{
"schema_version": 1,
"id": "PHYS-2026-08-13-01",
"canonical_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-08-13-01/",
"source_markdown_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-08-13-01.md",
"metadata": {
"schema_version": 1,
"id": "PHYS-2026-08-13-01",
"date": "2026-08-13",
"updated_at": "2026-08-13",
"title": "A singular self-energy does not determine the resistivity",
"summary": "Separate quasiparticle decay from current relaxation using Green functions, Ward identities, and the memory matrix, then derive how spatially random critical interactions can produce linear resistivity.",
"language": "en",
"entry_kind": "daily",
"status": "published",
"level": "research",
"user_difficulty": "unrated",
"domains": [
"condensed-matter",
"quantum-field-theory",
"statistical-mechanics",
"mathematical-physics"
],
"estimated_minutes": 105
},
"content_markdown": "\n## The claim under pressure\n\n**A singular electronic self-energy does not determine the d.c. resistivity.**\n\nA metal can lose its Landau quasiparticles while retaining a ballistic charge component. The reason is operator-theoretic: decay of a one-electron excitation and decay of the slow many-body mode that carries current are different questions.\n\nThe distinction becomes sharp near a metallic quantum critical point. A clean critical Fermi surface can have\n\n$$\n-\\operatorname{Im}\\Sigma^R(\\omega)\n\\propto |\\omega|^{2/3},\n$$\n\nyet an infinite ideal d.c. conductivity. Spatially random interactions can use the same critical fluctuations to relax momentum and produce\n\n$$\n\\rho(T)=\\rho_0+A T.\n$$\n\nNeither formula implies the other. The missing information is the operator that absorbs momentum.\n\n## 1. Three observables, three questions\n\nThe retarded Green function is\n\n$$\nG^R(\\mathbf k,\\omega)\n=\\frac{1}\n{\\omega-\\xi_{\\mathbf k}-\\Sigma^R(\\mathbf k,\\omega)}.\n$$\n\nIts spectral function,\n\n$$\nA(\\mathbf k,\\omega)\n=-\\frac{1}{\\pi}\\operatorname{Im}G^R(\\mathbf k,\\omega),\n$$\n\nasks whether adding or removing one electron creates a long-lived excitation.\n\nElectrical conductivity instead comes from a current-current correlator,\n\n$$\n\\sigma_{ij}(\\omega)\n=\\frac{1}{\\ii(\\omega+\\ii0^+)}\n\\left[\nK^R_{ij}(\\omega)-K^R_{ij}(0)\n\\right],\n$$\n\nwith the diamagnetic term included in the subtraction convention. It asks how long an imposed many-body current persists.\n\nThe momentum relaxation rate asks a third question:\n\n$$\n\\dot P_i=\\ii[H,P_i].\n$$\n\nIf this operator vanishes, internal collisions can redistribute momentum without removing it from the complete fluid.\n\n| Object | What it probes | What can broaden or relax it |\n| --- | --- | --- |\n| $A(\\mathbf k,\\omega)$ | one-electron addition or removal | every channel that destroys that excitation |\n| $\\sigma(\\omega)$ | current response | processes that degrade current |\n| $P$ | a candidate slow mode | only terms with $[H,P]\\neq0$ |\n\nConflating these rows is the source of the false inference that a linear spectral rate forces a linear resistivity.\n\n## 2. The quasiparticle pole test\n\nAt a Fermi momentum, a Landau quasiparticle requires more than a peak in a finite-resolution spectrum. In the normal state at zero temperature, the asymptotic conditions include\n\n$$\nZ_{\\mathbf k_F}\n=\\left[\n1-\\left.\n\\partial_\\omega\\operatorname{Re}\\Sigma^R(\\mathbf k_F,\\omega)\n\\right|_{\\omega=0}\n\\right]^{-1}>0,\n$$\n\nand\n\n$$\n\\frac{-\\operatorname{Im}\\Sigma^R(\\mathbf k_F,\\omega)}{|\\omega|}\n\\longrightarrow0.\n$$\n\nFor the patch theory of a two-dimensional Fermi surface coupled to a critical scalar,\n\n$$\n\\xi_{\\mathbf k}\n\\simeq v_F k_\\perp+\\frac{\\kappa}{2}k_\\parallel^2,\n$$\n\nthe large-$N$ saddle gives the scaling\n\n$$\n\\Sigma(\\ii\\omega_n)\n\\sim\n-\\ii B\\,\\operatorname{sgn}(\\omega_n)|\\omega_n|^{2/3}.\n$$\n\nThus\n\n$$\n\\frac{|\\Sigma(\\omega)|}{|\\omega|}\n\\sim |\\omega|^{-1/3}\\longrightarrow\\infty.\n$$\n\nThe bare frequency is subleading, the residue vanishes, and the quasiparticle pole is lost in this large-$N$ normal-state regime.\n\nA critical Fermi surface can still survive as a codimension-one singular locus of the zero-frequency Green function.\n\nWith disorder or a nonzero residual imaginary part, use a sign change or minimum of $\\operatorname{Re}G^{-1}(\\mathbf k,0)$. The literal equation $G^{-1}=0$ is not robust in every model.\n\nThis conclusion also has an infrared proviso. Pairing, density-wave order, confinement, or another instability may preempt the putative non-Fermi-liquid fixed point before the strict $\\omega\\to0$ limit is reached.\n\n## 3. The clean critical Fermi-surface model\n\nA useful large-flavor model is\n\n$$\n\\begin{aligned}\n\\mathcal L={}&\n\\sum_{\\alpha=1}^{N}\n\\psi_\\alpha^\\dagger\n\\left(\\partial_\\tau+\\varepsilon(-\\ii\\nabla)\\right)\n\\psi_\\alpha\\\\\n&+\\frac12\\sum_{\\gamma=1}^{M}\n\\left[(\\nabla\\phi_\\gamma)^2+s\\phi_\\gamma^2\\right]\n-\\frac1N g_{\\alpha\\beta\\gamma}\n\\phi_\\gamma\\psi_\\alpha^\\dagger\\psi_\\beta .\n\\end{aligned}\n$$\n\nTake $N,M\\to\\infty$ at fixed $M/N$. Flavor-random but spatially uniform Yukawa couplings lead schematically to\n\n$$\n\\Sigma\\sim g^2DG,\n\\qquad\n\\Pi\\sim -g^2GG,\n$$\n\ntogether with Dyson equations for the fermion and boson propagators.\n\nThis construction supplies a controlled self-consistent normal state. It does not remove translation symmetry because the flavor randomness is uniform in physical space.\n\nThe boson can absorb momentum from one electron and return it to another. The electronic distribution relaxes internally, but the total momentum of the electron-boson fluid need not decay.\n\n## 4. Why a dressed bubble is not enough\n\nIt is tempting to insert the dressed Green function into a conductivity bubble and identify\n\n$$\n\\tau_{\\rm tr}^{-1}\n\\stackrel{?}{=}\n-2\\operatorname{Im}\\Sigma^R.\n$$\n\nThat replacement discards the current vertex. A conserving calculation must treat self-energy and vertex corrections consistently.