Finite Frequency Noise in a Chiral Luttinger Liquid Coupled to Phonons

PHYSICAL REVIEW B 100, 155422 (2019)

Finite frequency noise in a chiral Luttinger liquid coupled to phonons

Edvin G. Idrisov Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg

(Received 30 April 2019; revised manuscript received 19 August 2019; published 21 October 2019)

We study transport between quantum Hall edge states at filling factor ν = 1 in the presence of electron- acoustic-phonon coupling. After a Bogoliubov-Valatin transformation, the low-energy spectrum of interacting electron-phonon system is presented. The electron-phonon interaction splits the spectrum into charged and neutral “downstream” and neutral “upstream” modes with different velocities. In the regimes of dc and periodic ac biases the tunneling current and nonequilibrium finite frequency nonsymmetrized noise are calculated perturbatively in tunneling coupling of the quantum point contact. We show that the presence of electron-phonon interaction strongly modifies noise and current relations compared to the free-fermion case.

DOI: 10.1103/PhysRevB.100.155422

I. INTRODUCTION classical skipping orbits. Electrons in these 1D chiral states are basically similar to photons: they propagate only in one Electron-phonon interactions play an important role in direction, and there is no intrinsic backscattering along the three-dimensional solid-state physics, for instance, to explain QH edge in the case of identical chiralities. However, elec- superconductivity [1,2]. However, intense theoretical and ex- trons satisfy Fermi-Dirac statistics and are strongly interacting perimental investigations have indicated that the static and particles. dynamical properties of electrons in one-dimensional (1D) In order to study chiral 1D edge excitations a low-energy Luttinger liquids are expected to be strongly modified by effective theory was introduced by Wen [21]. The key idea the presence of electron-phonon interaction as well [3–13]. behind this approach is to represent the fermionic field in Indeed, Martin and colleagues [3,4], using a Luttinger liquid terms of collective bosonic fields. This so-called bosonization description, calculated the exponents of correlation functions technique allows one to diagonalize the edge Hamiltonian of and discussed their remarkable sensitivity to the Wentzel- interacting 1D electrons with a linear spectrum and calculate Bardeen [14,15] singularity induced by the presence of acous- any equilibrium correlation functions. Wen’s bosonization tic phonons. Next, in Ref. [8] the transport in a suspended technique triggered better theoretical understanding of trans- metallic single-wall carbon nanotube in the presence of strong port properties of 1D chiral systems and explains a lot of electron-phonon interaction was investigated. It was shown experimental findings [22]. Particularly, the tunneling cur- that the differential conductance as a function of applied bias rent, nonequilibrium symmetric quantum noise between chiral voltage demonstrates three distinct types of phonon-assisted fractional QH edge states and the photoassisted current, and peaks. These peaks are best observed when the system is in the shot noise in the fractional QH effect were studied [23–27]. vicinity of the Wentzel-Bardeen singularity. Later, using func- It was shown that the differential conductance is nonzero tional bosonization, Galda et al. [10–12] studied the impact at finite temperature and manifests a power-law behavior, of electron-phonon coupling on electron transport through a G ∝ T 2(g−1), where T is the temperature and g depends on Luttinger liquid with an embedded scatterer. They investi- the filling factor of the fractional QH state. The exponent gated the directions of renormalization group flows which can g= ν,org = 1/ν, depending on the geometry of the tunneling be changed by varying the ratio of Fermi (electron-electron constriction, where ν is the filling factor. At zero temperature interaction strength) to sound (electron phonon strength) ve- it has been demonstrated that the tunneling current has the locities. A phase diagram was revealed with up to three fixed form I ∝ μ2g−1, where μ is an applied dc bias. In the lowest points: an unstable one with a finite value of conductance order of tunneling coupling at zero temperature it was shown and two stable ones, corresponding to an ideal metal or that the symmetric nonequilibrium noise is given by S(ω) ∝ 2g−1 ∗ insulator. The existence of only two stable fixed points (the σ =∓ |ω + σω0| , where ω0 = e μ/h¯ is the Josephson ideal metal or the ideal insulator) is guaranteed by the duality frequency [28] of the electron or quasiparticle and e∗ is the in a multichannel Luttinger liquid [16,17]. effective charge of the tunneling quasiparticle. Thus, an alge- Concurrently, the transport properties of electrons in chiral braic singularity is present at the Josephson frequency, which 1D systems has been a subject of intensive theoretical and depends on charge e∗. At finite temperature it was shown that experimental studies for a long time as well [18]. The ex- the ratio between nonequilibrium and equilibrium Johnson- perimental investigation of such systems was made possible Nyquist noises does not depend on the parameter g and has the mostly as a result of the discovery of integer and fractional familiar form S(ω, μ)/S(ω, 0) = (e∗μ/2T ) coth(e∗μ/2T ). quantum Hall (QH) effects in a two-dimensional electron gas In Ref. [29] it was already shown that the chiral QH edge (2DEG) [19,20]. The chiral electron states appearing at the state at filling factor ν = 1 is not a Fermi liquid in the presence edge of a 2DEG may be considered a quantum analog of of electron-phonon interaction. As a consequence the electron

