Abstract
We develop a time-dependent real-space renormalization-group approach which can be applied to Hamiltonians with time-dependent random terms. To illustrate the renormalization-group analysis, we focus on the quantum Ising Hamiltonian with random site- and time-dependent (adiabatic) transverse-field and nearest-neighbour exchange couplings. We demonstrate how the method works in detail, by calculating the off-critical flows and recovering the ground state properties of the Hamiltonian such as magnetization and correlation functions. The adiabatic time allows us to traverse the parameter space, remaining near-to the ground state which is broadened if the rate of change of the Hamiltonian is finite. The quantum critical point, or points, depend on time through the time-dependence of the parameters of the Hamiltonian. We, furthermore, make connections with Kibble–Zurek dynamics and provide a scaling argument for the density of defects as we adiabatically pass through the critical point of the system.
Highlights
An interacting quantum system evolving at zero temperature can demonstrate various forms of approach to equilibrium even with no loss of phase coherence
We address the problem of a random quantum spin chain, requiring control of more than one parameter as we pass through a quantum critical point
We address the issue of a generic time dependence in the quantum Hamiltonian (Eq (1)), focussing on developing a complete analysis of the state of the system at any given time, as well as the dynamics across a given quantum critical point
Summary
An interacting quantum system evolving at zero temperature can demonstrate various forms of approach to equilibrium even with no loss of phase coherence. This can be important in the development of a device to simulate the quantum critical point: Optimisation problems commonly in the NP-complete and NP-hard classes can be mapped onto an Ising model [28, 29] ( the architecture of, for example the D:Wave machines [30, 31], is designed to use a quantum adiabatic protocol, and run based on an adiabatic quantum Ising Hamiltonian) As such we will concentrate our analysis on an adiabatic Ising Hamiltonian with random transverse-fields and exchange couplings, but simplify the setup to consider an infinitely long one-dimensional chain of Ising spins.
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