{
"schema_version": 1,
"id": "PHYS-2026-07-28-01",
"canonical_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-07-28-01/",
"source_markdown_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-07-28-01.md",
"metadata": {
"schema_version": 1,
"id": "PHYS-2026-07-28-01",
"date": "2026-07-28",
"updated_at": "2026-07-28",
"title": "One static fixed point, two critical clocks",
"summary": "Why the same equilibrium phi-four theory can produce Model-A and Model-B response poles, different relaxation laws, and different dynamical critical exponents.",
"language": "en",
"entry_kind": "daily",
"status": "published",
"level": "graduate-advanced",
"user_difficulty": "unrated",
"domains": [
"statistical-mechanics",
"quantum-field-theory",
"condensed-matter"
],
"estimated_minutes": 55
},
"content_markdown": "\n## Key claim\n\n**A static RG fixed point does not determine critical relaxation by itself.**\n\nTwo systems can share the same equilibrium $\\phi^4$ theory and static susceptibility while conservation changes their response poles and dynamical exponents.\n\n## Theme\n\n**Static Wilson–Fisher universality versus dynamical critical slowing.**\n\n## Guiding question\n\nHow can two systems have\n\n$$\n\\chi(\\mathbf q,0)\\sim q^{-2+\\eta}\n$$\n\nyet exhibit different low-frequency response functions and different dynamical exponents $z$?\n\nThe equilibrium functional fixes static probabilities and correlations. A retarded response also needs an equation of motion, its conservation laws, kinetic coefficients, and noise.\n\n## Setup\n\nConsider a classical scalar order parameter in $d=4-\\epsilon$ spatial dimensions:\n\n$$\n\\mathcal F[\\phi;h]\n=\n\\int\\dd^dx\n\\left[\n\\frac12(\\nabla\\phi)^2\n+\\frac r2\\phi^2\n+\\frac{u}{4!}\\phi^4\n-h\\phi\n\\right].\n$$\n\nFor $\\epsilon>0$ and $u>0$, tuning $r$ to the critical surface leads to the Wilson–Fisher fixed point.\n\nThe equilibrium measure is\n\n$$\nP_{\\rm eq}[\\phi]\n=\n\\frac1Z\n\\exp\\left[-\\frac{\\mathcal F[\\phi]}{T}\\right],\n$$\n\nwhere $k_B=1$.\n\nThis measure contains no rule for how $\\phi$ relaxes.\n\nStart with nonconserved relaxational dynamics, Model A:\n\n$$\n\\partial_t\\phi(\\mathbf x,t)\n=\n-\\Gamma\n\\frac{\\delta\\mathcal F}{\\delta\\phi(\\mathbf x,t)}\n+\\zeta(\\mathbf x,t).\n$$\n\nThe white noise satisfies\n\n$$\n\\left\\langle\n\\zeta(\\mathbf x,t)\n\\zeta(\\mathbf x',t')\n\\right\\rangle\n=\n2\\Gamma T\\,\n\\delta^d(\\mathbf x-\\mathbf x')\n\\delta(t-t').\n$$\n\nThe drift and noise amplitude obey detailed balance and make $P_{\\rm eq}$ stationary.\n\nAt criticality, dynamic scaling takes the form\n\n$$\n\\chi^R(q,\\omega)\n=\nq^{-2+\\eta}\n\\Phi\\left(\\frac{\\omega}{q^z}\\right),\n$$\n\nup to nonuniversal metric factors.\n\nThe static exponent $\\eta$ does not determine $z$.\n\n## Analysis\n\n### Derivation: derive the Model-A pole\n\nSet $u=0$. In Fourier space,\n\n$$\n\\left[\n-\\ii\\omega+\\Gamma(r+q^2)\n\\right]\n\\phi(\\mathbf q,\\omega)\n=\n\\Gamma h(\\mathbf q,\\omega)\n+\\zeta(\\mathbf q,\\omega).\n$$\n\nThe retarded susceptibility is\n\n$$\n\\chi_A^R(q,\\omega)\n=\n\\frac{\n\\delta\\langle\\phi(\\mathbf q,\\omega)\\rangle\n}{\n\\delta h(\\mathbf q,\\omega)\n}\n=\n\\frac{\\Gamma}{\n\\Gamma(r+q^2)-\\ii\\omega\n}.\n$$\n\nEquivalently,\n\n$$\n\\chi_A^R(q,\\omega)\n=\n\\frac1{\nr+q^2-\\ii\\omega/\\Gamma\n}.\n$$\n\nFor $\\omega>0$, its dissipative part is\n\n$$\n\\chi_A''(q,\\omega)\n=\n\\frac{\n\\Gamma\\omega\n}{\n\\Gamma^2(r+q^2)^2+\\omega^2\n}.\n$$\n\nThe pole lies at\n\n$$\n\\omega_*=-\\ii\\Gamma(r+q^2),\n$$\n\nso\n\n$$\n\\tau_q\n=\n\\frac1{\\Gamma(r+q^2)}.\n$$\n\nAt the Gaussian critical point $r=0$,\n\n$$\n\\tau_q\\sim q^{-2},\n\\qquad\nz_A^{\\rm Gaussian}=2.