{
  "schema_version": 1,
  "id": "PHYS-2026-07-27-01",
  "canonical_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-07-27-01/",
  "source_markdown_url": "https://rimoooliii.github.io/physicsday/physics/PHYS-2026-07-27-01.md",
  "metadata": {
    "schema_version": 1,
    "id": "PHYS-2026-07-27-01",
    "date": "2026-07-27",
    "updated_at": "2026-07-28",
    "title": "Many-body path integrals beyond tunnelling",
    "summary": "How interacting path integrals connect auxiliary fields and collective saddles to sign problems, Keldysh contours, complex manifolds, generative sampling, and temporal tensor networks.",
    "language": "en",
    "entry_kind": "daily",
    "status": "published",
    "level": "graduate-advanced",
    "user_difficulty": "unrated",
    "domains": [
      "quantum-field-theory",
      "condensed-matter",
      "statistical-mechanics"
    ],
    "estimated_minutes": 70
  },
  "content_markdown": "\n## Key claim\n\n**A many-body path integral sums over more than particle trajectories.**\n\nIts integration variables may be coherent-state amplitudes, Grassmann fields, order parameters, gauge fields, noise histories, replicas, or auxiliary fields that were absent from the original Hamiltonian.\n\nInstantons form one saddle class within this functional integral.\n\n## Theme\n\n**Interacting path integrals: exact representations, collective fields, and nonperturbative saddles.**\n\n## Guiding question\n\nWhen you write an interacting quantum system as a path integral, which parts are:\n\n1. an exact rewriting of the operator problem;\n2. a controlled collective-field approximation;\n3. a numerical sampling problem;\n4. a real-time interference problem;\n5. a search for nonperturbative saddles?\n\nA Trotter identity, a saddle approximation, and a Monte Carlo estimator make different claims and fail for different reasons.\n\n## Setup: one model, several path integrals\n\nTake the attractive two-site Bose–Hubbard model at fixed total particle number $N$:\n\n$$\nH\n=-\\frac K2\n\\left(\na_L^\\dagger a_R+a_R^\\dagger a_L\n\\right)\n-\\frac g2\n\\sum_{\\alpha=L,R}\nn_\\alpha(n_\\alpha-1),\n\\qquad g>0.\n$$\n\nStart from the equilibrium operator identity\n\n$$\nZ=\\Tr\\ee^{-\\beta H}.\n$$\n\nInsert coherent-state resolutions of the identity between short imaginary-time steps. Continuum notation then gives\n\n$$\nZ\n=\\int_{\\rm periodic}\n\\mathcal D\\bar\\psi\\,\\mathcal D\\psi\\,\n\\ee^{-S_E[\\bar\\psi,\\psi]}.\n$$\n\nThe definition still includes the time lattice, measure, boundary conditions, and operator ordering. The displayed continuum integral abbreviates their controlled limit.\n\nAt fixed $N$, introduce the population imbalance and relative phase,\n\n$$\nz=\\frac{n_L-n_R}{N},\n\\qquad\n\\phi=\\phi_L-\\phi_R.\n$$\n\nOn the spin-coherent-state sphere, the classical Hamiltonian is, up to a constant,\n\n$$\nH_{\\rm cl}(z,\\phi)\n=-\\frac{KN}{2}\n\\sqrt{1-z^2}\\cos\\phi\n-\\frac{gN^2}{4}z^2.\n$$\n\nDefine\n\n$$\n\\Lambda=\\frac{gN}{K}.\n$$\n\nFor $\\Lambda>1$, mean-field theory has two degenerate minima,\n\n$$\nz=\\pm z_0,\n\\qquad\nz_0=\\sqrt{1-\\Lambda^{-2}},\n\\qquad\n\\phi=0.\n$$\n\nThe collective Euclidean action is\n\n$$\nS_E[z,\\phi]\n=\\int\\dd\\tau\n\\left[\n\\frac{\\ii N}{2}(1-z)\\dot\\phi\n+H_{\\rm cl}(z,\\phi)\n\\right],\n$$\n\nup to a Berry-phase gauge choice and boundary terms.\n\nThe Bose–Hubbard dimer supplies a concrete instanton example within a much wider path-integral framework.\n\n## The Path-Integral Ladder\n\n### Level 1: exact representation\n\nTrotter decomposition and coherent-state insertion rewrite an operator trace or amplitude as a sum over histories, without assuming a saddle point.\n\n### Level 2: exact change or enlargement of variables\n\nA Hubbard–Stratonovich transformation introduces an auxiliary field to linearize an interaction. The field need not correspond to a microscopic observable.\n\n### Level 3: controlled or uncontrolled truncation\n\nA stationary auxiliary field gives mean-field theory. Its Hessian gives Gaussian fluctuations, collective modes, and often RPA-type response.\n\nControl requires a small parameter, a large-$N$ limit, weak coupling, or a demonstrated separation of scales.\n\n### Level 4: nonperturbative sectors\n\nAn expansion about one vacuum misses instantons, vortices, skyrmions, bounces, and other complex saddles.\n\nBoundary conditions and the integration contour decide which saddles contribute. The stationary equations only identify candidates.