{ "schema_version": 1, "id": "PHYS-2026-07-25-01", "canonical_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-07-25-01/", "source_markdown_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-07-25-01.md", "metadata": { "schema_version": 1, "id": "PHYS-2026-07-25-01", "date": "2026-07-25", "updated_at": "2026-07-25", "title": "When logarithmic entanglement lies about universality", "summary": "The XXZ ferromagnetic endpoint shows how a Dicke state can imitate conformal entanglement scaling while its excitations remain nonrelativistic, and why MPS diagnostics need independent dynamical checks.", "language": "en", "entry_kind": "daily", "status": "published", "level": "graduate-advanced", "user_difficulty": "unrated", "domains": [ "condensed-matter", "quantum-field-theory", "quantum-information" ], "estimated_minutes": 55 }, "content_markdown": "\n## Key claim\n\n**Logarithmic entanglement does not, by itself, imply conformal criticality.**\n\nAn MPS can reproduce the logarithm and still misidentify the physics that generated it.\n\n## Theme\n\n**The XXZ ferromagnetic endpoint: when entanglement scaling lies about universality.**\n\n## Guiding question\n\nIn the spin-$1/2$ XXZ chain, can a CFT entropy formula and finite-entanglement scaling distinguish a Luttinger liquid from the exceptional point $\\Delta=-1$?\n\nThe collision joins three subjects:\n\n- integrable spin chains;\n- conformal field theory;\n- matrix product states.\n\nThe trap is unusually clean. The endpoint has logarithmic ground-state entanglement, but its low-energy dynamics are not relativistic.\n\n## Setup\n\nTake an even periodic chain,\n\n$$\nH(\\Delta)\n=J\\sum_{j=1}^{N}\n\\left(\nS_j^xS_{j+1}^x\n+S_j^yS_{j+1}^y\n+\\Delta S_j^zS_{j+1}^z\n\\right),\n\\qquad J>0.\n$$\n\nFor $-1<\\Delta\\leq1$, the thermodynamic low-energy theory is a $c=1$ compact boson, or Luttinger liquid.\n\nThe endpoint $\\Delta=-1$ is different. Apply a staggered $\\pi$ rotation about $z$:\n\n$$\nU=\\prod_{\\substack{j=1\\\\j\\ {\\rm even}}}^{N}\n\\exp\\left(\\ii\\pi S_j^z\\right).\n$$\n\nBecause every bond contains one even site,\n\n$$\nUH(-1)U^\\dagger\n=-J\\sum_j\\mathbf S_j\\cdot\\mathbf S_{j+1}.\n$$\n\nThus the endpoint is unitarily equivalent to the isotropic Heisenberg ferromagnet.\n\nIts ground-state manifold has maximal total spin $S=N/2$. Fixing $S^z_{\\rm tot}=0$ selects the half-filled Dicke state, up to the staggered onsite rotation.\n\nThat rotation factorizes as $U_A\\otimes U_{\\bar A}$ across any spatial cut. It therefore leaves the entanglement spectrum unchanged.\n\nFor a block of $L$ sites,\n\n$$\n\\left|D_N^{N/2}\\right\\rangle\n=\\sum_k\\sqrt{p_k}\\,\n\\left|D_L^k\\right\\rangle\n\\left|D_{N-L}^{N/2-k}\\right\\rangle,\n$$\n\nwith\n\n$$\np_k\n=\\frac{\n\\binom{L}{k}\n\\binom{N-L}{N/2-k}\n}{\n\\binom{N}{N/2}\n}.