Topological Uhlmann phase transitions for a spin-jparticle in a magnetic field

Uhlmann phase of coherent states and the Uhlmann-Berry correspondence

“…Our exact results of the Uhlmann phases of bosonic and fermionic coherent states suggest that they indeed carry geometric information, as expected by the concept of holonomy and analogy to the Berry phase. We will show that the Uhlmann phases of both cases decrease smoothly with temperature without a finite-temperature transition, in contrast to some examples with finite-temperature transitions in previous studies [22][23][24][25][26][27][28][29][30]. As temperature drops to zero, the Uhlmann phases of bosonic and fermionic coherent state approach the corresponding Berry phases.…”

“…The energy levels of system are characterized by Ĥ|n〉 = ħ hω(n+ 1 2 )|n〉 with n = 0, 1, 2, • • • . Previously studied examples of the Uhlmann phase of low-dimensional systems [22,26,28] and spin-j systems [24,25] are both in finite-dimensional Hilbert spaces. The bosonic harmonic oscillator will give an infinite-dimensional example.…”

“…To present an exact correspondence between the Uhlmann and Berry phases, we instead simplify the qutrit model to a spin-j paramagnet with j = 1, whose Uhlmann phase has been studied in Refs. [24,25]. A loop in the parameter space of the spin-1 model corresponds to a loop on the two-dimensional unit sphere S 2 .…”

“…There have been several approaches to address this problem [14][15][16][17][18][19][20][21], among which the Uhlmann phase has attracted much attention recently since it has been shown to exhibit topological phase transitions at finite temperatures in several 1D, 2D, and spin-j systems [22][23][24][25][26].…”

“…However, due to the complexity of the mathematical structure and physical interpretation, the knowledge of the Uhlmann phase is far less than that of the Berry phase in the literature. Moreover, only a handful of models allow analytical results of the Uhlmann phase to be obtained [22][23][24][25][26][27][28][29][30]. The Berry phase is purely geometric in the sense that it does not depend on any dynamical effect during the time evolution of the quantum system of interest [31].…”