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Quantum critical effects in the spectrum of excited states

Felipe Andrés Matus Ramírez

Published Jan 1, 2025• 0 citations
OpenAlex

Abstract

We explore quantum driving protocols that optimize ground-state fidelity in both closed and open quantum systems, using a model of interacting qubits governed by a Lipkin-Meshkov-Glick type of Hamiltonian. This model exhibits both ground-state and excited-state quantum phase transitions (QPTs and ESQPTs). For comparison, we also analyze a single-qubit system. The selected driving paths in parameter space traverse quantum critical regions, which have a large impact on the target ground-state fidelity. The influence of ESQPTs is recognized particularly in the case of an open system. We apply advanced theoretical techniques like quantum geometry, adiabatic perturbation theory and the method of hierarchical equations of motion.

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We explore quantum driving protocols that optimize ground-state fidelity in both closed and open quantum systems, using a model of interacting qubits governed by a Lipkin-Meshkov-Glick type of Hamiltonian. This model exhibits both ground-state and excited-state quantum phase transitions (QPTs and ESQPTs). For comparison, we also analyze a single-qubit system. The selected driving paths in parameter space traverse quantum critical regions, which have a large impact on the target ground-state fidelity. The influence of ESQPTs is recognized particularly in the case of an open system. We apply advanced theoretical techniques like quantum geometry, adiabatic perturbation theory and the method of hierarchical equations of motion.
An open fully connected system of qubits at nonzero temperature is driven within a finite time interval along various paths in the space of its control parameters. The driving leads across finite-size precursors of first- and second-order quantum phase transition from factorized to entangled ground-state phases, aiming at the preparation of the complex ground state of the system at the final parameter point with maximal fidelity. During the drive, the system is coupled to a heat bath at a constant temperature, the dynamics being determined in a nonperturbative way by the method of hierarchical equations of motion. It is shown that the presence of the heat bath in combination with specific patterns of avoided crossings affecting the ground and excited states in the parameter region around the quantum phase transition may considerably improve the fidelity of preparation of the target ground state.
Dynamical signatures of quantum phase transitions for excited states Jakub Dolejší Abstract We study the impact of quantum phase transitions (QPTs) and excited- state quantum phase transitions (ESQPTs) on the validity of the adiabatic approximation for a slowly varying Hamiltonian. We compare two cases, when the initial state is the ground state of the initial Hamiltonian and when the initial state is a statistical mixture of excited states induced by a finite temperature. We use the Lipkin-Meshkov-Glick model of a spin lattice and obtain an abruptly decreasing scaling law of the ground-state population with a growing system size N. We comment on the justifiability of using the Landau-Zener formula to make a quantitative prediction in the case of a first-order and a second-order QPT. To achieve a truly adiabatic evolution in the thermodynamic limit, one would need to perform the Hamiltonian change during an impossibly long time period. It is possible, however, to obtain the same adiabatic final state in a given finite time period by inducing the quantum evolution with another Hamiltonian specifically devised for this purpose, thus employing the so called adiabatic shortcut. We verify the validity of adiabatic shortcuts in the presence of QPTs and ESQPTs and study the costs of performing such adiabatic...
The basic Lipkin-Meshkov-Glick model displays a second-order ground-state quantum phase transition and an excited-state quantum phase transition (ESQPT). The inclusion of an anharmonic term in the Hamiltonian implies a second ESQPT of a different nature. We characterize this ESQPT using the mean field limit of the model. The alternative ESQPT, associated with the changes in the boundary of the finite Hilbert space of the system, can be properly described using the order parameter of the ground-state quantum phase transition, the energy gap between adjacent states, the participation ratio, and the quantum fidelity susceptibility.
This work is concerned with the excited state quantum phase transitions (ESQPTs) defined in Ann. Phys. 323, 1106 (2008). In many‐body models that exhibit such transitions, the ground state quantum phase transition (QPT) occurs in parallel with a singularity in the energy spectrum that propagates to higher energies as the control parameter increases beyond the QPT critical point. The analysis of the spectrum has been a main tool for the detection of these ESQPTs. Studies of the effects of this transition on the system dynamics are more limited. Here, we extend our previous works and show that the evolution of an initial state with energy close to the ESQPT critical point may be extremely slow. This result is surprising, because it may take place in systems with long‐range interactions, where the dynamics is usually expected to be very fast. A timely example is the one‐dimensional spin‐1/2 model with infinite‐range Ising interaction studied in experiments with ion traps. Its Hamiltonian has a U(2) algebraic structure. More generally, the slow dynamics described here occurs in two‐level bosonic or fermionic models with pairing interactions and a Hamiltonian exhibiting a QPT between its limiting and dynamical symmetries. In this work, we compare the results for , and 3.
We apply the method of transitionless quantum driving for time-dependent quantum systems to spin systems. For a given Hamiltonian, the driving Hamiltonian is constructed so that the adiabatic states of the original system obey the Schrödinger equation. For several typical systems such as the XY spin chain and the Lipkin-Meshkov-Glick model, the driving Hamiltonian is constructed explicitly. We discuss possible interesting situations when the driving Hamiltonian becomes time independent and when the driving Hamiltonian is equivalent to the original one. For many-body systems, a crucial problem occurs at the quantum phase transition point where the energy gap between the ground and first excited states becomes zero. We discuss how the defect can be circumvented in the present method.
Excited-state quantum phase transitions (ESQPTs) are generalizations of quantum phase transitions to excited levels. They are associated with local divergences in the density of states. Here, we investigate how the presence of an ESQPT can be detected from the analysis of the structure of the Hamiltonian matrix, the level of localization of the eigenstates, the onset of bifurcation, and the speed of the system evolution. Our findings are illustrated for a Hamiltonian with infinite-range Ising interaction in a transverse field. This is a version of the Lipkin-Meshkov-Glick (LMG) model and the limiting case of the one-dimensional spin-12\frac{1}{2}21​ system with tunable interactions realized with ion traps. From our studies for the dynamics, we uncover similarities between the LMG and the noninteracting XX models.