\n\nGauge invariance relates them through a Ward identity. In one common notation,\n\n$$\nq_\\mu\\Gamma^\\mu(k+q,k)\n=G^{-1}(k+q)-G^{-1}(k).\n$$\n\nThe identity prevents arbitrary combinations of dressed propagators and bare vertices. It encodes charge conservation inside the response calculation.\n\nFor the clean large-$N$ critical Fermi-surface model with a convex Fermi surface, Guo, Patel, Esterlis, and Sachdev found a Drude contribution and a zero coefficient for the proposed regular $1/\\omega^{2/3}$ optical term.\n\nThis is a result for that model and geometry, not a theorem for every clean non-Fermi liquid.\n\nNonconvex Fermi surfaces, extra slow modes, finite wavevector, and different operators can change the regular optical conductivity.\n\nThe general lesson is narrower and stronger: **the self-energy alone is not a conserving transport calculation.**\n\n## 5. Momentum protection in the memory matrix\n\nSuppose momentum is the dominant slow operator. A one-mode memory-matrix projection gives\n\n$$\n\\sigma(\\omega)\n\\simeq\n\\frac{\\chi_{JP}^2}\n{M_{PP}(\\omega)-\\ii\\omega\\chi_{PP}}\n+\\sigma_{\\rm inc}(\\omega).\n$$\n\nThe static susceptibilities are\n\n$$\n\\chi_{JP}=\\langle J;P\\rangle,\n\\qquad\n\\chi_{PP}=\\langle P;P\\rangle,\n$$\n\nand $M_{PP}$ measures the decay of the part of $P$ orthogonal to retained slow modes.\n\nIf\n\n$$\n[H,P]=0,\n\\qquad\n\\chi_{JP}\\neq0,\n$$\n\nthen $M_{PP}(0)=0$. The conductivity contains a Drude distribution,\n\n$$\n\\operatorname{Re}\\sigma(\\omega)\n\\supset \\pi D\\,\\delta(\\omega),\n$$\n\nand the ideal infinite-system d.c. resistivity vanishes.\n\nIn a Galilean continuum with a parabolic band,\n\n$$\n\\mathbf J=\\frac{e}{m}\\mathbf P,\n$$\n\nso the protected overlap is explicit. Electron-electron and electron-boson collisions can destroy quasiparticles without stopping the center of mass.\n\nThe regular term $\\sigma_{\\rm inc}$ need not vanish. It describes current orthogonal to the protected momentum component. Infinite d.c. conductivity and a nontrivial finite-frequency continuum can coexist.\n\n## 6. Translation symmetry is not sufficient by itself\n\nThe previous argument has two conditions, not one.\n\nFirst, a conserved extensive momentum or pseudomomentum must exist. A crystal has discrete translations. Umklapp can transfer momentum to the lattice, and phonons, disorder, boundaries, or commensurate processes can supply additional sinks.\n\nSecond, the conserved operator must overlap with electrical current. At charge neutrality or in a compensated system,\n\n$$\n\\chi_{JP}=0\n$$\n\ncan occur by symmetry. Translation invariance then does not protect charge transport through $P$, and a finite incoherent conductivity is possible.\n\nSeveral slow operators may also be comparable. The correct expression is then a matrix inverse,\n\n$$\n\\sigma(\\omega)\n=\\chi_{JA}\n\\left[M(\\omega)-\\ii\\omega\\chi\\right]^{-1}_{AB}\n\\chi_{BJ}\n+\\sigma_{\\rm fast}.\n$$\n\nReducing it to one momentum mode is a controlled approximation only when the other slow directions are unimportant.\n\n## 7. Forward scattering gives a useful heuristic\n\nFor elastic or quasielastic Born scattering through an angle $\\theta$,\n\n$$\n\\frac1{\\tau_{\\rm sp}}\n\\propto\\int d\\theta\\,W(\\theta),\n$$\n\nwhereas\n\n$$\n\\frac1{\\tau_{\\rm tr}}\n\\propto\\int d\\theta\\,W(\\theta)(1-\\cos\\theta).\n$$\n\nAt small angle,\n\n$$\n1-\\cos\\theta\\simeq\\frac{\\theta^2}{2}.\n$$\n\nForward scattering can therefore broaden a spectrum strongly while changing current weakly.\n\nThis angular formula is a kinetic illustration, not a derivation for a fully incoherent non-Fermi liquid. The Ward identity and slow-mode analysis are the safer statements when quasiparticles themselves do not exist.\n\n## 8. Potential disorder can separate marginal spectra from transport\n\nOrdinary potential disorder changes the critical boson. In the large-$N$ model analyzed in [the conductivity study](https://arxiv.org/abs/2207.08841), the low-energy propagator becomes diffusive in the relevant regime,\n\n$$\nD^{-1}(\\mathbf q,\\ii\\Omega_n)\n\\sim q^2+\\gamma|\\Omega_n|,\n\\qquad z=2.\n$$\n\nThe fermion self-energy is marginal in form,\n\n$$\n\\Sigma(\\ii\\omega_n)\n\\sim -\\ii\\omega_n\n\\log\\frac{\\Lambda}{|\\omega_n|}.\n$$\n\nYet the model's transport remains Fermi-liquid-like rather than becoming automatically $T$-linear. Schematically,\n\n$$\n\\rho(T)=\\rho_0+cT^2+\\cdots.\n$$\n\nThis is a concrete counterexample to the rule $\\rho\\propto-\\operatorname{Im}\\Sigma$. It should not be promoted to a claim about every disordered quantum critical metal.\n\n## 9. Harris disorder and random Yukawa coupling are related, not identical\n\nThe natural symmetry-preserving disorder near a transition is a random local tuning parameter,\n\n$$\ns\\longrightarrow s+\\delta s(\\mathbf r).\n$$\n\nFor static spatial disorder, the Harris criterion uses the spatial dimension $d$. Clean criticality is unstable when\n\n$$\nd\\nu<2.\n$$\n\nThe reason is a block estimate. Fluctuations of the local critical parameter in a correlation volume scale as\n\n$$\n\\delta s_\\xi\\sim\\xi^{-d/2},\n$$\n\nwhile the uniform detuning from criticality scales as\n\n$$\n|s-s_c|\\sim\\xi^{-1/\\nu}.\n$$\n\nDisorder dominates as $\\xi\\to\\infty$ when $d/2<1/\\nu$.\n\nA solvable strange-metal model instead uses a spatially random Yukawa coupling,\n\n$$\ng(\\mathbf r)=g+g'(\\mathbf r),\n$$\n\nwith\n\n$$\n\\overline{g'(\\mathbf r)}=0,\n\\qquad\n\\overline{g'(\\mathbf r)g'(\\mathbf r')}\n=g'^2\\delta^{(2)}(\\mathbf r-\\mathbf r').\n$$\n\nRandom mass is what “Harris disorder” literally names. Random Yukawa coupling is a tractable representation of disordered critical interactions.\n\nField redefinitions or coarse graining can relate the two in specified theories. The relation is not a universal microscopic identity.\n\n## 10. The random interaction is a momentum sink\n\nWrite the disordered interaction as\n\n$$\nH_{\\rm dis}\n=\\int d^2r\\,g'(\\mathbf r)O(\\mathbf r),\n\\qquad\nO=\\phi\\psi^\\dagger\\psi,\n$$\n\nwith flavor indices suppressed.