We develop a new formulation of the functional renormalization group (RG) for interacting fermions. Our approach unifies the purely fermionic formulation based on the Grassmannian functional integral, which has been used in recent years by many authors, with the traditional Wilsonian RG approach to quantum systems pioneered by Hertz [Phys. Rev. B 14, 1165], which attempts to describe the infrared behavior of the system in terms of an effective bosonic theory associated with the soft modes of the underlying fermionic problem. In our approach, we decouple the interaction by means of a suitable Hubbard-Stratonovich transformation (following the Hertzapproach), but do not eliminate the fermions; instead, we derive an exact hierarchy of RG flow equations for the irreducible vertices of the resulting coupled field theory involving both fermionic and bosonic fields. The freedom of choosing a momentum transfer cutoff for the bosonic soft modes in addition to the usual band cutoff for the fermions opens the possibility of new RG schemes. In particular, we show how the exact solution of the Tomonaga-Luttinger model (i.e., one-dimensional fermions with linear energy dispersion and interactions involving only small momentum transfers) emerges from the functional RG if one works with a momentum transfer cutoff. Then the Ward identities associated with the local particle conservation at each Fermi point are valid at every stage of the RG flow and provide a solution of an infinite hierarchy of flow equations for the irreducible vertices. The RG flow equation for the irreducible single-particle self-energy can then be closed and can be reduced to a linear integro-differential equation, the solution of which yields the result familiar from bosonization. We suggest new truncation schemes of the exact hierarchy of flow equations, which might be useful even outside the weak coupling regime.
We show that at low temperatures T an inhomogeneous radial magnetic field with magnitude B gives rise to a persistent magnetization current around a mesoscopic ferromagnetic Heisenberg ring. Under optimal conditions this spin current can be as large as gµB(T / ) exp 1/2 ], as obtained from leading-order spin-wave theory. Here g is the gyromagnetic factor, µB is the Bohr magneton, and ∆ is the energy gap between the ground state and the first spin-wave excitation. The magnetization current endows the ring with an electric dipole moment.PACS numbers: 75.10. Jm, 75.10.Pq, 75.30.Ds, 73.23.Ra The controlled fabrication of submicron devices has opened the door to a rich new field of theoretical and experimental physics. At low temperatures these devices are mesoscopic in the sense that their quantum states must be described by coherent wave functions extending over the entire system. Then the usual assumptions underlying the averaging procedure in statistical mechanics are not necessarily valid, and quantum-mechanical interference effects become important [1].A prominent example is persistent currents in mesoscopic normal metal rings threaded by a magnetic flux [1]. Although this phenomenon was predicted long ago [2,3], the experimental difficulties in measuring persistent currents in an Aharonov-Bohm geometry were only overcome in the past decade [4,5,6]. Surprisingly, for metallic rings in the diffusive regime the observed currents were much larger than predicted by theory [1]. On the other hand, in the ballistic regime [6] the order of magnitude of the observed current can be explained with a simple model of free fermions moving on a ring pierced by a magnetic flux φ. Then the stationary en-. ., are the allowed wavevectors for a ring with circumference L. Here φ 0 = hc/e is the flux quantum and m * is the effective mass of the electrons. In the simplest approximation, one may calculate the current I = −c∂Ω gc (φ)/∂φ at constant chemical potential µ from the flux-dependent part of the grand canonical potential Ω gc (φ). At finite temperature T , one obtains for spinless fermionswhere v n = k n /m * . For T ≪ µ the amplitude of the current is I max ≈ −ev F /L (where v F is the Fermi velocity), in agreement with experiment [6].In this Letter, we show that Heisenberg spin chains in inhomogeneous magnetic fields can be used to realize a spin current analogue of mesoscopic persistent currents in normal metal rings. Note that in the presence of spin-orbit coupling spin currents in spin chains can also be driven by inhomogeneous electric fields [8], due to the Aharonov-Casher effect [9]. As detailed later on, the magnetization current is carried by magnons and endows the ring with an electric dipole field, which is the counterpart of the magnetic dipole field associated with the persistent charge current in a normal metal ring. We find that for realistic parameters the spin analogues of the experiments in Refs. 4, 5, 6 require the detection of a potential drop on the order of nanovolts.Due to its relevance for informat...
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