscopic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The most studied models of this kind are of fermions near an Ising-nematic or Ising/ferromagnetic QCP (γ = 1/3) and near an antiferromagnetic or charge density wave QCP (γ = 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The same effective interaction emerges for dispersion-less fermions in a quantum dot coupled to Einstein bosons (the Yuakawa- SYK model) [57–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For this last case, the exponent γ is a continuous variable γ ∈ (0, 1), depending on the ratio of fermion and boson flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' An extension of the Yukawa- SYK model to γ ∈ (1, 2) has recently been proposed [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We follow these works and consider γ as a continuous variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We note that the value of γ is generally larger deep in a superconducting state because of feedback from superconductivity on the bosonic polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For simplicity, we neglect potential in-gap states associated with non-s-wave pairing symmetry and focus on the spectral function of fermions away from the nodal points and on features in the density of states (DoS) above the gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' An extension to models with in-gap states is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In previous studies of the γ-model, we focused on the novel superconducting behavior at γ > 1, when the pairing interaction is attractive on the Matsubara axis, while on the real axis ReV (Ω) is repulsive [62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We argued that this dichotomy gives rise to phase slips of the gap function on the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Here, we restrict ourselves to γ ≤ 1, when this physics is not present and, hence, does not interfere with the analysis of the validity of a quasiparticle description in a superconducting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Pairing gap and quasiparticle residue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For superconductivity mediated by a dynamical interaction, the paring gap ∆(ω) and the inverse quasiparticle residue Z(ω) are functions of the running real fermionic frequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We define the gap edge ∆ (often called the gap) from the condition ∆(ω) = ω at ω = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For our purposes, it is convenient to introduce D(ω) = ∆(ω)/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The gap edge is at |D| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The equation for D(ω) that we need to solve is ωB(ω)D(ω) = A(ω) + C(ω), (1) where B(ω) and A(ω) are regular functions of ω (see [64, 65]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The C(ω) term depends on the running D(ω), C(ω) = ¯gγ sin πγ 2 � ω 0 dΩ Ωγ D(ω − Ω) − D(ω) � D2(ω − Ω) − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (2) Its presence makes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S3) an integral equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The inverse residue Z(ω) is expressed via D(ω′) as Z(ω) = B(ω) + ¯gγ sin πγ 2 ω � ω 0 dΩ Ωγ 1 � D2(ω − Ω) − 1 (3) 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Spectral function A(k, ω) at T = 0 for four representative γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The broadening in the plots is intrinsic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (a-d): color- coded plot at negative ω, as measured by the ARPES intensity at T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (e-f): constant-k cuts of A(k, ω) at ξk = 0 and at ξk = ±4¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For γ < 1/2, the spectral function has a sharp quasiparticle peak at ω + ∆ ∝ ξ2 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For γ > 1/2, the peak moves to ω + ∆ ∝ |ξk|1/(1−γ) and broadens up, which eventually disappears (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' and is readily obtained once D(ω) is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' At γ = 0, which models a BCS superconductor, C(ω) = 0 and D(ω) = A(ω)/(ωB(ω)) is a regular function of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Near the gap edge at ω > 0, D(ω)−1 ∼ ω−∆ and Z(ω) ≈ Z(∆) ≡ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We assume and then verify that D(ω) remains regular in some range of γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Substituting D(ω)−1 ∼ ω−∆ into (S7) for γ > 0, we obtain C(ω)−C(∆) ∼ (ω−∆)3/2−γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We see that C(ω) is non-analytic near the gap edge, but for γ < 1/2, the exponent 3/2−γ is larger than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In this situation, the non-analytic term in C(ω) generates a non-analytic term in D(ω) of order (ω − ∆)3/2−γ, which is smaller than the regular ω−∆ term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Evaluating the prefactors, we obtain slightly above the gap edge, at ω = ∆ + δ D′(∆ + δ) = 1 + αδ + A cos[π(3/2 − γ)]δ3/2−γ, D′′(∆ + δ) = −A sin[π(3/2 − γ)]δ3/2−γ, (4) where α ∼ 1/¯g, A = � α 2 ¯gγ sin(πγ/2) ∆B(∆) J(γ, 1) and J(γ, ν) is expressed via Beta functions: J(γ, ν) = B(1 − γ, γ − 1 − ν 2) − B(1 − γ, γ − 1 + ν 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (5) For γ > 1/2, 3/2 − γ > 1, and the calculation of D(ω) has to be done differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We find after straightforward analysis that the leading δ-dependent term in D(∆ + δ) is non-analytic and of order δν, where ν is the solution of J(γ, ν) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The exponent ν ≈ 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='67(γ − 1/2) for γ ≈ 1/2 and ν ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='3 for γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The subleading term in D(∆ + δ) scales as δν+c, where c > 0 is approximately linear in γ − 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 2, we plot ν(γ) and c(γ) along with the numerical results of D(ω) for a representative γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The exponent ν extracted from this numerical D(ω) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='18, which matches perfectly with the analytical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The behavior at γ = 1/2 is special, and we discuss it in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Substituting D(∆ + δ) into the formula for Z(ω), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S8), we obtain Z′(∆ + δ)=Z(∆)+B cos(π(γ + ν/2 − 1))δ1−γ−ν/2,(6) Z′′(∆ + δ)=B sin(π(γ + ν/2 − 1))δ1−γ−ν/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (7) where B = ¯gγ sin πγ 2 ∆ √ 2α B(1 − γ, ν 2 + γ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For γ < 1/2, Z(ω) = Z(∆) + O(δ1/2−γ) is approximately a constant near the gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For γ > 1/2, the inverse residue diverges at the gap edge, indicating a qualitative change in the system behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Spectral function and DoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The spectral function and the DoS per unit volume are given by A(k, ω) = − 1 π ImGR(k, ω), N(ω) = 1 V � k A(k, ω) = NF ωIm � 1 ∆2(ω) − ω2 , (8) where the retarded Green’s function GR(k, ω) = −(ωZ(ω) + ξk)/(ξ2 k + (∆2(ω) − ω2)Z2(ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' ARPES intensity is proportional to A(k, ω)nF (ω), which at T = 0 selects negative ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' At γ = 0 (BCS limit), N(ω) ∼ 1/(ω−∆)1/2, and the spectral function has a δ-functional peak at ω = (∆2 + (ξk/Z)2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 2 (c,d), we show the DoS N(ω), obtained from the numerical solution of the full gap equation (S3) for representative γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='35 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We see that in both cases the DoS describes a gapped continuum, but there is a qualitative difference in the behavior near the gap edge: for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='35, N(ω) has the same 1/δ1/2 singularity as for γ = 0, and for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8 (e) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='35 (f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='45 1 1 A(k,w) Sk = 0 0> .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='. 0< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 0 0 10 6- 8 7 6 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 2 w/g w/g(g) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='65 (h) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='1 0 0 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 w/g w/g(a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='35 (b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='45 2 1 4 2 19 6 3 8 3 10 4 12 10 5 0 5 10 10 5 0 5 10 Sk/g Sk/g 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='65 (d) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 2 10 5 0 5 10 10 5 0 5 10 Sk/g Sk/g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5Max 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='4 the DOS behaves as 1/δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='59, which perfectly matches the analytical form δ−ν/2, given that ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='18 for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The spectral function A(k, ω) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For comparison with ARPES, we set ω to be negative: ω = −(∆ + δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For any γ, there is no frequency range, where A(k, ω) is a δ-function, simply because the bosonic mass vanishes at a QCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Still, for γ < 1/2, D(−(∆+δ))−1 ∝ δ and Z(−(∆+δ)) ≈ Z(−∆) = Z(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In this situation, the spectral weight on the Fermi surface, integrated over an infinitesimally small range around ω = −∆ immediately above the real axis, is finite, like in BCS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Away from the Fermi surface, the spectral function vanishes as |ω + ∆|1/2−γ at the gap edge and displays a quasiparticle peak at ω ≈ −(∆2 + (ξk/Z(∆))2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The peak is well defined at small δ as its width O(δ1/2−γ) is parametrically smaller than its frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This is the same behavior as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For γ > 1/2, the situation is qualitatively different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Now Z(−∆ − δ) diverges at δ → 0 and D(−∆ − δ) − 1 ∼ |δ|ν ≪ |δ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In this case, the integral of A(kF , ω) over an infinitesimally small range around ω = −∆ vanishes, which can be interpreted as a vanishing of a quasiparticle peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' At finite ξk, the spectral function diverges at the gap edge as 1/|ω + ∆|γ/2+γ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For γ slightly above 1/2, A(k, ω) is non-monotonic and possess a broad maximum at |ω + ∆| ∼ (ξk/¯gγ) 1 1−γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This is the same behavior as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For larger γ, the maximum disappears, and A(k, ω) monotonically decreases at |ω| > ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This is the same behavior as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For small ξk, the maximum disappears at γ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For