{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.many-body-quantum-matter.zero-sound-continuum-and-limit-paths",
  "title": "Zero sound, the particle-hole continuum, and paths to the origin",
  "scope": {
    "system": "isotropic spin-rotation-invariant three-dimensional normal Fermi liquid",
    "channel": "retarded density response of a neutral or statically screened system",
    "frequency_domain": "positive frequency; the negative-frequency mirror is omitted",
    "momentum_domain": "q much smaller than k_F",
    "state": "low temperature; panel (a) takes T=0 or tau_coll^(-1) -> 0",
    "units": "hbar = 1"
  },
  "scale_status": {
    "overall": "schematic",
    "panel_a": "The continuum and zero-sound rays use the equation-checked F0^s=1 slope ratio, but finite-q curvature and widths are omitted.",
    "panel_b": "Path angles and crossover-band width are schematic and do not encode sound speeds or a collision-kernel solution."
  },
  "assumptions": [
    "The collisionless zero-sound equation uses an F0^s-only isotropic Landau kernel.",
    "The first-sound formula assumes a Galilean-invariant liquid and retains F1^s.",
    "The collision integral conserves particle number, momentum, and energy.",
    "Landau parameters use the total density of states at the Fermi surface.",
    "The quasiparticle Fermi velocity is v_F^*, not the bare velocity."
  ],
  "axes": [
    {
      "panel": "a",
      "horizontal": "Q = q/k_F",
      "vertical": "W = omega/(v_F^* k_F)",
      "domain": "Q >= 0, W >= 0"
    },
    {
      "panel": "b",
      "horizontal": "q",
      "vertical": "omega",
      "domain": "q >= 0, omega >= 0"
    }
  ],
  "regions": [
    {
      "id": "one-particle-one-hole-continuum",
      "panel": "a",
      "support": "0 < W < Q at leading order in Q",
      "upper_boundary": "omega_ph^+(q) = q v_F^* + O(q^2)",
      "quadratic_dispersion_correction": "+q^2/(2m^*)",
      "meaning": "collisionless one-particle-one-hole spectral support and the associated retarded branch cut",
      "visual_encoding": "diagonal hatch bounded above by a dashed line"
    },
    {
      "id": "collision-crossover",
      "panel": "b",
      "condition": "omega tau_coll is of order one",
      "meaning": "strongly absorptive crossover whose detailed dispersion and damping depend on a conserving collision kernel",
      "not_a": "phase boundary",
      "visual_encoding": "pale horizontal band with dotted boundaries"
    }
  ],
  "curves": [
    {
      "id": "zero-sound",
      "panel": "a",
      "equation": "omega = s_0 q v_F^*",
      "root_equation": "1 = F0^s [(s_0/2) ln((s_0+1)/(s_0-1)) - 1]",
      "existence_in_displayed_truncation": "F0^s > 0 gives one real physical-sheet root s_0 > 1",
      "hierarchy": "omega tau_coll >> 1 and q ell_coll >> 1",
      "physical_status": "undamped only in the collisionless linearized one-pair theory; finite temperature, collisions, multipair states, and finite-q curvature can broaden it",
      "visual_encoding": "thick solid line"
    },
    {
      "id": "first-sound-guide",
      "panel": "b",
      "equation": "omega = c_1 q",
      "speed": "c_1/v_F^* = sqrt(((1+F0^s)(1+F1^s/3))/3)",
      "hierarchy": "omega tau_coll << 1 and q ell_coll << 1",
      "physical_status": "hydrodynamic mode produced by local equilibrium, not a pole of the collisionless kernel",
      "visual_encoding": "dash-dot arrow toward the origin"
    }
  ],
  "limit_paths": [
    {
      "id": "static",
      "order": "omega -> 0 at fixed q, then q -> 0",
      "result": "equilibrium compressibility or static susceptibility",
      "visual_encoding": "dashed L-shaped path"
    },
    {
      "id": "uniform-dynamic",
      "order": "q -> 0 at fixed nonzero omega, then omega -> 0",
      "result": "uniform dynamic response constrained by conservation laws",