\n$$\n\nThe Schmidt probabilities form a hypergeometric distribution. This gives an exact route to the logarithm without invoking CFT.\n\n## Analysis\n\n### Derivation — solve the endpoint in its native language\n\nSet $L=N/2$. The hypergeometric distribution has\n\n$$\n\\langle k\\rangle=\\frac{N}{4},\n\\qquad\n\\operatorname{Var}(k)\n=\\frac{N^2}{16(N-1)}\n\\sim\\frac{N}{16}.\n$$\n\nIts width is therefore\n\n$$\n\\sigma\\sim\\frac{\\sqrt N}{4}.\n$$\n\nNear its peak, $p_k$ approaches a discrete Gaussian. Its Shannon entropy is\n\n$$\nS_A\n=-\\sum_kp_k\\log p_k\n=\\frac12\\log\\left(2\\pi\\ee\\sigma^2\\right)+o(1).\n$$\n\nHence\n\n$$\nS_A=\\frac12\\log N+O(1).\n$$\n\nThe logarithm records the number of macroscopically plausible ways to distribute a fixed total magnetization between the two halves.\n\nIt does not come from a tower of relativistic local modes.\n\nThe distinction is visible in the excitation spectrum. For the rotated ferromagnet, a one-magnon excitation has\n\n$$\n\\varepsilon(q)\n\\propto1-\\cos q\n\\sim\\frac{q^2}{2}.\n$$\n\nThus\n\n$$\nz=2,\n$$\n\nwhereas an ordinary $(1+1)$-dimensional CFT requires linear low-energy dispersion and $z=1$.\n\n### Scope and assumptions — attack “logarithm means central charge”\n\nFor a periodic conformal chain, the von Neumann entropy of an interval of length $L$ is\n\n$$\nS_A(L,N)\n=\\frac{c}{3}\n\\log\\left[\n\\frac{N}{\\pi a}\n\\sin\\left(\\frac{\\pi L}{N}\\right)\n\\right]\n+s_0.\n$$\n\nHere $a$ is a short-distance cutoff. At $L=N/2$, the coefficient of $\\log N$ is $c/3$.\n\nIf the Dicke result were inserted blindly, one would infer\n\n$$\n\\frac{c_{\\rm eff}}{3}=\\frac12,\n\\qquad\nc_{\\rm eff}=\\frac32.\n$$\n\nThis number is an **effective fit coefficient**, not the central charge of an endpoint CFT. No relativistic CFT governs the endpoint.\n\nThe same functional form has arisen from a different mechanism: a collective conserved-charge fluctuation inside a highly degenerate ferromagnetic ground-state manifold.\n\nThis is not only a conceptual counterexample. Near $\\Delta=-1$, finite systems show a long crossover between the endpoint coefficient and the neighboring $c=1$ regime.\n\nEntropy fits can therefore return plausible but scale-dependent central charges.\n\nThe correct question is not merely\n\n$$\n\\text{“Is }S_A\\text{ logarithmic?”}\n$$\n\nIt is\n\n$$\n\\text{“Which degrees of freedom produce the logarithm?”}\n$$\n\n### Physical interpretation — what an MPS actually knows\n\nFirst separate two geometries.\n\nFor an open-boundary MPS cut across one virtual bond of dimension $\\chi$,\n\n$$\n\\operatorname{rank}\\rho_A\\leq\\chi,\n\\qquad\nS_A\\leq\\log\\chi.\n$$\n\nFor an interval in a periodic MPS, the bipartition crosses two virtual bonds:\n\n$$\n\\operatorname{rank}\\rho_A\\leq\\chi^2,\n\\qquad\nS_A\\leq2\\log\\chi.\n$$\n\nThese are exact algebraic bounds. They do not diagnose a universality class.