Ever since the introduction of resonating valence bond states by P.\

W.\ Anderson in 1970s, the search for exotic many--body quantum

behavior in peculiar states of matter known as quantum spin liquids

has been a central challenge in the field of condensed matter

physics. An important searching ground for such \emph{quantum--ness}

is the low energy physics of frustrated magnets. In a quantum

system, frustration---the inability of a physical system to achieve

a global state that consistently minimizes its energy

locally---enables the unavoidable quantum fluctuations to hybridize

an extensive number of classical states into highly entangled

quantum states at low temperatures. It can be argued that this is

the most practical approach to achieve genuine many--body quantum

behavior at large scales.

The fundamental source of frustration in many physical systems, is

the existence of local non--commuting terms. In the case of

antiferromagnet spin theories the simplest examples are triangles of

nearest--neighboring sites. Consequently, lattices with triangles as

building blocks have long been studied in hopes of discovering spin

liquids physics. In particular the kagome lattice, which is composed

of corner--sharing triangles, has been the center of at least four

decades of spin liquid research. This is due to the underconstrained

nature of system made out of corner--sharing triangles.

After the first three chapters, introduction in

chapter~\ref{chap:intro}, preliminary physics in

chapter~\ref{chap:prelim}, and a brief introduction to spin--liquids

in chapter~\ref{chap:QSLs}, in chapter~\ref{chap:kagome} I tackle a

simpler version of the long-standing problem of spin--$1/2$

Heisenberg antiferromagnets by considering a quasi--1D lattice

consisting entirely of corner--sharing triangles, \emph{kagome

strip}. I illustrate that the standard Heisenberg antiferromagnet

Hamiltonian over kagome strip is an extended gapless quantum phase,

that is well characterized by two fermionic/bosonic gapless modes

and power--law decaying spin and bond--energy correlations. I also

demonstrate that the correlation functions oscillate at tunably

incommensurate wave vectors. It turns out that this phase can be

identified by a particular marginal instability of a two-band spinon

Fermi surface coupled to an emergent U(1) gauge field. This

interpretation is supported by analytic Abelian bosonization and

with extensive numerical large--scale density matrix renormalization

group study as well as variational Monte Carlo calculations on

Gutzwiller ans\"atz wave functions. This intriguing result is the

first numerical demonstration of emergent fermionic spinons in a

simple SU(2) invariant nearest-neighbor Heisenberg model beyond the

strictly 1D (Bethe chain) limit.

The unexpected success of the fermionic spinons in describing the low

energy physics of kagome antiferromagnet, reintroduces the

questions about the validity of spinon physics as an effective

theory of quantum spin liquids. However, numerical methods are still

far behind the theoretical advances. And still even today, an

accurate description of the quantum spin liquid states using tensor

network methods is notoriously challenging. It is known that for

large quasi-1D systems, the density matrix renormalization group and

related methods usually require significant computational resources

and sometimes fail to converge to a satisfactory state. On the other

hand, variational wavefunctions acquired from the Gutzwiller

projection of gaussian fermionic theories have long served as both a

theoretical starting point for the construction of such spin liquid

states and as an inspiration for numerical variational Monte Carlo

(VMC) to calculate observables of interest. Noting this observation

I examine a different method by exploring the possibility of

constructing a matrix product state (MPS) representation for a

Gutzwiller--projected state from two given MPS representations of

gaussian fermionic theories in the \ref{chap:gutz} chapter.

I investigate the complexity of different approaches to achieve

Gutzwiller projection for MPSs and introduce the novel algorithm

which we call the Guzwiller zipper method. The performance of the

algorithm is tested against two copies of a single half-filled band

of spin-1/2 fermionic spinons. In a successful attempt to describe

the nature of spin liquid states on quasi-1D strips of triangular

and kagome-like lattices, I apply this method to two MPS of

multi-band fermionic spinon theories and compare with the complexity

of the traditional VMC approach. In particular, we methodically

disprove the conjecture of a spinon fermi surface spin liquid for

the triangular lattice.

Finally, I conclude this thesis by laying out a bird's eye view of the

current and future of spin liquid physics in kagome and triangular

lattices and pointing out numerous possible applications of the

novel Gutzwiller zipper method. In addition, I would also discuss its

possible extensions to more complicated tensor networks as well as

versions for higher spatial dimensions.