Proposal with an Incommensurate Charge Density Wave Ring

PHYSICAL REVIEW B 96, 094308 (2017)

Quantum time crystal by decoherence: Proposal with an incommensurate charge density wave ring

K. Nakatsugawa,1 T. Fujii,3 and S. Tanda1,2 1Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan 2Center of Education and Research for Topological Science and Technology, Hokkaido University, Sapporo 060-8628, Japan 3Department of Physics, Asahikawa Medical University, Asahikawa 078-8510, Japan (Received 23 April 2017; revised manuscript received 1 September 2017; published 20 September 2017)

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.

DOI: 10.1103/PhysRevB.96.094308

I. INTRODUCTION independent of its phase (i.e., position), which implies that the expected ground state of an ICDW ring is a superposition The original proposal of a quantum time crystal (QTC) of ICDWs with different phases. Ring-shaped crystals and given by Wilczek [1] and Li et al. [2] is a quantum mechanical ring-shaped (I)CDWs have been produced [7] [Fig. 2(b)]. The ground state which breaks time translation symmetry. In this presence of circulating CDW current [8] and Aharonov-Bohm QTC ground state there exists an operator Qˆ whose expectation oscillation (evidence of macroscopic wave function) [9]is value oscillates permanently with a well-defined “lattice con- verified experimentally. stant” P , that is, with a well-defined period. Volovik [3] relaxed We show in Sec. II that the charge density expectation the condition of permanent oscillation and proposed the value of an isolated ICDW ring with moment of inertia I is possibility of effective QTC, that is, a periodic oscillation in a periodic in time with period P = 4πI/h¯. This periodicity is metastable state such that the oscillation will persist for a finite a consequence of the uncertainty relation on the S1 (ring). duration τ P in the time domain and will eventually decay. Q However, this oscillation becomes unobservable at ground In this paper we promote Volovik’s line and consider the state because the ground state of an isolated ICDW ring is a possibility of a metastable QTC state without spontaneous plane wave state, i.e., a coherent superposition of ICDWs with symmetry breaking: We consider symmetry breaking by different phases. Therefore, in Sec. III we use the Caldeira- decoherence of a macroscopic quantum ground state (Fig. 1). Leggett model [10,11] to show that time translation symmetry Decoherence is defined as the loss of quantum coherence of a is broken by decoherence. More precisely, the superposition system coupled to its environment. Coupling to environment will inevitably introduce friction to the system such that the is broken by decoherence and the amplitude of the ICDW oscillation will eventually decay at t = τ . However, for oscillates periodically [Fig. 2(c)]. If the ICDW ring weakly damp couples to its environment then this state is a metastable ground t<τQ τdamp the oscillation period P is well defined. If friction is sufficiently weak such that P τ , then we have state. Therefore, our model forms an effective QTC with a finite Q length in space and in time. a model of effective QTC with lifetime τQ. Our model consists of a ring-shaped incommensurate Before developing our main arguments, we compare our charge density wave (ICDW ring) threaded by a fluctuating work to recent developments of QTC. In analogy with spatial magnetic flux [Fig. 2(a)]. A charge density wave (CDW) is a crystals, the original proposal of a QTC is based on the periodic (spatial) modulation of electric charge density which spontaneous breaking of time translation symmetry. However, occurs in quasi-one-dimensional crystals [5]: the periodic Bruno [12] and Watanabe and Oshikawa [13] theoretically modulation of the electric charge density occurs due to proved that spontaneous breaking of time translation symmetry electron-phonon interaction. If the ratio between the CDW cannot occur at ground state. Recently, it was shown that wavelength λ and the lattice constant a of the crystal is a simple there is a notion of spontaneous breaking of time translation fraction like 2, 5/2, etc., then the CDW is commensurate with symmetry in periodically driven (Floquet) states [14] and this the underlying lattice. A commensurate CDW cannot move idea was proved experimentally [15]. On the other hand, the freely because of commensurability pinning; i.e., the CDW periodic oscillation we consider in this paper is inherent to phase is pinned by the ion’s position in the crystal. On the ring systems with a macroscopic wave function. other hand, if λ/a is effectively an irrational number [5], then the CDW is incommensurate with the underlying lattice. An II. GROUND STATE OF AN ISOLATED ICDW RING ICDW ring with a radius of 10 μm, for instance, contains approximately 105 wavelengths [6], so it is possible that A. Classical theory of ICDW ring a/λ is very close to an irrational number. See the discussion It is well known that the electric charge density of a section for an elaboration of this assumption. The sliding of quasi-one-dimensional crystal becomes periodic by opening an ICDW without pinning is described by a gapless Nambu- a gap at the Fermi wave number kF and forms a charge Goldstone (phason) mode [5] and the energy of an ICDW is density wave (CDW) ground state with a wavelength λ = π/kF

FIG. 1. The concept of translation symmetry breaking by decoherence is illustrated. (a) A simplified version of the two-state system considered by Leggett et al. [4]. A particle can tunnel through a potential barrier and exist at two states simultaneously. However, if this system starts to interact with its surrounding environment, then the particle will localize at one of the states. (b) Similarly, the ground state of a free particle confined on a ring is a plane wave FIG. 2. (a) We consider an incommensurate CDW ring threaded state. Coupling to environment will localize the particle and break by a fluctuating magnetic flux. This figure shows a commensurate rotational symmetry. This “particle” corresponds to the phase of an CDW with λ/a = 2 because it is easier to visualize. The wave incommensurate charge density wave ring (ICDW ring) in our model (typically ∼105 wavelengths) represents the charge density and (Fig. 2). the dots represent the atoms of a quasi-one-dimensional crystal. (b) ICDW ring crystals such as monoclinic TaS3 ring crystals and NbSe3 ring crystals can be produced experimentally [7–9]. Our [5]. Consider a CDW formed on a ring-shaped quasi-one- model can be tested provided clean ring crystals with almost no dimensional crystal with radius R. The order parameter of this defects and impurities can be produced. (c) The ground state of CDW ring is a complex scalar (x,t) =|(x,t)| exp[iθ(x,t)], an isolated quantized ICDW ring is a superposition of periodically where |(x,t)| is the size of the energy gap at ±kF, θ(x,t)is oscillating ICDWs; hence the oscillation is unobservable. Coupling to the phase of the CDW, x ∈ [0,2πR) is the coordinate on the environment (fluctuating magnetic flux) will break the superposition crystal, and t is the time coordinate. The charge density is and the oscillation becomes apparent. given by Hamiltonian of the ICDW ring are, respectively, n(x,t) = n0 + n1 cos[2kFx + θ(x,t)], (1) 2πR ˙ = L ˙ = I ˙2 where n0 is the average charge density and n1 is the amplitude L0(θ) dx 0(θ) 2 θ , (2) of the wave. Bogachek et al. [16] derived the following La- 0 ˙ grangian density of the phase of a ring-shaped incommensurate ∂L0(θ) πθ (θ˙) = = Iθ,˙ (3) CDW (ICDW ring) threaded by a magnetic flux ∂θ˙ π 2 H (π ) = π θ˙ − L (θ˙) = θ , (4) ∂θ ∂θ N ∂θ 2 ∂θ 2 eA ∂θ 0 θ θ 0 2I L = 0 − 2 + 0 , c0 , ∂t ∂x 2 ∂t ∂x π ∂t ˙ = = 2 where θ dθ/dt and I hRv¯ F/c0 is the moment of inertia. We note that (2), (3), and (4) are time independent. where A is the magnetic vector potential, N0 = 2 2 2 v h¯ N(εF)/(2c ), N(εF) is the density of states of electrons F 0 B. Quantization of an isolated ICDW ring at the Fermi level per unit length and per spin√ direction, vF ∗ is the Fermi velocity of the crystal, c0 = m/m vF is the Next, we quantize the ICDW ring system. We show that a phason velocity, m∗ is the effective mass of electrons, andh ¯ quantized ICDW ring possesses an inherent oscillation which ˆ = 2 is the reduced Planck constant. We first consider an isolated originates from the uncertainty principle. Let H0 πˆθ /(2I) ICDW ring with A = 0. Assuming that N(εF) is equivalent andπ ˆθ be the Hamiltonian operator and angular momentum to the density of states of electrons on a one-dimensional operator of the ICDW ring, respectively. The macroscopic = = 2 ∈ H H line, that is, N(εF) 1/(πhv¯ F), we have N0 hv¯ F/(2πc0). quantum state ψ is defined in the Hilbert space An incommensurate CDW (ICDW) can slide freely because of positive square-integrable functions with the periodic of spatial translation symmetry, so the dynamics of an ICDW boundary condition ψ(θ + 2π) = ψ(θ). The canonical com- is understood by its phase θ. We further assume the rigid-body mutation relation [θ,ˆ πˆθ ] = ih¯ is not satisfactory because θˆ is model of ICDW; i.e., the ICDW ring does not deform locally a multivalued operator and is not well defined. Ohnuki and and the phase θ(x,t) = θ(t) is independent of position. Then, Kitakado [17] resolved this difficulty by using the unitary the Lagrangian, the canonical angular momentum, and the operator Wˆ and the self-adjoint angular momentum operator