\n\nNow consider an optimized uniform MPS for a genuine one-dimensional conformal ground state. Finite $\\chi$ induces an effective correlation length $\\xi_\\chi$.\n\nFor the entropy across one cut,\n\n$$\nS_\\chi\n\\simeq\\frac{c}{6}\\log\\xi_\\chi+s_\\chi,\n\\qquad\n\\xi_\\chi\\propto\\chi^\\kappa.\n$$\n\nThe standard finite-entanglement theory predicts\n\n$$\n\\kappa\n=\\frac{6}{\nc\\left(\\sqrt{12/c}+1\\right)\n}\n$$\n\nunder its CFT assumptions.\n\nThis theory is more than the inequality $S\\leq\\log\\chi$. It uses the universal distribution of entanglement-spectrum eigenvalues at a conformal critical point.\n\nFinite $\\chi$ is often described as introducing a mass-generating relevant perturbation. That description is a useful analogy, not an operator identity.\n\nA 2026 analysis finds that the effective perturbations selected by optimal tensor-network approximations can differ from those guessed from the most relevant CFT operator alone.\n\nThe geometry of the variational manifold also matters.\n\n## False claim to diagnose\n\n> If $S_\\chi$ is linear in $\\log\\xi_\\chi$ and the fitted slope gives $c\\approx1$, the XXZ state must be a Luttinger liquid.\n\nThe claim is tempting because this is a standard finite-entanglement diagnostic.\n\nBut the fit tests one relation over a finite range. It does not independently establish $z=1$, a conformal tower, or freedom from crossover contamination.\n\nThe $\\Delta=-1$ state supplies the counterexample:\n\n$$\nS_A\\sim\\frac12\\log N,\n\\qquad\n\\varepsilon(q)\\sim q^2.\n$$\n\nA defensible diagnosis requires mutually consistent evidence:\n\n- entropy scaling;\n- low-momentum dispersion or finite-size energy gaps;\n- MPS transfer-matrix spectra;\n- correlation-function exponents;\n- stability of fitted parameters as $N$ and $\\chi$ increase.\n\nThe entropy coefficient is evidence. It is not a verdict.\n\n## What follows — and what does not\n\nThe practical conflict is controlled by three competing lengths:\n\n$$\nN,\n\\qquad\n\\xi_\\chi,\n\\qquad\n\\xi_{\\rm cross}(\\Delta).\n$$\n\nThe endpoint crossover scale $\\xi_{\\rm cross}$ grows as $\\Delta\\to-1^+$.\n\nIf $N$ or $\\xi_\\chi$ remains below that scale, a calculation performed inside the $c=1$ phase can still look endpoint-like.\n\nIf the accessible scale exceeds it at fixed $\\Delta>-1$, the asymptotic Luttinger-liquid behavior can emerge.\n\nWriting only\n\n$$\nN\\to\\infty,\n\\qquad\n\\chi\\to\\infty,\n\\qquad\n\\Delta\\to-1^+\n$$\n\nhides this competition. One must specify the path through scale space.\n\nAn MPS does not classify universality directly. It provides a compressed state whose entropy, transfer matrix, correlations, symmetry sector, and convergence must be interpreted together.\n\n## Exercise\n\nTake the half-filled Dicke state $\\left|D_N^{N/2}\\right\\rangle$ and divide an even chain into two equal halves.\n\n1. Derive its Schmidt probabilities $p_k$.\n2. Approximate $p_k$ by a Gaussian and obtain $S_A=\\frac12\\log N+O(1)$.\n3. For a one-cut Schmidt truncation, find the rank needed to retain a fixed fraction $0\nHint 1\n\nUse the entropy of a discrete Gaussian:\n\n$$\nH\\simeq\\frac12\\log\\left(2\\pi\\ee\\sigma^2\\right).\n$$\n\n\n\n
