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We define geometric critical exponents for systems that undergo continuous second order classical and quantum phase transitions.
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R = 2 g [ ∂ ∂ x 2 ( g g 11 Γ 11 2 ) − ∂ ∂ x 1 ( g g 11 Γ 12 2 ) ] R=\frac{2}{\sqrt{g}}\left[\frac{\partial}{\partial x^{2}}\left(\frac{\sqrt{g}}{g_{11}}\Gamma^{2}_{11}\right)-\frac{\partial}{\partial x^{1}}\left(\frac{\sqrt{g}}{g_{11}}\Gamma^{2}_{12}\right)\right] , where g g is the determinant of the metric, and the Christoffel symbols are defined by Γ ν ρ μ = 1 2 g μ ζ ( ∂ g ζ ν ∂ x ρ + ∂ g ζ ρ ∂ x ν − ∂ g ν ρ ∂ x ζ ) \Gamma^{\mu}_{\nu\rho}=\frac{1}{2}g^{\mu\zeta}\left(\frac{\partial g_{\zeta\nu}}{\partial x^{\rho}}+\frac{\partial g_{\zeta\rho}}{\partial x^{\nu}}-\frac{\partial g_{\nu\rho}}{\partial x^{\zeta}}\right)
Cited in the paper.
Note however that the converse is not true : geodesics focusing at a point does not necessarily indicate a singularity there. This is most easily illustrated by considering a two-sphere, where great circles focus at the poles where there are no singularities
Cited in the paper.
When this is not the case, for example along the h = 1 h=1 transition line in the XY-spin chain model where the scalar curvature does not show a divergence, our method will not be applicable. For these cases, geodesics do not show any special behavior near the critical region
Cited in the paper.
This derivation is based on the fact that in two dimensions, the elements of the Ricci tensor are related to the metric by R μ ν = 1 2 g μ ν R R_{\mu\nu}=\frac{1}{2}g_{\mu\nu}R
Cited in the paper.