larger ξk, it disappears at smaller γ, first for positive ξk (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Comparison with ARPES The behavior shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 4 is our result in some range of γ > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For positive ξk (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=', outside the Fermi surface), the spectral function has a single non-dispersing maximum at the gap edge, except for the smallest ξk, while for negative ξk, A(k, ω) has a kink at the gap edge ω = −∆ and a dispersing maximum at ω = −∆ − O � |ξk|1/(1−γ)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This behavior is consistent with the ARPES data for Bi2201, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The data shows that the spectral function near the antinode, where our analysis is valid, displays an almost non-dispersing maximum at positive ξk, while for negative ξk it displays a non-dispersing kink at the same energy and a dispersing maximum at larger |ω|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We associate the non-dispersing feature at both positive and negative ξk with the gap edge ∆, and associate the dispersing maximum, observed in [66] at ξk < 0, with the dispersing maximum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Discussion and summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In this work, we analyzed the applicability of quasiparticle description of a superconducting state which emerges out of a non- Fermi liquid at a metallic QCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We considered the model with an effective dynamical 4-fermion interaction FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (a) Spectral function A(k, ω) at positive and negative ξk = ±4¯g at γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' To account for impurity scattering, we convoluted the spectral function with a Lorentzian of width ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='03¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (b) Spectral function at a set of discrete momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' It displays a non-dispersing gap edge singularity (green dots) and a dispersing maximum (blue circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This theoretical A(k, ω) is consistent with the ARPES data for Bi2201, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [66] (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' V (Ω) ∝ 1/Ωγ, mediated by a gapless boson at a QCP and analyzed the spectral function and the DoS for γ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Interaction V (Ω) gives rise to a non-Fermi liquid in the normal state with self-energy Σ(ω) ∝ ω1−γ and to pairing below some finite Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' A superconducting order gaps out low-energy excitations and, at a first glance, should restore fermionic coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We found, however, that this holds only for γ < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For larger γ the spectral function and the DoS exhibit qualitatively different behavior than that in a superconductor with coherent quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (different power-laws).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We argued that the quasiparticle peak broadens up and completely disappears for γ close to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Away from a QCP, a pairing boson is massive and at the lowest energies a Fermi-liquid description holds already in the normal state and continue to hold in a superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In particular, in the immediate vicinity of the gap edge, the system displays a BCS- like behavior for all γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Still, the system behavior over a broad frequency range is governed by the physics at a QCP, as numerous experiments on the cuprates and other correlated systems indicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We argued that our results are quite consistent with the ARPES data for Bi2201 [11, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We acknowledge with thanks useful conversations with a number of our colleagues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This work was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Department of Energy, Office of Science, Basic Energy Sciences, under Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' DE-SC0014402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Martin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Fiory, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Fleming, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Schneemeyer, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Waszczak, Physical Review B 41, 846 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' L¨ohneysen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Pietrus, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Portisch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Schlager, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Schr¨oder, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Sieck, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Trappmann, Physical review 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='05 0 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 w/g1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [61] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Schmalian, private communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [62] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Abanov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Chubukov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' B 103, 024522 