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H=vF σ⋅(k−b)+vF W(x) k⋅1H = v_F\,\boldsymbol{\sigma}\cdot(\mathbf{k}-\mathbf{b}) + v_F\,W(x)\,\mathbf{k}\cdot\mathbb{1}

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The tilt is where the geometry stops being a figure of speech. Once the tilt varies smoothly from place to place across a crystal, an electron propagating through it is, at leading order in the standard low-energy expansion, equivalent to a particle moving through a curved effective spacetime of the same mathematical form — a Painlevé–Gullstrand metric — that describes an object falling into a black hole, with the tilt itself playing the role of the metric’s off-diagonal, “flow” components. The paper writes the whole statement as a single Hamiltonian, fixing the Fermi velocity vFv_F , the Pauli matrices σ\boldsymbol{\sigma} acting on the electron’s internal degree of freedom, the momentum-space…

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σ\boldsymbol{\sigma}

Symbol boldsymbolσ

boldsymbolσ is one of the signed contributions combined to compute the quantity on the left.

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H=vF σ⋅(k−b)+vF W(x) k⋅1H = v_F\,\boldsymbol{\sigma}\cdot(\mathbf{k}-\mathbf{b}) + v_F\,W(x)\,\mathbf{k}\cdot\mathbb{1}

Equation 5 · Crossed Fields

A Strained Crystal Is a Designer Spacetime

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

The tilt is where the geometry stops being a figure of speech. Once the tilt varies smoothly from place to place across a crystal, an electron propagating through it is, at leading order in the standard low-energy expansion, equivalent to a particle moving through a curved effective spacetime of the same mathematical form — a Painlevé–Gullstrand metric — that describes an object falling into a black hole, with the tilt itself playing the role of the metric’s off-diagonal, “flow” components. The paper writes the whole statement as a single Hamiltonian, fixing the Fermi velocity vFv_F , the Pauli matrices σ\boldsymbol{\sigma} acting on the electron’s internal degree of freedom, the momentum-space…

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