094308-2 QUANTUM TIME CRYSTAL BY DECOHERENCE: PROPOSAL . . . PHYSICAL REVIEW B 96, 094308 (2017)

πˆθ defined by function of the ICDW ring diffuses due to the uncertainty principle, and then (ii) W = eiθ oscillates periodically. How- iθ ∂ψ(θ) θ|Wˆ |ψ =e ψ(θ), θ|πˆθ |ψ =−ih¯ , ever, the oscillation in (9) is not observable at the ground ∂θ ≡| | | | = 1 stateρ ˆ0 ψ0 ψ0 because θ ρˆ0 φ 2π and the θ integral which satisfy the commutation relation on H vanishes. Therefore, the ground state of an isolated ICDW ring is not yet a time crystal because of superposition. [ˆπθ ,Wˆ ] = h¯W.ˆ (5) ˆ H0 is a function ofπ ˆθ only; hence the complete orthonormal III. COUPLING TO ENVIRONMENT { }∞ H set ψl l=−∞ of√ momentum eigenstates spans and satisfies ilθ Now, suppose that the ICDW ring starts to interact with ψl(θ) = e / 2π. The eigenvalues ofπ ˆθ are quantized † its surrounding environment at t = 0. Then, we expect with ψ |πˆ |ψ =lh,l¯ ∈ Z. Wˆ and Wˆ are ladder operators l θ l decoherence of the phase θ. This interaction is modeled which satisfy Wψˆ = ψ + and Wψˆ = ψ − . Therefore, (5) l l 1 l l 1 using the Caldeira-Leggett model [10] which is a model is the one-dimensional version of the well-known angular quantum Brownian motion. It describes a particle coupled momentum algebra [18]. Time evolution is introduced via iHˆ t/h¯ −iHˆ t/h¯ to its environment. This environment is described as a set the Heisenberg picture:π ˆθ (t) = e 0 πˆθ e 0 and Wˆ (t) = ˆ − ˆ of noninteracting harmonic oscillators. First, the classical iH0t/h¯ ˆ iH0t/h¯ ˆ = e We .ˆπθ commutes with H0,soˆπθ (t) πˆθ .From solution θ(t) is calculated to study the dynamics of the ICDW the commutation relation (5) we obtain the following solutions ring. Next, this system is quantized to calculate the amplitude ˆ of W(t): of the charge density expectation value nˆ(x,t) . itπˆ /I − it itπˆ /I it Wˆ (t) = e θ Weˆ 2μ = Weˆ θ e 2μ , (6) A. Classical theory of ICDW ring with environment where μ = I/h¯. The two different expressions in (6)arise Let us consider the following Lagrangian of an ICDW ring from the noncommutativity between Wˆ and eitπˆθ /I .ForaQTC threaded by a fluctuating magnetic flux: we need a periodic expectation value at the ground state. So, N we define the time-dependent charge density operator 1 1 1 L˜ (θ,˙ q,q˙) = Iθ˙2 + A(q)θ˙ + mq˙2 − mω2q2 , n1 2ik x j j j nˆ(x,t) = n + [e F Wˆ (t) + H.c.](7) 2 = 2 2 0 2 j 1 and replace the classical charge density (1) by the expectation (10) value where qj are the normal coordinates of the fluctuation and = ˜ ˙ = ˜ n(x,t) =nˆ(x,t) . (8) π˜θ ∂L/∂θ and pj ∂L/∂q˙j are the canonical momenta of the ICDW ring and the environment, respectively. The Any states in H must be a linear superposition of {ψl}, fluctuating magnetic flux is given by = | |2 = that is ψ l∈Z clψl provided l cl 1. Therefore, the N ˆ expectation values W (t) and nˆ(x,t) are periodic with period A(q) = cq . P = 4πμ for any state ψ: j j=1 1 Wˆ (t) = tr[Wˆ (t)ˆρ + ρˆWˆ (t)] Classically, this magnetic flux will randomly change the phase 2 and the mechanical angular momentum Iθ˙ of the ICDW ring π 1 iθ it due to electromotive force. An equivalent Lagrangian obtained = dθe [e 2μ ρ(θ + t/μ,θ) 2 −π by a canonical transformation is ⎡ ⎤ − it N 2 + ρ(θ,θ − t/μ)e 2μ ]. 1 2 m ⎣ 2 2 Cj θ ⎦ L(θ,˙ R,R˙ ) = Iθ˙ + R˙ (θ)−ω Rj (θ)− , (9) 2 2 j j mω2 j j From the two different expressions of W(t)in(6) we can define (11) the Weyl form of the commutation relation [19] which is the Lagrangian of a bath of field particles R coupled − it j itπˆθ /I itπˆθ /I − Weˆ = e Weˆ μ ; pj cθ to the phase θ by springs. Rj (pj ,θ) =− , Pj (qj ) = mωj t mω q , and C = cω . Equations (10) and (11) are precisely hence the phase 2μ and the periodic oscillation with period j j j j 4πμ is a manifestation of the uncertainty principle. For the kinds of Lagrangian considered by Caldeira and Leggett, an alternative explanation, let us consider an electron with so we can use the results in [10] but with slight modifications effective mass m∗ confined in a finite space with volume L ∼ due to the periodicity of the ring. 2R. From the uncertainty principle, the momentum uncertainty of this particle is p ∼ h/¯ (4R). This means that the particle’s B. Classical solution = ∗ ∼ ∗ wave packet expands with velocity v p/ m h/¯ (4m R). The equation of motion of θ obtained from the Lagrangian Then, because of the periodic boundary condition, the phys- ix/λ (11) is the generalized Langevin equation [20] ical quantity W = e is periodic with period P = λ/v ∼ ∗ = ∗ = 2 = t 4πm R/(¯hkF) 4πm R/mvF 4πRvF/c0 4πμ. So, the Iθ¨(t) + 2 dταI(t − τ)θ(τ) = ξ(t) (12) origin of the periodicity is that (i) the macroscopic wave 0