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [63] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Abanov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Chubukov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' B 103, 184508 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [64] See the Supplementary Information for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [65] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Marsiglio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Schossmann, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Carbotte, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' B 37, 4965 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [66] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' He, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Hashimoto, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Karapetyan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Koralek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Hinton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Testaud, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Nathan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Yoshida, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Yao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Tanaka, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=', Science 331, 1579 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 7 Supplementary information for “Density of states and spectral function of a superconductor out of a quantum-critical metal” by Shang-Shun Zhang and Andrey V Chubukov GAP EQUATION ALONG THE REAL-FREQUENCY AXIS AND ITS SOLUTION We will use the approach pioneered by Marsiglio, Shossmann, and Carbotte [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In this approach, one first solves non-linear gap equation along the Matsubara axis, which can be done rather straightforwardly as the gap function ∆(ωm) can be chosen to be real for all frequencies and is a regular function of ωm even when the pairing boson is massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' One then uses this ∆(ωm) as an input for the equation for complex ∆(ω) along the real frequency axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The non-linear integral equation for D(ωm) = ωm∆(ωm) on the Matsubara axis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' and the equation for the inverse quasiparticle residue Z(ωm) = 1 + Σ(ωm)/ωm (Σ(ωm) is the fermionic self-energy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' have the form ωmD(ωm) = πT � ω′m (D(ω′ m) − D(ωm)) sgn(ω′ m) � 1 + D2(ω′m) V (ω − ω′ m),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S1) Z(ωm) = 1 + 1 ωm πT � ω′m sgn(ω′ m) � 1 + D2(ω′m) V (ω − ω′ m),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S2) where V (Ωm) = (¯g/|Ωm|)γ is the same as in the main text (¯g is an effective fermion-boson coupling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' and γ depends on the underlying microscopic model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='This set of equations has a non-zero solution D(ωm) below a finite pairing temperature Tp ∼ ¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S1 shows the numerical solution for ∆(ωm) at T = 10−6¯g ≪ Tp, for different γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We see from the figure that ∆(ωm) approaches a constant value at small frequencies and decays as ω−γ m at high frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This behavior holds for all γ and can be easily verified analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The gap equation along the real frequency axis is ωB(ω)D(ω) = A(ω) + C(ω), (S3) (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (1) in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This equation is obtained by using the spectral representation of an analytic function on the upper frequency half-plane f(iωm) = 1 π � dx Imf(x) x − iωm (S4) and, where possible, keeping D(ωm) as an input function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This approach was pioneered for electron-phonon interaction in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [3–6] for the electron-phonon problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Numerical resulls for the gap function ∆(ωm) along the Matsubara axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The calculation is performed at temperature T = 10−6¯g using the hybrid-frequency method [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 10 10° 6/(um)V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 10 102 100 104 wm /g8 20 40 60 80 100 120 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 D(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=') (a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:25 #5 D0 D00 20 40 60 80 100 120 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='=7g 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 2 Z(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=') (b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:25 Z0 Z00 1 2 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 (c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:5 1 2 3 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='=7g 0 2 4 6 (d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:5 0 1 2 3 4 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 (e) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:8 0 1 2 3 4 5 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='=7g 0 2 4 6 8 10 (f) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Numerical results for D(ω) and Z(ω) for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='25, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For our case, the functions A(ω) and B(ω) are directly expressed via D(ωm) along the Matsubara axis as A(ω) = 1 2 � ∞ 0 dωm D(ωm) � 1 + D2(ωm) × � ¯gγ (ωm + iω)γ + ¯gγ (ωm − iω)γ � , (S5) B(ω) = 1 + i 2ω � ∞ 0 dωm 1 � 1 + D2(ωm) × � ¯gγ (ωm + iω)γ − ¯gγ (ωm − iω)γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S6) and