Technical Skills
Analytical
Mean field theory: I have used mean field theories to construct effective one-particle Hamiltonians from interacting Hamiltonians. In particular, I have constructed Bardeen–Cooper–Schrieffer (BCS)-like gap equations for d-wave superconductors in the absence of disorder and Bogolubov-de Gennes (BdG) self-consistent equations in the presence of disorder.
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Schrieffer–Wolff transformation: A controlled perturbative way to handle strong electron-electron correlations is to use a unitary transformation called Schrieffer–Wolff transformation. This transformation projects out the high energy excitations of the Hamiltonian. This results in an effective Hamiltonian in low energy subspace with restrictions in the Hilbert space. Using this transformation, I derived an effective 'tJ'-like Hamiltonian from a strongly disordered and strongly correlated Hubbard model.
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Gutzwiller approximation: Gutzwiller projected wave functions provide variational ansatz to various strongly correlated problems. A simplified approach of analytically handling the Hilbert space restrictions due to these projected wave functions is called Gutzwiller approximation. I have implemented the Gutzwiller approximation in the presence of disorder in a strongly correlated d-wave superconductor.
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Ginzburg Landau theory of phase transitions: Close to phase transitions, the free energy of a system can be expressed in terms of an expansion of the order parameters associated with corresponding phases. A mean field approach to solve this free energy is to consider the minimum in the free energy. I have used a Ginzburg Landau mean field theory to study a system with competing order parameters.
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Renormalization group techniques: Mean field theories are inadequate in describing the order parameter fluctuations which are important in low dimensions close to the critical region. Renormalization group treatment is one of the efficient ways to treat these fluctuations. This technique is based on integrating out the fast momentum fluctuations iteratively to write an effective free energy, which captures the slow momentum or long wavelength fluctuations. I have used this technique to account for the fluctuations of the order parameters in an anisotropic non-linear sigma model in two spatial dimensions.
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Green's function formalism: Green's functions are often used to investigate quantum many-body systems. I am currently using Green's functions extensively in order to study phonon mediated superconductors and proximity effect in multi-band superconductors. In both of these systems, the Green's function formalism is immensely beneficial to study the frequency dependence of the self energy.
Numerical
Self consistent inhomogeneous Hartree-Fock-Bogoliubov solutions: BCS self-consistent gap equations can be solved analytically only for some specific interactions. Additionally, in the presence of disorder, the number of such self-consistent (BdG) equations scale as the system size. This constrains us from solving them analytically. Presence of Hartree- and Fock-shifts complicates the situation further. I have extensively solved these self-consistent equations numerically both in momentum space (with no disorder) and real space (with disorder).
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Functional renormalization group (fRG)+Mean field theory: There are some situations where mean-field theories fail to correctly describe the system. For example, near a van Hove singularity in two-dimension, the superconducting (particle-particle) instability is strongly influenced by particle-hole instabilities and vice versa. This feedback effect of different instabilities is not taken into account in biased calculations like RPA or MFT. There already exists unbiased functional renormalization group (fRG) approaches which take into account the feedback of the particle-particle and particle-hole instabilities on each other. fRG however is formulated in momentum space and hence cannot describe real space inhomogeneities. In collaboration with Prof. Andreas Schnyder and Dr. Pietro Maria Bonetti (both from Max Planck Institute of Solid State Research, Stuttgart, Germany), we have formulated a joint framework of fRG+MFT, already developed in momentum space, to real space for studying edge states of topological superconductors. The developed method is useful for studying the simulatenouas interplay of strong correlations, disorder, interfaces, and also topology.
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Classical Monte Carlo: Monte Carlo methods are useful in simulating models exactly. As an exercise, I have performed Monte Carlo simulations to determine thermal transitions in the Ising model and scalar field models with quartic interaction.
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Determinant quantum Monte Carlo: I have built confidence in determinant quantum Monte Carlo technique by solving a repulsive Hubbard model and matching the results with publicly available codes. Currently, I am developing a determinant quantum Monte Carlo code for a spin fermion model with two bands.