094308-3 K. NAKATSUGAWA, T. FUJII, AND S. TANDA PHYSICAL REVIEW B 96, 094308 (2017) with the dissipation kernel αI(t − τ), the memory function γ (t − τ), and the classical fluctuating force ξ(t) defined by I d α (t − τ) = Iγ(0)δ(t − τ) + γ (t − τ), I 2 dt N 2 Cj γ (t − τ) = cos ωj (t − τ), Imω2 j=1 j N P (0) ξ(t) = C R (0) cos ω t + j sin ω t . j j j mω j j=1 j The correlation function of the classical force is given by the noise kernel = − ξ(t)ξ(τ) env hα¯ R(t τ), N C2 hω¯ α (t − τ) = j coth j cos ω (t − τ), R 2mω 2k T j j j B · where the average env is taken with respect to the environment coordinate at equilibrium. It is convenient to define the spectral density function N C2 J = π j − (ω) δ(ω ωj ) (13) = 2−s = −8 2 mj ωj FIG. 3. These plots are shown with gs 1Hz and μ 10 j sec. (a) The fundamental solution G(t) of a classical ICDW ring s and assume the power-law spectrum J (ω) = Igs ω [21] with coupled to its environment is shown for sub-Ohmic (s<1), Ohmic = ≈ a cutoff frequency and 0 1) damping. Note that G(t) t for t written less than some time scale τdamp,s . The time t is normalized by 4πμ, so the horizontal axis gives the number of “lattice points.” (b) The Igs s hω¯ ˙ αR(t − τ) = ω coth cos ω(t − τ)dω. derivative of the fundamental solution, G(t), heuristically describes π 0 2kBT the velocity of the ICDW ring. Note that G˙ (t) ≈ 1fort τdamp,s . The classical solution of θ is 1 t θ(t) = G(t)θ˙(0) + G˙ (t)θ(0) + dτG(t − τ)ξ(t) (14) C. Quantization of ICDW ring coupled to environment I 0 The ICDW ring + environment system is quantized with the fundamental solution using the commutation relations [π˜ˆθ ,Wˆ ] = h¯Wˆ and 1 = G(t) = L−1 (t), [qˆj ,pˆk] ihδ¯ jk. Define the orthonormal position state z2 + zγˆ(z) | = N | q i=1 qi and the orthonormal momentum state N −1 | = | where L is the inverse Laplace transform. The Laplace p i=1 pi such that transform of the memory function γ (t) can be written [11] ∂ − | | = | | | =− | g 1/(2 s) q qˆj ψ qj q ψ , q pˆj ψ ih¯ q ψ , (16) = 2−s s−1 = s ∂qj γˆ(z) ωs z ,ωs πs , sin 2 and the inner product of |q and |p is defined as q|p = and G(t) takes the form of a generalized Mittag-Leffler √ 1 i · ∞ k N exp ( q p). The periodic boundary condition of the = x 2πh¯ h¯ function Eα,β (x) k=0 (αk+β) : ring implies that θ + 2πn|π˜θ =θ|π˜θ for some integer 2−s G(t) = tE − [−(ω t) ]. n; hence the angular momentum eigenstates are quantized 2 s,2 s −1/2 ψl|π˜ˆθ |ψl =lh,l¯ = 0, ± , ± 2,..., |π˜θ =h¯ |ψl , s = γ z = g ≡ γ √1 ilθ For Ohmic damping with 1 and ˆ( ) 1 2 ,we θ|ψl = e . Moreover, one can easily show that obtain 2π − 1 − e 2γt | = | G(t) = . (15) θ,p π˜θ ,q θ,R(θ) πθ ,P . (17) 2γ Wˆ is independent of the environmental coordinate. So, the G(t) and G˙ (t)areshowninFig.3. We note that G˙ (t) ≈ 1fort expectation value of Wˆ is less than some damping time scale τ . In other words, the damp,s fluctuating magnetic flux does not affect the dynamics of an 1 π π iθ(t) ˆ = | ˆ | ICDW ring for t<τdamp,s and e oscillates periodically with W(t) dθf dφfρ(θf,φf,t) φf W θf 2 − − period P ≈ 4πμ. Next, we quantize the ICDW ring + envi- π π π π ronment system to show that this oscillation is observable for a 1 + dθf dφf θf|Wˆ |φf ρ(φf,θf,t), finite time τQ and forms an effective QTC as a metastable state. 2 −π −π

094308-4 QUANTUM TIME CRYSTAL BY DECOHERENCE: PROPOSAL . . . PHYSICAL REVIEW B 96, 094308 (2017) π where the reduced density matrix of the ICDW ring is (see + = − Appendix A) r2 dθi ρ(θi f2(t),θi,0) −π ∈ n S2(θ,t) π π = −iμθf˙ (t)+ iμ f (t)f˙ (t)− (2πn,t;f (t),0) ρ(θf,φf,t) dθi dφi ρ(θi,φi,0) × e 2 2 2 2 cl 2 , −π −π l1,l2∈Z π − r = dθi ρ(θi,θi + f2(t),0) × J (θf + 2πl1,φf + 2πl2,t; θi,φi,0). (18) 2 −π n∈S2(θ,t) The exact form of J (θf,φf,t; θi,φi,0) for Ohmic dissipation − ˙ − iμ ˙ − × iμθf2(t) 2 f2(t)f2(t) cl(2πn,t;f2(t),0) s = 1 was calculated in [10]. For general damping with e , arbitrary s the computation of the reduced density matrix is with f1(t) = 2πnG˙ (t) − G(t)/μ, f2(t) = 2πnG˙ (t), essentially equivalent to [10] and we obtain S (θ,t) ={n ∈ Z|−π<θ+ f (t) <π}, and S (θ,t) = 1 1 2 {n ∈ Z|−π<θ+ f2(t) <π}. This is the most general form = 2 i + − − − ˆ J (θf,φf,t; θi,φi,0) F (t)exp h¯ S[ϕcl ,ϕcl ] [ϕcl ] . of the expectation value of W for an ICDW ring coupled to its environment. Although the derivation of (25) is exact, it is (19) not very insightful, so we make some approximations. + = 1 + − = − ϕcl 2 (θcl φcl) and ϕcl φcl θcl are the classical coordinates obtained from the Euler-Lagrange equation: D. Early-time approximation + Let us consider the classical solution (22) with ϕ = θf t cl − − and Ohmic damping (15); then we see immediately that Iϕ¨ (u) + 2 dτϕ (τ)α (τ − u) = 0, (20) cl cl I θ˙(t) ∼ θ˙(0)e−2γt. Therefore, we are interested in the range t u u 1/(2γ ) ≡ τ . For general dissipation, we can see from + + + − = damp,1 Iϕ¨cl (u) 2 dτϕcl (τ)αI(u τ) 0, (21) Fig. 3 that there exists a time scale τdamp,s such that G(t) ≈ t, 0 G˙ (t) ≈ 1fort τdamp,s . Then, writing t = 2πI(m + a)/h¯ whose solution is given in terms of boundary conditions for an integer m and 0 −2aπ,t) ={m}, and S2(θ,t) ={0}. There- fore, we conclude that S1(θ,t) is approximately the (m/2)th + = + + + ˙ ≈−˙ ϕcl (u) κi (u; t)ϕi κf (u; t)ϕf , (22) lattice point. Then, using f1(t) G(t)¯h/I we obtain the approximate form − = − − + − − ϕcl (u) κi (t u; t)ϕf κf (t u; t)ϕi , (23) π Wˆ (t) ≈ dθi ρ(θi,θi − G(t)/μ,0) G˙ (t) G(u) −π ∈ κ (u; t) = G˙ (u) − G(u),κ(u; t) = . (24) n S1(θ,t) i G(t) f G(t) G(t)G˙ (t) × exp iθ G˙ (t)−i − (2πn,t; f (t),0) The classical action and the noise action are given by i 2μ cl 1 + − = + − + − π S[ϕcl ,ϕcl ] Scl(ϕf ,ϕf ,t; ϕi ,ϕi ,0) + dθi ρ(θi + G(t)/μ,θi,0) + − + − =− − −π ∈ I[˙ϕ (t)ϕ ϕ˙ (0)ϕi ], n S1(θ,t) cl f cl [ϕ−] = (ϕ−,t; ϕ−,0) G(t)G˙ (t) cl cl f i × ˙ + − exp iθiG(t) i cl(2πn,t; f1(t),0) . t t 2μ = 1 − − − dτ dτ ϕ (τ)αR(τ τ )ϕ (s). − ≈ 2¯h 0 0 For t τdamp,s we have ϕcl (τ) τ/μand the noise action can be written F 2(t) is a normalization function such that tr[ρ(t)] =1 =1. ≈ The winding numbers l1 and l2 can be absorbed into the θf and cl(2πn,t; f1(t),0) T,s(t) φf integrals, respectively, by changing the domain of θf and 1 1 gs hω¯ φf from S to R . Then, taking care of the non-Hermiticity of = dωcoth ϒ(ω), Wˆ , we obtain 2πμ 0 2kBT s−4 2 2 + − ϒ(ω) = ω (2 + ω t − 2 cos ωt − 2ωt sin ωt). r (t) + r (t) Wˆ (t) = 1 1 , (25) r+(t) + r−(t) Taking the low-temperature limit h¯ →∞ such that 2 2 2kBT π coth h¯ → 1 we obtain + = − 2kBT r1 dθi ρ(θi f1(t),θi,0) s− s−3 1 s−1 1 −π ∈ 3 − 2 2 − n S1(θ,t) gs 1F2 2 ; 2 , 2 ; 4 t 1 T,s(t) =− −inπ−iμθf˙ (t)+ iμ f (t)f˙ (t)− (2πn,t;f (t),0) πμ s − 3 × e 1 2 1 1 cl 1 , 2 s−1 s−1 1 s+1 − 1 2 2 − π gs t 1F2 ; , ; t 1 − = + − 2 2 2 4 , r1 dθi ρ(θi,θi f1(t),0) 2πμ s − 1 −π n∈S1(θ,t) where F (a ; b ,b ; z) is the generalized hypergeometric iμ 1 2 1 1 2 −inπ−iμθf˙1(t)− f1(t)f˙1(t)−cl(2πn,t;f1(t),0) × e 2 , function. Deﬁne the decoherence time τdecoh,s such that