C(ω) is given by C(ω) = ¯gγ sin πγ 2 � ω 0 dΩ Ωγ D(ω − Ω) − D(ω) � D2(ω − Ω) − 1 , (S7) (Eqn (3) in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This function depends on the running D(ω − Ω), which makes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S3) an integral equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The inverse residue Z(ω) is expressed via D(ω′) as Z(ω) = B(ω) + ¯gγ sin πγ 2 ω � ω 0 dΩ Ωγ 1 � D2(ω − Ω) − 1 (S8) (Eqn (4) in the main text) and is readily obtained once D(ω) is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The gap equation along the real-frequency axis has an iterative structure in the sense that D(ω) depends on D(ω′) at ω′ < ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This allows us to solve this equation iteratively, using the low-frequency form D(ω) ≃ ∆(0)/ω as an input, with ∆(0) ≡ ∆(ωm = πT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S2 we show the results for D(ω) and Z(ω) for three representative values of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In all cases, D(ω) and Z(ω) are real below the gap edge ω = ∆ and are complex above the gap edge, where ∆ is defined as ∆(ω) = 1 at ω = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We see that for γ > 1/2, Z(ω) diverges at the gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We use this fact in the main text in the analysis of the spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The real and imaginary parts of the gap function near the gap edge ω = ∆ for γ = 1/2, obtained by solving the non-linear gap equation numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The leading term in D′(∆ − δ) − 1, shown in the inset of (a), is linear in δ∆ − ω, and the subleading scales as δ/| log |δ||, as is confirmed by the linear relation in panel (a), The imaginary part D′′ appears at negative δ above the gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The numerical result in panel (b) clearly shows the scaling relation D′′ ∼ δ/ log2 (|δ|/¯g), expected from the Kramers-Kronig relation with D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' THE CASE OF γ = 1/2 In the main text we argued that for γ < 1/2, the function D(∆ − δ) − 1 ∝ δ, where δ = ∆ − ω, and the correction scales as δ3/2−γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' More specifically, we found iteratively that D(∆ − δ) = 1 + δ ∞ � n=0 αnδnϵ (S9) where ϵ = 1/2−γ and α0 = O(1/¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The expression for α1 is presented in the main text, after Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' It is proportional to J(γ, 1) = B(1 − γ, γ − 3/2) − B(1 − γ, γ − 1/2), where B(a, b) is a Beta function (B(a, b) = Γ(a)Γ(b)/Γ(a + b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For small ϵ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=', for γ ≤ 1/2), α1 ∼ J(γ, 1) ∼ 1/ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For the next term in (S9) we find α2 ∼ 1/ϵ2, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We see that the perturbative expansion in δϵ in (S9) holds for (δ/¯g)ϵ/ϵ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Outside this range, all terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S9) are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' As γ approaches 1/2 from below and ϵ decreases, the perturbative regime shrinks to exponentially small δ < ¯g exp(−| log ϵ|/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' To understand the form of D(ω) outside the perturbative regime, we express (δ/¯g)ϵ as eϵ log (δ/¯g) and expand (S10) in powers of log (δ/¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We obtain D(∆ − δ) = 1 + δ ∞ � n=0 ˜αn(log δ ¯g )n (S10) where ˜α0 = α0 + α1 + α2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='., ˜α1 = ϵα1 + 2ϵα2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='., ˜α2 = 2ϵ2α2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='. We see that each ˜αn is a series, in which the first term is independent on ϵ, and the others diverge as powers of 1/ϵ, because ��1 ∼ 1/ϵ, α2 ∼ 1/ϵ2, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We now argue that singular parts of ˜αn can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The argument is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' First, in the calculations, the 1/ϵ divergencies originate from the divergence of J(1/2 − ϵ, 1) ≈ 1/(2ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This divergence is regularized by a finite boson mass, such that strictly at ϵ = 0, one has J(1/2, 1) = 0 instead of infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Second, if we assume that ˜α0 in (S10) remains finite at ϵ = 0 and substitute the trial D(∆ − δ) = 1 + δ˜α0 in the gap equation at γ = 1/2 and compute iteratively the next term in D(∆−δ), we find it in the form ˜α1δ log (δ/¯g) with a finite ˜α1 = √¯g˜α0/(4∆B(∆)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Extending the iterative analysis, we find that all ˜αn are finite at γ = 1/2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=', ϵ = 0), as we anticipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We didn’t manage to sum up analytically the logarithmic series in (S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The numerical solution for D(∆ − δ) for γ = 1/2 shows that D(∆ − δ) − 1 remains linear in δ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S2 (c)), and the corrections scale as 1/| log (|δ|/¯g)| (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S3 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' By Kramers-Kronig relation, this implies that at negative δ, when ω > ∆ is above the gap edge, the imaginary part of D(ω) scales as D ′′(∆ + |δ|) ∝ δ/ log2 (|δ|/¯g) (the same form is obtained by just noticing that log(−|δ|) = log(−(ω − ∆ + i0)) = log |δ| − iπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This form of D ′′(∆ + |δ|) is consistent with our numerical solution above