This chapter discusses the properties of tree tensor network states and the methods for evaluating the ground state and thermodynamic properties of quantum lattice models on a Bethe lattice or, more generally, a Husimi lattice. It starts with a brief discussion of the canonical form of a tree tensor network state. Then, a canonicalization scheme is proposed. To calculate the ground state through the imaginary time evolution, the full and simple update methods are introduced to renormalize the local tensors. Finally, as the correlation length of a quantum system is finite even at a critical point, an accurate and efficient method is described to compute the thermodynamic quantities of quantum lattice models on the Bethe lattice.

In previous chapters, we saw in detail the immense success of quantum theory in describing microscopic systems. We also saw that it leads to concrete predictions that are persistently being confirmed by experiment, even predictions that grossly violate all physical intuition that had been generated by 250 years of classical physics.

Newtonian mechanics was the first great synthesis of modern physics. It provided the main theory about the workings of the physical world from the date of its first presentation (1687) until the beginnings of the twentieth century.

The study of composite systems typically requires their analysis into simpler systems. In classical physics, the simplest systems are particles, that is, pointlike bodies that move in space. A particle is traditionally described by three position coordinates and three momenta, so the associated state space is ${ℝ}^6$. In Chapter 14, we will see that this description is incomplete: Particles also have an additional degree of freedom called spin.

This chapter discusses the properties of matrix product state (MPS). It starts with a simple proof that the wave function generated by DMRG is an MPS. Then three different but equivalent canonical forms or representations of MPS are introduced. An MPS generally has redundant gauge degrees of freedom on each bond linking two neighboring local tensors. One can convert it into a canonical form by taking a canonical transformation to remove the gauge redundancy in the local tensors. Finally, the implementation of symmetries, including both the U(1) and SU(2) symmetries, is discussed.

In classical mechanics, a particle of mass $m$ subject to a restoring force linear in displacement, $x$, from a potential minimum such that $F(x)=-{\kappa }_{0}x$, where ${\kappa }_0$ is the force constant, results in one-dimensional simple harmonic motion with an oscillation frequency $\omega =\sqrt{{\kappa }_0/m}$.

If one reads about science, writes about science, or teaches science, one should know about the whats, hows, whens, and whys of science. What is science? How is it done? When is science needed? Science seeks to understand and systematize the natural world. It does so experimentally, using test tubes, computers, and animals (including humans), among other things. Curiosity and necessity drive science. Since ancient times, people have wanted to understand and then manipulate their world. For example, science has provided the means to painlessly and noninvasively look into the human body through the development of X-rays, magnetic resonance imaging, computed tomography, and scintigraphy (radioisotopes). Electronics and materials science have enabled creation of cell phones. Chemistry has given us therapeutic drugs, Teflon, and Velcro. Physics and engineering have taken us to the Moon, Mars, and beyond. This broad scope of science makes it difficult, but not impossible, to define. This chapter provides a holistic view of science.

The fundamental kinematical symmetry is the invariance under transformations between inertial reference frames. In the regime of small velocities, this symmetry corresponds to the Galilei group. However, the Galilei symmetry is only approximate. The exact symmetry, in the absence of gravity, is defined by the Poincaré group, which we analyse here.

This chapter constructs the MPS representation of a quantum state in the continuous limit. It starts with an MPS representation for the corresponding state in a discretized lattice system. Then the limit of the discretized lattice constant going to zero is taken to obtain its continuous presentation. The formulas for determining the expectation values are also derived. Finally, we discuss the scheme of canonicalization and the method for optimizing the local tensors of the continuous MPS.

In Chapter 11, we saw that by Wigner’s theorem, symmetry transformations in quantum theory are represented by unitary or antiunitary operators. Then, we focused exclusively in the symmetry of space rotations. Since our focus was so narrow, we did not have to introduce the most appropriate language for the description of symmetries, namely group theory.

This chapter discusses the properties of infinite MPS and their associated transfer matrices. The formulas for determining the expectation values of physical observables are derived and expressed using the leading eigenvectors of the transfer matrix. The concept of the string order parameter is introduced and exemplified with the AKLT state, followed by a statement on the condition for the existence of string order. Furthermore, the procedure of canonicalizing an infinite MPS with one or more than one site in a unit cell is discussed.