094308-5 K. NAKATSUGAWA, T. FUJII, AND S. TANDA PHYSICAL REVIEW B 96, 094308 (2017)

the commensurability pinning energy is negligible, which is possible if λ/a cannot be expressed as a simple fraction like 2, 5/2, etc. In order to explain this, suppose that for some integer M 2, Ma/λ is an integer. In other words, the same atom-electron conﬁguration is obtained if we move the CDW = by M wavelengths and k+2MkF k (k is the energy of an electron with momentumhk ¯ ). Then, the energy required to move a CDW by a small phase φ from its equilibrium is [22] − ||2 e|| M 2 Mφ2 (φ; M) = , F W 2

where F is the Fermi energy, W is the bandwidth, || is the CDW gap width, and e is the elementary charge in natural units. We see that (φ; M) approaches zero rapidly for large M as the distinction between rational and irrational numbers becomes academic. For example, for a ring crystal with Na atoms and Nλ CDW wavelengths, we obtain λ/a = Na/Nλ. Therefore, for a large Na and a large Nλ, M can always be arbitrarily large (order of Na); hence the commensurability energy is completely negligible. In fact, we can experimentally make submicrometer-scale ICDW rings such that, we expect, Na and Nλ are not so large but M is very large such that commensurability energy is negligible. FIG. 4. (a) The amplitude of the charge density oscillation is Usually, superposition of ICDWs with different phases 2−s −8 shown for gs = 1Hz , μ = 10 sec, and = 1/μ. This oscil- is not observed because of impurity pinning. However, the lation is an effective QTC. In general, the oscillation amplitude is probability of impurity decreases with decreasing radius. Com- larger for super-Ohmic (s>1) damping but has a small lifetime τQ. mensurability effect may become significant for small radius On the other hand, the oscillation amplitude is small for sub-Ohmic (more precisely, for some small M). But, the origin of the time (s<1) damping but has a long τQ. (b) The charge density oscillation crystal periodicity in our model is the uncertainty principle, = = 0.8 is shown for super-Ohmic damping with s 1.2, g1.2 1Hz , which appears as a collective fluctuation of the ICDW phase. = −8 = μ 10 sec, and 1/μ. The time t is normalized such that If the commensurability effect becomes significant, then the the horizontal axis gives the number of “lattice points.” ICDW phase is expected to fluctuate periodically around some phase θ = θ0 determined by the ion’s position. This fluctuation T,s(τdecoh,s ) = 1. Numerical analysis shows that the order is expected to become apparent and oscillate periodically by of τdecoh,s does not change with <1/(2μ) and decreases coupling to environment (fluctuating magnetic flux). rapidly for >1/(2μ). If we set ∼ 1/μ and use the ground Next, we discuss the presence of an upper bound and a state ρ0, then for t τQ ≡ min{τdamp,s ,τdecoh,s } we have lower bound for the radius of the ICDW ring in our model. Decoherence-induced breaking of time translation symmetry ≈ + osc. n(x,t) n0 n1 (t) cos(2kFx), (26) occurs, in principle, only in mesoscopic systems: There is an upper bound for the CDW radius determined by the coupling G˙ (t)G(t) − strength γ = ω /2 and a lower bound given by the CDW nosc.(t) = n sinc[πG˙ (t)] cos e T,s(t). (27) s 1 1 2μ wavelength λ. We will first calculate the upper bound by replacing the environment with an equivalent LC circuit. Here, Equation (26) is the main result of this paper. It shows that the we focus on Ohmic damping because an approximate form of amplitude of an ICDW ring threaded by a (time-independent) γ can be calculated explicitly. In general, the coupling strength fluctuating magnetic flux oscillates for a finite time τQ and γ depends on the CDW radius R. In order to explicitly see the → forms an effective QTC. In the no-damping limitγ ˆ(z) 0 R dependence, suppose that the fluctuating magnetic flux in → → ˆ = we have G(t) t,G(t) 1 and recover W(t) 0. This our model comes from an external coil connected to a series charge density oscillation is shown in Fig. 4. of parallel capacitors. Then, the Lagrangian of the CDW + environment system is IV. DISCUSSION N N m First, we elaborate our assumption of ICDW ring. A L = L + MI I + I 2 − ω2Q2 . circuit 0 CDW j 2 j j j mathematical definition of ICDW is that λ/a is an irrational j=1 j=1 number. We note that a CDW formed on a macroscopic crystal is basically incommensurate because the wavelength L0 is the Lagrangian of an isolated ICDW ring, ICDW = eθ/π˙ of a CDW is given by λ = π/kF, where the Fermi wave is the CDW current induced by the fluctuating flux, Qj C number kF is usually an irrational number for an arbitrary is the net charge on the capacitor j with capacitance j , band filling. However, strictly speaking, λ/a of a finite-size m is the inductance of the coil, ωj = 1/ mCj , and M is system can never be irrational. A physical condition is that the mutual inductance between the coil and the CDW. We