\n\n### Level 5: computational representation\n\nResearchers may sample the discretized integral by Monte Carlo, reorganize it diagrammatically, deform its contour, or contract it as a tensor network. Formal exactness says nothing about computational cost.\n\n## Analysis\n\n### Derivation: interactions become fluctuating fields\n\nConsider a schematic fermionic action\n\n$$\nS[\\bar\\psi,\\psi]\n=\n\\bar\\psi G_0^{-1}\\psi\n+\\frac U2\\int_x\n\\mathcal O(x)^2.\n$$\n\nA continuous Hubbard–Stratonovich identity has the form\n\n$$\n\\exp\\left[\n-\\frac U2\\mathcal O^2\n\\right]\n\\propto\n\\int\\dd\\phi\\,\n\\exp\\left[\n-\\frac{\\phi^2}{2U}\n+\\ii\\phi\\mathcal O\n\\right],\n$$\n\nwith contours and factors adjusted to the sign of $U$ and the chosen channel.\n\nThe fermions are now quadratic and can be integrated out:\n\n$$\nZ\n=\\int\\mathcal D\\phi\\,\n\\ee^{-S_{\\rm eff}[\\phi]},\n$$\n\nwhere\n\n$$\nS_{\\rm eff}[\\phi]\n=\n\\frac{1}{2U}\\int_x\\phi(x)^2\n-\\Tr\\log\n\\left[\nG_0^{-1}+\\ii\\Gamma\\phi\n\\right].\n$$\n\nThe saddle equation\n\n$$\n\\frac{\\delta S_{\\rm eff}}{\\delta\\phi(x)}=0\n$$\n\nis a self-consistent mean-field or gap equation.\n\nExpanding\n\n$$\n\\phi=\\phi_\\star+\\delta\\phi\n$$\n\ngives\n\n$$\nS_{\\rm eff}\n=S_{\\rm eff}[\\phi_\\star]\n+\\frac12\n\\delta\\phi\\,\\Gamma^{(2)}\\,\\delta\\phi\n+O(\\delta\\phi^3).\n$$\n\nZeros or poles of the inverse fluctuation kernel $\\Gamma^{(2)}$ determine collective modes and instabilities.\n\nDepending on the chosen channel, the field $\\phi$ describes density, magnetization, pairing, or another collective degree of freedom.\n\nDifferent exact HS decompositions reproduce the same untruncated integral. Once they are truncated at saddle level, however, they can disagree.\n\nThis channel dependence is the practical content of the Fierz ambiguity.\n\n### Scope and assumptions: why the measure may not be positive\n\nAfter fermions are integrated out, auxiliary-field Monte Carlo often produces\n\n$$\nZ\n=\\int\\mathcal D\\phi\\,\n\\ee^{-S_B[\\phi]}\n\\det M_\\uparrow[\\phi]\n\\det M_\\downarrow[\\phi].\n$$\n\nSome symmetries force the determinant product to be nonnegative. The half-filled repulsive Hubbard model on a bipartite lattice is a standard example.\n\nWhen those conditions fail, the weight may change sign or acquire a complex phase.\n\nSampling the absolute value gives\n\n$$\n\\langle\\ee^{\\ii\\theta}\\rangle_{|w|}\n=\\frac{Z}{Z_{|w|}}\n\\sim\n\\ee^{-\\beta V\\Delta f}.\n$$\n\nIf $\\Delta f>0$, the signal-to-noise ratio decays exponentially with inverse temperature $\\beta$ and volume $V$.\n\nThe generic sign problem is NP-hard. The result does not make every instance equally hard: symmetry removes the problem in several important model classes.\n\nA Trotterized action retains discretization error until the time-step limit is taken or a continuous-time formulation is used.\n\nCoherent-state actions also contain Berry phases and endpoint subtleties. Replacing operators naively by commuting numbers can miss ordering corrections.\n\nNear a critical point, soft directions develop in the Hessian. A Gaussian expansion may then fail where collective fluctuations matter most.\n\nDeriving $S_{\\rm eff}$ can be exact. Claiming that one smooth, real saddle dominates requires an additional argument.\n\n### Physical interpretation: choose the contour to match the question\n\nImaginary time computes equilibrium quantities:\n\n$$\nZ=\\Tr\\ee^{-\\beta H}\n=\\int\\mathcal D\\phi\\,\n\\ee^{-S_E[\\phi]}.\n$$\n\nFrom it one obtains thermodynamics, Matsubara correlators, and ground-state projection. Real-time response and nonequilibrium evolution instead use a forward-and-backward Schwinger–Keldysh contour:\n\n$$\nZ\n=\\int\n\\mathcal D\\phi_+\\,\n\\mathcal D\\phi_-\\,\n\\ee^{\n\\ii S[\\phi_+]\n-\\ii S[\\phi_-]\n}.\n$$\n\nThe two fields track the ket and bra. Together they enforce normalization and causality.\n\nThe Keldysh rotation\n\n$$\n\\phi_{\\rm cl}\n=\\frac{\\phi_++\\phi_-}{2},\n\\qquad\n\\phi_{\\rm q}\n=\\phi_+-\\phi_-\n$$\n\nseparates classical histories from response or quantum components.\n\nIntegrating out an environment gives the reduced system an influence action:\n\n$$\nZ_{\\rm red}\n=\\int\n\\mathcal D\\phi_+\\,\n\\mathcal D\\phi_-\\,\n\\ee^{\n\\ii S[\\phi_+]\n-\\ii S[\\phi_-]\n+\\ii S_{\\rm IF}[\\phi_+,\\phi_-]\n}.\n$$\n\nThe influence functional is generally nonlocal in time. Its retarded part encodes dissipation; its Keldysh component encodes fluctuations and noise.\n\nA stochastic weak-noise process can also have\n\n$$\nP[x]\\asymp\n\\ee^{-I[x]/\\epsilon}.