2469-9950/2019/100(15)/155422(11) 155422-1 ©2019 American Physical Society EDVIN G. IDRISOV PHYSICAL REVIEW B 100, 155422 (2019) √ correlation function is modified. Particularly, it was demon- aˆk = L/2πkρˆ−k, which satisfy a standard commutation re- † strated that the ac conductance of the chiral channel exhibits lation [âk, aˆ ] = δkk , i.e., ω k resonances at longitudinal wave vectors q and frequencies = ω/ = ω/ =−ω/ 2π − related by q v1, q v2, and q v3, where v1, φˆ(x) = ϕˆ + 2ππˆ x + [eikxaˆ + e ikxaˆ†], (1) 0 0 Lk k k v2, v3 are renormalized Fermi and sound velocities. However, k>0 it was claimed that the dc Hall conductance is not modified where zero modes fulfill the canonical commutation relation by electron-phonon interaction and is given by G = e2/2πh¯. H [ˆπ , ϕˆ ] = i/L and L is the total size of the system. We Moreover, we would like to mention that according to the 0 0 consider the thermodynamic limit L →+∞; consequently, L Wiedemann-Franz law [30,31], the thermal Hall conductance drops out in the final results. is not changed as well, namely, KH = TGH L0, where T is 3 2 The total Hamiltonian includes three terms: the temperature and L0 = (π /3)(kB/e) is the free-fermion Lorenz number (see Appendix C). Hˆ = Hˆ0 + Hˆph + Hê−ph. (2) Motivated by the previous progress in Ref. [29], we take a Here the first term further step to investigate the noise properties of the tunneling vF 2 current between two ν = 1 edge states in the presence of Hˆ0 = dx[∂xφˆ(x)] (3) 4π strong electron-acoustic phonon interaction. Such a system may be realized in conventional electron optics experiments is the free part of the total Hamiltonian, where vF is the Fermi [32] and also in bilayer systems with a total filling factor velocity. The second term describes free phonons, equal to 2 [18]. At filling factor ν = 1 we deal with a 1D chiral free-fermion system and no electron-electron interac- 1 1 Hˆ = dx ζ v2(∂ dˆ)2 + ˆ 2 , (4) tion influences on tunneling current and noise. Therefore, any ph 2 s x ζ d modifications will arise due to electron-phonon interactions. where dˆ(x) is a phonon field operator, ˆ d (x) is the canonical We expect that one may investigate this experimentally as in ζ the works by Milliken et al. [33], where the authors measured conjugate to it, vs is the sound velocity, and is the linear the temperature dependence of the tunneling current between mass density of the crystal. The phonon field and its conjugate edge states with electron-phonon interaction. are given by the following sums of phonon creation and annihilation operators: The rest of the paper is organized as follows. In Sec. II, ζ | | we introduce the model of the system, starting with the vs k ikx † ˆ (x) = i e (bˆ − bˆ− ), Hamiltonian of all the constituting parts and the bosonization d 2L k k k prescription, and provide a Bogoliubov-Valatin (BV) transfor- (5) mation in order to calculate exactly the correlation function. 1 ikx † In Sec. III, we calculate perturbatively the tunneling current dˆ(x) = e (bˆ + bˆ− ). 2Lζ v |k| k k in the regime of dc bias. In Sec. IV, we calculate the finite k s frequency nonsymmetrized noise in the dc regime. Section V The third term takes into account the strong electron-phonon is devoted to the derivation of the tunneling current in the interaction regime of periodic ac bias. The result for finite frequency nonsymmetrized noise under the external ac bias is given in Hê−ph = D dxρˆ (x)∂xdˆ(x), (6) Sec. VI. We present our conclusions and future perspectives in Sec. VII. Details of calculations and additional information = πζ 3 where D vs is an electron-phonon coupling (deforma- are presented in the Appendixes. Throughout the paper, we set tion potential constant). Strictly speaking, phonons are not | |= = = e h¯ kB 1. 1D, and one needs to take into account all phonon modes. This means that apart from the intraedge interaction with electrons, Eq. (6), phonons can mediate the interedge interaction. The II. THEORETICAL MODEL latter can substantially change the physics of the QH edge state. However, here, according to Ref. [29], we assume that A. QH edge coupled to phonons one normal mode of the phonons is along the QH edge. Next, We start by introducing the total Hamiltonian of the system for phonons with a wavelength much larger than the cross of QH edge states at filling factor ν = 1 coupled to acoustic section of the edge strip, the perpendicular normal modes con- phonons. The relevant energy scales in the experiments with tribute as an overall constant in the adiabatic approximation such systems are sufficiently small compared to the Fermi and can be ignored. energy F , which suggests using the effective low-energy For further consideration it is convenient to rewrite the total theory of QH edge states [23]. The advantage of this approach Hamiltonian in Eq. (2) in momentum representation is that it allows us to take into account exactly the electron- ˆ = † + ˆ† ˆ + ˆ† ˆ electron and, particularly in our case, the strong electron- H vF kaˆk aˆk vsk[bkbk b−kb−k] phonon interactions. According to the effective theory, edge k>0 k>0 states can be described as collective fluctuations of the charge † + [v k(bˆ + bˆ− )â + H.c.], (7) densityρ ˆ (x). The charge density operator is expressed in c k k k k>0 terms of bosonic field φˆ(x), namely,ρ ˆ (x) = (1/2π )∂xφˆ(x). √ The boson field φˆ(x) can be written in√ terms of boson where vc = D/ πζvs and we ignore zero modes as well. † = / π ρ creation and annihilation operators,a ˆk L 2 k ˆk and Typically, the sound velocity is much smaller than the Fermi