094308-6 QUANTUM TIME CRYSTAL BY DECOHERENCE: PROPOSAL . . . PHYSICAL REVIEW B 96, 094308 (2017) immediately see that Lcircuit is equivalent to the Lagrangian by a macroscopic wave function with internal periodicity, the in Eq. (10) but with the interaction Lagrangian replaced by number of lattice points N in our model is numerous. And ˙ N Me ˙ = = 2 yet, the periodicity of Wˆ (t) seems to be universal for any θ j=1 π Q. Therefore, define pj mIj , Cj Meωj /π and we obtain the Lagrangian in Eq. (11) after the canonical wave functions on S1 (ring system). Therefore, our results transformation may be applicable to earlier models of QTC such as [1,2,25] and annular Josephson junctions [26]. Moreover, it was shown N + d very recently in [27] that the ground state of a 40Ca ring L = L − (MI + p )Q circuit dt CDW j j trap possesses rotational symmetry as the number of ions j is decreased. Our results predict that quantum oscillations may appear in such ring traps with the appropriate setup. = ˙ = =− − with Pj mRj mωj Qj and Rj (pj We also recall that Volovik’s proposal of metastable effective MICDW )/(mωj ). Assume that the radius of the coil QTC [3] is not restricted to ring systems. Therefore, time rcoil is much larger than the CDW radius R and that translation symmetry breaking by decoherence may occur in the coil and the CDW are concentric. Then, we obtain other systems coupled to time-independent environment, and = 2 M (μ0πR )/(2rcoil). Next, integrate (13) with respect without a periodic driving field. We also expect that many other = = = = to ω from ω 0toω 1/μ and obtain γ βR, incommensurate systems such as incommensurate spin density = 2 2 6 2 3 β (πμ0e ρc0)/(32¯hrcoilmvF ). Here, μ0 is the permeability waves [5], incommensurate mass density waves [28–30], or of free space and ρ is the density of states defined by possibly some dielectrics that exhibit incommensurate phases → ∼ j dωρ. If we assume that ρ like in the [31] may be used to test our results and to model QTC without Caldeira-Leggett model, then the order of β may change spontaneous symmetry breaking. depending on the parameters of the CDW and the parameters of the coil, but γ = βR is usually smaller than 1 Hz. Now, if the radius of the CDW ring is too large, then periodicity ACKNOWLEDGMENTS does not appear because the oscillation period exceeds the lifetime τQ of the time crystal. The upper bound of the We thank K. Ichimura, T. Matsuura, N. Hatakenaka, CDW radius R is given by the condition N>1, where A. Saxena, Y. Hasegawa, K. Yakubo, and T. Honma for N = min{τdamp,s ,τdeph,s }/(4πμ) is the number of oscillations. stimulating and valuable discussions. = For Ohmic damping (s 1) we have the√ approximate form −1 −1 τdamp,1 = γ and τdeph,1 = μγ = μγ τdamp,1.Let 1 μγ < 1, i.e., τdamp,1 >τdeph,1 (which is a valid assumption APPENDIX: QUANTUM BROWNIAN MOTION ON S −6 −6 because a typical value for μ with radius 10 mis10 s 1. Reduced density matrix on S1 and γ is usually smaller than 1 Hz); then, the upper bound to observe more than one oscillation is R< √c0 which Consider an ICDW ring A coupled to its environment B. 4π vF β Let ρ and ρ denote the density operators of A and B, is typically 1 mm. There is also a lower bound determined A B respectively. Let θ and φ be angular coordinates on the ring by the CDW wavelength λ: The radius should be large system A and let p ={p : k = 1,N } and s ={s : k = 1,N } enough to define a Fermi surface. This condition is given by k k be momentum coordinates of the bath B. Then, the density p = hπ/λ¯ h/R¯ . In other words, R should be much larger F matrix of the coupled system can be written than λ. Finally, we discuss how our model can be tested experimentally and discuss how our results may be applicable to ρ(θ,p,φ,s) =θ,p|ρAB (t)|φ,s other physical systems. Ring-shaped crystals and ICDW ring 2π ∞ crystals (such as monoclinic TaS3 ring crystals and NbSe3 ring = dθ dφ dp ds K(θ,p,t; θ ,p ,0) crystals) have been produced and studied by the Hokkaido 0 −∞ ∗ group [7–9]. Therefore, our model can be tested provided ring × K (φ,s,t; φ ,s ,0) θ ,p |ρAB (0)|φ ,s , crystals with almost no defects and impurities can be produced. The oscillation in (26) implies that the local charge density of where the ICDW ring oscillates with frequency ω = h/¯ 2I.Foraring 3 5 with diameter 2R = 1 μm, vF/c0 = 10 , and vF = 10 m/s, − i Htˆ we have ω = 108 Hz. The time dependence of the charge K(θ,p,t; θ ,p ,0) =θ,p|e h¯ |θ ,p (A1) density modulation can be measured using scanning tunneling microscopy (STM) [23] and/or using narrow-band noise with and vanishing threshold voltage [24]. We recall that the origin of the quantum oscillation in our ∗ i ˆ = | h¯ Ht| model is the uncertainty principle. If we were to consider a K (φ,s,t; φ ,s ,0) φ ,s e φ,s (A2) single particle with mass m∗ confined on a ring with radius R, then the particle’s wave packet expands with a velocity are recognized as Feynman propagators if we notice that |p v ∼ h/¯ (4m∗R) and eiθ oscillates with period P = 2πR/v. and |s are actually position states after canonical transfor- However, a charge density wave has an internal periodicity mation. The propagators (A1) and (A2) can be written using given by the wavelength λ. Then the period of oscillation path integrals by dividing the time t into N time steps of is P ∼ λ/v. Therefore, because an ICDW ring is described length = t/(N + 1), p = pN , p = p0, θ = θN , and θ = θ0.

094308-7 K. NAKATSUGAWA, T. FUJII, AND S. TANDA PHYSICAL REVIEW B 96, 094308 (2017)

For N →∞the propagator becomes N 2π N−1 ∞ N−1 N i K(θ,p,t; θ ,p ,0) = θ,p lim exp − Hˆ θ ,p = lim dθn dpn →0 h¯ →0 −∞ 0 n=1 n=1 n=1

× Kn(θn,pn,; θn−1,pn−1,0). The Hamiltonian operator Hˆ can be decomposed into the kinetic part Kˆ and the potential part Vˆ , i.e., Hˆ = Kˆ + Vˆ , which satisfy the eigenvalue equations

Kˆ |ψl,q =K(l,P) |ψl,q , Vˆ |θ,p =V(θ,R(θ)) |θ,p . (A3) Then we obtain ∞ = −iKˆ −iVˆ K(θn,pn,; θn−1,pn−1,0) dqn θn,pn exp ψln ,qn ψln ,qn exp θn−1,pn−1 −∞ h¯ h¯ ln ∞ ∞ = 1 dqn −iK − iV N exp (ln,Pn) (θn−1,Rn−1(θn−1)) 2π −∞ (2πh¯) h¯ h¯ l =−∞ n i × exp il (θ − θ − ) − q · (p − p − ) . n n n 1 h¯ n n n 1 = = Cj Next, deﬁne An j cqj,n j 2 Pj,n and obtain mωj ∞ ∞ = 1 dqn −iK − iV K(θn,pn,; θn−1,pn−1,0) N exp (ln,Pn) (θn−1,Rn−1(θn−1)) 2π −∞ (2πh¯) h¯ h¯ l =−∞ n i × exp i(l − A /h¯)(θ − θ − ) + P · (R (θ ) − R − (θ − )) . n n n n 1 h¯ n n n n 1 n 1

The sum over ln can be replaced by a sum of integrals using the Poisson resummation formula ∞ f (l) = f (ζ )exp(2πlζ)dζ. (A4) −∞ l∈Z l∈Z Then, ∞ ∞ ∞ = dζn dqn −iK − iV K(θn,pn,; θn−1,pn−1,0) N exp (ζn,Pn) (θn−1,Rn−1(θn−1)) −∞ 2π −∞ (2πh¯) h¯ h¯ l =−∞ n i × exp i(ζ − A /h¯)(θ − θ − + 2πl ) + P · (R (θ + 2πl ) − R − (θ − )) . n n n n 1 n h¯ n n n n n 1 n 1 For our Hamiltonian of an ICDW ring threaded by a ﬂuctuating magnetic ﬂux we have

2 − 2 (ζnh¯ An) 1 2 1 2 Cj θn−1 K(ζn,Pn) = + P ,V(θn−1,Rn−1(θn−1)) = mω Rj,n−1(θn−1) − . 2I 2m j,n 2 j mω2 j j j We note that that the potential V(θ,R(θ)) is rotationally invariant. That is, for an arbitrary rotation θ → θ + δ, the potential is V(θ + δ,R(θ + δ)) = V(θ,R(θ)).Letζ˜n = ζn − An/h¯. We use the fact that the phase space volume is conserved under canonical transformation. Then, ⎛ ⎞ ⎛ ⎞ N N−1 2π ∞ N ∞ ∞ ˜ ⎝ 1 ⎠ ⎝ dPn dζn ⎠ K(θ,p,t; θ ,p ,0) = lim dθn dRn(θn) → N mωj 0 0 −∞ −∞ (2πh¯) 2π j=1 n=1 n=1 ln=−∞ ⎧ ⎛ ⎞ ⎫ N ⎨ 2 ⎬ i⎝ h¯ 2 1 2 ⎠ i × exp − ζ˜ + P − V(θ − ,R − (θ − )) ⎩ h¯ 2I n 2m j,n h¯ n 1 n 1 n 1 ⎭ n=1 j N i × exp iζ˜ (θ − θ − + 2πl ) + P · [R (θ + 2πl ) − R − (θ − )] . n n n 1 n h¯ n n n n n 1 n 1 n=1

094308-8 QUANTUM TIME CRYSTAL BY DECOHERENCE: PROPOSAL . . . PHYSICAL REVIEW B 96, 094308 (2017)

