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Open Systems & Information Dynamics

Quantum Criticality of Ground and Thermal States in XX Model

Wonmin Son, Vlatko Vedral

Published Aug 13, 2009• 1 citation
Publisher

Abstract

Here, we compare the critical behaviour of the ground state and the thermal state of the XX model. We analyze full energy spectra and the exact form of eigenstates to reconstruct the ground state and the thermally excited state. With the solutions, we discuss the properties of quantum phase transition at zero temperature as well as the state in a thermal equilibrium.

DOI
10.1142/s1230161209000219

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Here, we compare the critical behaviour of the ground state and the thermal state of the XX model. We analyze full energy spectra and the exact form of eigenstates to reconstruct the ground state and the thermally excited state. With the solutions, we discuss the properties of quantum phase transition at zero temperature as well as the state in a thermal equilibrium.
Quantum phase transitions occur at zero temperature driven by quantum fluctuations instead of thermal fluctuations. They take place due to competing ground state phases that are accessible for different values of certain non-thermal parameters such as coupling constants, pressure or magnetic field. The chapter starts with a discussion of the main phenomenological features of this class of non-thermal transitions. In particular, it is argued that traces of these transitions can be detected at finite (but low) temperature. Then, examples of materials that show this behaviour are provided. Finally, the quantum Ising model is discussed and it is shown that a quantum model in d dimensions can be mapped to a classical model in d+1 dimensions.
We report a kind of phase transition which takes place in isolated quantum systems with a discrete symmetry. Different phases are separated by a critical energy and recognized by means of an order parameter, which can be experimentally measured. Below the critical energy, the symmetry is broken and the equilibrium state is non-thermal, keeping large amounts of memory of the initial condition. Above, the symmetry is restored and a thermal equilibrium state is reached. We identify two universality classes, one showing a second-order phase transition and represented by the Lipkin-Meshkov-Glick model, and another showing a first-order phase transition and represented by the Dicke model. This work is dedicated to the memory of Joaquín Retamosa .
We study the thermodynamics of the full version of the Dicke model, including all the possible values of the total angular momentum j, with both microcanonical and canonical ensembles. We focus on both the excited-state quantum phase transition, appearing in the microcanonical description of the maximum angular momentum sector, j=N/2, and the thermal phase transition, which occurs when all the sectors are taken into account. We show that two different features characterize the full version of the Dicke model. If the system is in contact with a thermal bath and is described by means of the canonical ensemble, the parity symmetry becomes spontaneously broken at the critical temperature. In the microcanonical ensemble, and despite that all the logarithmic singularities which characterize the excited-state quantum phase transition are ruled out when all the j sectors are considered, there still exists a critical energy (or temperature) dividing the spectrum into two regions: one in which the parity symmetry can be broken, and another in which this symmetry is always well defined.
Abstract The thermodynamical properties of a generalized Dicke model are calculated and related with the critical properties of its energy spectrum, namely the quantum phase transitions (QPT) and excited state quantum phase transitions (ESQPT). The thermal properties are calculated both in the canonical and the microcanonical ensembles. The latter deduction allows for an explicit description of the relation between thermal and energy spectrum properties. While in an isolated system the subspaces with different pseudospin are disconnected, and the whole energy spectrum is accessible, in the statistical ensemble the situation is radically different. The multiplicity of the lowest energy states for each pseudospin completely dominates the thermal behavior, making the set of degenerate states with the smallest pseudospin at a given energy the only ones playing a role in the thermal properties. As a result, the states in the region with positive thermal energy cannot be thermally populated because their negligible probability, making that energy region thermally unreachable at finite temperatures. The quantum phase transitions of the lowest energy states, from a normal to a superradiant phase, produce the thermal transition. The other critical phenomena, the ESQPTs occurring at excited energies, have no manifestation in the thermodynamics, although their effects could be seen in finite size corrections. A new superradiant phase is found, which only exists in the generalized model, and can be relevant in finite size systems.