\n\nUsing\n\n$$\n[P_i,O(\\mathbf r)]=-\\ii\\partial_iO(\\mathbf r),\n$$\n\nand integrating by parts gives, up to the convention for the translation generator,\n\n$$\n\\dot P_i\n=-\\int d^2r\\,\n(\\partial_i g'(\\mathbf r))O(\\mathbf r).\n$$\n\nThis equation supplies the missing operator statement. A spatially uniform $g$ changes the strength of internal scattering. A random $g'(\\mathbf r)$ also lets the frozen background absorb momentum.\n\nDisorder averaging restores translation invariance statistically. It does not restore momentum conservation in an individual sample. Confusing these statements would erase the physical relaxation mechanism.\n\nTo leading order in weak disorder, the memory element has the structure\n\n$$\nM_{P_iP_i}\n\\sim g'^2\\int\\frac{d^2q}{(2\\pi)^2}\\,\nq_i^2\n\\lim_{\\omega\\to0}\n\\frac{\\operatorname{Im}G^R_{OO}(\\mathbf q,\\omega)}{\\omega}.\n$$\n\nThe temperature exponent is set by the critical spectral weight of the operator that appears in $\\dot P$, not by the electron self-energy in isolation.\n\n## 11. The 2d-YSYK linear-resistivity regime\n\nThe spatially random interaction theory of Patel, Guo, Esterlis, and Sachdev finds, within its large-$N$ ensemble,\n\n$$\n\\rho(T)-\\rho_0\\propto g'^2T,\n$$\n\ntogether with\n\n$$\nC(T)\\sim T\\log(1/T)\n$$\n\nand an optical relaxation rate linear in frequency.\n\nThe linked signatures follow because the random interaction does two jobs. It couples to abundant critical spectral weight and appears directly in the momentum-relaxing force operator.\n\nThis theory does not establish one universal coefficient $A$ for all strange metals.\n\nThe slope depends on susceptibilities, velocities, disorder variance, the critical spectrum, and the definition of the extracted transport rate.\n\nA “Planckian” spectral width and a Planckian transport rate are also distinct claims. Inferring either from a fitted slope requires the relevant mass or Drude weight, not only a temperature exponent.\n\n## 12. Three clean transitions, one disordered description\n\n[Sachdev's 2024 lectures](https://arxiv.org/abs/2407.15919) compare three classes of metallic transition:\n\n1. a Fermi-surface deformation near a small-momentum order parameter;\n2. Fermi-surface reconstruction near spin-density-wave order;\n3. an FL$^*$-to-FL transition that changes Fermi volume and can involve emergent gauge structure.\n\nTheir clean theories are not microscopically equivalent. The lecture's proposed unification occurs after elastic scattering blurs sharp momentum selection beyond the mean free path and disorder produces a common diffusive critical sector.\n\nThe phrase “all clean transitions are perfect metals” belongs to the modeled clean quantum-critical limits.\n\nIt is not a theorem excluding Umklapp, compensated transport, phonons, or other current-relaxing operators in every material.\n\n## 13. The low-energy continuation is spatially inhomogeneous\n\nThe self-averaging 2d-YSYK saddle is best viewed as a controlled regime, not a guaranteed final fixed point.\n\nAn explicit treatment of random boson mass found localized overdamped bosonic eigenmodes at low energy.\n\nThe resulting regime retains nearly $T$-linear resistivity and develops quantum-Griffiths character rather than simply reverting to a clean metal.\n\nLarge-scale hybrid Monte Carlo has since found a gapless strange-metal phase with localized antiferromagnetic fluctuations and nearly Planckian linear resistivity in a model without a large-$N$ or replica construction.\n\nThese results strengthen the case that spatial randomness can organize strange transport. They remain model and numerical results, not a proof that disorder is required in every material.\n\nA [2026 preprint](https://arxiv.org/abs/2606.23582) extends the same localized-boson setting to pairing.\n\nLocalized critical modes create a random pairing vertex, superconducting puddles at higher temperature, and an extended pairing eigenstate at lower temperature.\n\nThe emerging picture is a crossover:\n\n$$\n\\text{clean critical metal}\n\\longrightarrow\n\\text{self-averaging random-interaction regime}\n\\longrightarrow\n\\text{localized collective modes and rare regions}.\n$$\n\nThe locations and even the existence of these arrows depend on the microscopic model.\n\n## 14. How to read experiments without identifying rates\n\nARPES measures the occupied part of $A(\\mathbf k,\\omega)$, multiplied by matrix elements and convolved with resolution. A broad line can diagnose short single-particle coherence, subject to the extraction model.\n\nOptical conductivity probes a current-current correlator. Separating a narrow Drude component from a regular continuum is essential. The optical memory function contains vertex and collective effects absent from an ARPES linewidth.\n\nD.c. transport adds the strict low-frequency limit, sample geometry, impurities, phonons, Umklapp, and contacts.\n\nIts exponent cannot be read from ARPES without a demonstrated bridge between one-particle decay and the force that relaxes the relevant slow mode.\n\nA useful inference chain is\n\n$$\n\\Sigma^R\n\\longrightarrow\nA(\\mathbf k,\\omega),\n$$\n\n$$\n\\{H,J,P,\\ldots\\}\n\\longrightarrow\n\\text{slow-mode content},\n$$\n\n$$\n\\dot P\n\\longrightarrow\nM_{PP}\n\\longrightarrow\n\\rho.\n$$\n\nThe first arrow does not skip to the last line.\n\n## The deliberately false inference\n\n**If $-\\operatorname{Im}\\Sigma^R(0,T)\\propto T$, then $\\rho(T)\\propto T$ with the same scattering rate.**\n\nThe claim assumes what it needs to prove. It identifies the one-electron lifetime with a transport lifetime, drops vertex corrections, and ignores conserved operators.\n\nIt becomes approximately useful only after the scattering geometry, current vertex, momentum sink, and slow-mode overlaps have been controlled.\n\n## Collision: five levels\n\n### Spectral level\n\n$|\\Sigma|\\gg|\\omega|$ destroys the Landau pole. This statement concerns the one-electron Green function.