Solve the ζ˜n and Pj,n integrals to obtain ⎛ ⎞ ⎛ ⎞ N N N−1 2π ∞ N ∞ ⎝ 1 ⎠ ⎝ I m ⎠ K(θ,p,t; θ ,p ,0) = lim dθn dRn(θn) → mωj 0 0 −∞ 2πih¯ 2πih¯ j=1 n=1 n=1 ln=−∞ N i I 2 i m 2 × exp (θ − θ − + 2πl ) + [R (θ + 2πl ) − R − (θ − )] h¯ 2 n n 1 n h¯ 2 n n n n 1 n 1 n=1 N i × exp − V(θ − ,R − (θ − )) . h¯ n 1 n 1 n 1 n=1

The integral including θN−1 is ∞ ∞ N I m 2π ∞ IN−1 = dθN−1 dRN−1(θN−1) 2πih¯ 2πih¯ 0 −∞ l =−∞ l − =−∞ N N 1 i I 2 i I 2 i m 2 × exp (θ − θ − + 2πl ) + (θ − − θ − + 2πl − ) exp [R (θ + 2πl ) − R − (θ − )] h¯ 2 N N 1 N h¯ 2 N 1 N 2 N 1 h¯ 2 N N N N 1 N 1 i m 2 i × exp [R − (θ − + 2πl − ) − R − (θ − )] exp − V(θ − ,R − (θ − )) . h¯ 2 N 1 N 1 N 1 N 2 N 2 h¯ N 1 N 1 N 1

The sum over lN−1 can be absorbed into the θN−1 integral by changing the domain of θN−1 integration from [0,2π)to(−∞,∞). This is done by the following procedure: (1) Transform lN to l˜N = lN + lN−1 and write θ˜N−1 = θN−1 + 2πlN−1 to obtain ∞ ∞ N I m 2π ∞ IN−1 = dθN−1 dRN−1(θN−1) 2πih¯ 2πih¯ 0 −∞ l =−∞ l − =−∞ N N 1 i I 2 i I 2 i m 2 × exp (θ˜ − θ − + 2πl˜ ) + (θ˜ − − θ − ) exp [R (θ + 2πl˜ ) − R − (θ˜ − )] h¯ 2 N N 1 N h¯ 2 N 1 N 2 h¯ 2 N N N N 1 N 1 i m 2 i × exp [R − (θ˜ − ) − R − (θ − )] exp − V(θ˜ − ,R − (θ˜ − )) . h¯ 2 N 1 N 1 N 2 N 2 h¯ N 1 N 1 N 1 2π → 2π(n+1) ˜ (2) Change the variable of integration 0 dθn 2πn dθn and change the domain of θN−1 integration from [0,2π)to (−∞,∞) and use the periodicity of V(θN−1,RN−1(θN−1)) to obtain ∞ N I m ∞ ∞ IN−1 = dθN−1 dRN−1(θN−1) 2πih¯ 2πih¯ −∞ −∞ l =−∞ N i I 2 i I 2 i m 2 × exp (θ − θ − + 2πl ) + (θ − − θ − ) exp [R (θ + 2πl ) − R − (θ − )] h¯ 2 N N 1 N h¯ 2 N 1 N 2 h¯ 2 N N N N 1 N 1 i m 2 i × exp [R − (θ − ) − R − (θ − )] exp − V(θ − ,R − (θ − )) . h¯ 2 N 1 N 1 N 2 N 2 h¯ N 1 N 1 N 1

Repeat this procedure for the integrals involving θn from n = N − 2ton = 1, and obtain ∞ ∞ N−1 ∞ N−1 N N I m K(θ,q,t; θ ,q ,0) = lim dθn dRn−1(θn−1) →0 −∞ −∞ 2πih¯ 2πih¯ l=−∞ n=1 n=1 n=1 i I 2 × exp (θ − θ − + 2πlδ ) h¯ 2 n n 1 n,N i m 2 i × exp [R (θ + 2πlδ ) − R − (θ − )] − V(θ − ,R − (θ − )) . h¯ 2 n n n,N n 1 n 1 h¯ n 1 n 1 n 1 Deﬁne

N 2πih¯ 2πih¯ A = , A = , (A5) A I B m

094308-9 K. NAKATSUGAWA, T. FUJII, AND S. TANDA PHYSICAL REVIEW B 96, 094308 (2017)

− − 1 ∞ N1 dθ 1 ∞ N1 dR (θ ) Dθ = n , DR(θ) = n n , (A6) A −∞ A A −∞ A A n=1 A B n=1 B 2 2 I θn − θn−1 + 2πlδn,N m Rn(θn + 2πlδn,N ) − Rn−1(θn−1) L (θ ,R (θ )) = + − V(θ − ,R − (θ − )), (A7) n n n n 2 2 n 1 n 1 n 1 where

L[θ,R(θ)] = lim Ln[θn,Rn(θn)] = L0[θ] + LB [θ,R(θ)], (A8) →0 I L [θ] = θ˙2, (A9) 0 2

N N 2 m 2 1 2 Cj θ LB [θ,R(θ)] = R˙j (θ) − mω Rj (θ) − (A10) 2 2 j mω2 j=1 j=1 j is the Lagrangian of the coupled system. The action S[θ,R(θ)] is given by t S[θ,R(θ)] = L[θ,R(θ)]ds = S0[θ] + SB [θ,R(θ)], (A11) 0 t S0[θ] = dτL[θ], (A12) 0 t SB [θ,R(θ)] = dτLB [θ,R(θ)]. (A13) 0 Now, suppose that we initially have the total density operator given by

ρˆAB (0) = ρˆA(0)ρ ˆB (0). (A14) Then, the reduced density matrix is ∞ ρ(θ,φ,t) = dR(θ)dQ(θ)δ(R(θ) − Q(θ))ρ(θ,p,φ,s,t) −∞ 2π

= dθ dφ J (θ,φ,t; θ ,φ ,0)ρA(θ ,φ ,0), (A15) 0 where ∞ ∞ θ+2πl φ+2πl ∗ i J (θ,φ,t; θ ,φ ,0) = Dθ Dφ exp (S0[θ] − S0[φ])F[θ,φ] (A16) h¯ l=−∞ l =−∞ θ φ is the propagator of the density matrix, ∞

F[θ,φ] = dR(θ)dQ(φ)dR (θ )dQ (φ )δ(R(θ) − Q(φ))ρB (p ,s ,0) −∞ R(θ) Q(φ) i × DR(θ) DQ(φ)exp {SB [θ,R(θ)] − SB [φ,Q(φ)]} R (θ ) Q (φ ) h¯ is the inﬂuence functional, and Q(φ) =−(sj − cφ)/mωj . Supposed that the density operatorρ ˆB can be written as a canonical ensemble at t = 0. That is,

exp(−βHˆB ) ρB = . (A17) trB [exp(−βHˆB )] Introduce the imaginary time τ =−ihβ¯ ; then the density matrix of the environment at t = 0is iτ p exp − HˆB s K (p ,τ; s ,0) ρ (p ,s ,0) = h¯ = B . B − iτ ˆ dqK (q,τ; q,0) trB exp h¯ HB B

094308-10 QUANTUM TIME CRYSTAL BY DECOHERENCE: PROPOSAL . . . PHYSICAL REVIEW B 96, 094308 (2017)

2. Momentum representation of the density matrix of a harmonic oscillator The propagator of a harmonic oscillator with the Lagrangian m 1 L = x˙2 − mωx2 (A18) 2 2 has a well-known solution i t K(x,t; x ,0) = Dx exp Lds (A19) h¯ 0 mω i mω = exp [(x2 + x 2) cos ωt − 2xx ] . (A20) 2πih¯ sin ωt h¯ 2sinωt

K(p,t; p ,0) is obtained by a Fourier transformation ∞ dxdx i K(p,t : p ,0) = K(x,t; x ,0) exp px − p x . (A21) −∞ 2πh¯ h¯ = mω cos ωt = mω Let A 2 sin ωt and B 2 sin ωt ; then F (t) i Ap2 + Ap 2 − 2Bpp K(p,t; p ,0) = √ exp , 4B2 − 4A2 h¯ 4B2 − 4A2 where m2ω2(1 − cos2 ωt) 4B2 − 4A2 = = m2ω2, (A22) sin2 ωt so, F (t) i K(p,t; p ,0) = exp S[p(t)/mω]. (A23) mω h¯

