Renormalization for Quantum Systems

Igor Viktorovich Tyutin

Wei-Ming Huang, Wei-Min Zhang

By solving the exact master equation of open quantum systems, we formulate the quantum thermodynamics from weak to strong couplings. The open quantum systems exchange matters, energies, and information with their reservoirs through quantum particle tunnelings that are described by the generalized Fano-Anderson Hamiltonians. We find that the exact solution of the reduced density matrix of these systems approaches a Gibbs-type state in the steady-state limit for the systems in arbitrary initial states as well as for both the weak and strong system-reservoir coupling strengths. When the couplings become strong, thermodynamic quantities of the system must be renormalized. The renormalization effects are obtained nonperturbatively after exactly tracing over all reservoir states through the coherent state path integrals. The renormalized system Hamiltonian is characterized by the renormalized system energy levels and interactions, corresponding to the quantum work done by the system. The renormalized temperature is introduced to characterize the entropy production counting the heat transfer between the system and the reservoir. We further find that only with the renormalized system Hamiltonian and other renormalized thermodynamic quantities can the exact steady state of the system be expressed as the standard Gibbs state. Consequently, the corresponding exact steady-state particle occupations in the renormalized system energy levels obey the Bose-Einstein and the Fermi-Dirac distributions for bosonic and fermionic systems, respectively. In the very weak system-reservoir coupling limit, the renormalized system Hamiltonian and the renormalized temperature are reduced to the original bare Hamiltonian of the system and the initial temperature of the reservoir. Thus, the conventional statistical mechanics and thermodynamics are thereby rigorously deduced from quantum dynamical evolution. In the last, this nonperturbative renormalization method is also extended to general interacting open quantum systems.

Cristina Manuel, R. Tarrach

M. Kolb

A real-space renormalization-group transformation for quantum systems is proposed, based on an equivalent classical representation. Classical real-space renormalization groups and in particular Monte Carlo renormalization groups can then be used to study quantum systems. This formulation is valid for zero temperature as well as finite temperature. The Ising model in a transverse field in one and two dimensions is treated by Monte Carlo renormalization group.

Rémi Jullien, K. A. Penson, Pierre M. Pfeuty +1 more

Gang Su

We review the development of numerical renormalization group methods,including Wilson's numerical renormalization group,White's density matrix renormalization group,and several recent rapidly developing tensor network state-based algorithms.Among these,particular emphasis is laid on the linearized tensor renormalization group method,which was recently proposed to accurately calculate the thermodynamic properties of quantum many-body lattice systems,and which may have broad applications in both one-and two-dimensional quantum systems.