155422-2 FINITE FREQUENCY NOISE IN A CHIRAL LUTTINGER … PHYSICAL REVIEW B 100, 155422 (2019) velocity. However, according to Ref. [29], it should be possi- Here the dimensionless 3 × 3 matrix has the following form: ⎡ ⎤ ble to see the effects of electron-phonon interactions in GaAs αββ heterojunctions. For instance, at magnetic fields of the order ⎣ ⎦ − = β 10, of 5 T and 2D electron densities of the order of 1015 m 2, A (9) −β 0 −1 the Fermi velocity might be of the order of the sound velocity ∼ × 3 / vs 5 10 m s[29,34]. where α = vF /vs and β = vc/vs. We call it a dynamical The Hamiltonian in Eq. (2) gives a complete description matrix. of our system. We note that Hamiltonian Hˆ is bilinear in Next, according to Ref. [35], the Hamiltonian of bosons in boson creation and annihilation operators; thus, the dynamics Eq. (7) is BV diagonalizable if its dynamical matrix is physi- associated with this Hamiltonian can be accounted for exactly cally diagonalizable (stable theory). The dynamical matrix is by solving linear equations of motion. However, the solution said to be stable if it is diagonalizable, and all its eigenvalues of the equation of motion has an inconvenient and complex are real. Therefore, we write the characteristic equation to find form. To overcome those difficulties, we use the alternative the eigenvalues, way below, applying the BV transformation to diagonalize the λ3 − αλ2 − λ + χ = , total Hamiltonian. 0 (10) where we introduced the notation χ = α − 2β2. To have three B. Bogoliubov-Valatin transformation distinct real eigenvalues the coefficients of the characteristic Using the operator identity [âbˆ, cˆ] = aˆ[bˆ, cˆ] + [â, cˆ]bˆ for cubic equation must satisfy the inequality (see Fig. 1) bosonic operators, the equations of motion can be derived g(α,β) = 18αχ + 4α3χ + (α2 + 4) − 27χ 2 > 0. (11) from Eq. (7): ⎡ ⎤ ⎡ ⎤ The eigenvalues of dynamical matrix (9) are provided in aˆ (t ) aˆ (t ) k k Appendix A. Using the ideas of Ref. [35], one can construct d ⎣ ˆ ⎦ =− ⎣ ˆ ⎦. bk (t ) ivskA bk (t ) (8) the BV transformation matrix, dt ˆ† ˆ† b−k (t ) b−k (t ) ⎡ ⎤ λ2− 2 λ2− 2 λ2− 2 ( 1 1) ( 2 1) ( 3 1) ⎡ ⎤ ⎢ λ2− 2+ β2λ λ2− 2+ β2λ − λ2− 2− β2λ ⎥ ⎢ ( 1 1) 4 1 ( 2 1) 4 2 ( 3 1) 4 3 ⎥ b11 b12 b13 ⎢ ⎥ β λ2−1 2 β λ2−1 2 β λ2−1 2 = ⎣ ⎦ = ⎢ ( 1 ) ( 2 ) ( 3 ) ⎥, B b21 b22 b23 ⎢ 2 2 2 ⎥ (12) λ1−1 λ2− + β2λ λ2−1 λ2− + β2λ λ3−1 − λ2− − β2λ ⎢ ( 1 1) 4 1 ( 2 1) 4 2 ( 3 1) 4 3 ⎥ b31 b32 b33 ⎣ ⎦ λ2− 2 λ2− 2 λ2− 2 −β ( 1 1) −β ( 2 1) −β ( 3 1) 2 2 2 λ1+1 λ2− + β2λ λ2+1 λ2− + β2λ λ3+1 − λ2− − β2λ ( 1 1) 4 1 ( 2 1) 4 2 ( 3 1) 4 3

