This chapter introduces the tensor network representation of the partition functions of classical statistical models. A classical statistical model may have different tensor network representations. Using simple examples, we show how to represent a classical statistical model as a tensor network model, for example, on the lattice where the Hamiltonian is defined or on its dual lattice. A tensor network representation of a model defined on a vertex-sharing lattice is also introduced. Finally, we discuss the duality properties of the q-state clock and q-state Potts models and determine their self-dual points using their tensor network models on the square lattice.

One of the most important concepts of classical mechanics is that of a closed system. A closed system is loosely defined as a system whose components interact only with each other, and it is characterized by phase space volume conservation and energy conservation – see Section 1.2.

We saw in Chapter 2 that Born’s statistical interpretation of the wave function was one of the building blocks of quantum theory. According to Born’s interpretation, the wave function of a particle at a given moment of time defined a probability density $|\psi (x){|}^2$ with respect to the position $x$. This result is generalized to state vectors of an Hilbert space and to general observables through the following procedure.

This chapter discusses the truncation criteria in the RG treatment of a non-Hermitian matrix, starting with a modified definition of the reduced density matrix using the leading left and right eigenvectors. As the reduced density matrix so defined is not Hermitian, there is no theorem to protect or guarantee that its eigenvalues are semi-positive definite. This non-Hermitian problem causes trouble in the determination of an optimized truncation scheme. Three truncation schemes for determining the RG transformation matrices are introduced, relying on the canonical diagonalization of the reduced density matrix, biorthonormalization, and lower-rank approximation of the environment density matrix, respectively. The canonical diagonalization scheme is optimal if the reduced density matrix is semi-positive definite. The scheme of biorthonormalization may not be optimal, but it is mathematically more stable.

In Chapter 1, we presented the fundamental principles of classical physics, and then we motivated and presented the fundamental principles of quantum physics. The two sets of principles are summarized and compared in Table 10.1.

Truncation of basis states is a vital step in the tensor network renormalization. This chapter introduces the concept of reduced density matrices and discusses the criterion of judging which state should be retained and which not in the basis truncation. In a Hermitian system, the reduced density matrix of a quantum state is semi-positive definite, and its eigenvalues measure the probabilities of the corresponding eigenvectors. Therefore, we should do the truncation according to the eigenvalues of the reduced density matrix. This criterion is equivalent to taking a Schmidt decomposition for the wave function of the quantum state and truncating the basis states according to their singular values. It is also equivalent to maximizing the fidelity of the targeted state before and after truncation. We also introduce the edge and bond density matrices and show that they have the same eigen-spectra as the reduced density matrix.

This chapter details the practical, theoretical, and philosophical aspects of experimental science. It discusses how one chooses a project, performs experiments, interprets the resulting data, makes inferences, and develops and tests theories. It then asks the question, "are our theories accurate representations of the natural world, that is, do they reflect reality?" Surprisingly, this is not an easy question to answer. Scientists assume so, but are they warranted in this assumption? Realists say "yes," but anti-realists argue that realism is simply a mental representation of the world as we perceive it, that is, metaphysical in nature. Regardless of one's sense of reality, the fact remains that science has been and continues to be of tremendous practical value. It would have to be a miracle if our knowledge and manipulation of the nature were not real. Even if they were, how do we know they are true in an absolute sense, not just relative to our own experience? This is a thorny philosophical question, the answer to which depends on the context in which it is asked. The take-home message for the practicing scientist is "never assume your results are true."