\n$$\n\nThe equilibrium correlation spectrum is\n\n$$\nC_A(q,\\omega)\n=\n\\frac{\n2\\Gamma T\n}{\n\\omega^2+\\Gamma^2(r+q^2)^2\n}.\n$$\n\nIt obeys the classical fluctuation–dissipation relation\n\n$$\nC_A(q,\\omega)\n=\n\\frac{2T}{\\omega}\n\\chi_A''(q,\\omega).\n$$\n\nThe correlation function and dissipative response therefore contain the same relaxation rate.\n\n### Scope and assumptions: conservation changes the clock\n\nThe functional $\\mathcal F[\\phi]$ does not select Model A.\n\nSuppose $\\phi$ is a conserved density. Local conservation requires\n\n$$\n\\partial_t\\phi+\\nabla\\cdot\\mathbf j=0.\n$$\n\nChoose the dissipative current\n\n$$\n\\mathbf j\n=\n-\\Gamma\\nabla\n\\frac{\\delta\\mathcal F}{\\delta\\phi}\n+\\boldsymbol\\xi.\n$$\n\nIts noise obeys\n\n$$\n\\left\\langle\n\\xi_i(\\mathbf x,t)\n\\xi_j(\\mathbf x',t')\n\\right\\rangle\n=\n2\\Gamma T\\,\n\\delta_{ij}\n\\delta^d(\\mathbf x-\\mathbf x')\n\\delta(t-t').\n$$\n\nThe order parameter then follows Model-B dynamics:\n\n$$\n\\partial_t\\phi\n=\n\\Gamma\\nabla^2\n\\frac{\\delta\\mathcal F}{\\delta\\phi}\n+\\zeta_B,\n\\qquad\n\\zeta_B=-\\nabla\\cdot\\boldsymbol\\xi.\n$$\n\nThe conserved noise vanishes at zero momentum:\n\n$$\n\\left\\langle\n\\zeta_B(\\mathbf q,t)\n\\zeta_B(-\\mathbf q,t')\n\\right\\rangle\n=\n2\\Gamma Tq^2\\delta(t-t').\n$$\n\nIn the Gaussian theory,\n\n$$\n\\chi_B^R(q,\\omega)\n=\n\\frac{\n\\Gamma q^2\n}{\n\\Gamma q^2(r+q^2)-\\ii\\omega\n}.\n$$\n\nIts static limit matches Model A:\n\n$$\n\\chi_B^R(q,0)\n=\n\\frac1{r+q^2}.\n$$\n\nIts pole does not:\n\n$$\n\\omega_*\n=\n-\\ii\\Gamma q^2(r+q^2).\n$$\n\nAt $r=0$,\n\n$$\n\\tau_q\\sim q^{-4},\n\\qquad\nz_B^{\\rm Gaussian}=4.\n$$\n\nThe extra $q^2$ comes from local conservation. At $q=0$, the total conserved order parameter cannot relax.\n\nAt the interacting fixed point,\n\n$$\nz_A=2+O(\\epsilon^2),\n$$\n\nwhile equilibrium Model B obeys\n\n$$\nz_B=4-\\eta.\n$$\n\nThe Model-B result assumes detailed balance and no additional slow field coupled to the order parameter.\n\n### Physical interpretation: separate restoring force from mobility\n\nWrite both linearized dynamics as\n\n$$\n\\left[\n-\\ii\\omega\n+\\mathcal K(q)(r+q^2)\n\\right]\\phi\n=\n\\mathcal K(q)h+\\zeta.\n$$\n\nThe static inverse susceptibility\n\n$$\nr+q^2\n$$\n\nis the thermodynamic restoring force.\n\nThe kinetic kernel is\n\n$$\n\\mathcal K_A(q)=\\Gamma\n$$\n\nfor Model A and\n\n$$\n\\mathcal K_B(q)=\\Gamma q^2\n$$\n\nfor Model B.\n\nThe static functional fixes the restoring force. Conservation fixes the small-$q$ structure of the kinetic kernel, which then sets the response pole.\n\nThe correspondence can be summarized as follows:\n\n| Structure | Model A | Model B |\n| --- | --- | --- |\n| Order parameter | Nonconserved | Conserved |\n| Kinetic kernel | $\\Gamma$ | $\\Gamma q^2$ |\n| Gaussian rate at $r=0$ | $\\Gamma q^2$ | $\\Gamma q^4$ |\n| Gaussian exponent | $z=2$ | $z=4$ |\n| Static susceptibility | $q^{-2}$ | $q^{-2}$ |\n\nThis correspondence has a clear boundary.\n\nCoupling the order parameter to conserved energy produces Model C. Coupling a conserved order parameter to momentum density leads toward Model H.\n\nColored noise, memory kernels, driving, or nonreciprocal couplings can also invalidate the equilibrium fluctuation–dissipation mapping.\n\n## False claim to diagnose\n\n> Once the static RG flow reaches the Wilson–Fisher fixed point, the low-frequency susceptibility is fixed uniquely because all microscopic kinetic coefficients are irrelevant.\n\nThe claim confuses numerical kinetic coefficients with the structure of the kinetic operator.\n\nWithin one dynamical universality class, $\\Gamma$ sets a nonuniversal time scale. But conservation changes $\\mathcal K(q)$ from a constant to a quantity that vanishes as $q^2$.