where λ1, λ2, and λ3 are real eigenvalues of Eq. (10). The C. Correlation function straightforward algebra confirms that the BV matrix diagonal- The important quantity which allows us to describe the izes the initial total bosonic Hamiltonian (7). Namely, we get transport properties of the system under consideration is the following quadratic form (the nonessential constant term the equilibrium two-point correlation function. It is defined is omitted): † as G(x1, t1; x2, t2 ) = ψˆ (x1, t1)ψˆ (x2, t2 ) , where, according ˆ = α†α + βˆ †βˆ + γ † γ , H 1 ˆ k ˆ k 2 k k 3 ˆ−k ˆ−k (13) to the bosonization technique, the fermionic field is given k>0 k>0 k>0 by ψˆ (x) ∝ exp[iφˆ(x)], where we omit the ultraviolet cutoff where i = vik, i = 1, 2, 3, and v1 = λ1vs > 0, v2 = λ2vs > 0, prefactor. The average is taken with respect to the equilibrium and v3 =−λ3vs > 0. One can check that the relations density matrixρ ˆ0 ∝ exp[(Hˆ0 + Hˆph )/T ], and T is tempera- + − = 2 + 2 − 2 = 1 2 3 F and b11 b12 b13 1 are invariants of ture. the BV transformation and the statistics of the system, i.e., Next, using the expansion of bosonic fields in terms of commutation relations of boson operators, is preserved after collective modes in Eq. (1) and the BV transformation in the BV transformation. The renormalized spectrum is given Eqs. (12)–(14), we finally arrive at the following result for in Figs. 2 and 3. From these figures we see that 1 and the fermion correlation function [29] (see Appendix B): 2 are strongly modified by electron-phonon interactions in comparison to free plasmons, collective excitation of the QH b2 iη 11 edge and free phonons, and 3 is slightly changed from its , , ∝ G(x1 t1; x2 t2 ) initial value s. The elements of the first row of the BV x1 − x2 − v1(t1 − t2 ) + iη transformation are given in Figs. 4–7. b2 η 12 According to the BV transformation the bosonic operator × i is expressed in terms of three independent bosonic terms [29], x1 − x2 − v2(t1 − t2 ) + iη namely, b2 − η/ 13 † i aˆk = b11αˆ k + b12βˆk + b13γˆ . (14) × , (15) −k x − x + v (t − t ) − iη This equation is the main result of this section. It will be 1 2 3 1 2 used in the next section to calculate the two-point correlation function. where η = a/2π and a is an ultraviolet cutoff.

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FIG. 3. The renormalized spectrum of 3/s as a function of renormalized Fermi energy F /s.Wesetβ = 0.1.

b2 / 12 × iaT 2v2 πT sinh [x1 − x2 − v2(t1 − t2 )] + iη v2 b2 −iaT/2v 13 × 3 . (16) FIG. 1. The condition of validation of the BV transformation. πT − + − − η sinh v [x1 x2 v3(t1 t2 )] i The light orange region satisfies the inequality g(α,β) < 0, and the 3 light blue region is given by g(α,β) > 0. The BV transformation is As one can mention, the correlation function splits into three valid in the light blue region. independent correlation functions with different velocities. Two of them with velocities v1 and v2 correspond to “down- The free-fermion two-point correlation function for stream” modes. One of these modes, v , carries a charge. ∝ η/ − − 1 the right-moving mode is proportional to i [x1 x2 The third correlation function is related to the “upstream” − + η vF (t1 t2 ) i ]. Here we see that the electron-phonon inter- mode and travels on the opposite side with respect to the action changes the correlation function sufficiently compared right-moving downstream modes. In the following sections to the free-fermion one. we use these correlators to derive the tunneling current and The finite temperature correlation function is obtained noise perturbatively in the tunneling coupling. by applying a conformal transformation [36]. To ap- ply the conformal transformation we go to Euclidean III. TUNNELING CURRENT IN THE dc REGIME time −iτE = t1 − t2 − (x1 − x2 )/v and then set vτE → (v/2πT )arctan(2πT τE ). It follows that The electron-phonon interaction is taken into account exactly; therefore, the tunneling has to be considered pertur- G(x1, t1; x2, t2 ) batively. The tunneling (backscattering) Hamiltonian of elec- b2 trons at the quantum point contact (QPC) located at point x is / 11 ∝ iaT 2v1 given by πT sinh [x1 − x2 − vα (t1 − t2 )] + iη † † v1 ˆ = ˆ + ˆ , ˆ = τψˆ ψˆ , HT A A A d (x) u(x) (17)

FIG. 2. The renormalized spectrum of 1/s and 2/s as a function of renormalized Fermi energy F /s. For comparison the FIG. 4. The squares of the ﬁrst two matrix elements of the BV spectrum of the uncoupled phonon and plasmon system is shown as transformation as a function of renormalized Fermi energy F /s [see well. We set β = 0.1. Eq. (12)]. We set β = 0.1.