      "visual_encoding": "dotted L-shaped path"
    },
    {
      "id": "collisionless-fixed-z",
      "order": "q and omega -> 0 at fixed z = (omega+i0^+)/(q v_F^*) while maintaining omega tau_coll >> 1",
      "result": "particle-hole continuum and a possible zero-sound pole",
      "finite_tau_caveat": "At fixed nonzero tau_coll the route terminates at the crossover; it reaches the origin only if tau_coll^(-1)/omega -> 0 as part of the limit.",
      "visual_encoding": "solid diagonal arrow ending above the crossover band"
    },
    {
      "id": "hydrodynamic-first-sound",
      "order": "q and omega -> 0 with omega tau_coll << 1 and q ell_coll << 1",
      "result": "first sound in the longitudinal density channel",
      "visual_encoding": "dash-dot diagonal arrow reaching the origin"
    }
  ],
  "verification_fixture": {
    "parameters": {
      "F0^s": 1.0,
      "F1^s": 0.0,
      "Q": 0.1
    },
    "results": {
      "s_0": 1.0443820337608,
      "c_1_over_v_F_star": 0.8164965809277,
      "W_zero_sound": 0.1044382033761,
      "W_continuum_edge": 0.1,
      "W_first_sound_guide": 0.0816496580928,
      "zero_sound_root_residual_absolute": 8.9e-16,
      "normalized_pole_residue": 0.1042167715,
      "normalized_pole_residue_definition": "Z_0/[N(0) q v_F^*]"
    },
    "display_note": "The line thickness is categorical and does not encode pole residue. Panel (b) does not compare these slopes."
  },
  "exclusions": [
    "No separated F0^s-only physical-sheet pole is shown for F0^s <= 0.",
    "The acoustic density-channel picture does not apply to an unscreened three-dimensional Coulomb system, where the charge mode is lifted to a plasmon.",
    "Finite-q recoil, thermal smearing, multipair spectral weight, lattice anisotropy, disorder, and detailed collision-kernel interpolation are omitted.",
    "At strictly zero temperature the hydrodynamic window closes unless another momentum-conserving equilibration mechanism is supplied."
  ],
  "visual_encoding": {
    "continuum": "hatching plus a dashed upper boundary and a direct label",
    "zero_sound": "thick solid line plus a direct label",
    "first_sound": "dash-dot line plus a direct label",
    "crossover": "pale band, dotted boundaries, and a direct qualification",
    "static_path": "dashed arrows",
    "uniform_dynamic_path": "dotted arrows"
  },
  "sources": [
    {
      "citation": "L. D. Landau, Oscillations in a Fermi Liquid, Soviet Physics JETP 5 (1957) 101-108, especially printed pp. 102-105",
      "url": "https://www.jetp.ras.ru/cgi-bin/dn/e_005_01_0101.pdf",
      "use": "collisionless kinetic equation, zero-sound condition, and distinction between ordinary and zero sound"
    },
    {
      "citation": "A. A. Abrikosov and I. M. Khalatnikov, The Theory of a Fermi Liquid (the Properties of Liquid He-3 at Low Temperatures), Reports on Progress in Physics 22 (1959) 329-367",
      "url": "https://doi.org/10.1088/0034-4885/22/1/310",
      "use": "conserving collisions and the hydrodynamic-to-collisionless crossover"
    },
    {
      "citation": "N. Dupuis, Fermi-Liquid Theory, Chapter 4 of Field Theory of Condensed Matter and Ultracold Gases, updated 2025, sections 4.3.3-4.3.4, printed pp. 302-307",
      "url": "https://www.lptmc.jussieu.fr/user/dupuis/chap_fl.pdf",
      "use": "response convention, continuum, zero-sound pole equation, and first-sound speed"
    }
  ],
  "semantic_equivalent": {
    "page_route": "/many-body-quantum-matter/fermi-liquids-beyond/response-zero-sound/",
    "section": "Four paths to the origin give four observables",
    "note": "The adjacent four-row table is the primary reflowing text equivalent for the limit paths; the page derivation supplies the equations and qualifications."
  }
}