On the Equilibrium States of Infinite Quantum Systems at T ≠ O

Rudolf Haag

Yuuya Chiba, Yasushi Yoneta, Ryusuke Hamazaki +1 more

The second law of thermodynamics for adiabatic operations -- constraints on state transitions in closed systems under external control -- is one of the fundamental principles of thermodynamics. On the other hand, it is recently established that even pure quantum states can represent thermal equilibrium. However, pure quantum states do not satisfy the second law in that they are not passive, i.e., work can be extracted from them if arbitrary unitary operations are allowed. It therefore remains unresolved how quantum mechanics can be reconciled with thermodynamics. Here, based on our key quantum-mechanical notions of thermal equilibrium and adiabatic operations, we address the emergence of the second law for adiabatic operations in the thermodynamics limit. We first introduce infinite-observable macroscopic thermal equilibrium (iMATE); a quantum state, including pure states, is in iMATE if the expectation values of all additive observables agree with their equilibrium values. We also introduce a macroscopic operation as unitary evolution generated by a time-dependent additive Hamiltonian, which is regarded as corresponding to adiabatic operations. Employing these concepts, we show that no extensive work can be extracted from any quantum state in iMATE through any macroscopic operations. Furthermore, we introduce a quantum-mechanical form of entropy density such that it agrees with thermodynamic entropy density for any quantum state in iMATE. We then prove that for any initial state in iMATE, this entropy density cannot be decreased by any macroscopic operations, followed by a time-independent relaxation process. Our theory thus proves two different forms of the second law, by adopting macroscopically reasonable classes of observables, equilibrium states, and operations. We also discuss the time scales of macroscopic operations in these results.

Pierre M. V. Résibois

Nicolaas Marinus Hugenholtz

Infinite systems, particularly infinite quantum systems are considered. It is pointed out that algebraic methods can be applied not only to statistical mechanics but also to quantum field theory. Inequivalent representations of canonical anticommutation relations, states and observables, observable algebras for certain simple cases, the ground state, the equilibrium state at T and O, the Heisenberg model with long-range interaction are covered. (JFP)

Anushya Chandran, Arun Nanduri, Steven S. Gubser +1 more

The quantum $O(N)$ model in the infinite-$N$ limit is a paradigm for symmetry breaking. Qualitatively, its phase diagram is an excellent guide to the equilibrium physics for more realistic values of $N$ in varying spatial dimensions ($d>1$). Here, we investigate the physics of this model out of equilibrium, specifically its response to global quenches starting in the disordered phase. If the model were to exhibit equilibration, the late-time state could be inferred from the finite-temperature phase diagram. In the infinite-$N$ limit, we show that not only does the model not lead to equilibration on account of an infinite number of conserved quantities, it also does not relax to a generalized Gibbs ensemble (GGE) consistent with these conserved quantities. Instead, an infinite number of new conservation laws emerge at late times and the system relaxes to an emergent GGE consistent with these. Nevertheless, we still find that the late-time states following quenches bear strong signatures of the equilibrium phase diagram. Notably, we find that the model exhibits coarsening to a nonequilibrium critical state only in dimensions $d>2$, that is, if the equilibrium phase diagram contains an ordered phase at nonzero temperatures.

Oscar E. Lanford, Derek W. Robinson