Non-local quantum superpositions of topological defects

Abstract

Topological defects (such as monopoles, vortex lines, or domain walls) mark locations where disparate choices of a broken symmetry vacuum elsewhere in the system lead to irreconcilable differences 1,2 . They are energetically costly (the energy density in their core reaches that of the prior symmetric vacuum) but topologically stable (the whole manifold would have to be rearranged to get rid of the defect). We show how, in a paradigmatic model of a quantum phase transition, a topological defect can be put in a nonlocal superposition, so that -in a region large compared to the size of its core -the order parameter of the system is "undecided" by being in a quantum superposition of conflicting choices of the broken symmetry. We dub such a topological Schrödinger cat state a 'Schrödinger kink', and devise a version of a double-slit experiment suitable for topological defects to describe one possible manifestation of the phenomenon. Coherence detectable in such experiments will be suppressed as a consequence of interaction with the environment. We analyze the environment-induced decoherence and discuss its role in symmetry breaking.Topological defects are the epitome of locality. An example of a defect occurs in the quantum Ising model where a lattice of spins interact ferromagnetically with their nearest neighbors, i.e., the Hamiltonian contains an interaction of the form −σ z n σ z n+1 for the n th spin on the lattice. The entire collection of spins achieves its lowest energy when they are all aligned. However, there are two choices for this lowest energy state, | · · · ↑↑↑↑ · · · or | · · · ↓↓↓↓ · · · . Both are energetically identical but each of them breaks the symmetry of the Hamiltonian, which has no preference between "up" and "down".A configuration that includes topological defects could arise, for example, in a driven phase transition, i.e., "a quench". In the Ising model, a quench can be induced when the transverse field strength g is changed in the HamiltonianWhen the external field is strong enough (g 1), it "wins" and all spins align with σ x . A decrease in g, however, will lead to a phase transition at g = 1 with the symmetry breaking term −σ z n σ z n+1 trying to align all the spins in their z-direction. The choice of whether the spins will point up or down is made locally. As a result of causality and the finite speed at which signals propagate, the size of these domains will be determined by rate of change of g 3-8 . This will lead to configurations, such as | · · · ↑↑↓↓ · · · , where the topological defect marks the location of a "switch" from one broken symmetry ground state to the other. In this one-dimensional example, the defect is a kink (a domain wall), but, in a sense, all the topological defects (monopoles, vortex lines, and so on) that exist in higher dimensions "look the same". A discussion of the consequences of this process for the post-transition state is beyond the scope of this work (but see, e.g., Refs. 9-11). We only note that the density of the resulting kinks will be finite.The...

Kibble-Zurek scaling of the irreversible entropy production

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Decoherence and relaxation of topological states in extended quantum Ising models

“…More recently, the quantum Ising model was studied in the non-equilibrium regime to investigate the dynamical behavior of quantum phase transitions, e.g. the quenching in a driven Ising chain [13][14][15][16][17][18], the Kibble-Zurek mechanism [19,20], the Loschmidt echo of a single impurity coupled to the Ising chain [21], the engineered quantum transfer [22], the quantum superposition of topological defects [23], the decoherence dynamics in the strong coupling regime [24] as well as the role of quantum correlations in quantum phase transitions [25][26][27]. Importantly, the generalized class of Ising models can be characterized by a topological number [28][29][30][31][32] and, in the topologically nontrivial phase, localized states can occur at the end of an open chain [1,4] or at the interface separating regions with different topological number [33].…”

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