\n$$\n\nIts optimal rare path obeys saddle equations much like a Euclidean instanton.\n\nThe histories mean different things. A stochastic path carries probability; a quantum path contributes a complex amplitude that can interfere with other saddles.\n\n## The Instanton Sector: one nonperturbative use\n\nReturn to the attractive Bose–Hubbard dimer.\n\nThe two mean-field minima are approximately\n\n$$\n\\ket{N,0},\n\\qquad\n\\ket{0,N}.\n$$\n\nA collective saddle connects $z=+z_0$ to $z=-z_0$ in Euclidean time.\n\nThe first-order Berry-phase action yields Hamilton equations in imaginary time rather than Newton’s equation for a particle in an inverted potential.\n\nThe phase $\\phi$ generally becomes complex along the connecting saddle. Eliminating it can produce a one-coordinate action for $z$, but with a coordinate-dependent kinetic term.\n\nAt large $N$,\n\n$$\n\\Delta E\n\\sim\nA(N,\\Lambda)\n\\ee^{-Nb(\\Lambda)}.\n$$\n\nHere\n\n$$\n\\hbar_{\\rm eff}\\sim\\frac1N.\n$$\n\nThe factor $N$ reflects collective semiclassical scaling, not $N$ independent particles paying identical tunnelling actions.\n\n### The microscopic Fock-space picture\n\nThe hopping operator transfers only one boson:\n\n$$\n\\ket{N-k+1,k-1}\n\\longrightarrow\n\\ket{N-k,k}.\n$$\n\nThe endpoint states first couple at order $N$ in $K$.\n\nFor the unperturbed interaction Hamiltonian,\n\n$$\nE_k-E_0=gk(N-k).\n$$\n\nThe virtual states near $k=N/2$ are the most energetically expensive. The collective instanton packages this chain of matrix elements and denominators into a saddle action and prefactor.\n\n### When the one-instanton picture fails\n\nNear $\\Lambda=1$, a collective mode softens. The fluctuation determinant becomes singular, so the standard dilute-instanton expansion is not uniform.\n\nCoupling to a gapless environment may generate a memory kernel:\n\n$$\nS_{\\rm diss}\n\\sim\n\\int\\dd\\tau\\,\\dd\\tau'\\,\n\\frac{\n[z(\\tau)-z(\\tau')]^2\n}{\n(\\tau-\\tau')^2\n}.\n$$\n\nThe bath can change the tunnelling exponent, destroy coherence, or produce a localization transition.\n\nMultiple complex saddles can also interfere. The saddle equations find candidates; the original contour determines which ones contribute.\n\n## Frontier Map: beyond one saddle\n\n### 1. Complex contours and Lefschetz thimbles\n\nHolomorphic flow deforms the original integration manifold into complex field space without crossing singularities.\n\nPhase fluctuations often shrink near a thimble. Hubbard-model studies report substantial improvement where the undeformed average sign is exponentially small.\n\nResidual phases, Jacobians, multimodal sampling, singularities, and interference among several thimbles can still dominate the calculation.\n\n### 2. Symmetry-designed sign-free formulations\n\nSome auxiliary-field decompositions produce determinant pairs related by antiunitary symmetry:\n\n$$\n\\det M_2[\\phi]\n=\n\\det M_1[\\phi]^*.\n$$\n\nThen\n\n$$\n\\det M_1\\det M_2\n=\n|\\det M_1|^2\n\\geq0.\n$$\n\nFor these model classes, Monte Carlo weights are nonnegative exactly. Researchers are enlarging the classes through Majorana representations, symmetry criteria, basis changes, and redesigned interactions.\n\n### 3. Generative and flow-based sampling\n\nNormalizing flows learn an invertible map from a simple reference distribution to field configurations.\n\nReweighting or a Metropolis correction can keep the resulting sampler asymptotically exact.\n\nGauge-equivariant flows can build local symmetry into the architecture. Proof-of-principle work has reached four-dimensional $SU(3)$ lattice gauge theory.\n\nTraining cost, mode coverage, volume scaling, topology freezing, and transfer between couplings still limit these methods.\n\nLearned proposals may explore the measure faster, but they leave the partition function unchanged and do not automatically cure a complex phase.\n\n### 4. Temporal tensor networks\n\nA discretized path integral forms a tensor network in spacetime. For an open system, TEMPO represents the influence functional as a matrix product operator along the time direction.\n\nThis compresses bath memory when temporal correlations have manageable bond dimension.\n\nRecent extensions treat Grassmann influence functionals for fermionic impurity models. 2026 preprints extend the framework to more general bosonic couplings and superconducting fermionic baths.\n\nTemporal entanglement or operator-space complexity sets the cost. Long memory and strong nonequilibrium correlations can force rapid bond-dimension growth.