\nHint 2\n\nA fixed fraction of a Gaussian lies within a window of width $O(\\sigma)$ around its mean.\n\n
\n\n**Oral check 1.** Why is $S\\leq\\log\\chi$ exact for one MPS cut, while $\\xi_\\chi\\propto\\chi^\\kappa$ is not?\n\n**Oral check 2.** Which most directly separates the endpoint from the Luttinger liquid: the entropy coefficient, $\\langle S^z\\rangle$, or the low-$q$ dispersion?\n\n
\nSolution\n\nCounting configurations with $k$ up spins in the left half gives\n\n$$\np_k\n=\\frac{\n\\binom{N/2}{k}\n\\binom{N/2}{N/2-k}\n}{\n\\binom{N}{N/2}\n}.\n$$\n\nThis distribution has\n\n$$\n\\sigma^2\n=\\frac{N^2}{16(N-1)}\n\\sim\\frac{N}{16}.\n$$\n\nTherefore,\n\n$$\nS_A\n\\simeq\\frac12\\log\\left(2\\pi\\ee\\sigma^2\\right)\n=\\frac12\\log N+O(1).\n$$\n\nTo retain any fixed fraction $0\n\n## Check your understanding\n\nIs entanglement scaling a property of the infrared theory, the selected state sector, or the variational representation?\n\nName one measurement that separates those three layers in this example.\n\nYou may also reply with “deeper,” “too easy,” “too hard,” or your attempted derivation.\n\n## Further Reading\n\n- [Entanglement entropy scaling of the XXZ chain](https://arxiv.org/abs/1306.5828)\n- [Permutation operators, entanglement entropy, and the XXZ limit $\\Delta\\to-1^+$](https://arxiv.org/abs/1011.4706)\n- [Theory of finite-entanglement scaling at one-dimensional quantum critical points](https://arxiv.org/abs/0812.2903)\n- [On the origin of finite entanglement scaling](https://arxiv.org/abs/2607.15124)\n\n## Connections and next step\n\n- Pressure point: a logarithmic entropy is not a unique fingerprint of CFT.\n- Exact layer: the Dicke Schmidt spectrum and MPS rank bounds.\n- Dynamical layer: $z=2$ at the ferromagnetic endpoint versus $z=1$ in the Luttinger liquid.\n- Numerical layer: $N$, $\\xi_\\chi$, and $\\xi_{\\rm cross}$ compete near $\\Delta=-1$.\n- Revisit: extract $c$, $z$, and the Luttinger parameter from independent observables in three runs.\n", "content_sha256": "fa02621552e1e0bfa37c80ed51c8c5e46fcc11e2ce9a06a0dbcc1613941c8c70", "formula_count": 95, "formulas": [ { "index": 1, "display": false, "tex": "1/2", "line": 32, "sha256": "d939926f05444b0f4495fb9629ecbfa80d99a8ec1a20d06800ca5a3d5f4fd276" }, { "index": 2, "display": false, "tex": "\\Delta=-1", "line": 32, "sha256": "4825da6c6d2f977e77ca0f8fb26974c725619c15222af4cc10edd96d24f08ecd" }, { "index": 3, "display": true, "tex": "H(\\Delta)\n=J\\sum_{j=1}^{N}\n\\left(\nS_j^xS_{j+1}^x\n+S_j^yS_{j+1}^y\n+\\Delta S_j^zS_{j+1}^z\n\\right),\n\\qquad J>0.", "line": 