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Dynamic quantum phase transitions (DQPT) following quantum quenches exhibit singular behavior of the overlap between the initial and evolved states. Here we present a formalism to incorporate a geometric phase into quench dynamics of mixed quantum states, a process named the Uhlmann quench, based on the Uhlmann parallel transport. To overcome the incompatibility between the Uhlmann parallel-transport condition and Hamiltonian dynamics, we formulate the evolution of purification of the density matrix in a form which not only respects the dynamics according to the density matrix but also follows the Uhlmann parallel-transport condition to generate a geometric phase after a quantum quench. For cyclic processes exemplified by a spin-1/2 system, geometric DQPTs (GDQPTs) can emerge with both singular behavior in the dynamic analogue of the free energy and jumps of the geometric phase. Moreover, the Uhlmann phase reflecting the holonomy is generated at the end of each cycle. The Uhlmann quench thus paves the way for investigating the interplay between quantum dynamics and geometric processes in mixed states., Comment: 10 pages, 1 figure

The quantum geometric tensor (QGT) reveals local geometric properties and associated topological information of quantum states. Here a generalization of the QGT to mixed quantum states at finite temperatures based on the Sj$\ddot{\text{o}}$qvist distance is developed. The resulting Sj$\ddot{\text{o}}$qvist QGT is invariant under gauge transformations of individual spectrum levels of the density matrix. A Pythagorean-like relation connects the distances and gauge transformations, which clarifies the role of the parallel-transport condition. The real part of the QGT naturally decomposes into a sum of the Fisher-Rao metric and Fubini-Study metric, allowing a distinction between different contributions to the quantum distance. The imaginary part of the QGT is proportional to a weighted summation of the Berry curvatures, which leads to a geometric phase for mixed states under certain conditions. We present three examples of different dimensions to illustrate the temperature dependence of the QGT and a discussion on possible implications., Comment: 11 pages, 4 figures

We show that phase transitions in the quantum $q$-state clock model for $q \leq 4$ can be characterized by an enhanced decay behavior of the Loschmidt echo via a small quench. The quantum criticality of the quantum $q$-state clock model is numerically investigated by the finite-size scaling of the first minimum of the Loschmidt echo and the short-time average of the rate function. The equilibrium correlation-length critical exponents are obtained from the scaling laws which are consistent with previous results. Furthermore, we study dynamical quantum phase transitions by analyzing the Loschmidt echo and the order parameter for any $q$ upon a big quench. For $q \leq 4$, we show that dynamical quantum phase transitions can be described by the Loschmidt echo and the zeros of the order parameter. In particular, we find the rate function increases logarithmically with $q$ at the first critical time. However, for $q > 4$, we find that the correspondence between the singularities of the Loschmidt echo and the zeros of the order parameter no longer exists. Instead, we find that the Loschmidt echo near its first minimum converges, while the order parameter at its first zero increases linearly with $q$., Comment: 13 pages, 5 figures

In this paper, we investigate the spin and tunneling dynamics of a spin-orbit-coupled noninteracting Bose-Einstein condensate in a periodically driven non-Hermitian double-well potential. Under high-frequency driving, we obtain the effective time-averaged Hamiltonian by using the standard time-averaging method, and analytically calculate the Floquet quasienergies, revealing that the parity-time (PT)-breaking phase transition appears even for arbitrarily small non-Hermitian parameters when the spin-orbit coupling strength takes half-integer value, irrespective of the values of other parameters used. When the system is PT-symmetric with balanced gain and loss, we find numerically and analytically that in the broken PT-symmetric regions, there will exist the net spin current together with a vanishing atomic current, if we drop the contribution of the exponential growth of the norm to the current behaviors. When the system is non-PT-symmetric, though the quasienergies are partial complex, a stable net spin current can be generated by controlling the periodic driving field, which is accompanied by a spatial localization of the condensate in the well with gain. The results deepen the understanding of non-Hermitian physics and could be useful for engineering a variety of devices for spintronics., Comment: 12 pages, 6 figures

Motivated by the framework constructed by Brugnano and Casulli $[$SIAM J. Sci. Comput. 30: 463--472, 2008$]$, we analyze the finite termination property of the generalized Netwon method (GNM) for solving the absolute value equation (AVE). More precisely, for some special matrices, GNM is terminated in at most $2n + 2$ iterations. A new result for the unique solvability and unsolvability of the AVE is obtained. Numerical experiments are given to demonstrate the theoretical analysis., Comment: 11 pages

Recent advances in localized implicit functions have enabled neural implicit representation to be scalable to large scenes. However, the regular subdivision of 3D space employed by these approaches fails to take into account the sparsity of the surface occupancy and the varying granularities of geometric details. As a result, its memory footprint grows cubically with the input volume, leading to a prohibitive computational cost even at a moderately dense decomposition. In this work, we present a learnable hierarchical implicit representation for 3D surfaces, coded OctField, that allows high-precision encoding of intricate surfaces with low memory and computational budget. The key to our approach is an adaptive decomposition of 3D scenes that only distributes local implicit functions around the surface of interest. We achieve this goal by introducing a hierarchical octree structure to adaptively subdivide the 3D space according to the surface occupancy and the richness of part geometry. As octree is discrete and non-differentiable, we further propose a novel hierarchical network that models the subdivision of octree cells as a probabilistic process and recursively encodes and decodes both octree structure and surface geometry in a differentiable manner. We demonstrate the value of OctField for a range of shape modeling and reconstruction tasks, showing superiority over alternative approaches., Comment: 13 pages, 9 figures, NeurIPS 2021