Rigorous bounds on dynamical response functions and time-translation  symmetry breaking

Medenjak, Marko; Prosen, Tomaž; Zadnik, Lenart

doi:10.21468/scipostphys.9.1.003

Cited by 33 publications

(36 citation statements)

References 42 publications

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“…This suggests that the general Euler-scale structure holds not just at large wavelengths and low frequencies, but rather as an expansion around given wavelengths and frequencies: if τ t and ι x are the usual, homogeneous time and space translation isomorphisms, then e iωt τ t and e ikx ι x also are appropriate time and space translation isomorphisms, satisfying all required properties. These give access to hydrodynamic quantities near to frequency ω and wavenumber k. This is relevant in respect of recent works [91][92][93]. Our results therefore establish that the linearised Euler-scale structure is extremely universal.…”

Section: The Problem Of Finite-dimensionality Is Nontrivial In Non-integrable Models Perhapssupporting

confidence: 77%

Hydrodynamic Projections and the Emergence of Linearised Euler Equations in One-Dimensional Isolated Systems

Doyon

2022

Commun. Math. Phys.

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One of the most profound questions of mathematical physics is that of establishing from first principles the hydrodynamic equations in large, isolated, strongly interacting many-body systems. This involves understanding relaxation at long times under reversible dynamics, determining the space of emergent collective degrees of freedom (the ballistic waves), showing that projection occurs onto them, and establishing their dynamics (the hydrodynamic equations). We make progress in these directions, focussing for simplicity on one-dimensional systems. Under a model-independent definition of the complete space of extensive conserved charges, we show that hydrodynamic projection occurs in Euler-scale two-point correlation functions. A fundamental ingredient is a property of relaxation: we establish ergodicity of correlation functions along almost every direction in space and time. We further show that to every extensive conserved charge with a local density is associated a local current and a continuity equation; and that Euler-scale two-point correlation functions of local conserved densities satisfy a hydrodynamic equation. The results are established rigorously within a general framework based on Hilbert spaces of observables. These spaces occur naturally in the $$C^*$$ C ∗ algebra description of statistical mechanics by the Gelfand–Naimark–Segal construction. Using Araki’s exponential clustering and the Lieb–Robinson bound, we show that the results hold, for instance, in every nonzero-temperature Gibbs state of short-range quantum spin chains. Many techniques we introduce are generalisable to higher dimensions. This provides a precise and universal theory for the emergence of ballistic waves at the Euler scale and how they propagate within homogeneous, stationary states.

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Section: The Problem Of Finite-dimensionality Is Nontrivial In Non-integrable Models Perhapssupporting

confidence: 77%

Hydrodynamic Projections and the Emergence of Linearised Euler Equations in One-Dimensional Isolated Systems

Doyon

2022

Commun. Math. Phys.

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“…For spin-1/2 XXZ model at root of unity, we have two sets of commuting charges Z m and Y n , while they do not commute with operators in the other set. The non-commutability between Z m and Y n has consequences in the thermodynamic limit, leading to oscillatory behaviour of auto-correlation functions [52,53]. The relation between the oscillatory behaviour of correlation functions in the thermodynamic limit to the hidden Onsager algebra symmetries in those models still needs further consideration.…”

Section: Discussionmentioning

confidence: 99%

“…Motivated by the exact correspondence between the Onsager generators of O Q r n and conserved charges (Z and Y charges) in the XX case (q = exp(iπ/2) = i), cf. (45) and (53), it is natural to generalise similar relations for the spin-1/2 XXZ model at arbitrary root of unity (q = exp (iπ 1 / 2 )), despite that the operators Q r n , Z n and Y n are no longer able to be expressed in local densities but quasilocal ones [33][34][35][36]. Another motivation to the following conjectures is that the structure of descendant towers and exact (Fabricius-McCoy) strings are of no difference between XX model and XXZ model at other root of unity.…”

Section: Conjectures On Hidden Onsager Algebra Symmetries In Spin-1/2 Xxz Models At Root Of Unitymentioning

confidence: 99%

Conjectures on hidden Onsager algebra symmetries in interacting quantum lattice models

Yuan

2021

SciPost Phys.

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We conjecture the existence of hidden Onsager algebra symmetries in two interacting quantum integrable lattice models, i.e. spin-1/2 XXZ model and spin-1 Zamolodchikov-Fateev model at arbitrary root of unity values of the anisotropy. The conjectures relate the Onsager generators to the conserved charges obtained from semi-cyclic transfer matrices. The conjectures are motivated by two examples which are spin-1/2 XX model and spin-1 U(1)-invariant clock model. A novel construction of the semi-cyclic transfer matrices of spin-1 Zamolodchikov-Fateev model at arbitrary root of unity values of the anisotropy is carried out via the transfer matrix fusion procedure.

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“…Thus, measuring the deviance of stationary states (or time-periodic states) from equilibrium in terms of passivity is an interesting prospect to look for modest nonequilibrium quantum time crystals. In passing, we mention another framework of investigation of breakdown of time-translation symmetry advocated by [34]. Of course, it does not discuss the usual sense of SSB of time-translation symmetry.…”

Section: Quantum Time Crystals Beyond the Case Of Equilibriummentioning

confidence: 99%

Nonexistence of spontaneous symmetry breakdown of time-translation symmetry: a review

Moriya¹

2022

Preprint

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The time invariance of equilibrium states has been established in the C * -algebraic framework assuming an infinite-volume Heisenberg time evolution as C * -dynamics. This fundamental property of equilibrium states implies that spontaneous breakdown of time-translation symmetry is impossible in general quantum systems. In particular, any non-trivial order, such as periodic, quasi-periodic, and chaotic order cannot appear in the time direction of equilibrium states for general quantum models, meaning that genuine quantum time crystals are strictly forbidden. We compare our statement on the impossibility of quantum time crystals with the result by Watanabe-Oshikawa which has been considered as a milestone in the study on quantum time crystals. Our no-go statement based on the Kubo-Martin-Schwinger condition for C * -dynamics is not only mathematically rigorous but also more general than the main result of Watanabe-Oshikawa and its improvement by Watanabe-Oshikawa-Koma based on temporal correlation functions under cut-off quantum time evolutions.

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Rigorous bounds on dynamical response functions and time-translation symmetry breaking

Cited by 33 publications

References 42 publications

Hydrodynamic Projections and the Emergence of Linearised Euler Equations in One-Dimensional Isolated Systems

Hydrodynamic Projections and the Emergence of Linearised Euler Equations in One-Dimensional Isolated Systems

Conjectures on hidden Onsager algebra symmetries in interacting quantum lattice models

Nonexistence of spontaneous symmetry breakdown of time-translation symmetry: a review

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