Geometric Phases Characterise Operator Algebras and Missing Information

TL;DR

The paper builds a concrete link between geometric (Berry) phases and the algebraic structure of von Neumann algebras, showing that a vanishing geometric phase enables a well-defined trace while nonzero phases signal missing information across observer patches. Starting from simple bipartite spin systems and using the Sz SZK (entanglement-orbit) construction, it ties the phase to entanglement properties and entanglement temperature, then generalises to infinite-spin systems where the phase controls the existence of traces and the algebra type (II vs III). In holographic AdS/CFT, time-translation phases reveal a common centre in the left-right algebras of an eternal black hole and encode how 1/N corrections move the system from type III to type II, enabling a trace in the latter. The work further interprets geometric phases as indicators of missing information for local observers, both with and without entanglement, and discusses multiple Berry-phase variants (Virasoro, gauge, modular) in wormhole spacetimes, culminating in prospects for Hawking-Page transitions and symmetry-resolved entanglement geometry.

Abstract

We show how geometric phases may be used to fully describe quantum systems, with or without gravity, by providing knowledge about the geometry and topology of its Hilbert space. We find a direct relation between geometric phases and von Neumann algebras. In particular, we show that a vanishing geometric phase implies the existence of a well-defined trace functional on the algebra. We discuss how this is realised within the AdS/CFT correspondence for the eternal black hole. On the other hand, a non-vanishing geometric phase indicates missing information for a local observer, associated to reference frames covering only parts of the quantum system considered. We illustrate this with several examples, ranging from a single spin in a magnetic field to Virasoro Berry phases and the geometric phase associated to the eternal black hole in AdS spacetime. For the latter, a non-vanishing geometric phase is tied to the presence of a centre in the associated von Neumann algebra.

Figures (4)

• Figure 1: Visualisation of the duality between the eternal black hole in AdS spacetime and the two boundary CFTs entangled in the TFD state Maldacena:2001krVanRaamsdonk:2010pwMaldacena:2013xja. On the left-hand side, the eternal black hole in an AdS spacetime is shown in global coordinates. The dashed lines represent the black hole horizons. The wavy lines are the future and past singularities. The two wedges attached to the singularities represent the black hole interior. At the left and right boundaries of the AdS spacetime, marked in blue and green respectively, the left and right boundary CFTs are defined. The left and right asymptotic symmetry groups $G_{L/R}$ constitute global symmetries of the respective CFT. The eternal black hole geometry is invariant under the diagonal subgroup $G_D$ of the full asymptotic symmetry group $G_L\times G_R$. The dual description of the eternal black hole is depicted on the right-hand side. The two CFTs, defined on the blue and green planes that represent the left and right asymptotic boundaries, are entangled in the TFD state.
• Figure 2: Left panel: a visualisation of the concept of a fibre bundle. At each point $x_i$ of the base manifold, fibres $F_{x_i}$ are attached. For principal fibre bundles, on which we focus in this paper, the fibres are isomorphic to some group $G$. Right panel: a path, closed in the base manifold, may no longer be closed when uplifted in the fibre direction. The mismatch of the endpoints of the uplifted path, marked in green, is the holonomy. In physics, the holonomy is more commonly referred to as geometric phase or Berry phase Berry:1984jvSimon:1983mh. In this work, we use this concept to calculate geometric phases both for small and large spin systems. We connect these results to holonomies of the fibre bundle obtained by quantisation of the moduli space of classical solutions in AdS/CFT.
• Figure 3: The three differently coloured spheres shown in this plot represent the geometry of the base manifold for three different values of the entanglement. The blue sphere has maximal volume corresponding to $\alpha=\frac{\pi}{2}$ and vanishing entanglement. The green sphere is the base manifold for an intermediate value of entanglement determined by the value of $\alpha$ (for the plot we used $\alpha=\frac{\pi}{8}$). The small black sphere represents the states of (almost) maximal entanglement (in the plot, $\alpha\sim 10^{-3}$).
• Figure 4: Visualisation of different time evolutions of the TFD state. In the central panel, the holographic dual of the usual TFD state \ref{['eq:TFD']} at the $\delta=0$ slice is depicted. The blue line represents the slice at which \ref{['eq:TFD']} is defined. The left panel shows the holographic dual to the TFD state time-evolved by $H_L+H_R$, which changes the state in the CFT. The panel on the right shows the time evolution by $H_L-H_R$, which leaves the CFT state invariant.