Topological Uhlmann phase transitions for a spin-$j$ particle in a magnetic field

The generalization of the geometric phase to the realm of mixed states is known as the Uhlmann phase. Recently, applications of this concept to the field of topological insulators have been made and an experimental observation of a characteristic critical temperature at which the topological Uhlmann phase disappears has also been reported. Here we study the case of the Uhlmann phase of a paradigmatic system such as the spin-$j$ particle in the presence of a slowly rotating magnetic field at finite temperature in an exact analytical form. We find that the Uhlmann phase is given by the argument of a complex-valued second-kind Chebyshev polynomial of order $2j$. Correspondingly, the Uhlmann phase displays $2j$ singularities, occurring at the roots of such polynomials which define the critical temperatures at which the system undergoes topological order transitions. Appealing to the argument principle of complex analysis, each topological order is characterized by a winding number, which happens to be $2j$ for the ground state and decreases by unity each time the increasing temperature passes through a critical value. We hope this study encourages experimental verification of this phenomenon of thermal control of topological properties, as has already been done for the spin-$1/2$ particle.

Figure 1

Uhlmann topological phases ${\mathrm{\Phi }}_U^{\left(j\right)}$ (solid) and Chebyshev polynomials ${(-1)}^{2j}{U}_{2j}$ (dashed) as functions of $\beta B$, for $\theta =\pi /2$. The left column presents the phase for the half-integer values (a) $j=1/2$, (c) =3/2, and (e) =5/2. The right column displays the phase for the integer values (b) $j=1$, (d) =2, and (f) =3. The integers between the zeros of the polynomials indicate associated winding numbers, corresponding to Uhlmann numbers $n_U^{\left(j\right)}$ (see text in Sec. 3b).

Figure 2

Argand diagram of $z\left(\theta \right)$ for several values of $\beta B$. Considering the case $j=3/2$, the points marked on the real axis are the roots of polynomial $U_3\left(z\right)$. At high enough temperatures, the curve $z\left(\theta \right)$ (solid purple) does not enclose any root. As the temperature reduces, the curve expands and progressively encloses the roots. The Uhlmann numbers are given by the number of roots inside $z\left(\theta \right)$.

Figure 3

Uhlmann number as a function of temperature, for some values of the spin number $j$. The steps are located at a set of values of ${\beta }_{k}B$, which define critical temperatures $T_{c,k}^{\left(j\right)}$, with $k=1,...,2j$.

Figure 4

Uhlmann phases ${\mathrm{\Phi }}_U^{\left(j\right)}(\theta ,\beta B)$ for $j=1/2,1,...,3$. There are $2j$ singularities along the $\theta =\pi /2$ line. The Uhlmann number characterizing a topological order can be obtained visually by encircling one or more singularities with a simple closed curve and counting the number of times an isochromatic line is repeated (see text).