Condensed matter at eleven parameters: the reverse paradigm on a lattice
Abstract
The eleven-parameter relativity has a compact momentum sector that closes, for a body in vacuum, at the Planck momentum M P c = 6.5 kg m s -1 , where nothing is built. An electron in a crystal has a crystal momentum bounded by πℏ/a, twenty-five orders of magnitude lower, and the reverse paradigm of the series, the momentum as the base and the position as derived, is its kinematics word for word, on a torus in place of a sphere. This paper states the dictionary. The Bloch oscillation is the kinematics of a bounded momentum. The Wannier centre is the derived position, defined modulo the lattice constant that is the crystal's resolution as a frame body. The Berry geometry of a band is the abelian shadow of the reverse paradigm's gauging over momentum space, with the Chern number for its charge. The derived positions of a band do not commute, by the cell times the charge over 2π, and they have an exponentially localised Wannier basis exactly when the charge vanishes. A Dirac material's electrons are massless bodies with a light speed of their own, v F ≃ c/300 in graphene, with the lattice constant for the Planck length and the bandwidth, to a factor of two, for the Planck energy. A gap gives them a mass and with it the eleven's acceleration bound m * v 3 F /ℏ, reached in the field ∆ 2 /(4eℏv F ), which sets the scale of the Zener breakdown field. In the two-band model the Landau-Zener exponent is the Schwinger exponent πE c S /E exactly, E c S being that field, and tunnel diodes have been operating at a few tenths of it since 1958. The curvature of the Zener plot of a gapped monolayer at that field tests the analogue at its bound, and the two-band model gives Kane's straight line. The chirality of a Weyl node is the sixteenth charge, its anomaly the pumping of the helicity block of the thermal element and the intervalley time its relaxation. Strain is the compensator made visible. The superconductor's condensate is the compensator with a dynamics, whose Schwinger field, twenty volts per metre in aluminium, brings it to its depairing velocity in one gap time. The Barnett and Einstein-de Haas effects are the rotation block, the London moment its reading of the Cooper pair's mass to a part in 10 4 , and the Grüneisen parameter is the trace that denies the phonon gas its dilatation block. The maximal temperature ∆/(4πk B ) of a semiconductor is the temperature at which its intrinsic density reaches 10 17 cm -3 . A Lorentzian hot spot in the Dirac fluid of graphene is the conformal soliton, whose first picosecond would read the special conformal block of the thermal element, which no relativistic fluid has shown in real time. A flat band is a crystal at the Planck scale of the reverse paradigm. All of it is an analogue, exact in the kinematics of its model Hamiltonians, approximate in its thermal element and silent on the constants, and the paper says which of the sixteen charges a crystal keeps.
Related
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BibTeX
@misc{reynaud2026condensed,
title = {{Condensed matter at eleven parameters: the reverse paradigm on a lattice}},
author = {Patrick B. Reynaud},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05769920v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Condensed matter at eleven parameters: the reverse paradigm on a lattice
AU - Patrick B. Reynaud
PY - 2026
DA - 2026/09/29
PB - HAL
UR - https://hal.science/hal-05769920v2
AB - The eleven-parameter relativity has a compact momentum sector that closes, for a body in vacuum, at the Planck momentum M P c = 6.5 kg m s -1 , where nothing is built. An electron in a crystal has a crystal momentum bounded by πℏ/a, twenty-five orders of magnitude lower, and the reverse paradigm of the series, the momentum as the base and the position as derived, is its kinematics word for word, on a torus in place of a sphere. This paper states the dictionary. The Bloch oscillation is the kinematics of a bounded momentum. The Wannier centre is the derived position, defined modulo the lattice constant that is the crystal's resolution as a frame body. The Berry geometry of a band is the abelian shadow of the reverse paradigm's gauging over momentum space, with the Chern number for its charge. The derived positions of a band do not commute, by the cell times the charge over 2π, and they have an exponentially localised Wannier basis exactly when the charge vanishes. A Dirac material's electrons are massless bodies with a light speed of their own, v F ≃ c/300 in graphene, with the lattice constant for the Planck length and the bandwidth, to a factor of two, for the Planck energy. A gap gives them a mass and with it the eleven's acceleration bound m * v 3 F /ℏ, reached in the field ∆ 2 /(4eℏv F ), which sets the scale of the Zener breakdown field. In the two-band model the Landau-Zener exponent is the Schwinger exponent πE c S /E exactly, E c S being that field, and tunnel diodes have been operating at a few tenths of it since 1958. The curvature of the Zener plot of a gapped monolayer at that field tests the analogue at its bound, and the two-band model gives Kane's straight line. The chirality of a Weyl node is the sixteenth charge, its anomaly the pumping of the helicity block of the thermal element and the intervalley time its relaxation. Strain is the compensator made visible. The superconductor's condensate is the compensator with a dynamics, whose Schwinger field, twenty volts per metre in aluminium, brings it to its depairing velocity in one gap time. The Barnett and Einstein-de Haas effects are the rotation block, the London moment its reading of the Cooper pair's mass to a part in 10 4 , and the Grüneisen parameter is the trace that denies the phonon gas its dilatation block. The maximal temperature ∆/(4πk B ) of a semiconductor is the temperature at which its intrinsic density reaches 10 17 cm -3 . A Lorentzian hot spot in the Dirac fluid of graphene is the conformal soliton, whose first picosecond would read the special conformal block of the thermal element, which no relativistic fluid has shown in real time. A flat band is a crystal at the Planck scale of the reverse paradigm. All of it is an analogue, exact in the kinematics of its model Hamiltonians, approximate in its thermal element and silent on the constants, and the paper says which of the sixteen charges a crystal keeps.
ER -