Therefore, using τ =−ihβ¯ , β = 1/kB T , νj = hω¯ j β, and i sinh x = sin ix we have N 2 + 2 1 mωj mωj p j s j p j s j p | exp(−iτHˆB /h¯)|s = exp − cosh ν − 2 (A24) mω 2πh¯ sinh ν 2¯h sinh ν m2ω2 m2ω2 j=1 j j j j j and ∞ N − ˆ = | − ˆ | = 1 trB [exp( iτHB /h¯)] dq q exp( iτHB /h¯) q νj . −∞ 2sinh j=1 2 We are assuming that the environment is not coupled to the system at t = 0, so R = p /mω and Q = s /mω . Therefore, ⎛ ⎞ j j j j j j N 1 ρ (p ,s ,0) = ⎝ ⎠ρ (R ,Q ,0), (A25) B mω B j=1 j where N ν mω mω ρ (R ,Q ,0) = 2sinh j j exp − j R 2 + Q 2 cosh ν − 2R Q . (A26) B 2 2πh¯ sinh ν 2¯h sinh ν j j j j j j=1 j j = + = − Let us rewrite this density matrix into a simpler form. Deﬁne x0j Rj Qj and x0j Rj Qj . Then, using 2 + 2 − = 1 − 2 + 1 + 2 Rj Qj cosh νj 2R j Q j 2 (cosh νj 1)x0j 2 (cosh νj 1)x0j (A27) and the identity cosh ν − 1 sinh ν ν j = j = tanh j , (A28) sinh νj cosh νj + 1 2 we obtain N mω mω x2 ρ (R ,Q ,0) = j exp − j 0j + x 2μ , (A29) B πhμ¯ 4¯h μ 0j j j=1 j j ν μ = coth j . (A30) j 2

094308-11 K. NAKATSUGAWA, T. FUJII, AND S. TANDA PHYSICAL REVIEW B 96, 094308 (2017)

3. The inﬂuence functional F[θ,φ] Now, we calculate the density functional F[θ,φ] explicitly. We will write R and Q instead of R[θ] and Q[φ] for simplicity, but keep in mind that they depend on θ and φ, respectively. Then, the inﬂuence functional can be written as ∞ R Q F = N D D ∗ [θ,φ] dxt dxt dx0dx0(1/2) δ(xt )ρB (R ,Q ,0) R Q −∞ R Q ⎧ ⎫ ⎨ ⎬ t C2 i m 2 2 1 2 2 2 j 2 2 × exp ds R˙ − Q˙ − mω R − Q + Cj (θRj − φQj ) − (θ − φ ) . (A31) ⎩h¯ 2 j j 2 j j j 2mω2 ⎭ 0 j j

To evaluate this we introduce the variables ϕ = θ + φ, ϕ = θ − φ, (A32) = + = − xj Rj Qj ,xj Rj Qj . Then, the inﬂuence functional becomes ∞ F = N D D [θ,φ] dxt dxt dx0dx0(1/2) δ(xt )ρB (R ,Q ,0) x x −∞ ⎧ ⎫ ⎨ ⎬ t C2 i m 1 2 Cj j × exp ds x˙j x˙ − mω xj x + (ϕx + ϕ xj ) − ϕ ϕ ⎩h¯ 2 j 2 j j 2 j 2mω2 ⎭ 0 j j ⎧ ⎫ ⎨ ⎬ ∞ t C2 = N im − − i j dxt dxt dx0dx0(1/2) δ(xt )ρB (R ,Q ,0) exp (xtj x˙tj x0j x˙0j ) ds 2 ϕ ϕ −∞ ⎩ 2¯h h¯ 2mω ⎭ j 0 j j ⎧ ⎫ ⎨ ⎬ i t C × DxDx exp ds −g(x )x + j ϕx , ⎩h¯ j j 2 j ⎭ 0 j where m 1 C g(x ) = x¨ + mω2x − j ϕ . (A33) j 2 j 2 j j 2

We note that the classical solution xj,cl satisﬁes the Euler-Lagrange equation = = = g(xj,cl) 0,g(xj (0)) 0,g(xj (t)) 0. (A34)

The path integral over x can be done ﬁrst. Calling this integral Ix,x we obtain ⎧ ⎫ ⎨ t ⎬ i Cj I = DxDx exp ds −g(x )x + ϕx (A35) x,x ⎩h¯ j j 2 j ⎭ 0 j ∞ N−1 N = 1 dxdx i − + Cj exp g(xn,j )xn,j ϕxj (A36) |A |2 −∞ |A |2 h¯ 2 B n=1 B n=1 j ∞ N−1 N−1 N = 1 dx i Cj δ g(xn,j ) exp ϕxn,j . (A37) |A |2 −∞ |A |2 2πh¯ h¯ 2 B n=1 B j n=1 n=1

|A|−2 is absorbed into the delta functions. The interpretation of the delta function is to insert the classical solution xj,cl into xj = which satisﬁes g(xj,cl) 0. The outcome is ⎛ ⎞ t 1 ⎝ i Cj ⎠ I = exp ds ϕx . (A38) x,x |A |2 h¯ 2 cl B 0 j

094308-12 QUANTUM TIME CRYSTAL BY DECOHERENCE: PROPOSAL . . . PHYSICAL REVIEW B 96, 094308 (2017)

Therefore, the inﬂuence functional is ∞ F = N i m − [θ,φ] dxt dxt dx0dx0(1/2) δ(xt )ρB (R ,Q ,0) exp (xtj x˙tj x0j x˙0j ) −∞ h¯ 2 j ⎛ ⎞ 1 i t C2 C × exp ⎝ ds − j ϕ ϕ + j ϕx ⎠. |A |2 h¯ 2mω2 2 cl B 0 j j

The xt integral gives a delta function of x˙t : ⎛ ⎞ ∞ t C2 F = N ⎝ i − j + Cj + i m ⎠ [θ,φ] dxt dx0dx0(1/2) δ(xt )δ(x˙t )ρB (R ,Q ,0) exp ds 2 ϕ ϕ ϕxcl x0j x˙0j . −∞ h¯ 2mω 2 h¯ 2 0 j j j

Insert Eq. (A29) for the density matrix and do the x0 integral; then we obtain ∞ t C2 x˙ 2 F = i − j + Cj − mμj ωj 0j + 2 [θ,φ] dxt dx0δ(xt )δ(x˙t ) exp ds 2 ϕ ϕ ϕxcl 2 x0j . −∞ h¯ 2mω 2 4¯h ω j 0 j j = The two delta functions δ(xt ) and δ(x˙t ) can be interpreted as boundary conditions on the classical solution of g(x (s)) 0.

The result of doing the remaining integrals would be to substitute the classical solution of xcl, x˙0j , and x0j which satisfy these boundary conditions. The solution to the classical solution of g(x) = 0, that is

C x¨ + ω2x − j ϕ = 0, (A39) j j j m is t C2ϕ (s) x˙ =− j − + − − tj − xj (τ) sin ωj (τ s)ds xtj cos ωj (t τ) sin ωj (t τ). (A40) τ mωj ωj For our boundary condition this solution reduces to t =− Cj ϕ (s) − xj (τ) sin ωj (τ s)ds, (A41) τ mωj t =− Cj ϕ (s) − x˙j (τ) cos ωj (τ s)ds. (A42) τ m Therefore, the result of integration is ⎧ ⎫ ⎨ ⎬ i C2 t t 2 F[θ,φ] = exp − j ϕ(τ) δ(τ − s) − sin ω (τ − s) ϕ (s)dτds ⎩ h¯ 2mω ω j ⎭ j j 0 τ j ⎡ ⎤ μ t t C2 × exp ⎣− j j ϕ (τ) cos ω (τ − s)ϕ (s)dsdτ⎦. 4¯h mω j j 0 0 j

Since s and τ enter into the second integral symmetrically it can be rewritten t t t τ E(τ,s)dsdτ = 2 E(τ,s)dsdτ. (A43) 0 0 0 0 The ﬁrst integral can be written t t t τ E(τ,s)dsdτ = E(s,τ)dsdτ. (A44) 0 τ 0 0 Then, the inﬂuence functional becomes F[θ,φ] = exp i[θ,φ], (A45)