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FIG. 5. The square of the third matrix element of the BV trans- FIG. 7. The relations between the ﬁrst row elements of the BV β = . formation as a function of renormalized Fermi energy F /s [see matrix in Eq. (12). We set 0 1. Eq. (12)]. We set β = 0.1. obtains where τ (generally a complex number) is the tunneling cou- ψˆ = , † pling constants and i(x), i u d, is an electron annihilation I = dt [Aˆ (t ), Aˆ(0)] . (21) operator [37,38]intheith channel. The vertex operator Aˆ in Eq. (17) mixes the up (u) and down (d) channels and transfers The perturbative result is valid as long as the tunneling current electrons from one channel to another (see Fig. 8). We I is small compared to the Hall current I = G μ, where μ is ˆ = ˆ˙ = ˆ , ˆ H H introduce the tunneling current operator J Nd i[HT Nd ], a dc bias applied between upper and lower chiral channels (see ˆ = ψˆ † ψˆ where Nd dx d (x) d (x) is the number of electrons in the Fig. 8). In our model there is no interaction between upper and down channel. After simple algebra we obtain the tunneling lower channels; that is, they are independent, and therefore, current operator the correlation functions in Eq. (21) split into the product of Jˆ = i(Aˆ† − Aˆ). (18) two single-particle correlators, namely, In the interaction representation the average current is given =|τ|2 iμt ψˆ † ψˆ ψˆ ψˆ † I dte [ u (t ) u(0) d (t ) (0) by the expression d † † − ψˆ ψˆ ψˆ ψˆ † , I = Uˆ (t, −∞)Jˆ(t )Uˆ (t, −∞) , (19) d (0) d (t ) u(0) u (t ) ] (22) where the average is taken with respect to the dc biased ground where, for deﬁniteness, we set x = 0 and the average now state in QH edges and free phonons and at the lowest order of is taken with respect to the equilibrium density matrixρ ˆ0 tunneling coupling the evolution operator is given by from Sec. II. Next, without loss of generality we consider the case of positive bias μ>0. The sign of the bias μ t1 simply determines the direction of the current. Substituting Uˆ (t1, t2 ) ≈ 1 − i dtHˆT (t ). (20) t2 the correlation functions from Eq. (15) into Eq. (22) and employing the integral We evaluate the average current by perturbatively expanding the evolution operator to the lowest order in the tunneling eη+iy 2π amplitude. It is clear that the average tunneling current can lim dy = , (23) η→+0 (η + iy)b [b] be written as a commutator of vertex operators; namely, one where y = μt is a dimensionless variable of integration, we obtain the expression for tunneling current at zero tempera-

FIG. 8. Schematic representation of the setup: A QPC (dashed red line) with tunneling amplitude τ connects up (u) and down (d) chiral edge channels of integer QH at ﬁlling factor ν = 1. Each channel is coupled to acoustic phonons. The bias μ is applied to FIG. 6. The sum of squares of the ﬁrst two BV matrix elements the upper incoming edge channel, while the down edge channel is [see Eq. (12)]. We set β = 0.1. Note that the sum is not equal to 1. grounded.

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Fermi liquid. Namely, at high temperatures from Eq. (26)we obtain

I ∝ T b−2μ, μ/T 1, (27)

which demonstrates an Ohmic behavior, and at low temperatures we get the expression

I ∝ μb−1,μ/T 1, (28)

with non-Ohmic behavior, speciﬁc for a non-Fermi liquid. Next, apart from the current or conductance it is useful to measure the time-dependent ﬂuctuations of tunneling current, noise. For instance, it is known that measuring shot noise FIG. 9. The tunneling current Iμ versus applied dc voltage μ at allows direct access to fractional charges of Laughlin quasi- zero temperature for different even values of b [see Eq. (24)]. particles [39]. The calculation of noise is the subject of the next sections. ture: π IV. FINITE AND ZERO-FREQUENCY NOISE 2 2 b−1 Iμ ∝ D|τ| μ , (24) [b] IN THE dc REGIME

−2b2 −2b2 −2b2 A. Finite frequency noise where D = (2πv1) 11 (2πv2 ) 12 (2πv3) 13 is the renormalized density of states in the presence of electron-phonon Experimentally, the measured spectral noise density of = 2 + 2 + 2 current fluctuations strongly depends on how the detector op- interaction, b 2(b11 b12 b13), [b] is the Euler gamma function, and we omit the prefactor which depends on the erates [40]. We consider the finite frequency nonsymmetrized ultraviolet cutoff. The dependence of the tunneling current on noise. In the absence of time-dependent external fields the applied dc bias is presented in Figs. 9 and 10. It is a monotonic correlation function in Eqs. (15) and (16) depends on the function. time difference. This is true as well for finite frequency noise For the case of finite temperature an analytical continuation defined as on the complex plane is applied to Eq. (22). Then shifting the ω integration variable πTt = u + iπ/2 (this does not affect the S(ω) = dtei t δJˆ(t )δJˆ(0) , (29) singularities of integrand) and using the integral μ μ 2 where the average is taken with respect to the dc biased ground i π u 1 + e T − (b i π ) δ ˆ = lim du = 2b 1 2 T , (25) state in QH edges and free phonons as in Eq. (21) and J(t ) η→+0 coshb[u + iη] [b] Jˆ(t ) − Jˆ(t ) . This has become a measurable quantity in re- we get the final result for the tunneling current, cent experiments [41]. We evaluate Eq. (29) perturbatively in the lowest order of tunneling coupling and get the following μ 2 μ 1 (b + i ) expression for finite frequency nonsymmetrized noise in terms I ∝ 2D|τ|2(2πT )b sinh 2 πT . (26) 2T 2πT [b] of vertex operators, defined in Eq. (17), namely, In the noninteracting free-fermion case at b = 2, the cur- ω S(ω) = dtei t [ Aˆ†(t )Aˆ(0) + Aˆ(t )Aˆ†(0) ]. (30) rent is given by the well-known Landauer-Buttiker formula 2 I =|τ| μ/2πvF , and the QPC is in the Ohmic regime. How- ever, a non-Ohmic behavior could be observed for the non- One can mention that to obtain the symmetrized noise one needs to simply construct an even function of nonsymmetrized noise, i.e., [S(ω) + S(−ω)]/2. Evaluating Eq. (30) at zero temperature, we obtain the result for finite frequency nonsymmetrized noise: 2π S ω ∝ D|τ|2 |ω + σμ|b−1θ ω + σμ , ( ) ( ) (31) [b] σ =∓

