  Many of the exciting features of the Standard Model of the elementary
particles are inherently non-perturbative. A theoretical understanding of many
physics aspects beyond the Standard Model of elementary particles also requires
a non-perturbative framework. One such framework involves discretizing quantum
field theories on a spacetime lattice. We can use this lattice regularization
method to study supersymmetric versions of physics beyond the Standard Model.
In this thesis, we discuss the spacetime lattice setup, and with the examples
of different models, we will see the numerical capability of this tool in
exploring field theory regimes that are not accessible through perturbation
theory. We use an efficient version of the Monte Carlo algorithm to update the
field configurations in the path integral and eventually reach the equilibrium
configurations. A version of the gauge/gravity conjecture connects weakly
coupled gravitational theories to strongly coupled field theories. We will
mainly focus on the non-conformal analogs of the conjecture in lower
dimensions. This thesis mainly discusses the numerical simulation results of
two lower-dimensional models. One is the bosonic version of the BMN matrix
quantum mechanics and the other is a two-dimensional Yang-Mills theory
containing four supersymmetries. Our numerical results suggest that the phase
diagram of bosonic BMN model smoothly interpolates between the bosonic BFSS and
the gauged Gaussian model, with first-order deconfinement phase transition at
all couplings. Our simulation results for two-dimensional Yang-Mills theory
that contains four supersymmetries show that this model admits a deconfinement
phase transition in the limit of a large number of colors. We also show that
the nature of the transition looks similar to its maximally supersymmetric
cousin in the weak coupling regime.
