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arXiv:2111.08127

Time-crystalline long-range order in chiral fermionic vacuum

Nobuyuki Ōkuma

Published Nov 15, 2021• 0 citations
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Abstract

It is widely believed that there is no macroscopic time-crystalline order in the ground states of short-range interacting systems. In this paper, we consider a time-dependent correlation function for an order operator with a spatially discontinuous weight in a one-dimensional chiral fermionic system. Although both the Hamiltonian and the order parameter are composed of spatially local operators, the time-dependent correlation function diverges logarithmically in equal time intervals. This result implies a breakdown of an inequality that claims the absence of time-crystalline long-range order in the ground states, unless the upper-bound constant is set to be infinity. This behavior is due to the combination of the discontinuity of the order operator and the infinite dimensionality of quantum field theory. In the language of bosonization, it can also be related to the divergence of a space-time-resolved bosonic correlation function.

DOI
10.48550/arxiv.2111.08127

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It is widely believed that there is no macroscopic time-crystalline order in the ground states of short-range interacting systems. In this paper, we consider a time-dependent correlation function for an order operator with a spatially discontinuous weight in a one-dimensional chiral fermionic system. Although both the Hamiltonian and the order parameter are composed of spatially local operators, the time-dependent correlation function diverges logarithmically in equal time intervals. This result implies a breakdown of an inequality that claims the absence of time-crystalline long-range order in the ground states, unless the upper-bound constant is set to be infinity. This behavior is due to the combination of the discontinuity of the order operator and the infinite dimensionality of quantum field theory. In the language of bosonization, it can also be related to the divergence of a space-time-resolved bosonic correlation function.
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Topological states of matter, such as quantum Hall states, are characterized by the absence of local order parameters. In the continuum, their topological order has been reinterpreted as the condensation of nonlocal composite bosons, but this condensation and the resulting long-range correlations have so far eluded direct confirmation beyond the simplest trial states. Here, we apply tensor-network methods to lattice Hamiltonians hosting quantum Hall states, revealing this exotic order directly in interacting ground states and demonstrating both the long-range correlations and the condensation of composite bosons. These correlations define a nonlocal order parameter that identifies phase transitions between trivial and topological phases, for both bosonic and fermionic models. The corresponding operators are accessible through site-resolved, local measurements on an extensive number of particles, making them directly probable in quantum simulators. Our work opens new avenues for microscopic studies of topological quantum matter beyond the reach of traditional solid-state approaches.
Higher-order topological crystalline phases in low-dimensional interacting quantum systems represent a challenging and largely unexplored research topic. Here, we derive a Hamiltonian describing fermions interacting through correlated hopping processes that break chiral invariance, but preserve both inversion and time-reversal symmetries. In this way, we show that our one-dimensional model gives rise to an interacting second-order topological insulating phase that supports gapped edge states. The topological nature of such an interacting phase turns out to be revealed by both long-range order of a nonlocal string correlation function and by even degeneracy of the entanglement spectrum. For strong interactions we instead find that the topological crystalline phase is destroyed and replaced by a singlet superconducting phase. The latter, characterized by local fermionic pairing, turns out to appear both in a homogeneous and in a phase separated form. Relevantly, the derived one-dimensional model and the second-order topological insulator can be explored and investigated in atomic quantum simulators.
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The off-diagonal ordering of one- and two-dimensional Bose particle systems of finite thickness and cross-section is considered in the presence and absence of a symmetry-breaking field. It is shown rigorously, by applying Bogoliubov's inequality to a subdomain of the system, that no spontaneous ordering can occur; several definitions of long-range order are discussed. Explicit bounds on the order-order correlation function, integrated over a subdomain, are obtained which indicate how the short-range order decays with distance. Conditions under which information on the pointwise behavior of the correlation function can be inferred are also discussed. The inequalities are assessed numerically for real situations.
A natural generalization of the Heisenberg uncertainty principle inequality holding for non-Hermitian operators is presented and applied to the fractional quantum Hall effect (FQHE). This inequality was used in a previous paper to prove the absence of long-range order in the ground state of several one-dimensional (1D) systems with continuous-group symmetries. In this paper, we use it to rule out the occurrence of Bose-Einstein condensation in the bosonic representation of the FQHE wave function proposed by Girvin and MacDonald. We show that the absence of off-diagonal long-range order in this two-dimensional (2D) problem is directly connected with the q2{\mathit{q}}^{2}q2 behavior of the static structure function S(q) at small momenta.