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We introduce a non-linear differential flow equation for density matrices that provides a monotonic decrease of the free energy and reaches a fixed point at the Gibbs thermal state.
T. Shi, J. I. Cirac, and E. Demler, arXiv:1904.00932
1904
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T. Shi, E. Demler, and J. I. Cirac, Annals of Physics 390
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I. Esterlis, S. Kivelson, and D. Scalapino, Phys. Rev. B 99
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Note that in principle this procedure may lead to the flow stopping at a local rather than the global minimum. However, the experience so far shows that this can be overcome by taking different initial values for ξ ( 0 ) \xi(0)
Cited in the paper.
Strictly speaking, this is the only solution as long as we impose that ρ \rho has full rank
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For the bosonic case, the state Φ + \Phi^{+} is not normalizable. However, since we use the covariant matrix formalism, this problem can be easily circumvented
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See Supplemental Material
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