\n\nThe calculation compresses correlations among histories instead of sampling each history independently.\n\n## False claim to diagnose\n\n> Once an interacting system has been written as a path integral, the many-body problem has been solved; all that remains is approximation detail.\n\nThe rewriting can be exact while evaluation remains exponentially hard.\n\nThe action may be complex, nonlocal, multimodal, singular, or dominated by several interfering sectors.\n\nEven the choice of variables can move difficulty between the action, measure, contour, and observable.\n\nThe narrower claim is:\n\n> A path integral reorganizes the many-body problem through fields, topology, saddles, stochastic samples, or tensor contractions. None is guaranteed to be efficient.\n\n## What follows — and what does not\n\nFive layers are involved:\n\n| Layer | Object | Status |\n| --- | --- | --- |\n| Operator theory | $\\Tr\\ee^{-\\beta H}$ or a real-time amplitude | Exact starting problem |\n| Functional representation | Coherent, Grassmann, auxiliary, or contour fields | Exact when the measure and regulator are specified |\n| Effective description | Saddle, gradient expansion, or low-energy field | Controlled only with a scale argument |\n| Nonperturbative sector | Instantons, defects, complex saddles | Contribution depends on contour and boundary data |\n| Numerical realization | Monte Carlo, flows, diagrams, tensor contraction | Accuracy and cost are method-dependent |\n\nThe many-body instanton occupies the fourth row and cannot substitute for the other four.\n\n## Exercise: recover the instanton exponent microscopically\n\nWork in the strong-attraction regime\n\n$$\ngN\\gg K.\n$$\n\nStarting from $\\ket{N,0}$, compute the leading off-diagonal matrix element connecting it to $\\ket{0,N}$ using $N$th-order degenerate perturbation theory.\n\nShow that\n\n$$\n\\boxed{\n\\Delta_N\n=\n2\n\\frac{\n(K/2)^N N!\n}{\ng^{N-1}[(N-1)!]^2\n}\n}.\n$$\n\nAt fixed\n\n$$\n\\Lambda=\\frac{gN}{K}\\gg1,\n$$\n\nuse Stirling’s approximation to extract the leading exponential behavior.\n\nThen answer the transfer question:\n\n> Which part of this result would be missed by keeping only one Gaussian expansion around either minimum?\n\n<details>\n<summary>Hint 1</summary>\n\nThe $k$th hopping matrix element along the minimal Fock-space path is\n\n$$\n-\\frac K2\\sqrt{k(N-k+1)}.\n$$\n\n</details>\n\n<details>\n<summary>Hint 2</summary>\n\nUse\n\n$$\n\\prod_{k=1}^{N-1}k(N-k)\n=[(N-1)!]^2.\n$$\n\n</details>\n\n**Oral check 1.** Why does the coupling first appear at order $N$, even though the Hamiltonian is only quadratic in creation and annihilation operators?\n\n**Oral check 2.** In auxiliary-field Monte Carlo, are the sampled variables electron trajectories, microscopic observables, or decoupling fields?\n\n**Oral check 3.** Why can a Keldysh path integral not be interpreted as an ordinary probability distribution over histories?\n\n<details class=\"solution\">\n<summary>Solution</summary>\n\nThere is one minimal monotonic sequence:\n\n$$\n\\ket{N,0}\n\\to\n\\ket{N-1,1}\n\\to\\cdots\\to\n\\ket{0,N}.\n$$\n\nThe product of hopping matrix elements is\n\n$$\n\\prod_{k=1}^{N}\n\\left[\n-\\frac K2\n\\sqrt{k(N-k+1)}\n\\right]\n=\n\\left(-\\frac K2\\right)^N N!.\n$$\n\nThe $N-1$ intermediate denominators are\n\n$$\nE_0-E_k=-gk(N-k).\n$$\n\nTherefore,\n\n$$\n|t_{\\rm eff}|\n=\n\\frac{\n(K/2)^N N!\n}{\ng^{N-1}[(N-1)!]^2\n}.\n$$\n\nThe symmetric and antisymmetric combinations split by\n\n$$\n\\Delta_N=2|t_{\\rm eff}|.\n$$\n\nSubstitute\n\n$$\ng=\\frac{\\Lambda K}{N}.\n$$\n\nThen\n\n$$\n\\Delta_N\n=\nK\\,2^{1-N}\n\\frac{\nN^N\n}{\n\\Lambda^{N-1}(N-1)!\n}.\n$$\n\nStirling’s formula gives\n\n$$\n\\boxed{\n\\log\\Delta_N\n=\n-N[\\log(2\\Lambda)-1]\n+O(\\log N)\n}.\n$$\n\nThus\n\n$$\n\\Delta_N\n\\sim\n\\operatorname{poly}(N)\n\\exp\\left\\{\n-N[\\log(2\\Lambda)-1]\n\\right\\}.\n$$\n\nThe exponent is the strong-coupling limit of the collective instanton action.\n\nThe algebraic corrections belong to the prefactor and to finite-$N$ matching.\n\nA Gaussian expansion around one minimum describes perturbative fluctuations confined to that basin. It cannot generate the exponentially small matrix element between distinct saddles.\n\nThe auxiliary-field variables in Monte Carlo are decoupling fields. Their relation to observables depends on the chosen channel and source derivatives.\n\nA Keldysh weight is complex:\n\n$$\n\\ee^{\\ii S[\\phi_+]-\\ii S[\\phi_-]}.