Marek M. Rams

Assisted Finite-Rate Adiabatic Passage Across a Quantum Critical Point: Exact Solution for the Quantum Ising Model

The dynamics of a quantum phase transition is inextricably woven with the formation of excitations, as a result of critical slowing down in the neighborhood of the critical point. We design a transitionless quantum driving through a quantum critical point, allowing one to access the ground state of the broken-symmetry phase by a finite-rate quench of the control parameter. The method is illustrated in the one-dimensional quantum Ising model in a transverse field. Driving through the critical point is assisted by an auxiliary Hamiltonian, for which the interplay between the range of the interaction and the modes where excitations are suppressed is elucidated.

Quantum fidelity in the thermodynamic limit.

We study quantum fidelity, the overlap between two ground states of a many-body system, focusing on the thermodynamic regime. We show how a drop in fidelity near a critical point encodes universal information about a quantum phase transition. Our general scaling results are illustrated in the quantum Ising chain for which a remarkably simple expression for fidelity is found.

Exact results for fidelity susceptibility of the quantum Ising model: the interplay between parity, system size, and magnetic field

We derive an exact closed-form expression for fidelity susceptibility of even- and odd-sized quantum Ising chains in a transverse field. To this aim, we diagonalize the Ising Hamiltonian and study the gap between its positive and negative parity subspaces. We derive an exact closed-form expression for the gap and use it to identify the parity of the ground state. We point out the misunderstanding in some of the former studies of fidelity susceptibility and discuss its consequences. Last but not least, we rigorously analyze the properties of the gap. For example, we derive analytical expressions showing its exponential dependence on the ratio between the system size and the correlation length.

Transfer matrices and excitations with matrix product states

We investigate the relation between static correlation functions in the ground state of local quantum many-body Hamiltonians and the dispersion relations of the corresponding low energy excitations using the formalism of tensor network states. In particular, we show that the Matrix Product State Transfer Matrix (MPS-TM) - a central object in the computation of static correlation functions - provides important information about the location and magnitude of the minima of the low energy dispersion relation(s) and present supporting numerical data for one-dimensional lattice and continuum models as well as two-dimensional lattice models on a cylinder. We elaborate on the peculiar structure of the MPS-TMs eigenspectrum and give several arguments for the close relation between the structure of the low energy spectrum of the system and the form of static correlation functions. Finally, we discuss how the MPS-TM connects to the exact Quantum Transfer Matrix (QTM) of the model at zero temperature. We present a renormalization group argument for obtaining finite bond dimension approximations of MPS, which allows to reinterpret variational MPS techniques (such as the Density Matrix Renormalization Group) as an application of Wilsons Numerical Renormalization Group along the virtual (imaginary time) dimension of the system.

Scaling of ground-state fidelity in the thermodynamic limit: XY model and beyond

We study ground-state fidelity defined as the overlap between two ground states of the same quantum system obtained for slightly different values of the parameters of its Hamiltonian. We focus on the thermodynamic regime of the XY model and the neighborhood of its critical points. We describe extensively fidelity when it is dominated by the universal contribution reflecting the quantum criticality of the phase transition. We show that proximity to the multicritical point leads to anomalous scaling of fidelity. We also discuss fidelity in a regime characterized by pronounced oscillations resulting from the change in either the system size or the parameters of the Hamiltonian. Moreover, we show when fidelity is dominated by non-universal contributions, study fidelity in the extended Ising model, and illustrate how our results provide additional insight into dynamics of quantum phase transitions. Special attention is given to studies of fidelity from the momentum space perspective. All our main results are obtained analytically. They are in excellent agreement with numerics.

Variational tensor network renormalization in imaginary time: Benchmark results in the Hubbard model at finite temperature

A Gibbs operator

Matrix product state renormalization

{e}^{\ensuremath{-}\ensuremath{\beta}H}

A quantum phase transition in a quantum external field: Superposing two magnetic phases

for a two-dimensional (2D) lattice system with a Hamiltonian

Engineering non-equilibrium quantum phase transitions via causally gapped Hamiltonians

H

Truncating an exact matrix product state for the XY model: Transfer matrix and its renormalization

can be represented by a 3D tensor network, with the third dimension being the imaginary time (inverse temperature)