\n\n### Conserving-response level\n\nThe current vertex must satisfy the same conservation laws as the self-energy. A dressed bubble with an inconsistent bare vertex can give a false transport exponent.\n\n### Slow-mode level\n\nIf a conserved $P$ has $\\chi_{JP}\\neq0$, conductivity contains a ballistic component even when the spectrum is incoherent.\n\n### Momentum-sink level\n\nUmklapp, disorder, phonons, boundaries, and random interactions enter through $\\dot P$. Their operator content fixes what can relax the protected current.\n\n### Disordered-critical level\n\nRandom critical interactions can make the force-force correlator scale linearly with $T$. Localized bosonic modes can replace the self-averaging description deeper in the infrared.\n\n## Do this now\n\nConsider a circular two-dimensional Fermi surface. Let a critical mode produce the forward-peaked probability\n\n$$\nW(\\theta)=\\frac{W_0}{\\theta^2+\\theta_0^2},\n\\qquad\n\\theta_0\\ll\\theta_c\\ll1.\n$$\n\n### Target A: separate two kinetic rates\n\nEvaluate, up to numerical constants,\n\n$$\n\\frac1{\\tau_{\\rm sp}}\n\\sim\n\\int_{-\\theta_c}^{\\theta_c}d\\theta\\,W(\\theta),\n$$\n\nand\n\n$$\n\\frac1{\\tau_{\\rm tr}}\n\\sim\n\\int_{-\\theta_c}^{\\theta_c}d\\theta\\,\nW(\\theta)(1-\\cos\\theta).\n$$\n\nFind the leading dependence on $\\theta_0$ and interpret the ratio.\n\n### Target B: derive the force operator\n\nTake\n\n$$\nH_{\\rm dis}=\\int d^2r\\,g'(\\mathbf r)O(\\mathbf r),\n$$\n\nwith\n\n$$\n\\overline{g'(\\mathbf r)g'(\\mathbf 0)}\n=g'^2\\delta^{(2)}(\\mathbf r).\n$$\n\nDerive $\\dot P_i$ and identify where momentum goes.\n\n### Target C: extract the thermal exponent\n\nSuppose the disorder-averaged critical kernel is\n\n$$\n\\mathcal K(q,T)\n\\equiv\n\\lim_{\\omega\\to0}\n\\frac{\\operatorname{Im}G^R_{OO}(q,\\omega)}{\\omega}\n=T^{-1}\\Phi(q/\\sqrt T),\n$$\n\nwhere $\\Phi$ is finite and rapidly decaying. Determine the scaling of\n\n$$\nM_{P_iP_i}\n\\sim\ng'^2\\int d^2q\\,q_i^2\\mathcal K(q,T).\n$$\n\nWhen does this imply the same scaling for resistivity?\n\n### Target D: audit an experimental claim\n\nAn ARPES fit reports\n\n$$\n-\\operatorname{Im}\\Sigma^R(0,T)=\\alpha k_BT.\n$$\n\nList the additional operator and response information needed before concluding that the d.c. transport rate is $\\alpha k_BT/\\hbar$.\n\n### Oral check 1\n\nState the two conditions under which momentum produces a protected Drude contribution.\n\n### Oral check 2\n\nWhy does statistical translation invariance after disorder averaging fail to restore momentum conservation in a sample?\n\n\nHints
\n\n- Use $1-\\cos\\theta\\simeq\\theta^2/2$.\n- Integrate $[P_i,O]=-\\ii\\partial_iO$ by parts.\n- In the memory integral set $q=\\sqrt T\\,x$.\n- In two dimensions, isotropy gives $q_i^2\\to q^2/2$ after angular averaging.\n\n \n\n\nSolution outline
\n\nThe spectral rate is\n\n$$\n\\frac1{\\tau_{\\rm sp}}\n\\sim\nW_0\\int\\frac{d\\theta}{\\theta^2+\\theta_0^2}\n\\sim\\frac{W_0}{\\theta_0}.\n$$\n\nFor transport,\n\n$$\n\\frac1{\\tau_{\\rm tr}}\n\\sim\n\\frac{W_0}{2}\n\\int_{-\\theta_c}^{\\theta_c}d\\theta\\,\n\\frac{\\theta^2}{\\theta^2+\\theta_0^2}\n\\sim W_0\\left[\\theta_c-O(\\theta_0)\\right].\n$$\n\nHence\n\n$$\n\\frac{\\tau_{\\rm tr}^{-1}}{\\tau_{\\rm sp}^{-1}}\n\\sim\\theta_0\\theta_c\\longrightarrow0.\n$$\n\nSharper forward scattering destroys the one-particle state increasingly fast without a comparable increase in current relaxation.\n\nFor the random interaction,\n\n$$\n\\dot P_i\n=-\\int d^2r\\,\n(\\partial_i g'(\\mathbf r))O(\\mathbf r),\n$$\n\nup to the sign convention for $P_i$. Momentum is transferred to the frozen spatial background.\n\nFor the kernel, set $q=\\sqrt T\\,x$. Then\n\n$$\nd^2q=T\\,d^2x,\n\\qquad\nq_i^2=T x_i^2,\n\\qquad\n\\mathcal K=T^{-1}\\Phi(x).\n$$\n\nTherefore\n\n$$\nM_{P_iP_i}\n\\sim g'^2T\n\\int d^2x\\,x_i^2\\Phi(x)\n\\propto g'^2T.\n$$\n\nIn the one-slow-mode, weak-relaxation limit,\n\n$$\n\\rho\\simeq\\frac{M_{PP}}{\\chi_{JP}^2}.\n$$\n\nLinear $M_{PP}$ then gives linear resistivity if $\\chi_{JP}$ is nonzero and nonsingular and no parallel incoherent channel changes the d.c. inversion.\n\nThe ARPES claim still needs the current vertex, the momentum-relaxing operator, slow-mode overlaps, the Drude weight or optical mass, scattering anisotropy, and vertex corrections.\n\nWithout these, equality of the coefficients is unsupported.\n\n \n\n## Exit ticket\n\nA metal has\n\n$$\n-\\operatorname{Im}\\Sigma^R(0,T)\\propto T,\n\\qquad\n\\rho(T)=\\rho_0+AT.\n$$\n\nGive two inequivalent microscopic explanations for the shared exponent. For each, write the operator that relaxes current and name one experiment that could distinguish it from the other explanation.\n\nA strong answer should address:\n\n1. random critical interactions versus phonons or Umklapp;\n2. the overlap $\\chi_{JP}$;\n3. Drude weight and optical memory function;\n4. momentum-space anisotropy in ARPES;\n5. disorder dependence of $A$;\n6. the possibility of localized collective modes.\n\n## Research checks worth doing next\n\n1. **Ward audit.** Verify that the self-energy and vertex approximation satisfy the relevant Ward identity before extracting a transport exponent.\n2. **Slow-mode census.** Compute all susceptibilities $\\chi_{JA}$ for plausible conserved or nearly conserved operators.\n3. **Force spectroscopy.** Identify $\\dot P$ microscopically and measure the spectral weight of the operator that appears in it.\n4. **Disorder scaling.** Test whether the linear slope tracks controlled changes in interaction randomness or only the residual resistivity.\n5. **Optical separation.** Follow Drude weight and regular spectral weight independently as temperature changes.\n6. **Harris crossover.** Estimate the scale at which random mass overtakes the clean critical theory.\n7. **Localization test.** Search for spatially localized overdamped collective modes using local magnetic or spectroscopic probes.\n8. **Pairing correlation.** Test whether regions with enhanced low-energy bosonic weight also host enhanced local pairing.