\n$$\n\nIt sums amplitudes with interference and therefore is not an ordinary positive history probability.\n\n</details>\n\n## Check your understanding\n\nTake one interacting problem you know and identify:\n\n1. the exact path-integral variables;\n2. any auxiliary or collective fields;\n3. the small parameter behind a saddle expansion;\n4. the main numerical obstruction;\n5. whether the relevant contour is Euclidean, Keldysh, or complex-deformed.\n\nWithout item 3, the approximation has no stated control parameter.\n\nYou may also reply with “deeper,” “too easy,” “too hard,” or your perturbative derivation.\n\n## Further Reading\n\n- [Computational complexity and fundamental limitations of the fermion sign problem](https://arxiv.org/abs/cond-mat/0408370)\n- [Taming the finite-density Hubbard sign problem with Lefschetz thimbles](https://arxiv.org/abs/1906.02726)\n- [Normalizing flows for lattice gauge theory in arbitrary spacetime dimension](https://arxiv.org/abs/2305.02402)\n- [Efficient non-Markovian dynamics using time-evolving matrix product operators](https://arxiv.org/abs/1711.09641)\n- [Grassmann TEMPO for fermionic path-integral simulations](https://arxiv.org/abs/2410.11541)\n- [Tensor-network influence functionals for general Gaussian bosonic baths](https://arxiv.org/abs/2603.23432)\n- [Grassmann TEMPO for superconducting baths](https://arxiv.org/abs/2604.23301)\n\n## Connections and next step\n\n- Pressure point: a path integral is a representation, not an automatic solution.\n- Exact layer: Trotterization, coherent states, and Hubbard–Stratonovich identities.\n- Approximation layer: mean field, Gaussian fluctuations, and collective effective actions.\n- Nonperturbative layer: instantons, defects, Berry phases, and complex saddles.\n- Computational layer: sign-free symmetry, contour deformation, generative sampling, and temporal tensor networks.\n- Revisit: derive the Keldysh influence action of an Ohmic bath in three runs.\n",
  "content_sha256": "8a2c7c3ab92cc3e265fbdbd9127ce9c56ceaf7ac1356c9889cdb67f7c129417f",
  "formula_count": 76,
  "formulas": [
    {
      "index": 1,
      "display": false,
      "tex": "N",
      "line": 46,
      "sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
    },
    {
      "index": 2,
      "display": true,
      "tex": "H\n=-\\frac K2\n\\left(\na_L^\\dagger a_R+a_R^\\dagger a_L\n\\right)\n-\\frac g2\n\\sum_{\\alpha=L,R}\nn_\\alpha(n_\\alpha-1),\n\\qquad g>0.",
      "line": 48,
      "sha256": "e51fb437b5294e36ffde6e97f566498500979860d773b6c1cc23591321b3a57a"
    },
    {
      "index": 3,
      "display": true,
      "tex": "Z=\\Tr\\ee^{-\\beta H}.",
      "line": 62,
      "sha256": "bbec622042328e6dc7b471c149f825afa715c2d5740fef7dfbf9a269e8334487"
    },
    {
      "index": 4,
      "display": true,
      "tex": "Z\n=\\int_{\\rm periodic}\n\\mathcal D\\bar\\psi\\,\\mathcal D\\psi\\,\n\\ee^{-S_E[\\bar\\psi,\\psi]}.",
      "line": 68,
      "sha256": "df435c40e7187fbefcabbaaf2953da298be87fa33b90896b32d340f85960aef0"
    },
    {
      "index": 5,
      "display": false,
      "tex": "N",
      "line": 77,
      "sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
    },
    {
      "index": 6,
      "display": true,
      "tex": "z=\\frac{n_L-n_R}{N},\n\\qquad\n\\phi=\\phi_L-\\phi_R.",
      "line": 79,
      "sha256": "8420827093be00b08953256d5b3752bd9f2db4de8c6a2c8a0990f04cf838bebf"
    },
    {
      "index": 7,
      "display": true,
      "tex": "H_{\\rm cl}(z,\\phi)\n=-\\frac{KN}{2}\n\\sqrt{1-z^2}\\cos\\phi\n-\\frac{gN^2}{4}z^2.",
      "line": 87,
      "sha256": "c4993455f0449b7d299ee9d8f15dccd559f5fb56e6748a0de565947aa0740f10"
    },
    {
      "index": 8,
      "display": true,
      "tex": "\\Lambda=\\frac{gN}{K}.",
      "line": 96,
      "sha256": "96b6fca75421e873cfac066830e577a6f1d533f07e45d89bfbf8b9f01edc34c5"
    },
    {
      "index": 9,
      "display": false,
      "tex": "\\Lambda>1",
      "line": 100,
      "sha256": "4f0a9d77ed0bd1079d9f41f5603a84084e8c38311f017fb79d13e9d33842f198"
    },
    {
      "index": 10,
      "display": true,
      "tex": "z=\\pm z_0,\n\\qquad\nz_0=\\sqrt{1-\\Lambda^{-2}},\n\\qquad\n\\phi=0.",
      "line": 102,
      "sha256": "d9d2abdcfef6f6a41f19c22cf97492d1540f4aa41fe6fa965d8c86926124c6e6"
    },
    {
      "index": 11,
      "display": true,
      "tex": "S_E[z,\\phi]\n=\\int\\dd\\tau\n\\left[\n\\frac{\\ii N}{2}(1-z)\\dot\\phi\n+H_{\\rm cl}(z,\\phi)\n\\right],",
      "line": 112,
      "sha256": "5314e8d0b4359f80a500f861c6fcc0eeb43c69a881402004b160f1e18839c242"
    },
    {
      "index": 12,