where θ(x) is the Heaviside function. The normalized ﬁnite frequency nonsymmetrized noise is plotted in Fig. 11.From Eq. (31) one can notice that at ω μ, we obtain S(ω) ∝ Iμ, which is the classical shot noise result of the next section. As expected, it does depend on the interaction parameter b.Note that at ω =∓μ there are knife-edged singularities in ﬁnite frequency noise. In the case of a noninteracting fermionic FIG. 10. The tunneling current Iμ versus applied dc voltage μ at liquid at b = 2 these singularities are caused by the Pauli zero temperature for different odd values of b [see Eq. (24)]. exclusion principle [28].

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FIG. 11. The normalized nonsymmetrized finite frequency noise FIG. 12. The normalized tunneling current I/I is plotted versus S(ω)/Iμ is plotted versus the dimensionless frequency ω/μ > 0 the dimensionless dc component of bias μ/ > 0 at zero tempera- (emission noise) at zero temperature and fixed dc bias μ>0[see ture and fixed >0[seeEq.(40)]. Here we set b = 0.6, b = 1.2, Eq. (31)]. We set b = 0.6andb = 1.2. μ1/ = 2, and I is given by Eq. (24) by simply replacing μ by frequency .

For finite temperatures we substitute the correlation func- where the ac part of V (t ) averages to zero over one period tion from Eq. (16) into Eq. (30) and get T = 2π/and the number of electrons per pulse is q = μ/. ω+σμ ω+σμ exp 1 (b + i ) 2 One can consider more exotic shapes of the drive, such as a S ω ∝ D|τ|2 πT b 2T 2 πT . “train” of Lorentzian pulses (q =∓1, ∓2,...) in the context ( ) (2 ) π σ =∓ 2 T [b] of levitons, the time-resolved minimal excitation states of a (32) Fermi sea, recently detected in a 2DEG [42–45]. The time-averaged tunneling current between two edges in Away from singularities at |ω ∓ μ|/2T 1 we recover the case of ac voltage is given by the formula Eq. (31). At zero frequency this result coincides with Eq. (34) in the next section. I = 2Re[I], (36) where the quantity in square brackets has the form B. Zero-frequency noise T t t 1 i V (t˜)dt˜ † Experimentally, the zero-frequency spectral noise density I = dt dt e t Aˆ t , Aˆ t T [ ( ) ( )] (37) is studied as well [39]. Moreover, in the limit of zero fre- 0 −∞ quency the nonsymmetrized and symmetrized noises coin- and the average is taken with respect to the equilibrium cide. Unlike the tunneling current, the zero-frequency noise density operator ρ . After substituting the exact form of vertex is given by the anticommutator of vertex operators 0 operators from Eq. (17) into Eq. (37) and taking into account that the up and down edges are independent, we get S(0) = dt {Aˆ†(t ), Aˆ(0)} . (33) T t μ 1 μ − 1 I = i (t t ) i − A straightforward calculation similar to what we have done to dt dt e e [sin( t ) sin( t )] T 0 −∞ evaluate the tunneling current in Eq. (26)gives × [ ψˆ †(t )ψˆ (t ) ψˆ (t )ψˆ †(t ) μ u u d d S(0)/I = coth , (34) − ψˆ †(t )ψˆ (t ) ψˆ (t )ψˆ †(t ) ]. (38) 2T d d u u where the tunneling current I is given by Eq. (26). This Further progress in Eq. (38) is possible using the property expression is independent of the interaction parameter b.The of Bessel functions of the first kind, current fluctuations satisfy a classical (Poissonian) shot noise ∞ form. This is related to uncorrelated tunneling of electrons exp[iξ sin ϕ] = Jn(ξ )exp[inϕ], (39) through a QPC. The result is identical to the case of non- n=−∞ interacting electrons in the Landauer-Buttiker approach and ξ shows that the electron-phonon interaction is not important; where Jn( ) gives the Bessel function of the first kind and n is thus, successive electrons tunnel infrequently. an integer. Substituting Eqs. (15) and (39) into Eq. (38) and perform- ing the time integration, we obtain the final result for the V. TUNNELING CURRENT IN THE ac REGIME tunneling current at zero temperature,

