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arXiv:1510.01830

Incommensurate Charge Density Wave as Quantum Space-Time Crystal

Keiji Nakatsugawa, Toshiyuki Fujii, Satoshi Tanda

Published Oct 7, 2015• 0 citations
arXivHTMLPDF

Abstract

A quantum time crystal (QTC) is a novel quantum mechanical ground state that was recently proposed by Wilczek. Although many QTC models have been proposed, it is not clear yet if such states are possible. We propose the idea that an incommensurate charge density wave (ICDW) forms a quantum space crystal (QSC), which is an extension of a QTC in the spatial domain. Consequently, a rotating ring-shaped ICDW naturally forms a quantum space-time crystal (QSTC), which combines the two concepts of QTC and QSC. The breaking of space-time translation symmetry is understood using time-dependent density matrices. Furthermore, we show that this model can be observed in real systems such as TaS$_3$ ring crystals at finite temperature. Our results suggest that QTC/QSTC can exist.

DOI
10.48550/arxiv.1510.01830

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A quantum time crystal (QTC) is a novel quantum mechanical ground state that was recently proposed by Wilczek. Although many QTC models have been proposed, it is not clear yet if such states are possible. We propose the idea that an incommensurate charge density wave (ICDW) forms a quantum space crystal (QSC), which is an extension of a QTC in the spatial domain. Consequently, a rotating ring-shaped ICDW naturally forms a quantum space-time crystal (QSTC), which combines the two concepts of QTC and QSC. The breaking of space-time translation symmetry is understood using time-dependent density matrices. Furthermore, we show that this model can be observed in real systems such as TaS3_33​ ring crystals at finite temperature. Our results suggest that QTC/QSTC can exist.
We show that time translation symmetry of a ring system with a macroscopic quantum ground state is broken by decoherence. In particular, we consider a ring-shaped incommensurate charge density wave (ICDW ring) threaded by a fluctuating magnetic flux: the Caldeira-Leggett model is used to model the fluctuating flux as a bath of harmonic oscillators. We show that the charge density expectation value of a quantized ICDW ring coupled to its environment oscillates periodically. The Hamiltonians considered in this model are time independent unlike ``Floquet time crystals'' considered recently. Our model forms a metastable quantum time crystal with a finite length in space and in time.
Recently, Wilczek proposed a fascinating new concept of time crystals, which spontaneously break the continuous time translation symmetry, in analogy with ordinary crystals which break the continuous spatial translation symmetry. Li et al. soon followed with a concrete proposal for an experimental realization and observation of a (space-)time crystal using trapped ions in a ring threaded by an Aharonov-Bohm flux. However, the very existence, even as a matter of principle, of time crystals is rather controversial. Bruno proved a no-go theorem for time crystals for a specific class of Hamiltonians, which covers the original proposal. A similar observation was made for a charge density wave in a superconducting ring by Nozieres. However, since these arguments were not fully general, several new realizations of time crystals, which avoid these no-go arguments, were proposed. In fact, a part of the confusion can be attributed to the lack of a precise mathematical definition of time crystals. Here, we first propose a definition of time crystals in the equilibrium, which is a natural generalization of that of ordinary crystals and can be formulated precisely also for time crystals. We then prove generally the absence of time crystals defined as such, in the ground state of an arbitrary Hamiltonian which consists only of short-range interactions. While the nature of the time crystal is less understood, it is natural to expect that time crystals would be most easily realized, if at all, at zero temperature, as in the case of ordinary crystals. Thus, we focus on the question whether time crystals can exist as the ground state, as in Wilczek’s original proposal. Naively, time crystals would be defined in terms of the ground-state expectation value of an observable O(t). If it exhibits a periodic time dependence, the system may be regarded as a time crystal. However, the very definition of the ground state Ĥ|0〉 = E0|0〉 immediately implies that the expectation value of any Heisenberg operator O(t) ≡ eiĤtO(0)e−iĤt is time-independent. Yet it is too early to reject the idea of time crystals just from this observation, since a similar argument would preclude ordinary (spatial) crystals. One might naively define crystals from a spatially modulating expectation value of the density operator ρ(~x) = e−i ~ P ·~xρ(~0)ei ~ P ·~x. The unique ground state of the Hamiltonian in a finite box is nevertheless symmetric and hence ~ P |0〉 = 0, implying that 〈0|ρ(~x)|0〉 is constant over space.