\n\nThat power of $q$ survives coarse-graining because it expresses a conservation law.\n\nA defensible statement is:\n\n> Static universality fixes equilibrium scaling. Dynamic universality additionally requires the slow variables, conservation laws, and noise structure.\n\n## What follows — and what does not\n\nFour statements that sound similar have different logical status:\n\n| Statement | Status |\n| --- | --- |\n| Given $\\mathcal F$ and detailed balance, the equilibrium measure fixes static correlations. | Equilibrium statement |\n| Given a linear Langevin equation, its kinetic kernel fixes the Gaussian response pole. | Exact within the Gaussian dynamics |\n| Model A and Model B can share the Wilson–Fisher static fixed point. | RG classification |\n| They must share the same $z$. | False |\n\nThe static fixed point classifies equal-time long-distance fluctuations. The dynamical fixed point classifies how the slow variables approach equilibrium.\n\n## Exercise\n\nUse the Gaussian functional\n\n$$\n\\mathcal F_0\n=\n\\frac12\n\\int\\dd^dx\n\\left[\n(\\nabla\\phi)^2+r\\phi^2\n\\right]\n-\\int\\dd^dx\\,h\\phi.\n$$\n\n1. Derive $\\chi_A^R(q,\\omega)$ from Model-A dynamics.\n2. Find the frequency at which $\\chi_A''(q,\\omega)$ is maximal.\n3. At $r=0$, put $\\chi_A^R$ into the form\n\n $$\n \\chi_A^R(q,\\omega)\n =\n q^{-2}\n \\Phi_A\\left(\n \\frac{\\omega}{\\Gamma q^2}\n \\right)\n $$\n\n and read off $z_A$.\n\n4. Repeat the scaling analysis for Model B.\n5. Verify that both models have the same $\\chi^R(q,0)$.\n6. Derive the factor $q^2$ in the Model-B noise covariance from $\\zeta_B=-\\nabla\\cdot\\boldsymbol\\xi$.\n\n\nHint 1
\n\nFor\n\n$$\nf(\\omega)=\\frac{\\omega}{a^2+\\omega^2},\n$$\n\nsolve $\\dd f/\\dd\\omega=0$.\n\n \n\n\nHint 2
\n\nRead $z$ from the dimensionless combination of $q$ and $\\omega$ in the scaling function.\n\n \n\n**Oral check 1.** Can two systems have identical equilibrium susceptibilities but parametrically different relaxation times?\n\n**Oral check 2.** Which factor in Model B records local conservation?\n\n\nSolution
\n\nFor Model A,\n\n$$\n\\chi_A^R(q,\\omega)\n=\n\\frac{\\Gamma}{\n\\Gamma(r+q^2)-\\ii\\omega\n},\n$$\n\nand\n\n$$\n\\chi_A''(q,\\omega)\n=\n\\frac{\n\\Gamma\\omega\n}{\n\\Gamma^2(r+q^2)^2+\\omega^2\n}.\n$$\n\nDifferentiating with respect to $\\omega$ gives\n\n$$\n\\omega_{\\rm peak}\n=\n\\Gamma(r+q^2).\n$$\n\nAt $r=0$,\n\n$$\n\\chi_A^R(q,\\omega)\n=\nq^{-2}\n\\frac1{\n1-\\ii\\omega/(\\Gamma q^2)\n}.\n$$\n\nThus\n\n$$\nz_A^{\\rm Gaussian}=2.\n$$\n\nFor Model B,\n\n$$\n\\chi_B^R(q,\\omega)\n=\n\\frac{\n\\Gamma q^2\n}{\n\\Gamma q^2(r+q^2)-\\ii\\omega\n}.\n$$\n\nEquivalently,\n\n$$\n\\chi_B^R(q,\\omega)\n=\n\\frac1{\nr+q^2-\\ii\\omega/(\\Gamma q^2)\n}.\n$$\n\nAt $r=0$,\n\n$$\n\\chi_B^R(q,\\omega)\n=\nq^{-2}\n\\frac1{\n1-\\ii\\omega/(\\Gamma q^4)\n}.\n$$\n\nTherefore,\n\n$$\nz_B^{\\rm Gaussian}=4.\n$$\n\nSetting $\\omega=0$ gives\n\n$$\n\\chi_A^R(q,0)\n=\n\\chi_B^R(q,0)\n=\n\\frac1{r+q^2}.\n$$\n\nFinally,\n\n$$\n\\zeta_B(\\mathbf q,t)\n=\n-\\ii\\mathbf q\\cdot\n\\boldsymbol\\xi(\\mathbf q,t).\n$$\n\nUsing the current-noise covariance,\n\n$$\n\\left\\langle\n\\zeta_B(\\mathbf q,t)\n\\zeta_B(-\\mathbf q,t')\n\\right\\rangle\n=\n2\\Gamma Tq^2\\delta(t-t').\n$$\n\nThe same $q^2$ that suppresses uniform noise also slows the conserved mode.\n\n \n\n## Check your understanding\n\nWhich part of $\\chi^R(q,\\omega)$ follows from the static $\\phi^4$ fixed point, and which part requires a dynamical universality class?\n\nAnswer by naming one static datum and one dynamical datum.\n\nYou may also reply with “deeper,” “too easy,” “too hard,” or your derivation.