46, "sha256": "5f3888676f67d8830e611a5450d56c54234bf8b1c238b9a24cca21de38401340" }, { "index": 4, "display": false, "tex": "-1<\\Delta\\leq1", "line": 57, "sha256": "f4ad080d947c8e4f2457cab7a1cf215157fd419f8aa080511889b2fb5ed95532" }, { "index": 5, "display": false, "tex": "c=1", "line": 57, "sha256": "682607c9a4877b48b14ae0b08d6a64e0010dc31282128f16b2d0056039c0f68c" }, { "index": 6, "display": false, "tex": "\\Delta=-1", "line": 59, "sha256": "4825da6c6d2f977e77ca0f8fb26974c725619c15222af4cc10edd96d24f08ecd" }, { "index": 7, "display": false, "tex": "\\pi", "line": 59, "sha256": "f1cf04093d08b6dc0d58a87396405cb9093a22e9ed62a95bb95ed46e5da97097" }, { "index": 8, "display": false, "tex": "z", "line": 59, "sha256": "594e519ae499312b29433b7dd8a97ff068defcba9755b6d5d00e84c524d67b06" }, { "index": 9, "display": true, "tex": "U=\\prod_{\\substack{j=1\\\\j\\ {\\rm even}}}^{N}\n\\exp\\left(\\ii\\pi S_j^z\\right).", "line": 61, "sha256": "1205196f1761ad0804f94e33263547116656770af15d849801065d351c2f1bb9" }, { "index": 10, "display": true, "tex": "UH(-1)U^\\dagger\n=-J\\sum_j\\mathbf S_j\\cdot\\mathbf S_{j+1}.", "line": 68, "sha256": "b1439d7a632121fe7051847f4c5d4c9543a15599b1f9fc689d2698b895913c7b" }, { "index": 11, "display": false, "tex": "S=N/2", "line": 75, "sha256": "49f6995e5d4845ddf63c647acc5aa6f97871baab3a2865d6bd782c5d3a3ee465" }, { "index": 12, "display": false, "tex": "S^z_{\\rm tot}=0", "line": 75, "sha256": "fc8f0024b81926c7b98b77abcea516b77023ed7d7e7d021439231fc007bfc267" }, { "index": 13, "display": false, "tex": "U_A\\otimes U_{\\bar A}", "line": 77, "sha256": "90fbfded6a6181127b41fd3fce53d59d0fd423f0f06c83cd1732c3a40cf9a902" }, { "index": 14, "display": false, "tex": "L", "line": 79, "sha256": "72dfcfb0c470ac255cde83fb8fe38de8a128188e03ea5ba5b2a93adbea1062fa" }, { "index": 15, "display": true, "tex": "\\left|D_N^{N/2}\\right\\rangle\n=\\sum_k\\sqrt{p_k}\\,\n\\left|D_L^k\\right\\rangle\n\\left|D_{N-L}^{N/2-k}\\right\\rangle,", "line": 81, "sha256": "4b392c6889ed401127703e95192c5c010a6bd27dc82756100b38611b080f524c" }, { "index": 16, "display": true, "tex": "p_k\n=\\frac{\n\\binom{L}{k}\n\\binom{N-L}{N/2-k}\n}{\n\\binom{N}{N/2}\n}.", "line": 90, "sha256": "ffa4a630f83eaab6140ff13d20b8334e6ddcdbb52e7ed4269d5e8b946f30a82c" }, { "index": 17, "display": false, "tex": "L=N/2", "line": 106, "sha256": "f8cd4c4ace8a489316eaec3457b0d3f001de4274142428467ebe72f4410b3fbd" }, { "index": 18, "display": true, "tex": "\\langle k\\rangle=\\frac{N}{4},\n\\qquad\n\\operatorname{Var}(k)\n=\\frac{N^2}{16(N-1)}\n\\sim\\frac{N}{16}.", "line": 108, "sha256": "ef9c223363d5ac7264c15daeea0a4726b1bc46af5c90ed6308d8a32940beb692" }, { "index": 19, "display": true, "tex": "\\sigma\\sim\\frac{\\sqrt N}{4}.", "line": 118, "sha256": "9052a3d334428887a0d3131dbacdc151505482fc8a0dce8026621ddb5fff2356" }, { "index": 20, "display": false, "tex": "p_k", "line": 