In analogy with crystalline solids around us, Wilczek recently proposed the idea of "time crystals" as phases that spontaneously break the continuous time translation into a discrete subgroup. The proposal stimulated further studies and vigorous debates whether it can be realized in a physical system. However, a precise definition of the time crystal is needed to resolve the issue. Here we first present a definition of time crystals based on the time-dependent correlation functions of the order parameter. We then prove a no-go theorem that rules out the possibility of time crystals defined as such, in the ground state or in the canonical ensemble of a general Hamiltonian, which consists of not-too-long-range interactions.
A model demonstrating the existence of a thermodynamically stable quantum time-space crystal has been proposed and studied. This state is characterized by an order parameter periodic in both real and imaginary times. The average of the order parameter over phases of the oscillations vanishes but correlation functions of two or more order parameters show nondecaying oscillations. An alternative interpretation of the results is based on the concept of an operator order parameter introduced for this purpose. The model studied here has been suggested previously, in particular, for describing the pseudogap state in superconducting cuprates. Although many properties of the time-space crystal considered here are close to those of a well known DDW state, static magnetic moments oscillating at $(\ensuremath{\pi},\ensuremath{\pi})$ do not exist. Instead, $\ensuremath{\delta}$ peaks at finite energies are predicted in the cross-section of inelastic spin-polarized neutron scattering.
A constructive theoretical platform for the description of quantum space-time crystals uncovers for NNN interacting and ring-confined rotating particles the existence of low-lying states with proper space-time crystal behavior. The construction of the corresponding many-body trial wave functions proceeds first via symmetry breaking at the mean-field level followed by symmetry restoration using projection techniques. The ensuing correlated many-body wave functions are stationary states and preserve the rotational symmetries, and at the same time they reflect the point-group symmetries of the mean-field crystals. This behavior results in the emergence of sequences of select magic angular momenta Lm{L}_{m}Lm​. For angular-momenta away from the magic values, the trial functions vanish. Symmetry breaking beyond the mean-field level can be induced by superpositions of such good-Lm{L}_{m}Lm​ many-body stationary states. We show that superposing a pair of adjacent magic angular momenta states leads to formation of special broken-symmetry states exhibiting quantum space-time-crystal behavior. In particular, the corresponding particle densities rotate around the ring, showing undamped and nondispersed periodic crystalline evolution in both space and time. The experimental synthesis of such quantum space-time-crystal wave packets is predicted to be favored in the vicinity of ground-state energy crossings of the Aharonov-Bohm-type spectra accessed via an externally applied, natural or synthetic, magnetic field. These results are illustrated here for Coulomb-repelling fermionic ions and for a lump of contact-interaction attracting bosons.
Spontaneous symmetry breaking can lead to the formation of time crystals, as well as spatial crystals. Here we propose a space-time crystal of trapped ions and a method to realize it experimentally by confining ions in a ring-shaped trapping potential with a static magnetic field. The ions spontaneously form a spatial ring crystal due to Coulomb repulsion. This ion crystal can rotate persistently at the lowest quantum energy state in magnetic fields with fractional fluxes. The persistent rotation of trapped ions produces the temporal order, leading to the formation of a space-time crystal. We show that these space-time crystals are robust for direct experimental observation. We also study the effects of finite temperatures on the persistent rotation. The proposed space-time crystals of trapped ions provide a new dimension for exploring many-body physics and emerging properties of matter.