\n\n## Further Reading\n\n- [Theory of dynamic critical phenomena](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.49.435)\n- [Model-A dynamical exponent through fourth order in the epsilon expansion](https://arxiv.org/abs/0808.1347)\n- [Dynamic scaling of order-parameter fluctuations in Model B](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.108.074004)\n\n## Connections and next step\n\n- Conceptual primitive: static fixed point versus dynamical fixed point.\n- Fields connected: static RG, Langevin dynamics, and linear response.\n- Toy system: scalar $\\phi^4$ theory with Model-A and Model-B dynamics.\n- Decisive structure: $\\mathcal K_A(q)=\\Gamma$ versus $\\mathcal K_B(q)=\\Gamma q^2$.\n- Assumption challenged: static universality uniquely fixes critical dissipation.\n- Next step: dynamic RG and renormalization of $\\Gamma$.\n- Avoid repeating soon: basic analytic continuation and isolated quasiparticle decay.\n",
"content_sha256": "73a1e28f95a0e4cfebbd75a6f7803d7eb8ab4bdda92cd481e87d933ef9f49bfc",
"formula_count": 103,
"formulas": [
{
"index": 1,
"display": false,
"tex": "\\phi^4",
"line": 24,
"sha256": "ab8c353106812d45b6f7e5039b564e326450e2554cef53f68e707816b222658e"
},
{
"index": 2,
"display": true,
"tex": "\\chi(\\mathbf q,0)\\sim q^{-2+\\eta}",
"line": 34,
"sha256": "67fe98af53633d814d7144088c1c16291915977387fe31f23b37b494054365d0"
},
{
"index": 3,
"display": false,
"tex": "z",
"line": 38,
"sha256": "594e519ae499312b29433b7dd8a97ff068defcba9755b6d5d00e84c524d67b06"
},
{
"index": 4,
"display": false,
"tex": "d=4-\\epsilon",
"line": 44,
"sha256": "3b668192671ebc2f9b01c00ffca007e44ad4301a5d849ae3d2666811f732d5bb"
},
{
"index": 5,
"display": true,
"tex": "\\mathcal F[\\phi;h]\n=\n\\int\\dd^dx\n\\left[\n\\frac12(\\nabla\\phi)^2\n+\\frac r2\\phi^2\n+\\frac{u}{4!}\\phi^4\n-h\\phi\n\\right].",
"line": 46,
"sha256": "e89a7088e3a4a66e5a08745bc82d363c280248e9f9759fb91e8309422ca77551"
},
{
"index": 6,
"display": false,
"tex": "\\epsilon>0",
"line": 58,
"sha256": "4c7785f41b6ba845fbf6fe3db625563505eedd141a7e6a016df047b5d14c0cc7"
},
{
"index": 7,
"display": false,
"tex": "u>0",
"line": 58,
"sha256": "db7dd4e33fa27156741033e6e749007dea01c1cb39facdadbaf31ece9d9b1936"
},
{
"index": 8,
"display": false,
"tex": "r",
"line": 58,
"sha256": "454349e422f05297191ead13e21d3db520e5abef52055e4964b82fb213f593a1"
},
{
"index": 9,
"display": true,
"tex": "P_{\\rm eq}[\\phi]\n=\n\\frac1Z\n\\exp\\left[-\\frac{\\mathcal F[\\phi]}{T}\\right],",
"line": 62,
"sha256": "7a29d8b5a370870ecfb1e1047724d2e2892226c5fcb03af1a860c9dcfb22210b"
},
{
"index": 10,
"display": false,
"tex": "k_B=1",
"line": 69,
"sha256": "b6828eb971aaeda4a0bdec94e40c5923ebbbaa5640f2a1663fb5c7583961ff2c"
},
{
"index": 11,
"display": false,
"tex": "\\phi",
"line": 71,
"sha256": "10ce0200b450ea955833553e717821c7721818b8a95cffd7e54b46e08918b512"
},
{
"index": 12,
"display": true,
"tex": "\\partial_t\\phi(\\mathbf x,t)\n=\n-\\Gamma\n\\frac{\\delta\\mathcal F}{\\delta\\phi(\\mathbf x,t)}\n+\\zeta(\\mathbf x,t).",
"line": 75,
"sha256": "08b7a93df5084d607cbeb584f0468315d99230d042e5947bea2a5d37d50a956f"
},
{
"index": 13,
"display": true,
"tex": "\\left\\langle\n\\zeta(\\mathbf x,t)\n\\zeta(\\mathbf x',t')\n\\right\\rangle\n=\n2\\Gamma T\\,\n\\delta^d(\\mathbf x-\\mathbf x')\n\\delta(t-t').",
"line": 85,
"sha256": "460c901d144022485d417f603a968e7314908816add8d651e5415c6c7b695fc4"
},
{
"index": 14,
"display": false,
"tex": "P_{\\rm eq}",
"line": 96,
"sha256": "d67cfb473bcdd5f66e922275ee8bc19a6443b49097ec4888613bb1918b5687b1"
},
{
"index": 15,
"display": true,
"tex": "\\chi^R(q,\\omega)\n=\nq^{-2+\\eta}\n\\Phi\\left(\\frac{\\omega}{q^z}\\right),",
"line": 100,
"sha256": "fdbb5ee004132db439ee62c92eed51c9065ef3d2113fb0b17a9615c1fc81d63c"
},
{
"index": 16,
"display": false,
"tex": "\\eta",
"line": 109,