\n\n## Further reading\n\n- [Large $N$ theory of critical Fermi surfaces II: conductivity](https://arxiv.org/abs/2207.08841): clean Drude transport, cancellation of the proposed anomalous optical term for a convex Fermi surface, and the potential-disorder counterexample.\n- [Universal theory of strange metals from spatially random interactions](https://arxiv.org/abs/2203.04990): the large-$N$ 2d random-Yukawa model with linear resistivity and $T\\log(1/T)$ specific heat.\n- [Lectures on the quantum phase transitions of metals](https://arxiv.org/abs/2407.15919): clean metallic criticality, Harris disorder, and the proposed 2d-YSYK unification.\n- [Memory matrix theory of magnetotransport in strange metals](https://arxiv.org/abs/1502.04704): transport without quasiparticles organized by slow momentum and diffusive modes.\n- [Localization of overdamped bosonic modes and transport in strange metals](https://arxiv.org/abs/2312.06751): localized collective modes, quantum-Griffiths behavior, and nearly linear resistivity.\n- [Strange metals and Planckian transport in a gapless phase from spatially random interactions](https://arxiv.org/abs/2410.05365): hybrid Monte Carlo evidence beyond the large-$N$ construction.\n- [Linear Resistivity from Spatially Random Interactions and the Uniqueness of Yukawa Coupling](https://arxiv.org/abs/2507.09442): a 2025–2026 preprint classification within a specified family of random scalar couplings.\n- [Influence of Harris disorder on quantum-critical superconductivity](https://arxiv.org/abs/2606.23582): a 2026 preprint on localized critical modes, pairing puddles, and an extended pairing instability.\n\n## What to retain\n\n- The self-energy controls a one-electron propagator; conductivity is a conserving two-particle response.\n- A missing quasiparticle pole does not imply a missing ballistic charge component.\n- Momentum protects conductivity only when it is conserved and overlaps with current.\n- Vertex corrections carry conservation-law information that a dressed bubble can miss.\n- Random spatial interactions differ qualitatively from stronger uniform interactions because they enter $\\dot P$.\n- Harris disorder literally denotes random local detuning; random Yukawa coupling is a related effective model.\n- Linear resistivity in 2d-YSYK follows from the force-force correlator in a stated regime, not from a slogan about a linear self-energy.\n- Localized overdamped bosons provide a lower-energy continuation with rare-region physics and possible inhomogeneous pairing.\n- ARPES and transport rates may share an exponent without sharing a microscopic lifetime.\n\nNext: derive the full two-mode memory matrix for momentum and an imbalance current, then determine when an incoherent channel shorts the momentum-drag contribution.\n",
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"tex": "-\\operatorname{Im}\\Sigma^R(\\omega)\n\\propto |\\omega|^{2/3},",
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"tex": "G^R(\\mathbf k,\\omega)\n=\\frac{1}\n{\\omega-\\xi_{\\mathbf k}-\\Sigma^R(\\mathbf k,\\omega)}.",
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"tex": "A(\\mathbf k,\\omega)\n=-\\frac{1}{\\pi}\\operatorname{Im}G^R(\\mathbf k,\\omega),",
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"sha256": "1f41addd90921420a3c7fe5871a71cc9713e68b55008afe1fabe78be2e6fc5cd"
},
{
"index": 5,
"display": true,
"tex": "\\sigma_{ij}(\\omega)\n=\\frac{1}{\\ii(\\omega+\\ii0^+)}\n\\left[\nK^R_{ij}(\\omega)-K^R_{ij}(0)\n\\right],",
"line": 63,
"sha256": "b0b27f87812c5295aa9c8096c4318389ab284a1a6ec56ea524335c4170979c2c"
},
{
"index": 6,
"display": true,
"tex": "\\dot P_i=\\ii[H,P_i].",
"line": 75,
"sha256": "4015f4c721e2c7da45afe0f10c27e917bbf649351feaf56b4ce0cd9d36e15822"
},
{
"index": 7,
"display": false,
"tex": "A(\\mathbf k,\\omega)",
"line": 83,
"sha256": "f9f05f7fa8838eb9b8e4a999b0099fc891321ffc3462a8fca40cbd8ae0b4f4ce"
},
{
"index": 8,
"display": false,
"tex": "\\sigma(\\omega)",
"line": 84,
"sha256": "43c7bc907c28f5db9352e23cb225ad24032c2295356c3dea44e1a8ccc58172e7"
},
{
"index": 9,
"display": false,
"tex": "P",
"line": 85,
"sha256": "5c62e091b8c0565f1bafad0dad5934276143ae2ccef7a5381e8ada5b1a8d26d2"
},
{
"index": 10,
"display": false,
"tex": "[H,P]\\neq0",
"line": 85,
"sha256": "dd1f372368d8eedbcd8af76f6075a19068779b6f5d2d06c8528a31daee26747b"
},
{
"index": 11,
"display": true,
"tex": "Z_{\\mathbf k_F}\n=\\left[\n1-\\left.\n\\partial_\\omega\\operatorname{Re}\\Sigma^R(\\mathbf k_F,\\omega)\n\\right|_{\\omega=0}\n\\right]^{-1}>0,",
"line": 93,
"sha256": "a14ed5ed4d0f426185943d37c2d5758aafc831c1ea6eeead0295d82c1f85f1d9"
},
{
"index": 12,
"display": true,
"tex": "\\frac{-\\operatorname{Im}\\Sigma^R(\\mathbf k_F,\\omega)}{|\\omega|}\n\\longrightarrow0.",
"line": 104,
"sha256": "4872c5658578063d1c52b50da886886e0833bd85a2be4159ad5b1d06b8b56bb2"
},
{
"index": 13,
"display": true,
"tex": "\\xi_{\\mathbf k}\n\\simeq v_F k_\\perp+\\frac{\\kappa}{2}k_\\parallel^2,",
"line": 111,
"sha256": "fb54e837e04af59a473a0223ee505ed53ee088484d4a8e51ec8adf7a0657e478"
},
{
"index": 14,
"display": false,
"tex": "N",
"line": 116,
"sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
},
{
"index": 15,
"display": true,
"tex": "\\Sigma(\\ii\\omega_n)\n\\sim\n-\\ii B\\,\\operatorname{sgn}(\\omega_n)|\\omega_n|^{2/3}.",
"line": 118,
"sha256": "93ae9a699550678fd96d03b9d1682aa9b82ee5247f485c84280ff4bf5ae6975b"
},
{
"index": 16,
"display": true,
"tex": "\\frac{|\\Sigma(\\omega)|}{|\\omega|}\n\\sim |\\omega|^{-1/3}\\longrightarrow\\infty.",
"line": 126,
"sha256": "03d26d9f0a05154f55fc94bbf87ef0684f82d2dae177e987f4bf75dc039ae1c3"
},
{
"index": 17,
"display": false,
"tex": "N",
"line": 131,
"sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
},
{
"index": 18,
"display": false,
"tex": "\\operatorname{Re}G^{-1}(\\mathbf k,0)",
"line": 135,
"sha256": "de3ecc55883ba5a831e652fd7b4bee0000e591e842c306d5ae165aaa69575fec"
},
{
"index": 19,
"display": false,
"tex": "G^{-1}=0",
"line": 135,
"sha256": "571ab98115a131720a303bb961cd809108f5a7207ab9a77f266c376487b9ef41"