122, "sha256": "dc7ba1c49968c37c0fec8cb5588f9003e3c4041467b378dd3723242603e8b0b5" }, { "index": 21, "display": true, "tex": "S_A\n=-\\sum_kp_k\\log p_k\n=\\frac12\\log\\left(2\\pi\\ee\\sigma^2\\right)+o(1).", "line": 124, "sha256": "607155e15a1cad7c07184224938cb27362d80f023117e7ab92731128622ba3df" }, { "index": 22, "display": true, "tex": "S_A=\\frac12\\log N+O(1).", "line": 132, "sha256": "90f797e9b9232624ba817520d760800ba3ea9aa0f8b2689676731d74ea780971" }, { "index": 23, "display": true, "tex": "\\varepsilon(q)\n\\propto1-\\cos q\n\\sim\\frac{q^2}{2}.", "line": 142, "sha256": "4ca914754ffb5d5732c97a27222dc329b0cf7ac5375b6fd6178eec58d3345992" }, { "index": 24, "display": true, "tex": "z=2,", "line": 150, "sha256": "48000ae3cd0005778db61fa7904e3db5a32f1150c51cfeb8910651270a773ccc" }, { "index": 25, "display": false, "tex": "(1+1)", "line": 154, "sha256": "b5d8662b9e593119eff10b6ca8657bf76d4b036400832a0f4cf321718d9c1dd1" }, { "index": 26, "display": false, "tex": "z=1", "line": 154, "sha256": "9a545057d452f075f6ddfd3cbdb6d8ff4711b645e07ab13898b49316e7804ca0" }, { "index": 27, "display": false, "tex": "L", "line": 158, "sha256": "72dfcfb0c470ac255cde83fb8fe38de8a128188e03ea5ba5b2a93adbea1062fa" }, { "index": 28, "display": true, "tex": "S_A(L,N)\n=\\frac{c}{3}\n\\log\\left[\n\\frac{N}{\\pi a}\n\\sin\\left(\\frac{\\pi L}{N}\\right)\n\\right]\n+s_0.", "line": 160, "sha256": "68aac809306833dcca83de6d40a43e556a2a26abc874f8d3ecb1873a41fa36d6" }, { "index": 29, "display": false, "tex": "a", "line": 170, "sha256": "ca978112ca1bbdcafac231b39a23dc4da786eff8147c4e72b9807785afee48bb" }, { "index": 30, "display": false, "tex": "L=N/2", "line": 170, "sha256": "f8cd4c4ace8a489316eaec3457b0d3f001de4274142428467ebe72f4410b3fbd" }, { "index": 31, "display": false, "tex": "\\log N", "line": 170, "sha256": "8fb578943dbcccf44699508273a7a9272275ae05c1624dcd12dacdf12beaaa25" }, { "index": 32, "display": false, "tex": "c/3", "line": 170, "sha256": "907d716e5845120e64a9e1cb660bcc0ef44933d879092ff66fa3af69a7c732d9" }, { "index": 33, "display": true, "tex": "\\frac{c_{\\rm eff}}{3}=\\frac12,\n\\qquad\nc_{\\rm eff}=\\frac32.", "line": 174, "sha256": "34ecf3c6d13826412aefd208a01d4c78f7915ec2aca887af9b46e02347a694f8" }, { "index": 34, "display": false, "tex": "\\Delta=-1", "line": 184, "sha256": "4825da6c6d2f977e77ca0f8fb26974c725619c15222af4cc10edd96d24f08ecd" }, { "index": 35, "display": false, "tex": "c=1", "line": 184, "sha256": "682607c9a4877b48b14ae0b08d6a64e0010dc31282128f16b2d0056039c0f68c" }, { "index": 36, "display": true, "tex": "\\text{“Is }S_A\\text{ logarithmic?”}", "line": 190, "sha256": "9e7f92701cc6050229d52682cfcc0b075355cffda987014f466c721a372c19ce" }, { "index": 37, "display": true, "tex": "\\text{“Which degrees of freedom produce the logarithm?”