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arXiv:quant-ph/0312017 · Interdisciplinary Information Sciences

On Energy-Momentum Spectrum of Stationary States with Nonvanishing Current on 1-d Lattice Systems

Takayuki Miyadera

Published Jan 1, 2004• 0 citations
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Abstract

On one-dimensional two-way infinite quantum lattice system, a property of translationally invariant stationary states with nonvanishing current expectation is investigated. We consider GNS representation with respect to such a state, on which we have a group of space-time translation unitary operators. We show that spectrum of the unitary operators, energy-momentum spectrum with respect to the state, has a singularity at the origin.

DOI
10.4036/iis.2004.23

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On one-dimensional two-way infinite quantum lattice system, a property of translationally invariant stationary states with nonvanishing current expectation is investigated. We consider GNS representation with respect to such a state, on which we have a group of space-time translation unitary operators. We show that spectrum of the unitary operators, energy-momentum spectrum with respect to the state, has a singularity at the origin.
Abstract We rigorously examine, in generality, the ergodic properties of quantum lattice models with short range interactions, in the C∗C^*C∗ C ∗ algebra formulation of statistical mechanics. Ergodicity results, in the context of group actions on C∗C^*C∗ C ∗ algebras, assume that the algebra is asymptotically abelian, which is not generically the case for time evolution. The Lieb-Robinson bound tells us that, in a precise sense, the spatial extent of any time-evolved local operator grows linearly with time. This means that the algebra of observables is asymptotically abelian in a space-like region, and implies a form of ergodicity outside the light-cone. But what happens within it? We show that the long-time limit of the n-th moment of a ray-averaged observable, along space-time rays of almost every speed, converges to the n-th power of its expectation in the state (i.e. its ensemble average). Thus ray averages do not fluctuate in the long time limit. This is a statement of ergodicity, and holds in any state that is invariant under space-time translations and that satisfies weak clustering properties in space. The ray averages can be performed in a way that accounts for oscillations, showing that ray-averaged observables cannot sustainably oscillate in the long time limit. We also show that in the GNS representation of the algebra of observables, for any KMS state with the above properties, the long-time limit of the ray average of any observable converges (in the strong operator topology) to the ensemble average times the identity, again along space-time rays of almost every speed. This is a strong version of ergodicity, and indicates that, as operators, observables get “thinner” almost everywhere within the light-cone. A similar statement holds under oscillatory averaging.
The three-dimensional potential equation, motivated by representations of quantum mechanics, is investigated in four different scenarios: (i) In the usual Euclidean space E3\mathbb{E}_{3}E3​ where the potential is singular but invariant under the continuous inhomogeneous orthogonal group IO(3)\mathcal{I}O(3)IO(3). The invariance under the translation subgroup is compared to the corresponding unitary transformation in the Schrödinger representation of quantum mechanics. This scenario is well known but serves as a reference point for the other scenarios. (ii) Next, the discrete potential equation as a partial difference equation in a three-dimensional lattice space is studied. In this arena the potential is non-singular but invariance under IO(3)\mathcal{I}O(3)IO(3) is broken. This is the usual picture of lattice theories and numerical approximations. (iii) Next we study the six-dimensional continuous phase space. Here a phase space representation of quantum mechanics is utilized. The resulting potential is singular but possesses invariance under IO(3)\mathcal{I}O(3)IO(3). (iv) Finally, the potential is derived from the discrete phase space representation of quantum mechanics, which is shown to be an \emph{exact} representation of quantum mechanics. The potential function here is both non-singular and possesses invariance under IO(3)\mathcal{I}O(3)IO(3), and this is proved via the unitary transformations of quantum mechanics in this representation.