"sha256": "59a94416a2a319caf40a348286e620ef3594226a2eac1efe842231962d1e157d"
},
{
"index": 17,
"display": false,
"tex": "z",
"line": 109,
"sha256": "594e519ae499312b29433b7dd8a97ff068defcba9755b6d5d00e84c524d67b06"
},
{
"index": 18,
"display": false,
"tex": "u=0",
"line": 115,
"sha256": "953a947a0a53b228142b57831935a639b7fb7deae9d3192750598cd54f5cad24"
},
{
"index": 19,
"display": true,
"tex": "\\left[\n-\\ii\\omega+\\Gamma(r+q^2)\n\\right]\n\\phi(\\mathbf q,\\omega)\n=\n\\Gamma h(\\mathbf q,\\omega)\n+\\zeta(\\mathbf q,\\omega).",
"line": 117,
"sha256": "d8038864f774bd2fdd79586b8c4636f6297c5b445209aa3caa783ad1471776a9"
},
{
"index": 20,
"display": true,
"tex": "\\chi_A^R(q,\\omega)\n=\n\\frac{\n\\delta\\langle\\phi(\\mathbf q,\\omega)\\rangle\n}{\n\\delta h(\\mathbf q,\\omega)\n}\n=\n\\frac{\\Gamma}{\n\\Gamma(r+q^2)-\\ii\\omega\n}.",
"line": 129,
"sha256": "2a9a9abf829de747d52bbc33c77862c87018654d1b3a7b78c8480fb7716c937f"
},
{
"index": 21,
"display": true,
"tex": "\\chi_A^R(q,\\omega)\n=\n\\frac1{\nr+q^2-\\ii\\omega/\\Gamma\n}.",
"line": 145,
"sha256": "19fa26c9d7f593830e3ef6ac55c0b99b20e2d47c95b1b09553d534349d9666ab"
},
{
"index": 22,
"display": false,
"tex": "\\omega>0",
"line": 153,
"sha256": "58b71a083071f2f883cf2398b31d606fd438b70e4a0037b6aebbf4f8ad6a2395"
},
{
"index": 23,
"display": true,
"tex": "\\chi_A''(q,\\omega)\n=\n\\frac{\n\\Gamma\\omega\n}{\n\\Gamma^2(r+q^2)^2+\\omega^2\n}.",
"line": 155,
"sha256": "1c1171256d0eac84f5316100d2947d16cb0fce597ae880dba57325857ac82ecf"
},
{
"index": 24,
"display": true,
"tex": "\\omega_*=-\\ii\\Gamma(r+q^2),",
"line": 167,
"sha256": "c5f0b2b52e6766b8dd88893106c622f1ec3920fd086d2844200912848a1b64b6"
},
{
"index": 25,
"display": true,
"tex": "\\tau_q\n=\n\\frac1{\\Gamma(r+q^2)}.",
"line": 173,
"sha256": "527351593813317cd1d681f6a7c68a1bfe641f33f3f5ff47d220fb8d5c2fa1a6"
},
{
"index": 26,
"display": false,
"tex": "r=0",
"line": 179,
"sha256": "d3d30bd2fde29232f8d504f67d77bf6f6666b626295e01154eabcab8ab9d7b88"
},
{
"index": 27,
"display": true,
"tex": "\\tau_q\\sim q^{-2},\n\\qquad\nz_A^{\\rm Gaussian}=2.",
"line": 181,
"sha256": "3c2f8c97edba47753b1c7acb82a4e9f3f64164e9eb2a1dc1c82b4d2fcfc2bcba"
},
{
"index": 28,
"display": true,
"tex": "C_A(q,\\omega)\n=\n\\frac{\n2\\Gamma T\n}{\n\\omega^2+\\Gamma^2(r+q^2)^2\n}.",
"line": 189,
"sha256": "84e4502202f04793892449d8d24018ead131ae3213763c63744e8b9f2c2d5802"
},
{
"index": 29,
"display": true,
"tex": "C_A(q,\\omega)\n=\n\\frac{2T}{\\omega}\n\\chi_A''(q,\\omega).",
"line": 201,
"sha256": "db5fcf93293cc91102f20ebe5339589ee5c3605904e06ea6f3fc69e4ef7bd971"
},
{
"index": 30,
"display": false,
"tex": "\\mathcal F[\\phi]",
"line": 212,
"sha256": "779756364e5a45e59eafde64174e5de887c3326a82e7ec456231d88c330727b7"
},
{
"index": 31,
"display": false,
"tex": "\\phi",
"line": 214,
"sha256": "10ce0200b450ea955833553e717821c7721818b8a95cffd7e54b46e08918b512"
},
{
"index": 32,
"display": true,
"tex": "\\partial_t\\phi+\\nabla\\cdot\\mathbf j=0.",
"line": 216,
"sha256": "f33e717aaf438e6a2f6bc4714d13d96c4a581d79c872b2aa66e7bad5cced5ac3"
},
{
"index": 33,
"display": true,
"tex": "\\mathbf j\n=\n-\\Gamma\\nabla\n\\frac{\\delta\\mathcal F}{\\delta\\phi}\n+\\boldsymbol\\xi.",
"line": 222,
"sha256": "d28055f5ee38b421165e44ff0cd7c5f1eceda12620c7d185aac7d8437ba6e313"
},
{
"index": 34,
"display": true,
"tex": "\\left\\langle\n\\xi_i(\\mathbf x,t)\n\\xi_j(\\mathbf x',t')\n\\right\\rangle\n=\n2\\Gamma T\\,\n\\delta_{ij}\n\\delta^d(\\mathbf x-\\mathbf x')\n\\delta(t-t').",
"line": 232,
"sha256": "137d4f3dc0fc47d469e9e8fd1ec7f30df647034e5323c20f059c8b72e5aa4b43"
},
{
"index": 35,
"display": true,
"tex": "\\partial_t\\phi\n=\n\\Gamma\\nabla^2\n\\frac{\\delta\\mathcal F}{\\delta\\phi}\n+\\zeta_B,\n\\qquad\n\\zeta_B=-\\nabla\\cdot\\boldsymbol\\xi.",
"line": 246,
