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arXiv:2310.01493 · Physical review. B./Physical review. B

Pseudogap phases, chiral anomaly, and topological order with quantum loop entanglement

Predrag Nikolić

Published Aug 26, 2024• 1 citation
arXivHTMLPDF

Abstract

A many-body quantum system whose topological defects are conserved, abundant, and mobile is a correlated quantum liquid. Since topological defects can be classified by homotopy groups, each homotopy identifies a class of quantum liquids. Here we explore the quantum liquids based on the ${\ensuremath{\pi}}_{3}({S}^{2})$ homotopy group, i.e., Hopf fibration. Their topologically nontrivial dynamics emerges from the interlinking between magnetic flux or skyrmion loops in the charge and spin sectors respectively. We lay down a field theory foundation for analyzing such states by naturally incorporating the well-known framing regularization into the theory, and constructing the appropriate topological Lagrangian terms. We show that at least two strongly correlated phases of interlinked loops can exist in $d=3$ spatial dimensions at zero and low-finite temperatures. These phases are closely related to the chiral quantum anomaly and do not have an obvious topological order, but they are distinguished from the trivial disordered phase by a generalization of the Wilson loop operator. In $d=4$ spatial dimensions, interlinked loops are able to produce topological order at zero temperature, featuring charge, angular momentum, and braiding fractionalization. We discuss some possible experimental signatures of loop entanglement in the quantum noise of charge currents.

DOI
10.1103/physrevb.110.075148

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Topological states of matter are, generally, quantum liquids of conserved topological defects. We establish this by constructing and analyzing topological field theories which introduce gauge fields to describe the dynamics of singularities in the original field configurations. Homotopy groups are utilized to identify topologically protected singularities, and the conservation of their protected number is captured by a topological action term that unambiguously obtains from the given set of symmetries. Stable phases of these theories include quantum liquids with emergent massless Abelian and non-Abelian gauge fields, as well as topological orders with long-range quantum entanglement, fractional excitations, boundary modes, and unconventional responses to external perturbations. This paper focuses on the derivation of topological field theories and basic phenomenological characterization of topological orders associated with homotopy groups ${\ensuremath{\pi}}_{n}({S}^{n}), n\ensuremath{\ge}1$. These homotopies govern monopole and hedgehog topological defects in d=n+1d=n+1d=n+1 dimensions, and enable the generalization of both weakly interacting and fractional quantum Hall liquids of vortices to d>2d>2d>2. Hedgehogs have not been in the spotlight so far, but they are particularly important defects of magnetic moments because they can be stimulated in realistic systems with spin-orbit coupling, such as chiral magnets and d=3d=3d=3 topological materials. We predict topological orders in systems with $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{Spin}(d)$ symmetry in which fractional electric charge attaches to hedgehogs. Monopoles, the analogous defects of charge or generic U(1) currents, may bind to hedgehogs via Zeeman effect, or effectively emerge in purely magnetic systems. The latter can lead to spin liquids with different topological orders than that of the resonant valence bond spin liquid. Charge fractionalization of quarks in atomic nuclei is also seen as possibly arising from the charge-hedgehog attachment.
We explore the stability of certain many-body quantum states which may exist at zero or finite temperatures, may lack long-range order and even topological order, and still are thermodynamically distinct from uncorrelated disordered phases. We sharply characterize such states by the conservation of topological charge, or equivalently confinement of instantons, using a generalization of the Wilson loop and the correlation length of an emergent gauge field. Our main conclusions are (i) topological orders can exist at finite temperatures, (ii) relativistic liquids of topological defects can also exist as stable phases at finite temperatures, and (iii) there are two universality classes of instanton suppression. We also relate the instanton dynamics to the problem of the pseudogap state in underdoped cuprates. A universal experimental signature of the instanton deconfinement transition is a change of the quantum noise spectrum, which can perhaps be measured in some situations, for example, via a quantum anomaly, or indirectly detected with a specific heat jump. The method of analysis is a functional renormalization group that generalizes the Coulomb gas treatment of Kosterlitz and Thouless to arbitrary interactions and dimensions. In particular, we construct an exact nonperturbative technique for confining interactions between instantons that introduce irreparable infrared divergences in the standard perturbative approaches.
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