      "display": false,
      "tex": "N",
      "line": 139,
      "sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
    },
    {
      "index": 13,
      "display": true,
      "tex": "S[\\bar\\psi,\\psi]\n=\n\\bar\\psi G_0^{-1}\\psi\n+\\frac U2\\int_x\n\\mathcal O(x)^2.",
      "line": 157,
      "sha256": "e83f24c404285677200eda83f3b6beac6c28530631eee9b12ad1672f63a4bbf6"
    },
    {
      "index": 14,
      "display": true,
      "tex": "\\exp\\left[\n-\\frac U2\\mathcal O^2\n\\right]\n\\propto\n\\int\\dd\\phi\\,\n\\exp\\left[\n-\\frac{\\phi^2}{2U}\n+\\ii\\phi\\mathcal O\n\\right],",
      "line": 167,
      "sha256": "b22bb93293c23afebb07b2c5f499155c35af25316ab5c884a18f807cf45d6084"
    },
    {
      "index": 15,
      "display": false,
      "tex": "U",
      "line": 179,
      "sha256": "a25513c7e0f6eaa80a3337ee18081b9e2ed09e00af8531c8f7bb2542764027e7"
    },
    {
      "index": 16,
      "display": true,
      "tex": "Z\n=\\int\\mathcal D\\phi\\,\n\\ee^{-S_{\\rm eff}[\\phi]},",
      "line": 183,
      "sha256": "c16bbda5fe55b45a9157f1c7dce9c3d195d38a5dfc9245f89ae7e24d794ac445"
    },
    {
      "index": 17,
      "display": true,
      "tex": "S_{\\rm eff}[\\phi]\n=\n\\frac{1}{2U}\\int_x\\phi(x)^2\n-\\Tr\\log\n\\left[\nG_0^{-1}+\\ii\\Gamma\\phi\n\\right].",
      "line": 191,
      "sha256": "979ed40f91e2b648edaab999474866aee9d611bb2befd68b82136e3b391f135b"
    },
    {
      "index": 18,
      "display": true,
      "tex": "\\frac{\\delta S_{\\rm eff}}{\\delta\\phi(x)}=0",
      "line": 203,
      "sha256": "b50a79b3cebba61220e5d1f225ed7d7fc7045f193fa465e85d913c6fd0d9c415"
    },
    {
      "index": 19,
      "display": true,
      "tex": "\\phi=\\phi_\\star+\\delta\\phi",
      "line": 211,
      "sha256": "f785291857b663e77e798f554b235821ac4fa1d0134f2ae4e78188affc882d12"
    },
    {
      "index": 20,
      "display": true,
      "tex": "S_{\\rm eff}\n=S_{\\rm eff}[\\phi_\\star]\n+\\frac12\n\\delta\\phi\\,\\Gamma^{(2)}\\,\\delta\\phi\n+O(\\delta\\phi^3).",
      "line": 217,
      "sha256": "4f7d330aa844e8eecc428863d4efa1c54bfc447feb5501af2137e96389414398"
    },
    {
      "index": 21,
      "display": false,
      "tex": "\\Gamma^{(2)}",
      "line": 225,
      "sha256": "0dfbc117c2988fe848eee7b211d007cc1a4d1e1170b82b8021660b6b3adea985"
    },
    {
      "index": 22,
      "display": false,
      "tex": "\\phi",
      "line": 227,
      "sha256": "10ce0200b450ea955833553e717821c7721818b8a95cffd7e54b46e08918b512"
    },
    {
      "index": 23,
      "display": true,
      "tex": "Z\n=\\int\\mathcal D\\phi\\,\n\\ee^{-S_B[\\phi]}\n\\det M_\\uparrow[\\phi]\n\\det M_\\downarrow[\\phi].",
      "line": 237,
      "sha256": "44f02652a12559105857e83d1af9f940ef82329ee2cc5e5743bed167d41943b0"
    },
    {
      "index": 24,
      "display": true,
      "tex": "\\langle\\ee^{\\ii\\theta}\\rangle_{|w|}\n=\\frac{Z}{Z_{|w|}}\n\\sim\n\\ee^{-\\beta V\\Delta f}.",
      "line": 251,
      "sha256": "7197f2c781da192b084f30f5aa97345188c6437e54e7b8a51820b03b06e4e437"
    },
    {
      "index": 25,
      "display": false,
      "tex": "\\Delta f>0",
      "line": 258,
      "sha256": "6ff2b13fcc393980cf6d9c01be57521cfb3b3ccda581657b6a727285c63c438b"
    },
    {
      "index": 26,
      "display": false,
      "tex": "\\beta",
      "line": 258,
      "sha256": "f3f3804480e8551ac91a68ccc198310e514ec1424346595aca42a3a558ddee0a"
    },
    {
      "index": 27,
      "display": false,
      "tex": "V",
      "line": 258,
      "sha256": "de5a6f78116eca62d7fc5ce159d23ae6b889b365a1739ad2cf36f925a140d0cc"
    },
    {
      "index": 28,
      "display": false,
      "tex": "S_{\\rm eff}",
      "line": 268,
      "sha256": "e9babde828dd862a8c862cab6860ea20889a79a778a92412a560713c3b6690fa"
    },
    {
      "index": 29,
      "display": true,
      "tex": "Z=\\Tr\\ee^{-\\beta H}\n=\\int\\mathcal D\\phi\\,\n\\ee^{-S_E[\\phi]}.",
      "line": 274,
      "sha256": "f02f25ec06f9b8a573d2e69eea39e6de04ef4d5cc6ed80c4de8de891ad6ab6d5"
    },
    {
      "index": 30,
      "display": true,
      "tex": "Z\n=\\int\n\\mathcal D\\phi_+\\,\n\\mathcal D\\phi_-\\,\n\\ee^{\n\\ii S[\\phi_+]\n-\\ii S[\\phi_-]\n}.",
      "line": 282,
      "sha256": "3818b6b247594f32d769669b0a8b283fe08d5cd2c8ae1aa6a8d935bda1019a90"
    },
    {
      "index": 31,
      "display": true,
      "tex": "\\phi_{\\rm cl}\n=\\frac{\\phi_++\\phi_-}{2},\n\\qquad\n\\phi_{\\rm q}\n=\\phi_+-\\phi_-",
      "line": 297,
      "sha256": "87cc693893c9aabb8ecfb984540b41ace11f80b6fcd608852df6cd4007085f5c"
    },
    {