"sha256": "ae613571f9380dd22725af54b7364b47d1361536ee263064d385f0f406a20e13"
},
{
"index": 36,
"display": true,
"tex": "\\left\\langle\n\\zeta_B(\\mathbf q,t)\n\\zeta_B(-\\mathbf q,t')\n\\right\\rangle\n=\n2\\Gamma Tq^2\\delta(t-t').",
"line": 258,
"sha256": "b6994c8fbaa7d2f580b93b9d7fc47756bb1c376f6dcfd080a2748105d6c84edf"
},
{
"index": 37,
"display": true,
"tex": "\\chi_B^R(q,\\omega)\n=\n\\frac{\n\\Gamma q^2\n}{\n\\Gamma q^2(r+q^2)-\\ii\\omega\n}.",
"line": 269,
"sha256": "3f3a83d56e9d2ff1d5bdcb790464c53280dea21610a63518317d0bd095127c72"
},
{
"index": 38,
"display": true,
"tex": "\\chi_B^R(q,0)\n=\n\\frac1{r+q^2}.",
"line": 281,
"sha256": "b375f0852daf9f13069c2cadb025caea20c4b867f898fe27b0016c4819b85022"
},
{
"index": 39,
"display": true,
"tex": "\\omega_*\n=\n-\\ii\\Gamma q^2(r+q^2).",
"line": 289,
"sha256": "1008bd5de29240a2db9292f85c70ed2faa1d0cc7a1a570e0ac69357fe39de43a"
},
{
"index": 40,
"display": false,
"tex": "r=0",
"line": 295,
"sha256": "d3d30bd2fde29232f8d504f67d77bf6f6666b626295e01154eabcab8ab9d7b88"
},
{
"index": 41,
"display": true,
"tex": "\\tau_q\\sim q^{-4},\n\\qquad\nz_B^{\\rm Gaussian}=4.",
"line": 297,
"sha256": "2e7f445673e54243b2817f7a1807d7569ed0bcd4f10666d0abbc4aeb040da613"
},
{
"index": 42,
"display": false,
"tex": "q^2",
"line": 303,
"sha256": "be871d9e7245f709a6d83f34c1ca1efb0ab1509493cb84b57914fbc0b4d79ab1"
},
{
"index": 43,
"display": false,
"tex": "q=0",
"line": 303,
"sha256": "e5fdc81ec9e003f6d05562253517a9bdac74f2c6e2a7912bb7766492c3967a94"
},
{
"index": 44,
"display": true,
"tex": "z_A=2+O(\\epsilon^2),",
"line": 307,
"sha256": "89dee5a37fc7c829f72c1ea09af2bf5b5e806a5dc7ac12d6a569cf89f94d0001"
},
{
"index": 45,
"display": true,
"tex": "z_B=4-\\eta.",
"line": 313,
"sha256": "4569497b3f0e740efadfbf007a6b5d2636300dae3da2f4ca6f25567686af1883"
},
{
"index": 46,
"display": true,
"tex": "\\left[\n-\\ii\\omega\n+\\mathcal K(q)(r+q^2)\n\\right]\\phi\n=\n\\mathcal K(q)h+\\zeta.",
"line": 323,
"sha256": "23a156dacbcb9c7c24b2034b3c13558a93bb44e345ae1fac76cb5328c38d2ce9"
},
{
"index": 47,
"display": true,
"tex": "r+q^2",
"line": 334,
"sha256": "578fac9e2ac68512ad23e0ada9b711e1134afb23a4c0df52bf8a3f32d72c3787"
},
{
"index": 48,
"display": true,
"tex": "\\mathcal K_A(q)=\\Gamma",
"line": 342,
"sha256": "4565b642b2233e39565b4b09c98eb2008ca13e380f613523f02a5f014fefffea"
},
{
"index": 49,
"display": true,
"tex": "\\mathcal K_B(q)=\\Gamma q^2",
"line": 348,
"sha256": "54636dc9719244ce70ff92b7e6b7ac68785cdfbb34807a93e0c8a57966c3f2a8"
},
{
"index": 50,
"display": false,
"tex": "q",
"line": 354,
"sha256": "8e35c2cd3bf6641bdb0e2050b76932cbb2e6034a0ddacc1d9bea82a6ba57f7cf"
},
{
"index": 51,
"display": false,
"tex": "\\Gamma",
"line": 361,
"sha256": "57885e4c75965b23b196228446c46b00876b0951643235b678d18199f6dce5ec"
},
{
"index": 52,
"display": false,
"tex": "\\Gamma q^2",
"line": 361,
"sha256": "4c71be5843cfb1c0363d7c58b040316f08f5214d5be7745f3e40e4be941541c6"
},
{
"index": 53,
"display": false,
"tex": "r=0",
"line": 362,
"sha256": "d3d30bd2fde29232f8d504f67d77bf6f6666b626295e01154eabcab8ab9d7b88"
},
{
"index": 54,
"display": false,
"tex": "\\Gamma q^2",
"line": 362,
"sha256": "4c71be5843cfb1c0363d7c58b040316f08f5214d5be7745f3e40e4be941541c6"
},
{
"index": 55,
"display": false,
"tex": "\\Gamma q^4",
"line": 362,
"sha256": "e66215647f80ac49ccb4b29d0555e83098c5e0fd959d167d096f2657292d142f"
},
{
"index": 56,
"display": false,
"tex": "z=2",
"line": 363,
"sha256": "92d95b7c7b24cdc511bdfa2a508b8b92e1ed85ba741279e15fba3d8a8e2728d8"
},
{
"index": 57,
"display": false,
"tex": "z=4",
"line": 363,
"sha256": "87bc371bd226d06afa98dabd089d98119929881e6d4c3bf68694fe2916412abe"
},
{
"index": 58,
"display": false,
"tex": "q^{-2}",
"line": 364,
"sha256": "4be67f362e168229e9877c92cbc0c87ddaae34b45d192c8fbac1ea0f48227fe2"