},
{
"index": 20,
"display": false,
"tex": "\\omega\\to0",
"line": 137,
"sha256": "001ecb3ce7d7a0e0f2ba09bb141986f708288eb9a9bae7a768cd8e1bab1eddf8"
},
{
"index": 21,
"display": true,
"tex": "\\begin{aligned}\n\\mathcal L={}&\n\\sum_{\\alpha=1}^{N}\n\\psi_\\alpha^\\dagger\n\\left(\\partial_\\tau+\\varepsilon(-\\ii\\nabla)\\right)\n\\psi_\\alpha\\\\\n&+\\frac12\\sum_{\\gamma=1}^{M}\n\\left[(\\nabla\\phi_\\gamma)^2+s\\phi_\\gamma^2\\right]\n-\\frac1N g_{\\alpha\\beta\\gamma}\n\\phi_\\gamma\\psi_\\alpha^\\dagger\\psi_\\beta .\n\\end{aligned}",
"line": 143,
"sha256": "b46bf90139b052891e6d12dd73aa53972fe01c741114e7928307c96aa92b77d3"
},
{
"index": 22,
"display": false,
"tex": "N,M\\to\\infty",
"line": 157,
"sha256": "57db6be7b9bab069b7cef62c1de72eb88106fc44747bf9fb98329313e9b55d54"
},
{
"index": 23,
"display": false,
"tex": "M/N",
"line": 157,
"sha256": "923965f882d7e8195f843dfa2acb4274516e5562ba3e70bfa181a67fbcab5642"
},
{
"index": 24,
"display": true,
"tex": "\\Sigma\\sim g^2DG,\n\\qquad\n\\Pi\\sim -g^2GG,",
"line": 159,
"sha256": "3f2de1e6144163eac2beccd51ce204def748acc1a893423818e6d47c32491465"
},
{
"index": 25,
"display": true,
"tex": "\\tau_{\\rm tr}^{-1}\n\\stackrel{?}{=}\n-2\\operatorname{Im}\\Sigma^R.",
"line": 175,
"sha256": "9da666bc338349ee70b6de39da3657ff4d6110c82d02ef29b1c7b354da058dfe"
},
{
"index": 26,
"display": true,
"tex": "q_\\mu\\Gamma^\\mu(k+q,k)\n=G^{-1}(k+q)-G^{-1}(k).",
"line": 185,
"sha256": "7e0b9d0bd0d23c89ba3d68ef68d5316656202609fccc4a34d65165cf30df518e"
},
{
"index": 27,
"display": false,
"tex": "N",
"line": 192,
"sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
},
{
"index": 28,
"display": false,
"tex": "1/\\omega^{2/3}",
"line": 192,
"sha256": "96dc1eb398d0c415cda9342821d7f81a6737368824a4f19bf0145b661ef766a1"
},
{
"index": 29,
"display": true,
"tex": "\\sigma(\\omega)\n\\simeq\n\\frac{\\chi_{JP}^2}\n{M_{PP}(\\omega)-\\ii\\omega\\chi_{PP}}\n+\\sigma_{\\rm inc}(\\omega).",
"line": 204,
"sha256": "d5affbe618442c692c38c52b844ac165f80c88690a82961eb4d9e116dee823ac"
},
{
"index": 30,
"display": true,
"tex": "\\chi_{JP}=\\langle J;P\\rangle,\n\\qquad\n\\chi_{PP}=\\langle P;P\\rangle,",
"line": 214,
"sha256": "cc8af58a50d7cbc0d33e508d2f995a0e7799cc8e95cdeeb5ed3cc37c9f0f3792"
},
{
"index": 31,
"display": false,
"tex": "M_{PP}",
"line": 220,
"sha256": "8f579ca6bb2fe0db3dd555011fab889cf9b35494cab8e36b7797e7159504fcd9"
},
{
"index": 32,
"display": false,
"tex": "P",
"line": 220,
"sha256": "5c62e091b8c0565f1bafad0dad5934276143ae2ccef7a5381e8ada5b1a8d26d2"
},
{
"index": 33,
"display": true,
"tex": "[H,P]=0,\n\\qquad\n\\chi_{JP}\\neq0,",
"line": 224,
"sha256": "9c6710f37b4c15a0f39d640fe3b05fc5f888416e8af9e45ce4d1271e1ea125c1"
},
{
"index": 34,
"display": false,
"tex": "M_{PP}(0)=0",
"line": 230,
"sha256": "1347aa48bc21aaa704083df288b660af794d1b60f81a658d5347dbdec00d51fb"
},
{
"index": 35,
"display": true,
"tex": "\\operatorname{Re}\\sigma(\\omega)\n\\supset \\pi D\\,\\delta(\\omega),",
"line": 232,
"sha256": "a47d8875d867bafde9b04f688d8b1522bf4daa3ab722dae5d2030019824d4dc9"
},
{
"index": 36,
"display": true,
"tex": "\\mathbf J=\\frac{e}{m}\\mathbf P,",
"line": 241,
"sha256": "30cbdaad58f508c326d4ee0220ccf7f5e7fe548922b24cc743afae7481775d17"
},
{
"index": 37,
"display": false,
"tex": "\\sigma_{\\rm inc}",
"line": 247,
"sha256": "909ecf5f49144ce7d23976a085d91ee87fc0dad41b12b38bc40f2a654e49b7d3"
},
{
"index": 38,
"display": true,
"tex": "\\chi_{JP}=0",
"line": 257,
"sha256": "e4f9126de914ca8963ffc4632fbc337c569e4aa9d3f068f1fb24fa60357e10a6"
},
{
"index": 39,
"display": false,
"tex": "P",
"line": 261,
"sha256": "5c62e091b8c0565f1bafad0dad5934276143ae2ccef7a5381e8ada5b1a8d26d2"
},
{
"index": 40,
"display": true,
"tex": "\\sigma(\\omega)\n=\\chi_{JA}\n\\left[M(\\omega)-\\ii\\omega\\chi\\right]^{-1}_{AB}\n\\chi_{BJ}\n+\\sigma_{\\rm fast}.",
"line": 265,
"sha256": "ff72cb4b75e2cdbeb1e466d6ddb9432ea3c8a7a485c49523774599d9ed374168"
},
{
"index": 41,
"display": false,
"tex": "\\theta",
"line": 277,
"sha256": "2b4af44cb17349bb9452292a447fe70d48c9ce08e68dd2f74e9628cf8ac625b8"
},
{
"index": 42,
"display": true,
"tex": "\\frac1{\\tau_{\\rm sp}}\n\\propto\\int d\\theta\\,W(\\theta),",
"line": 279,
"sha256": "9b188f6c0856e803abfc76dd23fc2f8de8c6c61e40d3b20b742a40d3a14989f4"
},
{
"index": 43,
"display": true,
"tex": "\\frac1{\\tau_{\\rm tr}}\n\\propto\\int d\\theta\\,W(\\theta)(1-\\cos\\theta).",
"line": 286,
"sha256": "6dbddfb4ea9291a23ffb4efbfc95eb3792d8ac5b7ab2daf872228a26c8f8e940"
},
{
"index": 44,
"display": true,
"tex": "1-\\cos\\theta\\simeq\\frac{\\theta^2}{2}.",
"line": 293,
"sha256": "c105e97c2d5581f41b749e6f81f82712077840d2a477a2d5f21e501cc5ca9c33"
},
{
"index": 45,
"display": false,
"tex": "N",
"line": 303,
"sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
},
{
"index": 46,
"display": true,
"tex": "D^{-1}(\\mathbf q,\\ii\\Omega_n)\n\\sim q^2+\\gamma|\\Omega_n|,\n\\qquad z=2.",
"line": 305,
"sha256": "32ef01368c9f46f923f3c65e4f613b91d751b8e9348c77c8226926023b76eb9d"
},
{
"index": 47,
"display": true,
"tex": "\\Sigma(\\ii\\omega_n)\n\\sim -\\ii\\omega_n\n\\log\\frac{\\Lambda}{|\\omega_n|}.",
"line": 313,
"sha256": "0e0701c7284de8d4f4e872be8386f3ca76f4fe81d585b82351a6f1ac55188f41"
},
{
"index": 48,
"display": false,
"tex": "T",
"line": 319,
"sha256": "e632b7095b0bf32c260fa4c539e9fd7b852d0de454e9be26f24d0d6f91d069d3"
},
{
"index": 49,
"display": true,
"tex": "\\rho(T)=\\rho_0+cT^2+\\cdots.",
"line": 321,
"sha256": "af70671395983e61ef23a5a694e00c389f317555bdf7977b7837e53a3ad04858"
},
{
"index": 50,
"display": false,
"tex": "\\rho\\propto-\\operatorname{Im}\\Sigma",
"line": 325,
"sha256": "c36561246ba5b839fc72e869e90017e050e4f67db69b96e2f0d14bd16918172e"
},
{
"index": 51,
"display": true,
"tex": "s\\longrightarrow s+\\delta s(\\mathbf r).",
"line": 331,
"sha256": "f6279bfc9028145ef15203ad591db8e7c9949128f19fcd6955809b4b4e42de67"
},
{
"index": 52,
"display": false,
"tex": "d",
"line": 335,
"sha256": "18ac3e7343f016890c510e93f935261169d9e3f565436429830faf0934f4f8e4"
},
{
"index": 53,
"display": true,
"tex": "d\\nu<2.",
"line": 337,