      "index": 32,
      "display": true,
      "tex": "Z_{\\rm red}\n=\\int\n\\mathcal D\\phi_+\\,\n\\mathcal D\\phi_-\\,\n\\ee^{\n\\ii S[\\phi_+]\n-\\ii S[\\phi_-]\n+\\ii S_{\\rm IF}[\\phi_+,\\phi_-]\n}.",
      "line": 309,
      "sha256": "066c9914290fd4817ec5aa57a70b203334183ff1a4db81bfae45a4f4800410fd"
    },
    {
      "index": 33,
      "display": true,
      "tex": "P[x]\\asymp\n\\ee^{-I[x]/\\epsilon}.",
      "line": 325,
      "sha256": "0a7bcc53f03d6c7c134b576092298878027903eab99cb6c18de948eca6c24105"
    },
    {
      "index": 34,
      "display": true,
      "tex": "\\ket{N,0},\n\\qquad\n\\ket{0,N}.",
      "line": 340,
      "sha256": "e9ce8389d402897af45dee776ba39e0485cec0e175fe9972c45a7e094572acbb"
    },
    {
      "index": 35,
      "display": false,
      "tex": "z=+z_0",
      "line": 346,
      "sha256": "fa0b27f136d1a6d1da880a65d91bcfe4bc4a99d69209e154670296cb78f67342"
    },
    {
      "index": 36,
      "display": false,
      "tex": "z=-z_0",
      "line": 346,
      "sha256": "1c2663984452dc5061af32421089e96967a916ce9c445771df5abce1676a574e"
    },
    {
      "index": 37,
      "display": false,
      "tex": "\\phi",
      "line": 350,
      "sha256": "10ce0200b450ea955833553e717821c7721818b8a95cffd7e54b46e08918b512"
    },
    {
      "index": 38,
      "display": false,
      "tex": "z",
      "line": 350,
      "sha256": "594e519ae499312b29433b7dd8a97ff068defcba9755b6d5d00e84c524d67b06"
    },
    {
      "index": 39,
      "display": false,
      "tex": "N",
      "line": 352,
      "sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
    },
    {
      "index": 40,
      "display": true,
      "tex": "\\Delta E\n\\sim\nA(N,\\Lambda)\n\\ee^{-Nb(\\Lambda)}.",
      "line": 354,
      "sha256": "8dcf92a4ead4c14947af3f737cca69712513f61416347cbd9c28921fb74c375d"
    },
    {
      "index": 41,
      "display": true,
      "tex": "\\hbar_{\\rm eff}\\sim\\frac1N.",
      "line": 363,
      "sha256": "b63d634f38f259268304d740ca39c652dfa936bbb7e8338bc394682fecd0fb15"
    },
    {
      "index": 42,
      "display": false,
      "tex": "N",
      "line": 367,
      "sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
    },
    {
      "index": 43,
      "display": false,
      "tex": "N",
      "line": 367,
      "sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
    },
    {
      "index": 44,
      "display": true,
      "tex": "\\ket{N-k+1,k-1}\n\\longrightarrow\n\\ket{N-k,k}.",
      "line": 373,
      "sha256": "b7777b4ac384c68352f3458f037b0ad68ea60b6bcaa29c7d228b5e46ce3d3df6"
    },
    {
      "index": 45,
      "display": false,
      "tex": "N",
      "line": 379,
      "sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
    },
    {
      "index": 46,
      "display": false,
      "tex": "K",
      "line": 379,
      "sha256": "86be9a55762d316a3026c2836d044f5fc76e34da10e1b45feee5f18be7edb177"
    },
    {
      "index": 47,
      "display": true,
      "tex": "E_k-E_0=gk(N-k).",
      "line": 383,
      "sha256": "2eb7816934db090bad670fc178d6dc0a1ff77cdacfd616e3d3be2517dc6834fd"
    },
    {
      "index": 48,
      "display": false,
      "tex": "k=N/2",
      "line": 387,
      "sha256": "f99c4f356f4ce0c8d8ad6fed765140e303c165864c8c36bf6dc4750028fcf808"
    },
    {
      "index": 49,
      "display": false,
      "tex": "\\Lambda=1",
      "line": 391,
      "sha256": "bbb5611c14b06d273400fac2af994bfc255045b9ac97021daddadeee49b1f0cb"
    },
    {
      "index": 50,
      "display": true,
      "tex": "S_{\\rm diss}\n\\sim\n\\int\\dd\\tau\\,\\dd\\tau'\\,\n\\frac{\n[z(\\tau)-z(\\tau')]^2\n}{\n(\\tau-\\tau')^2\n}.",
      "line": 395,
      "sha256": "dc6adb913f311bf3d478e800a641f13960341034d471b5f55aa4cb43bd71888d"
    },
    {
      "index": 51,
      "display": true,
      "tex": "\\det M_2[\\phi]\n=\n\\det M_1[\\phi]^*.",
      "line": 424,
      "sha256": "7e497f9d21fe764b846739f1afa5794a8d1b50f695e1afdcbdbdfbe26606dec2"
    },
    {
      "index": 52,
      "display": true,
      "tex": "\\det M_1\\det M_2\n=\n|\\det M_1|^2\n\\geq0.",
      "line": 432,
      "sha256": "64818d10a6ea0c160e91b46ff5dbf056e5b700df44d266866dbc5d90419628a9"
    },
    {
      "index": 53,
      "display": false,
      "tex": "SU(3)",
      "line": 447,
      "sha256": "abab2a6ad8a3295458debe48e135a02ecbc0298a1517470918e19f31112fce8a"
    },
    {
      "index": 54,
      "display": false,
      "tex": "\\Tr\\ee^{-\\beta H}",
      "line": 485,
      "sha256": "7a24fbec8d2b2b1a3470447c17c3e0b8d807b968a11daadc59b6b2055a4ec34a"