},
{
"index": 59,
"display": false,
"tex": "q^{-2}",
"line": 364,
"sha256": "4be67f362e168229e9877c92cbc0c87ddaae34b45d192c8fbac1ea0f48227fe2"
},
{
"index": 60,
"display": false,
"tex": "\\Gamma",
"line": 378,
"sha256": "57885e4c75965b23b196228446c46b00876b0951643235b678d18199f6dce5ec"
},
{
"index": 61,
"display": false,
"tex": "\\mathcal K(q)",
"line": 378,
"sha256": "a796c58a0147e4afab567a9ee9b0657ad28749d7d51ee55120a9339230b609c7"
},
{
"index": 62,
"display": false,
"tex": "q^2",
"line": 378,
"sha256": "be871d9e7245f709a6d83f34c1ca1efb0ab1509493cb84b57914fbc0b4d79ab1"
},
{
"index": 63,
"display": false,
"tex": "q",
"line": 380,
"sha256": "8e35c2cd3bf6641bdb0e2050b76932cbb2e6034a0ddacc1d9bea82a6ba57f7cf"
},
{
"index": 64,
"display": false,
"tex": "\\mathcal F",
"line": 392,
"sha256": "77dd4d71ed008fa68519847b611063f25f9110f6ca806f91d00b0e6ac5e7e326"
},
{
"index": 65,
"display": false,
"tex": "z",
"line": 395,
"sha256": "594e519ae499312b29433b7dd8a97ff068defcba9755b6d5d00e84c524d67b06"
},
{
"index": 66,
"display": true,
"tex": "\\mathcal F_0\n=\n\\frac12\n\\int\\dd^dx\n\\left[\n(\\nabla\\phi)^2+r\\phi^2\n\\right]\n-\\int\\dd^dx\\,h\\phi.",
"line": 403,
"sha256": "bae148bebbee2e253179df2ecd6d8487911a8afedcc31faba3edc02dc343c25a"
},
{
"index": 67,
"display": false,
"tex": "\\chi_A^R(q,\\omega)",
"line": 414,
"sha256": "eee1932cf597538ca993394eb0152260625bfb1867f31e3d51e2f7ab18f0734b"
},
{
"index": 68,
"display": false,
"tex": "\\chi_A''(q,\\omega)",
"line": 415,
"sha256": "37df76ceeb1dbe06bebded0404e2bf965f1c14b710800e709ca68eb1a82f41e9"
},
{
"index": 69,
"display": false,
"tex": "r=0",
"line": 416,
"sha256": "d3d30bd2fde29232f8d504f67d77bf6f6666b626295e01154eabcab8ab9d7b88"
},
{
"index": 70,
"display": false,
"tex": "\\chi_A^R",
"line": 416,
"sha256": "b9cb046576f3c9503da59952d714e8742e10a55d165506a0cd92f7bfac24353a"
},
{
"index": 71,
"display": true,
"tex": "\\chi_A^R(q,\\omega)\n=\nq^{-2}\n\\Phi_A\\left(\n\\frac{\\omega}{\\Gamma q^2}\n\\right)",
"line": 418,
"sha256": "fa110ce61624509028e3f276caaf5f7f747f3d61e99f36e0f0d27c67ff2c57fe"
},
{
"index": 72,
"display": false,
"tex": "z_A",
"line": 427,
"sha256": "bfe73a955498b7842e07eabe92a2e04940d207de2e633eb7290607dfec5749ae"
},
{
"index": 73,
"display": false,
"tex": "\\chi^R(q,0)",
"line": 430,
"sha256": "b4801ca9eb9963bc526cda885666321a586b6ee50ce2d7e52b50f937907a6d12"
},
{
"index": 74,
"display": false,
"tex": "q^2",
"line": 431,
"sha256": "be871d9e7245f709a6d83f34c1ca1efb0ab1509493cb84b57914fbc0b4d79ab1"
},
{
"index": 75,
"display": false,
"tex": "\\zeta_B=-\\nabla\\cdot\\boldsymbol\\xi",
"line": 431,
"sha256": "93706ccd4fa530444cc8eadc6ee8d005fe349142d14334624679e9ffd2e64cd8"
},
{
"index": 76,
"display": true,
"tex": "f(\\omega)=\\frac{\\omega}{a^2+\\omega^2},",
"line": 438,
"sha256": "44131d07aa82ff44d9ba05d7cf95c2d87c86efe5cc091bcf093481292733fa2b"
},
{
"index": 77,
"display": false,
"tex": "\\dd f/\\dd\\omega=0",
"line": 442,
"sha256": "ea5390e2de81aa4a0ef6ceaa3d93608d5a8f0c23d36bd3395285b1971d287718"
},
{
"index": 78,
"display": false,
"tex": "z",
"line": 449,
"sha256": "594e519ae499312b29433b7dd8a97ff068defcba9755b6d5d00e84c524d67b06"
},
{
"index": 79,
"display": false,
"tex": "q",
"line": 449,
"sha256": "8e35c2cd3bf6641bdb0e2050b76932cbb2e6034a0ddacc1d9bea82a6ba57f7cf"
},
{
"index": 80,
"display": false,
"tex": "\\omega",
"line": 449,
"sha256": "11baa595827a4e0fd17af205816653eb40ea4c8410ba94a26118d3e60292d4e4"
},
{
"index": 81,
"display": true,
"tex": "\\chi_A^R(q,\\omega)\n=\n\\frac{\\Gamma}{\n\\Gamma(r+q^2)-\\ii\\omega\n},",
"line": 462,
"sha256": "19ffaa40283d5d3dc392a8c5524bd3c6ac6d74c2503b62a9ebd40d5f44dad944"
},
{
"index": 82,
"display": true,
"tex": "\\chi_A''(q,\\omega)\n=\n\\frac{\n\\Gamma\\omega\n}{\n\\Gamma^2(r+q^2)^2+\\omega^2\n}.",