To investigate the time-dependent driven case, we split the +∞ 2π voltage applied to the upper edge contact into the dc and ac I ∝ D|τ|2 J2(μ /)|μ + n|b−1sgn(μ + n). [b] n 1 parts, namely, n=−∞

V (t ) = μ + μ1 cos(t ), (35) (40)

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The normalized tunneling current is plotted in Fig. 12.At zero drive frequency, μ1/ →∞, and positive dc bias μ we recover the result in Eq. (24) in Sec. III. Because of the ac drive here we observe the singularities which strongly depend on the electron-phonon interaction parameter b and are related by a new energy scale, the drive frequency . Identical steps as in the case of zero temperature bring us to the result for finite temperatures, namely, +∞ ∝ D|τ|2 π b × 2 μ / I 2 (2 T ) Jn ( 1 ) n=−∞ 1 μ+n 2 μ + n (b + i ) × sinh 2 πT . (41) 2T 2πT [b] FIG. 13. The normalized unsymmetrized finite frequency noise μ / →∞ ω / ω/ > At zero drive frequency and 1 , observing that S( ) I is plotted versus the dimensionless frequency 0 ∞ 2 μ> > =−∞ J (μ1/) → 1, we recover the result provided in (emission noise) at zero temperature and fixed 0and 0[see n n = . = . μ / = μ/ = Eq. (26) in Sec. III. We do not provide the plot at finite tem- Eq. (43)]. Here we set b 0 6, b 1 2, 1 2, 3, and I perature, which is less interesting than the zero-temperature is given by Eq. (24) by simply replacing μ by frequency . case. However, one has to mention that at high temperatures T max{μ, } the singularities are expected to smear out, and a thermal broadening appears. Consequently, the singu- B. Zero-frequency noise larities are restored at low temperatures. The time-averaged zero-frequency nonsymmetrized noise is obtained from Eq. (42) and is given by the following VI. FINITE AND ZERO-FREQUENCY NOISE anticommutator of vertex operators: IN THE ac REGIME T t A. Finite frequency noise 1 i V (t˜)dt˜ † S = dt dt e t {Aˆ t + t / , Aˆ t − t / } . (0) T ( 2) ( 2) The time-averaged phonon-assisted finite frequency non- 0 symmetrized noise is given by the Wigner transformation (45) defined as T At zero temperature and after the time integration in 1 τ τ ωτ S(ω) = dτ dτ S τ + ,τ − ei , (42) Eq. (45) we obtain the following expression for zero- T 2 2 0 frequency noise: where we introduced the “center of mass” τ = (t + t )/2 τ = − ∞ and “relative” t t time coordinates. The integrand is 2π ∝ D|τ|2 2 μ / |μ + |b−1. given by the current fluctuation correlator in the time domain, S(0) Jn ( 1 ) n (46) [b] namely, S(t, t ) = δJˆ(t )δJˆ(t ) , with δJˆ(t ) = Jˆ(t ) − Jˆ(t ) , n=−∞ where the average is taken with respect to the ground state of QH edges with free phonons. Equation (42) is nothing but This result coincides with Eq. (43) for finite frequency noise at the Fourier transform of S(t, t ) in terms of the relative time zero frequency. The normalized zero-frequency noise at zero coordinate τ . temperature is plotted in Fig. 14. The straightforward calculations give the following result for the zero-temperature case: +∞ 2π S(ω) ∝ D|τ|2 J2(μ /) [b] n 1 n=−∞ σ =∓ ×|ω + σ (μ + n)|b−1θ[ω + σ (μ + n)]. (43) The normalized finite frequency nonsymmetrized noise at zero temperature is plotted in Fig. 13. At finite temperature we get the following result for finite frequency nonsymmetrized noise: +∞ ω ∝ D|τ|2 π b 2 μ / S( ) (2 T ) Jn ( 1 ) n=−∞ σ =∓ 1 ω+σ (μ+n) 2 ω + σ (μ + n) b + i FIG. 14. The normalized unsymmetrized zero-frequency noise × exp 2 πT . 2T 2πT [b] S(0)/I is plotted versus the dimensionless dc component of bias μ/ > 0 at zero temperature and fixed >0[seeEq.(46)]. Here (44) we set b = 0.6, b = 1.2, μ1/ = 2, and I is given by Eq. (24)by At zero frequency we recover the result in the next section. simply replacing μ by frequency .