094308-13 K. NAKATSUGAWA, T. FUJII, AND S. TANDA PHYSICAL REVIEW B 96, 094308 (2017) where the inﬂuence phase [θ,φ] can be written as t τ t τ i 1 i[θ,φ] =− dτdsϕ(s)αI (τ − s)ϕ (τ) − dsdτϕ (τ)αR(τ − s)ϕ (s), (A46) h¯ 0 0 h¯ 0 0 2 Cj hω¯ j αR(τ − s) = coth cos ωj (τ − s), (A47) 2mωj 2kB T j C2 2 α (τ − s) = j δ(τ − s) + sin ω (τ − s) . (A48) I 2mω ω j j j j Therefore, we obtain the density matrix propagator J (θ,φ,t; θ ,φ ,0),

J (θf,φf,t; θi,φi,0) (A49) θf φf ∗ i + − − = Dθ Dφ exp S[ϕ ,ϕ ] − [ϕ ] + , (A50) ϕ = (θ + φ)/2 θi φi h¯ ϕ− = φ − θ the classical action S[ϕ+,ϕ−]is t S[ϕ,ϕ ] =− dτIϕ˙+(τ)˙ϕ−(τ) 0 t s − + + 2 dτ dsϕ (τ)αI(τ − s)ϕ (s), 0 0 and the [ϕ−]isgivenby t t − 1 − − [ϕ ] = dτ dsϕ (τ)αR(τ − s)ϕ (s). 2¯h 0 0 The path integral in (A50) can be done exactly and we obtain i J (θ ,φ ,t; θ ,φ ,0) = F 2(t)exp S[ϕ+,ϕ−] − [ϕ−] . (A51) f f i i h¯ cl cl cl ± ϕcl are the classical coordinates obtained from the Euler-Lagrange equation t − + − − = Iϕ¨cl (u) 2 dτϕcl (τ)αI(τ u) 0, (A52) u u + + + − = Iϕ¨cl (u) 2 dτϕcl (τ)αI(u τ) 0, (A53) 0 ± = ± ± = ± whose solution is given in terms of boundary conditions ϕi ϕ (0),ϕf ϕ (t): + = + + + ϕcl (u) κi (u; t)ϕi κf (u; t)ϕf , (A54) − = − − + − − ϕcl (u) κi (t u; t)ϕf κf (t u; t)ϕi , (A55) G˙ (t) G(u) κ (u; t) = G˙ (u) − G(u),κ(u; t) = . (A56) i G(t) f G(t) Then, the classical action reduces to + − =− + − − − − S[ϕcl ,ϕcl ] I[˙ϕcl (t)ϕf ϕ˙cl (0)ϕi ]. (A57)

[1] F. Wilczek, Phys.Rev.Lett.109, 160401 (2012). [5] G. Grüner, Density Waves in Solids (Addison-Wesley Publishing [2] T. Li, Z.-X. Gong, Z.-Q. Yin, H. T. Quan, X. Yin, P.Zhang, L.-M. Company, Reading, Massachusetts, 1994); S. Kagoshima, H. Duan, and X. Zhang, Phys. Rev. Lett. 109, 163001 (2012). Nagasawa, and T. Sambongi, One-Dimensional Conductors [3]G.E.Volovik,JETP Lett. 98, 491 (2013). (Springer-Verlag, 1988); P.Monceau, Adv. Phys. 61, 325 (2012). [4]A.J.Leggett,S.Chakravarty,A.T.Dorsey,M.P.A.Fisher,A. [6] A. Zettl, G. Grüner, and A. H. Thompson, Phys. Rev. B 26, 5760 Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1 (1987). (1982).

094308-14 QUANTUM TIME CRYSTAL BY DECOHERENCE: PROPOSAL . . . PHYSICAL REVIEW B 96, 094308 (2017)

[7] S. Tanda, T. Tsuneta, Y. Okajima, K. Inagaki, K. Yamaya, and [19] B. Hall, Quantum Theory for Mathematicians, Graduate Texts N. Hatakenaka, Nature (London) 417, 397 (2002). in Mathematics (Springer, New York, 2013). [8] T. Matsuura, K. Inagaki, and S. Tanda, Phys. Rev. B 79, 014304 [20] P. Hänggi, Generalized Langevin equations: A useful tool (2009). for the perplexed modeller of nonequilibrium ﬂuctuations? [9] M. Tsubota, K. Inagaki, T. Matsuura, and S. Tanda, Europhys. in Stochastic Dynamics, Lecture Notes in Physics, Vol. 484 Lett. 97, 57011 (2012). (Springer, Berlin, 1997), pp. 15–22. [10] A. Caldeira and A. Leggett, Physica A 121, 587 (1983). [21] H. Grabert, P. Schramm, and G.-L. Ingold, Phys. Rep. 168, 115 [11] U. Weiss, Quantum Dissipative Systems (World Scientiﬁc (1988); P.Schramm and H. Grabert, J. Stat. Phys. 49, 767 (1987). Publishing Company, Singapore, 1999). [22] P. A. Lee, T. M. Rice, and P. W. Anderson, Solid. State Commun. [12] P. Bruno, Phys.Rev.Lett.111, 070402 (2013). 14, 703 (1974). [13] H. Watanabe and M. Oshikawa, Phys. Rev. Lett. 114, 251603 [23] K. Nomura and K. Ichimura, J. Vac. Sci. Technol. A 8, 504 (2015). (1990). [14] C. W. von Keyserlingk and S. L. Sondhi, Phys. Rev. B 93, [24] M. Ido, Y. Okajima, and M. Oda, J. Phys. Soc. Jpn. 55, 2106 245146 (2016); N. Y. Yao, A. C. Potter, I.-D. Potirniche, and A. (1986). Vishwanath, Phys. Rev. Lett. 118, 030401 (2017); D. V. Else, B. [25] F. Wilczek, Phys. Rev. Lett. 111, 250402 (2013). Bauer, and C. Nayak, Phys. Rev. X 7, 011026 (2017). [26] A. V. Ustinov, T. Doderer, R. P. Huebener, N. F. Pedersen, B. [15] S. Choi, J. Choi, R. Landig, G. Kucsko, H. Zhou, J. Isoya, F. Mayer, and V. A. Oboznov, Phys. Rev. Lett. 69, 1815 (1992). Jelezko, S. Onoda, H. Sumiya, V. Khemani, C. von Keyserlingk, [27] H.-K. Li, E. Urban, C. Noel, A. Chuang, Y. Xia, A. Ransford, B. N. Y. Yao, E. Demler, and M. D. Lukin, Nature (London) 543, Hemmerling, Y. Wang, T. Li, H. Häffner, and X. Zhang, Phys. 221 (2017); J. Zhang, P. W. Hess, A. Kyprianidis, P. Becker, Rev. Lett. 118, 053001 (2017). A. Lee, J. Smith, G. Pagano, I.-D. Potirniche, A. C. Potter, A. [28] V. J. Emery and J. D. Axe, Phys.Rev.Lett.40, 1507 (1978). Vishwanath, N. Y. Yao, and C. Monroe, ibid. 543, 217 (2017). [29] I. U. Heilmann, J. D. Axe, J. M. Hastings, G. Shirane, A. [16] E. N. Bogachek, I. V. Krive, I. O. Kulik, and A. S. Rozhavsky, J. Heeger, and A. G. MacDiarmid, Phys. Rev. B 20, 751 Phys. Rev. B 42, 7614 (1990). (1979). [17] Y. Ohnuki and S. Kitakado, J. Math. Phys. 34, 2827 (1993). [30] R. V. Pai and R. Pandit, Phys.Rev.B71, 104508 (2005). [18] J. J. Sakurai, Modern Quantum Mechanics, rev. ed. (Addison- [31] R. Blinc and A. P. Levanyuk, Incommensurate Phases in Wesley Publishing company, Reading, Massachusetts, 1993). Dielectrics: Parts 1 and 2 (North Holland, Amsterdam, 1986).

094308-15