"line": 472,
"sha256": "1c1171256d0eac84f5316100d2947d16cb0fce597ae880dba57325857ac82ecf"
},
{
"index": 83,
"display": false,
"tex": "\\omega",
"line": 482,
"sha256": "11baa595827a4e0fd17af205816653eb40ea4c8410ba94a26118d3e60292d4e4"
},
{
"index": 84,
"display": true,
"tex": "\\omega_{\\rm peak}\n=\n\\Gamma(r+q^2).",
"line": 484,
"sha256": "cb2bb13327f467c1855298cb3537969821ab28e5bbb8463808353804afb75fca"
},
{
"index": 85,
"display": false,
"tex": "r=0",
"line": 490,
"sha256": "d3d30bd2fde29232f8d504f67d77bf6f6666b626295e01154eabcab8ab9d7b88"
},
{
"index": 86,
"display": true,
"tex": "\\chi_A^R(q,\\omega)\n=\nq^{-2}\n\\frac1{\n1-\\ii\\omega/(\\Gamma q^2)\n}.",
"line": 492,
"sha256": "e8beb12cfa26609a9e2e73a086e35dd88325b7bfa35f81ac5d9cafbf0e88ef70"
},
{
"index": 87,
"display": true,
"tex": "z_A^{\\rm Gaussian}=2.",
"line": 503,
"sha256": "40739a84192f6d7259702ed367f74ada59d6efd470ec05f3edda15be1e4e5d06"
},
{
"index": 88,
"display": true,
"tex": "\\chi_B^R(q,\\omega)\n=\n\\frac{\n\\Gamma q^2\n}{\n\\Gamma q^2(r+q^2)-\\ii\\omega\n}.",
"line": 509,
"sha256": "3f3a83d56e9d2ff1d5bdcb790464c53280dea21610a63518317d0bd095127c72"
},
{
"index": 89,
"display": true,
"tex": "\\chi_B^R(q,\\omega)\n=\n\\frac1{\nr+q^2-\\ii\\omega/(\\Gamma q^2)\n}.",
"line": 521,
"sha256": "8d1ed81089f094c6677fae3b51925f358ef14a931c2fd8b2de866deadc678658"
},
{
"index": 90,
"display": false,
"tex": "r=0",
"line": 529,
"sha256": "d3d30bd2fde29232f8d504f67d77bf6f6666b626295e01154eabcab8ab9d7b88"
},
{
"index": 91,
"display": true,
"tex": "\\chi_B^R(q,\\omega)\n=\nq^{-2}\n\\frac1{\n1-\\ii\\omega/(\\Gamma q^4)\n}.",
"line": 531,
"sha256": "7696188f5d740467b2eac7e39bb1d6432ab681dde41bc9bc4e1ffd8409fee6f2"
},
{
"index": 92,
"display": true,
"tex": "z_B^{\\rm Gaussian}=4.",
"line": 542,
"sha256": "5231d384da9c6207fb137b5b1b7a21d12e83788e7040c09a070535ec9a7d5eaa"
},
{
"index": 93,
"display": false,
"tex": "\\omega=0",
"line": 546,
"sha256": "823c15d0519e391335eb8399e7b1a6eae7ab8cbacc7ce3cd100101e87beeea99"
},
{
"index": 94,
"display": true,
"tex": "\\chi_A^R(q,0)\n=\n\\chi_B^R(q,0)\n=\n\\frac1{r+q^2}.",
"line": 548,
"sha256": "ea687d337e0f79e239286ac0d5f82662a99c6ef7d00e05d843df496846bc5bcf"
},
{
"index": 95,
"display": true,
"tex": "\\zeta_B(\\mathbf q,t)\n=\n-\\ii\\mathbf q\\cdot\n\\boldsymbol\\xi(\\mathbf q,t).",
"line": 558,
"sha256": "7bcec60805d2053d7fd08caea3225cac7be9394f6cc94be764d61d14886d41e0"
},
{
"index": 96,
"display": true,
"tex": "\\left\\langle\n\\zeta_B(\\mathbf q,t)\n\\zeta_B(-\\mathbf q,t')\n\\right\\rangle\n=\n2\\Gamma Tq^2\\delta(t-t').",
"line": 567,
"sha256": "b6994c8fbaa7d2f580b93b9d7fc47756bb1c376f6dcfd080a2748105d6c84edf"
},
{
"index": 97,
"display": false,
"tex": "q^2",
"line": 576,
"sha256": "be871d9e7245f709a6d83f34c1ca1efb0ab1509493cb84b57914fbc0b4d79ab1"
},
{
"index": 98,
"display": false,
"tex": "\\chi^R(q,\\omega)",
"line": 582,
"sha256": "5b78edbe9020e6794284f2885a6c3661f8aaf3f70d37cb22fb7a801238064bb2"
},
{
"index": 99,
"display": false,
"tex": "\\phi^4",
"line": 582,
"sha256": "ab8c353106812d45b6f7e5039b564e326450e2554cef53f68e707816b222658e"
},
{
"index": 100,
"display": false,
"tex": "\\phi^4",
"line": 598,
"sha256": "ab8c353106812d45b6f7e5039b564e326450e2554cef53f68e707816b222658e"
},
{
"index": 101,
"display": false,
"tex": "\\mathcal K_A(q)=\\Gamma",
"line": 599,
"sha256": "4565b642b2233e39565b4b09c98eb2008ca13e380f613523f02a5f014fefffea"
},
{
"index": 102,
"display": false,
"tex": "\\mathcal K_B(q)=\\Gamma q^2",
"line": 599,
"sha256": "54636dc9719244ce70ff92b7e6b7ac68785cdfbb34807a93e0c8a57966c3f2a8"
},
{
"index": 103,
"display": false,
"tex": "\\Gamma",
"line": 601,
"sha256": "57885e4c75965b23b196228446c46b00876b0951643235b678d18199f6dce5ec"
}
]
}