Browsing by Subject "Quantum spin chain"

Correlation functions in a superconformal field theory are strictly constrained by conformal symmetry. Notably, one-point functions of conformal operators always vanish. However, when a defect is inserted into the spacetime of the field theory, certain one-point functions become non-zero due to the broken conformal symmetry, highlighting the special properties of the defect. One interesting type of defect is the domain wall, which separates spacetime into two regions with distinct vacua. The domain wall version of $\mathcal{N}=4$ supersymmetric Yang-Mills (SYM) theory has been extensively studied in recent years. In this context, the supersymmetric domain wall preserves integrability, allowing one to evaluate one-point functions in the defect field theory using integrability techniques. As an analogous study of the domain wall version of $\mathcal{N}=4$ SYM theory, this thesis focuses on the ABJM theory with a 1/2-BPS domain wall, meaning that the domain wall preserves half the original supersymmetry. We first review integrability methods, e.g. the Coordinate Bethe ansatz and the Algebraic Bethe ansatz for $\mathfrak{su}(2)$ Heisenberg spin chain. The spectrum of the spin chain can be determined by solving sets of the Bethe equations. Moreover, the Rational $Q$-system is examined, which solves the Bethe equations efficiently and eliminates all nonphysical solutions automatically. On the field theory side, we first review the original ABJM theory and its spectral integrability following J. A. Minahan's work in 2009. There exists an underlying quantum $\mathfrak{su}(4)$ spin chain with alternating even and odd sites, whose Hamiltonian can be identified with the two-loop dilation operator of ABJM theory in the planar limit. This correspondence allows us to find the spectrum of ABJM theory using the Bethe ansatz. We study the $\mathfrak{su}(4)$ alternating spin chain and demonstrate the procedure for constructing eigenstates of ABJM theory. Finally, we study the tree-level one-point functions in the domain wall version of ABJM theory. We derive the classical solutions for the scalar fields that describe a domain wall and explicitly demonstrate how the domain wall preserves half of the supersymmetry. With these classical solutions, we define a domain wall version of ABJM theory. Then, we introduce the so-called Matrix Product State, which is a boundary state in the spin chain's Hilbert space. The domain wall can be identified with an integrable matrix product state, leading to a compact determinant formula for the one-point functions in spin chain language. Consequently, we can evaluate one-point functions explicitly using the Bethe ansatz and boundary integrability.