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Springer eBooks

Invariance properties of the renormalization group

M. S. Green

Published Dec 11, 2005• 1 citation
Publisher
DOI
10.1007/bfb0034504

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The property of quantum many-body systems under spatial reflection and the relevant physics of renormalization group (RG) procedure are revealed. By virtue of the matrix product state (MPS) representation, various attributes for translational invariant systems associated with spatial reflection are manifested. We demonstrate subsequently a conservation rule of the conjugative relation for reflectional MPS pairs under RG transformations and illustrate further the property of the fixed points of RG flows. Finally, we show that a similar rule exists with respect to the target states in the density matrix renormalization group algorithm.
The property of quantum many-body systems under spatial reflection and the relevant physics of the renormalization group (RG) procedure are revealed. By virtue of the matrix product state (MPS) representation, various attributes for translational invariant systems associated with spatial reflection are manifested. We demonstrate subsequently a conservation rule of the conjugative relation for reflectional MPS pairs under RG transformations and illustrate further the property of the fixed points of RG flows. Finally, we show that a similar rule exists with respect to the target states in the density matrix renormalization group algorithm.
The renormalization-group (RG) approach proposed earlier by Shankar for interacting spinless fermions at T=0 is extended to the case of nonzero temperature and spin. We study a model with SU(N)-invariant short-range effective interaction and rotationally invariant Fermi surface in two and three dimensions. We show that the Landau interaction function of the Fermi liquid, constructed from the bare parameters of the low-energy effective action, is RG invariant. On the other hand, the physical forward-scattering vertex is found as a stable fixed point of the RG flow. We demonstrate that in d=2 and 3, the RG approach to this model is equivalent to Landau's mean-field treatment of the Fermi liquid. We discuss subtleties associated with the symmetry properties of the scattering amplitude, the Landau function, and the low-energy effective action. Applying the RG to response functions, we find the compressibility and the spin susceptibility as fixed points.