}", "line": 196, "sha256": "5c7304f3ef41ff5b912e32d55fb4df0974d2ab1ef45bbe2640f349c6d9a39d9f" }, { "index": 38, "display": false, "tex": "\\chi", "line": 204, "sha256": "8d50b5beb7fe1e79d493a28bd7b5833f3403ebe867c4ac17d3967b67cfa06141" }, { "index": 39, "display": true, "tex": "\\operatorname{rank}\\rho_A\\leq\\chi,\n\\qquad\nS_A\\leq\\log\\chi.", "line": 206, "sha256": "15f5aeca0e6c8864c6427773ef0141e0744054c49f516cde6766812fec1dd56c" }, { "index": 40, "display": true, "tex": "\\operatorname{rank}\\rho_A\\leq\\chi^2,\n\\qquad\nS_A\\leq2\\log\\chi.", "line": 214, "sha256": "b65e477f0601704c6fbbddaf7db4e42dee41e77f1e3a3025f3f5e66ece8d1e00" }, { "index": 41, "display": false, "tex": "\\chi", "line": 222, "sha256": "8d50b5beb7fe1e79d493a28bd7b5833f3403ebe867c4ac17d3967b67cfa06141" }, { "index": 42, "display": false, "tex": "\\xi_\\chi", "line": 222, "sha256": "fcb8524642e49da0f5a000690edbee37d00f17a09062f8135549ea017b84bc15" }, { "index": 43, "display": true, "tex": "S_\\chi\n\\simeq\\frac{c}{6}\\log\\xi_\\chi+s_\\chi,\n\\qquad\n\\xi_\\chi\\propto\\chi^\\kappa.", "line": 226, "sha256": "4b4fa67dac1e66281fb3527a21e3cc0355b0b3c12ff6c37beebfd819d772ef82" }, { "index": 44, "display": true, "tex": "\\kappa\n=\\frac{6}{\nc\\left(\\sqrt{12/c}+1\\right)\n}", "line": 235, "sha256": "9b26b74aad0e4bb4c765e8449c34c7ebc114df144501fa893a47d86e2bf9c3b8" }, { "index": 45, "display": false, "tex": "S\\leq\\log\\chi", "line": 244, "sha256": "27e5bdde469066c11aad72a267e434b9a6762de46ea61f3be26ddebd7ddfc895" }, { "index": 46, "display": false, "tex": "\\chi", "line": 246, "sha256": "8d50b5beb7fe1e79d493a28bd7b5833f3403ebe867c4ac17d3967b67cfa06141" }, { "index": 47, "display": false, "tex": "S_\\chi", "line": 254, "sha256": "6694a281d05a71d41ad272a6614839b92fbb1c111f7f172a90c0dc4f7cf6502d" }, { "index": 48, "display": false, "tex": "\\log\\xi_\\chi", "line": 254, "sha256": "cc2eef3a3f4427c96c21ad2d84082b06af1136c2011aad3c3e4f0d6fc0fc597a" }, { "index": 49, "display": false, "tex": "c\\approx1", "line": 254, "sha256": "c506449bfb65eeb6e531ff3b10d307adfa44bcbbc8eadef3d241dd65bf0f136e" }, { "index": 50, "display": false, "tex": "z=1", "line": 258, "sha256": "9a545057d452f075f6ddfd3cbdb6d8ff4711b645e07ab13898b49316e7804ca0" }, { "index": 51, "display": false, "tex": "\\Delta=-1", "line": 260, "sha256": "4825da6c6d2f977e77ca0f8fb26974c725619c15222af4cc10edd96d24f08ecd" }, { "index": 52, "display": true, "tex": "S_A\\sim\\frac12\\log N,\n\\qquad\n\\varepsilon(q)\\sim q^2.", "line": 262, "sha256": "41aca5886d3b525dfa865a0b6c50f1b1552eb79cd3561319ba98c4cea465a6c3" }, { "index": 53, "display": false, "tex": "N", "line": 274, "sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b" }, { "index": 54, "display": false, "tex": "\\chi", "line": 274, "sha256": "8d50b5beb7fe1e79d493a28bd7b5833f3403ebe867c4ac17d3967b67cfa06141" }, { "index": 55, "display": true, "tex": "N,\n\\qquad\n\\xi_\\chi,\n\\qquad\n\\xi_{\\rm cross}(\\Delta).", "line": 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