Unitary renormalization group for correlated electrons

Mukherjee, Anirban (2020) Unitary renormalization group for correlated electrons. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Anirban Mukherjee (14RS016)) 14RS016.pdf - Submitted Version Restricted to Repository staff only Download (13MB)

Abstract

This thesis reports the development of an analytic unitary tensor network renormalization group (URG) for models of strongly correlated electrons/spins. The RG proceeds via an iterative procedure involving the unitary disentanglement of the highest energy (UV) electronic degree of freedom from its low energy (IR) counterparts in a single step. The unitary transformations generate RG ow in the space of effective Hamiltonians, manifested in a hierarchy of RG ow equations for the 2n-point scattering vertex tensors. The ow equations support a non-trivial dependence on the renormalized self/correlation energy of the many-body configurations. All orders of loop expansion are non-perturbatively resummed in the vertex ows via the denominator of the RG ow functions; this is seen to lead to a non-trivial stable fixed point at which the RG ows are arrested. Further, the denominator of the RG ow equations possess an additional frequency dependence originating in the non-commutativity between different parts of the many-body Hamiltonian. Contributions from relevant higher-order scattering diagrams generated at every RG step can be tracked in a controlled manner. At weak coupling, the URG ow equations match those obtained from the perturbative truncations involved in the continuous unitary transformation (CUT) RG and functional RG (FRG) treatments of models of strongly correlated electrons such as the 2D Hubbard model on the square lattice. Importantly, the URG ows yield stable field points in the IR that are not accessible from other RG approaches. We have applied the URG technique to various paradigmatic models of correlated electrons and spins, including the 2d Hubbard model on the square lattice at and away from half-filling, the XXZ spin-1=2 antiferromagnet on the 2d Kagome lattice, a general one-band problem of interacting fermions in 2d, the generalised Sachdev Ye model for electrons, the 1d Hubbard model and the single impurity Kondo model. In these models, we have obtained a variety of stable fixed point theories and analysed their low-energy spectrum and eigenstates. In this way, the unitary RG formalism has revealed a multitude of exotic quantum many-body phenomena in these models: quantum criticality, topologically ordered spin and charge quantum liquids, pseudogapped phases, marginal Fermi liquids, charge fractionalization of excitations, many-body localization and thermalisation. We now list a set of important questions whose answers we attempt in the thesis. First, can we construct a non-perturbative unitary renormalization group for correlated electrons on a lattice? Does such a formulation generate a Hamiltonian renormalization group and an entanglement renormalization group concomitantly? Does the RG ow generically lead to a stable fixed point with an analytically tractable effective Hamiltonian? As the RG procedure is unitary, can we demonstrate dynamical spectral weight transfer (even as we show that the total spectral weight, or the f-sum rule, is conserved)? Specifically, can we show that the propagating degrees of freedom emergent from an instability of Fermi surface (e.g., Cooper pairs) help account for the spectral weight lost by the gapless excitations of the normal state (e.g., Landau quasiparticles)? By reconstructing the URG ow backwards from the IR ground state, can we track the evolution of correlation functions and entanglement features towards the UV?

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