"sha256": "854f65483a50016994aec033b505a9a56095679d322c8a0f9581b60fec854e8d"
},
{
"index": 54,
"display": true,
"tex": "\\delta s_\\xi\\sim\\xi^{-d/2},",
"line": 343,
"sha256": "b587aef6229ee27f75a924229b1644214301ad16a448c990f16b6a8bf26b9937"
},
{
"index": 55,
"display": true,
"tex": "|s-s_c|\\sim\\xi^{-1/\\nu}.",
"line": 349,
"sha256": "c59f5acfd394b82aa8e79557ae9e233fbf03e0e8779f543a15031cf3ad8d8059"
},
{
"index": 56,
"display": false,
"tex": "\\xi\\to\\infty",
"line": 353,
"sha256": "d79f7c2105581312f09264f227bb67d6363061840d040d48826eb2aef17e0cd2"
},
{
"index": 57,
"display": false,
"tex": "d/2<1/\\nu",
"line": 353,
"sha256": "e7e98224e45b50b847ed5f59ea49f1cd43949e1793b37d195c4aa6cca5e3a6c4"
},
{
"index": 58,
"display": true,
"tex": "g(\\mathbf r)=g+g'(\\mathbf r),",
"line": 357,
"sha256": "2cb9c03289d25c8accb89a6ff0c388b8c8ba7d55e32e0bc428b3b3a5ef59ce30"
},
{
"index": 59,
"display": true,
"tex": "\\overline{g'(\\mathbf r)}=0,\n\\qquad\n\\overline{g'(\\mathbf r)g'(\\mathbf r')}\n=g'^2\\delta^{(2)}(\\mathbf r-\\mathbf r').",
"line": 363,
"sha256": "64ea2229e7e64180a01d1b9cc9d5c5bd8f76361d3c79f52848a20f72d9e2582d"
},
{
"index": 60,
"display": true,
"tex": "H_{\\rm dis}\n=\\int d^2r\\,g'(\\mathbf r)O(\\mathbf r),\n\\qquad\nO=\\phi\\psi^\\dagger\\psi,",
"line": 378,
"sha256": "03d83ef55c35d80a95694a65cf3744ed307b8c12d50cd246e191fbbbc184215c"
},
{
"index": 61,
"display": true,
"tex": "[P_i,O(\\mathbf r)]=-\\ii\\partial_iO(\\mathbf r),",
"line": 389,
"sha256": "36f0638c9203c030e7b481d031ed375b3bad7ddef7d9966c6f8e911a7d7bd395"
},
{
"index": 62,
"display": true,
"tex": "\\dot P_i\n=-\\int d^2r\\,\n(\\partial_i g'(\\mathbf r))O(\\mathbf r).",
"line": 395,
"sha256": "600493d7b7966188eba2f11aeb9d07fd2ddeea9300e5447da9492ffebf0d7dcc"
},
{
"index": 63,
"display": false,
"tex": "g",
"line": 401,
"sha256": "cd0aa9856147b6c5b4ff2b7dfee5da20aa38253099ef1b4a64aced233c9afe29"
},
{
"index": 64,
"display": false,
"tex": "g'(\\mathbf r)",
"line": 401,
"sha256": "af4e97de21627a3c63bceb42179ede00738b7c83cb39099a3b7a3ce605d63238"
},
{
"index": 65,
"display": true,
"tex": "M_{P_iP_i}\n\\sim g'^2\\int\\frac{d^2q}{(2\\pi)^2}\\,\nq_i^2\n\\lim_{\\omega\\to0}\n\\frac{\\operatorname{Im}G^R_{OO}(\\mathbf q,\\omega)}{\\omega}.",
"line": 407,
"sha256": "906b3ed976afbe546fb68ae76c40e39244922f2c16348a17c921dc6ac455e305"
},
{
"index": 66,
"display": false,
"tex": "\\dot P",
"line": 415,
"sha256": "35679b068dabbae248a3381dc8c531de29bb451999ada92c0c883023bec06e99"
},
{
"index": 67,
"display": false,
"tex": "N",
"line": 419,
"sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
},
{
"index": 68,
"display": true,
"tex": "\\rho(T)-\\rho_0\\propto g'^2T,",
"line": 421,
"sha256": "ef4c81e9123a7cf56fd9159658e7463a096026b919a73904fc4d7487f0ba6dfe"
},
{
"index": 69,
"display": true,
"tex": "C(T)\\sim T\\log(1/T)",
"line": 427,
"sha256": "0aec2986f6c62377e192eb82352c56d620bbe2867d7d1d893ccf89f37231bb82"
},
{
"index": 70,
"display": false,
"tex": "A",
"line": 435,
"sha256": "559aead08264d5795d3909718cdd05abd49572e84fe55590eef31a88a08fdffd"
},
{
"index": 71,
"display": false,
"tex": "^*",
"line": 447,
"sha256": "2ce8c6c9cddec9ab28a7a901577c78dbc2760e9121444281776175c7964676cf"
},
{
"index": 72,
"display": false,
"tex": "T",
"line": 461,
"sha256": "e632b7095b0bf32c260fa4c539e9fd7b852d0de454e9be26f24d0d6f91d069d3"
},
{
"index": 73,
"display": false,
"tex": "N",
"line": 463,
"sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
},
{
"index": 74,
"display": true,
"tex": "\\text{clean critical metal}\n\\longrightarrow\n\\text{self-averaging random-interaction regime}\n\\longrightarrow\n\\text{localized collective modes and rare regions}.",
"line": 473,
"sha256": "a29574c626d6b99f554796d7449bc67153920c187460d9edd3bc922873860bbb"
},
{
"index": 75,
"display": false,
"tex": "A(\\mathbf k,\\omega)",
"line": 485,
"sha256": "f9f05f7fa8838eb9b8e4a999b0099fc891321ffc3462a8fca40cbd8ae0b4f4ce"
},
{
"index": 76,
"display": true,
"tex": "\\Sigma^R\n\\longrightarrow\nA(\\mathbf k,\\omega),",
"line": 495,
"sha256": "35e20ba6e8a4b8cdeb42c8a3ab11ee65bca604f009b0d2178ba382adf1745254"
},
{
"index": 77,
"display": true,
"tex": "\\{H,J,P,\\ldots\\}\n\\longrightarrow\n\\text{slow-mode content},",
"line": 501,
"sha256": "07106981a2f6aa7520369af67036d44990ea3a410ef3f1668aaf9054e0c43c27"
},
{
"index": 78,
"display": true,
"tex": "\\dot P\n\\longrightarrow\nM_{PP}\n\\longrightarrow\n\\rho.",
"line": 507,
"sha256": "4ed8e16a00e36d4002fa55aa10e668cccbc66c783fb8a762331d1742bc255659"
},
{
"index": 79,
"display": false,
"tex": "-\\operatorname{Im}\\Sigma^R(0,T)\\propto T",
"line": 519,
"sha256": "2b0d85a2c36b49d3a0e6042cc213cdf9bc29e9527bcecbd1dbb2f19dd30864fe"
},
{
"index": 80,
"display": false,
"tex": "\\rho(T)\\propto T",
"line": 519,
"sha256": "20cbcde56ab240e01834649bff66e64914206bfd7b79cd35ac3a24e77ebc36bd"
},
{
"index": 81,
"display": false,
"tex": "|\\Sigma|\\gg|\\omega|",
"line": 529,
"sha256": "7495974af685485c795f936f9790b42098fc2b5a722496a3dcda13b265ac481f"
},
{
"index": 82,
"display": false,
"tex": "P",
"line": 537,
"sha256": "5c62e091b8c0565f1bafad0dad5934276143ae2ccef7a5381e8ada5b1a8d26d2"
},
{
"index": 83,
"display": false,
"tex": "\\chi_{JP}\\neq0",
"line": 537,
"sha256": "ab1141d488604507fb478017e845ad3e8f8a325d1cf08c667269c8a20a17319d"
},
{
"index": 84,
"display": false,
"tex": "\\dot P",
"line": 541,
"sha256": "35679b068dabbae248a3381dc8c531de29bb451999ada92c0c883023bec06e99"
},
{
"index": 85,
"display": false,
"tex": "T",
"line": 545,
"sha256": "e632b7095b0bf32c260fa4c539e9fd7b852d0de454e9be26f24d0d6f91d069d3"
},
{
"index": 86,
"display": true,
"tex": "W(\\theta)=\\frac{W_0}{\\theta^2+\\theta_0^2},\n\\qquad\n\\theta_0\\ll\\theta_c\\ll1.",
"line": 551,
"sha256": "54861bf9ff11b611d018c860d666a1dfec22bb42d138bc91dbc89265d701f04c"
},
{
"index": 87,
"display": true,
"tex": "\\frac1{\\tau_{\\rm sp}}\n\\sim\n\\int_{-\\theta_c}^{\\theta_c}d\\theta\\,W(\\theta),",
"line": 561,
"sha256": "52ace7b41564e6abaf0eef0b463147ce4649f2ca1b3cbaa7050491592334bea4"
},
{
"index": 88,
"display": true,