the gap edge, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The solution clearly shows that the ratio D ′′(∆ + |δ|)/δ decreases at the smallest δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We next use the result for D(ω) to obtain the inverse quasi-particle residue near the gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Substituting D(∆ − δ) ≈ 1 + ˜α0δ into (S8), we obtain at γ = 1/2 Z(∆ − δ) = 1 2∆ � ¯g ˜α0 | log δ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S11) 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' I log(8)|12 000 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 GO 8 5 10 1og2 [8]15(a) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 ×10-4 gap edge 1 5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='843 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8435 / 3 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 G0000(b) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 13 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='510 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (a) The spectral function A(k, ω) for γ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (b) Constant ξk cuts along the blue lines in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Analytically continuing this function to negative δ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=', to ω above the threshold, we obtain Z(∆ + |δ|) = 1 2∆ � ¯g ˜α0 (| log |δ|| + iπ) , (S12) Note that the imaginary part of Z(ω) jumps to a finite value at ω infinitesimally above the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This behavior is consistent with the numerical solution for Z(ω), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S2 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Finally, we use the results for D(ω) and Z(ω) and compute the spectral function near the gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' On the Fermi surface, the spectral function at negative ω and |ω| > ∆ takes the form A(kF , ω) ∝ 1 |ω + ∆| log(¯g/|ω + ∆|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S13) Slightly away from the Fermi surface, the spectral function has a peak at |ω| = ∆+δk where δk ∼ (ξ2 k/¯g)/ log2(|¯g/ξk|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The peak width scales as δk/ log(|¯g/ξk|) and is logarithmically smaller than the energy variation |ω| − ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Also, for any non-zero ξk, the spectral function jumps at the gap edge to a finite value of order 1/ξ2 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S4 we show the numerical result for the spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' It is consistent with the behavior we just described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' UNIVERSAL FORM OF THE SPECTRAL FUNCTION AT 1/2 < γ < 1 For frequencies ω near the gap edge and for momenta near the Fermi surface, when ξk is much smaller than |ωZ(ω)|, a straightforward calculation shows that for γ > 1/2, the spectral function can be expressed as a scaling function of ξk/|ω + ∆|1−γ¯gγ (we set ω < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Namely, A(k, ω) ∝ 1 |ω + ∆| ν 2 +1−γ Φ � ξk |ω + ∆|1−γ¯gγ � , (S14) where Φ(x) ≡ x2 + Q2 γ sin[π(ν − c)]/ sin(πc) � x2 + Q2γ cos(2πγ) �2 + Q4γ sin2(2πγ) (S15) with Qγ = sin(πγ/2)B(1 − γ, ν/2 + γ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S5, we plot the dimensionless function Φ(x) for different values of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' At x ≫ 1, Φ(x) ∼ 1/x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' at x ≪ 1, Φ(x) ∼ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For γ < γc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='9, function Φ(x) contains a local maximum at x2 ∗ ∼ � (u − v)2 + w2 − u, (S16) where u = Q2 γ sin[π(ν − c)]/ sin(πc), v = Q2 γ cos(2πγ), and w = Q2 γ sin(2πγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This maximum can be interpreted as an over-damped, but still existing quasi-particle peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' At γ > γc, the function Φ(x) monotonically decreases with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In this case, the quasiparticle description breaks down completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 10 5 0 5 10 Sk/g2 1 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='4 2.' metadata={'source': 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+page_content='3 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='1 )(x) 0:5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 1:0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content="c ' 0:9 FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The function Φ(x), Eqn (S15), for different values of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We emphasize that this behavior holds only for small enough ξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For larger ξk, the spectral function does depend on the sign of ξk and as γ increases, the quasiparticle behavior gets completely destroyed first for positive ξk and then, at larger γ, for negative ξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Marsiglio, M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Wu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Abanov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Wang, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Chubukov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' B 102, 024525 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [3] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Marsiglio and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Carbotte, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' B 43, 5355 (1991), for more recent results see F.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Chubukov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' B 99, 144512 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'}