    },
    {
      "index": 55,
      "display": true,
      "tex": "gN\\gg K.",
      "line": 497,
      "sha256": "b7b30788994eda0508d74d5ca8db29346d0f7fd7a1975507d33a3e5a9bdd36e7"
    },
    {
      "index": 56,
      "display": false,
      "tex": "\\ket{N,0}",
      "line": 501,
      "sha256": "b0dc95e9c0fa4b3a2122c1605b015810e400f39f157bb50191d952042a594a3d"
    },
    {
      "index": 57,
      "display": false,
      "tex": "\\ket{0,N}",
      "line": 501,
      "sha256": "6e1a792bac8027e2161ab79746889d21e958384201634084404f2c3c582a4a31"
    },
    {
      "index": 58,
      "display": false,
      "tex": "N",
      "line": 501,
      "sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
    },
    {
      "index": 59,
      "display": true,
      "tex": "\\boxed{\n\\Delta_N\n=\n2\n\\frac{\n(K/2)^N N!\n}{\ng^{N-1}[(N-1)!]^2\n}\n}.",
      "line": 505,
      "sha256": "bfe6fbbda637d48365dbc6e17acbae8a631b60ffe942ed8b916ff51e0342df41"
    },
    {
      "index": 60,
      "display": true,
      "tex": "\\Lambda=\\frac{gN}{K}\\gg1,",
      "line": 520,
      "sha256": "47e2a8e3ffb8652b0bcb3783329921d297a57f29be83a2cc92ad93c2c516557f"
    },
    {
      "index": 61,
      "display": false,
      "tex": "k",
      "line": 533,
      "sha256": "8254c329a92850f6d539dd376f4816ee2764517da5e0235514af433164480d7a"
    },
    {
      "index": 62,
      "display": true,
      "tex": "-\\frac K2\\sqrt{k(N-k+1)}.",
      "line": 535,
      "sha256": "cd905aef9bd0dbd32699f19ccf3be2257923102bad89ecc28382629e7fb88242"
    },
    {
      "index": 63,
      "display": true,
      "tex": "\\prod_{k=1}^{N-1}k(N-k)\n=[(N-1)!]^2.",
      "line": 546,
      "sha256": "46a811d6cc0d0b9b3b5bcc9cd12c77c1087b64040138e72f7d64edccd3204593"
    },
    {
      "index": 64,
      "display": false,
      "tex": "N",
      "line": 553,
      "sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
    },
    {
      "index": 65,
      "display": true,
      "tex": "\\ket{N,0}\n\\to\n\\ket{N-1,1}\n\\to\\cdots\\to\n\\ket{0,N}.",
      "line": 564,
      "sha256": "2ae4b4f4571e110903d1736da20bfdb6b99a7783b3e3810d10d82e9ce2e32015"
    },
    {
      "index": 66,
      "display": true,
      "tex": "\\prod_{k=1}^{N}\n\\left[\n-\\frac K2\n\\sqrt{k(N-k+1)}\n\\right]\n=\n\\left(-\\frac K2\\right)^N N!.",
      "line": 574,
      "sha256": "9816d904cee2cb3975633763525da6ce87b91f7b7b0e7e2e72ece7513d6da159"
    },
    {
      "index": 67,
      "display": false,
      "tex": "N-1",
      "line": 584,
      "sha256": "a5b94796d549d06b1bc9f0126a05bcabd2bbf0be5a997f761cb2196a3d390053"
    },
    {
      "index": 68,
      "display": true,
      "tex": "E_0-E_k=-gk(N-k).",
      "line": 586,
      "sha256": "b3979b0a356b6d9d825566619a0cdf6c4d1508c01e678dc97ad58a6a75fed632"
    },
    {
      "index": 69,
      "display": true,
      "tex": "|t_{\\rm eff}|\n=\n\\frac{\n(K/2)^N N!\n}{\ng^{N-1}[(N-1)!]^2\n}.",
      "line": 592,
      "sha256": "7c64e60ba0e2d01087c1727146353af79a7477d610f66d8f64ec9727d0579736"
    },
    {
      "index": 70,
      "display": true,
      "tex": "\\Delta_N=2|t_{\\rm eff}|.",
      "line": 604,
      "sha256": "0701918e05c5d018a91b469fc1aec52836342bf11c5f9a1fc9e8d080ef2e8fd0"
    },
    {
      "index": 71,
      "display": true,
      "tex": "g=\\frac{\\Lambda K}{N}.",
      "line": 610,
      "sha256": "d82313466001867d5bf94a70e77fe405094de7b5e461b24de4a0554de177e7bd"
    },
    {
      "index": 72,
      "display": true,
      "tex": "\\Delta_N\n=\nK\\,2^{1-N}\n\\frac{\nN^N\n}{\n\\Lambda^{N-1}(N-1)!\n}.",
      "line": 616,
      "sha256": "c7bfa305a600313e3a3a1143dcb8d8b5b8c7fbf24d20a15f6bd967c2fb84cabe"
    },
    {
      "index": 73,
      "display": true,
      "tex": "\\boxed{\n\\log\\Delta_N\n=\n-N[\\log(2\\Lambda)-1]\n+O(\\log N)\n}.",
      "line": 629,
      "sha256": "1a214ebd2b3f9f41a81408d66655b16a5520ffe75961c5ec350da8f8e7fd929c"
    },
    {
      "index": 74,
      "display": true,
      "tex": "\\Delta_N\n\\sim\n\\operatorname{poly}(N)\n\\exp\\left\\{\n-N[\\log(2\\Lambda)-1]\n\\right\\}.",
      "line": 640,
      "sha256": "5e4a76c9276bd513189d78ccae45452b53f654bbbdf540c95410b07c53675d5f"
    },
    {
      "index": 75,
      "display": false,
      "tex": "N",
      "line": 651,
      "sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
    },
    {
      "index": 76,
      "display": true,
      "tex": "\\ee^{\\ii S[\\phi_+]-\\ii S[\\phi_-]}.",
      "line": 659,
      "sha256": "db107175c8bad61ff8bda64d822d7fc71612ab77cbd390efb2f8ea134334f9af"
    }
  ]
}