"tex": "\\frac1{\\tau_{\\rm tr}}\n\\sim\n\\int_{-\\theta_c}^{\\theta_c}d\\theta\\,\nW(\\theta)(1-\\cos\\theta).",
"line": 569,
"sha256": "27feae6bbfb547f2535601be0325af7f77beddd84c4a71dcb1df7617a997e76e"
},
{
"index": 89,
"display": false,
"tex": "\\theta_0",
"line": 576,
"sha256": "c32bc1bd750febf9fe25093c14a8e1be4f1b992a623e5dc3d4a82e99d45c6714"
},
{
"index": 90,
"display": true,
"tex": "H_{\\rm dis}=\\int d^2r\\,g'(\\mathbf r)O(\\mathbf r),",
"line": 582,
"sha256": "392548fb4b917e2834f67b453dc925e5742b735a8c18e0ce2ac4a2bd5ccd86ba"
},
{
"index": 91,
"display": true,
"tex": "\\overline{g'(\\mathbf r)g'(\\mathbf 0)}\n=g'^2\\delta^{(2)}(\\mathbf r).",
"line": 588,
"sha256": "79b1095c22d8f1065ac5e17fb989aead6b20e7686e31f61738ac4e3c86abd5aa"
},
{
"index": 92,
"display": false,
"tex": "\\dot P_i",
"line": 593,
"sha256": "d78d41cbced9c25d4f9b462614505c563217523cd53b5a5f06008ffb29147e37"
},
{
"index": 93,
"display": true,
"tex": "\\mathcal K(q,T)\n\\equiv\n\\lim_{\\omega\\to0}\n\\frac{\\operatorname{Im}G^R_{OO}(q,\\omega)}{\\omega}\n=T^{-1}\\Phi(q/\\sqrt T),",
"line": 599,
"sha256": "51d18ba80f35941bfcd0bd4730ebcc6daa28268ede7abbdbf4f4002083bed813"
},
{
"index": 94,
"display": false,
"tex": "\\Phi",
"line": 607,
"sha256": "9a2ae80e5efa1bd70e667dcfdc783a2dff45edef3b6d6bd9d23b93b2a07a810f"
},
{
"index": 95,
"display": true,
"tex": "M_{P_iP_i}\n\\sim\ng'^2\\int d^2q\\,q_i^2\\mathcal K(q,T).",
"line": 609,
"sha256": "7d2510cf10d2e615c2f305541a8a135b2387948581ba4903ee5bf777db1293c5"
},
{
"index": 96,
"display": true,
"tex": "-\\operatorname{Im}\\Sigma^R(0,T)=\\alpha k_BT.",
"line": 621,
"sha256": "fa5dc456daed6f6db4ec83d98d422d4cfb6b61b01bba1b65f323c0b23055c71a"
},
{
"index": 97,
"display": false,
"tex": "\\alpha k_BT/\\hbar",
"line": 625,
"sha256": "1198d8bd253ec81049359ffccd175238e69e8a0598e6cb60e1b5bda99eebcac7"
},
{
"index": 98,
"display": false,
"tex": "1-\\cos\\theta\\simeq\\theta^2/2",
"line": 638,
"sha256": "697421d59ae370c7844e3d1ad6aba1ec622fdbf41547de97490129db6e30e150"
},
{
"index": 99,
"display": false,
"tex": "[P_i,O]=-\\ii\\partial_iO",
"line": 639,
"sha256": "6e50d72e5f6eb36f1e0e66d0788db383d2730ec2d463558b0ab88c3ff273cdec"
},
{
"index": 100,
"display": false,
"tex": "q=\\sqrt T\\,x",
"line": 640,
"sha256": "be428234816f9580d13761eba15214d33a9cc6c0b1882c02f38f13e34fb2ba22"
},
{
"index": 101,
"display": false,
"tex": "q_i^2\\to q^2/2",
"line": 641,
"sha256": "5acdb91e6883f5190a756c3e2b569740a22d7fb69577bfeb200b8fe97fac3637"
},
{
"index": 102,
"display": true,
"tex": "\\frac1{\\tau_{\\rm sp}}\n\\sim\nW_0\\int\\frac{d\\theta}{\\theta^2+\\theta_0^2}\n\\sim\\frac{W_0}{\\theta_0}.",
"line": 650,
"sha256": "d6a8d48ae3fdefc5cbcebe8027eb353f9d7bc35f3c87172ada3a4fc3e9639831"
},
{
"index": 103,
"display": true,
"tex": "\\frac1{\\tau_{\\rm tr}}\n\\sim\n\\frac{W_0}{2}\n\\int_{-\\theta_c}^{\\theta_c}d\\theta\\,\n\\frac{\\theta^2}{\\theta^2+\\theta_0^2}\n\\sim W_0\\left[\\theta_c-O(\\theta_0)\\right].",
"line": 659,
"sha256": "920d9a5a0e49d56cab52351777bcfdaf1d87434bcef3fb076fe466bd646ba71a"
},
{
"index": 104,
"display": true,
"tex": "\\frac{\\tau_{\\rm tr}^{-1}}{\\tau_{\\rm sp}^{-1}}\n\\sim\\theta_0\\theta_c\\longrightarrow0.",
"line": 670,
"sha256": "618dc3fffe6c96553b658fe7f47666c3d093563c787677733bfe0bd13f27edf2"
},
{
"index": 105,
"display": true,
"tex": "\\dot P_i\n=-\\int d^2r\\,\n(\\partial_i g'(\\mathbf r))O(\\mathbf r),",
"line": 679,
"sha256": "1a7805ef6740b45825456a13849f13ebce2e5fbff1b23ae1ee70d6302ea275d4"
},
{
"index": 106,
"display": false,
"tex": "P_i",
"line": 685,
"sha256": "429e7f1087bafa6e1d16fa2ffc456540254f75f4249b5c5a801cd2afd469fa77"
},
{
"index": 107,
"display": false,
"tex": "q=\\sqrt T\\,x",
"line": 687,
"sha256": "be428234816f9580d13761eba15214d33a9cc6c0b1882c02f38f13e34fb2ba22"
},
{
"index": 108,
"display": true,
"tex": "d^2q=T\\,d^2x,\n\\qquad\nq_i^2=T x_i^2,\n\\qquad\n\\mathcal K=T^{-1}\\Phi(x).",
"line": 689,
"sha256": "97658717e03933dcbf286451e4c4d1f9f4b82239949f9324fa53f158390273f0"
},
{
"index": 109,
"display": true,
"tex": "M_{P_iP_i}\n\\sim g'^2T\n\\int d^2x\\,x_i^2\\Phi(x)\n\\propto g'^2T.",
"line": 699,
"sha256": "9cd060b54626294cd7830c5aa6c678b16805112509104c2973648dbab6c3b0c9"
},
{
"index": 110,
"display": true,
"tex": "\\rho\\simeq\\frac{M_{PP}}{\\chi_{JP}^2}.",
"line": 708,
"sha256": "cd074bd0bf03ad2fede523bccfee4247de2795140d26a46ff9124407afd1c4f8"
},
{
"index": 111,
"display": false,
"tex": "M_{PP}",
"line": 712,
"sha256": "8f579ca6bb2fe0db3dd555011fab889cf9b35494cab8e36b7797e7159504fcd9"
},
{
"index": 112,
"display": false,
"tex": "\\chi_{JP}",
"line": 712,
"sha256": "ba2356d1b8b32696ce36d0c7521ffaf8b53c9df0faa42fa61e71305038b61118"
},
{
"index": 113,
"display": true,
"tex": "-\\operatorname{Im}\\Sigma^R(0,T)\\propto T,\n\\qquad\n\\rho(T)=\\rho_0+AT.",
"line": 724,
"sha256": "0a6643ce1b01c7c813574c8581060e29478825f0cec66262aefc2fdd7e376d78"
},
{
"index": 114,
"display": false,
"tex": "\\chi_{JP}",
"line": 735,
"sha256": "ba2356d1b8b32696ce36d0c7521ffaf8b53c9df0faa42fa61e71305038b61118"
},
{
"index": 115,
"display": false,
"tex": "A",
"line": 738,
"sha256": "559aead08264d5795d3909718cdd05abd49572e84fe55590eef31a88a08fdffd"
},
{
"index": 116,
"display": false,
"tex": "\\chi_{JA}",
"line": 744,
"sha256": "57b8499ff9b8a96f9ce36559f3885f385e7dc9910ce9be38474bd06030fb0e12"
},
{
"index": 117,
"display": false,
"tex": "\\dot P",
"line": 745,
"sha256": "35679b068dabbae248a3381dc8c531de29bb451999ada92c0c883023bec06e99"
},
{
"index": 118,
"display": false,
"tex": "N",
"line": 754,
"sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
},
{
"index": 119,
"display": false,
"tex": "N",
"line": 755,
"sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
},
{
"index": 120,
"display": false,
"tex": "T\\log(1/T)",
"line": 755,
"sha256": "076a40c10a5b44d4efde58d5aaed36471d7744bbc7a3fab23a62ddcf0bfb8400"
},
{
"index": 121,
"display": false,
"tex": "N",
"line": 759,
"sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
},
{
"index": 122,
"display": false,
"tex": "\\dot P",
"line": 769,
"sha256": "35679b068dabbae248a3381dc8c531de29bb451999ada92c0c883023bec06e99"
}
]
}