This chapter introduces the quantum transfer matrix renormalization group (QTMRG). It is a method of studying the thermodynamic and correlation functions of one-dimensional quantum lattice models. The RG transformation matrices are determined using the criteria presented in the preceding chapter and used to update the transfer matrix and other physical quantities. The spin-1/2 and spin-1 antiferromagnetic Heisenberg models are used to demonstrate the accuracy and efficiency of the method.

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.

Presents the basic postulates of quantum mechanics in terms of the density matrix instead of the usual state vector formalism in the case of an isolated system. Extends it to the case of open systems with the help of the reduced density matrix formalism, and to the case of an imperfect state preparation described by a statistical mixture. Introduces the concept of quantum state purity to characterize the degree of mixture of the state, and shows that one can always "purify" a density matrix by going into a Hilbert space of larger dimension.

Typically, we are interested in a small system (a few particles, or some region in space), entangled with its surroundings. Exact equations (Dyson or Nakajima–Zwanzig) for the reduced system dynamics are readily derived using suitable projection operators. They are rarely solvable, however, since full account is taken for mutual influences between the system and its environment. Nevertheless, for weak mutual coupling, environment-induced dynamics in the (small) system can be much slower than system-induced dynamics in the (large) environment. This justifies the Born–Markov approximation, leading to closed equations for the system. In Hilbert space, we demonstrate the emergence of irreversible dynamics and exponential decay for pure sates. In Liouville space, we derive the Redfield equation. Invoking the secular (or, rotating-wave) approximation, we derive Pauli’s master equations, which properly account for relaxation to equilibrium. Rates of spontaneous emission, coherence transfer (Bloch equations), and pure dephasing are derived and analyzed for a dissipative qubit.