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For finite temperature the straightforward calculation of the the case of dc bias the ratio between shot noise and current integral over the time variable results in is independent of b and has the form S(0)/I = coth (μ/2T ). Thus, the current fluctuations satisfy a classical (Poissonian) +∞ shot noise form. This is related to uncorrelated tunneling S(0) ∝ 2D|τ|2(2πT )b J2(μ /) n 1 of electrons through the QPC. The result is identical to the n=−∞ case of noninteracting electrons in the Landauer-Buttiker ap- μ + 1 + μ+n 2 proach. Finally, we would like to point out that Eqs. (40), (41), n 2 (b i πT ) × cosh . (47) (46), and (47) can be combined into one expression, namely, T πT b ∞ 2 2 [ ] = 2 μ / μ + μ + ac n=−∞ Jn ( 1 ) dc( n ), where dc( n ) Here we can also mention that the result for finite frequency is given by Eqs. (24), (26), and (34), respectively. This is a noise given by Eq. (44)atω →+0 transforms into Eq. (47). general formula which applies to transport quantities, such as current, heat current, and shot noise under ac bias [46]. In conclusion, we have shown that the presence of electron- VII. CONCLUSION phonon interaction strongly modifies the analytical structures In this paper we have studied the impact of strong electron- of noise and current compared to the free-fermion case. As a acoustic-phonon interaction on QH edge state transport. future perspective, it is appropriate to investigate the tunneling We have discussed the case of filling factor ν = 1; thus, current and noise in two-QPC and resonant-level QH systems no electron-electron interaction influences transport proper- with Laughlin states in the presence of electron-phonon inter- ties. Using Wen’s effective low-energy theory of QH edge action, the periodic train of Lorentzian voltage pulses in the states and applying a BV transformation, we were able to context of levitonic physics [45]. take into account the electron-phonon interaction exactly. It was demonstrated that the equilibrium two-point correlation ACKNOWLEDGMENTS function splits into three independent correlation functions with different velocities. Namely, two of them correspond We are grateful to S. Groenendijk for a careful and critical to downstream modes, and the third one is related to the reading of the manuscript and T. L. Schmidt, T. Martin, and upstream mode, in agreement with Ref. [29]. The presence of C. Glattli for fruitful discussions. We acknowledge financial electron-phonon interaction strongly modifies the equilibrium support from the Fonds National de la Recherche Luxembourg propagator; that is, it destroys the Fermi liquid behavior under Grants ATTRACT 7556175 and INTER 11223315. in comparison with noninteracting electrons. It was already shown that dc Hall conductance of a single-branch QH edge APPENDIX A: EIGENVALUES OF THE with electron-phonon interaction is not varied and is given 2 CHARACTERISTIC EQUATION by quantum conductance, GH = e /h [29]. In this paper we state that according to the Wiedemann-Franz law, the In this Appendix we present the eigenvalues of Eq. (10) thermal Hall conductance is not modified as well, namely, from the main text. The three distinct real eigenvalues of this KH = TGH L0, where L0 is the free-fermion Lorentz number cubic equation are given by (see Appendix C). ⎛ We studied the tunneling current for a QPC in the lowest √ ⎜ ± ξ order of tunneling coupling under dc and ac biases. Per- 1⎜ 1 i 3 1 λ1,2 = ⎝α − ! turbative results were obtained for the arbitrary interaction 3 2 3 ξ + ξ 2 − ξ 3 parameter b, which depends on electron-phonon coupling 2 2 1 and Fermi and sound velocities. We observed that at high √ ! " ∓ temperatures μ/T 1 the tunneling current is proportional 1 i 3 3 2 3 − − ξ + ξ − ξ , (A1) to applied dc bias, I ∝ T b 2μ, which is the manifestation of 2 2 2 1 Ohmic behavior of the QPC. However, at low temperatures, ⎛ ⎞ − μ/T 1, we got a non-Ohmic behavior, I ∝ μb 1, specific ! ⎜ ξ ⎟ for a non-Fermi liquid. In contrast, in the case of ac drive at 1⎜ 1 3 2 3⎟ λ3 = ⎝α + ! + ξ2 + ξ − ξ ⎠, zero temperature we observed singularities in the tunneling 3 2 1 3 ξ + ξ 2 − ξ 3 current. These singularities strongly depend on the electron- 2 2 1 phonon interaction parameter b and are related by a new 2 3 2 energy scale, the drive frequency . One has to mention where ξ1 = α + 3, ξ2 = α − 9α + 27β , and α = vF /vs, that at high temperatures T max{μ, } the singularities are with β = vc/vs. These eigenvalues are important to get the expected to smear out and to reveal the thermal broadening. renormalized spectrum of phonons and plasmons, collective Consequently, at low temperatures the singularities are re- excitations in the QH edge state (see Fig. 2 in the main text). stored. In addition to current we studied the nonequilibrium finite APPENDIX B: CORRELATION FUNCTION AT frequency nonsymmetrized noise of tunneling current in dc ZERO TEMPERATURE and ac regimes. Here the correlations caused by electron- phonon interaction are responsible for algebraic singulari- In this Appendix we calculate the correlation func- ties, ω =∓μ, ω =∓(μ + n) at zero temperature, which tion of right-moving fermions G(x, t; xt ) at filling factor depend on b, the electron-phonon interaction parameter. In ν = 1 in the presence of electron-phonon interaction at zero

155422-9 EDVIN G. IDRISOV PHYSICAL REVIEW B 100, 155422 (2019) temperature. Using the bosonization technique, we can write Substituting the relation for bosonic operators in terms of BV- transformed new bosonic operators from Eq. (14), −iφˆ(x1,t1 ) iφˆ(x2,t2 ) M(x1t1;x2t2 ) G(x1t1; x2t2 ) ∝ e e =e , (B1) where, in the Gaussian approximation and for equal bosonic −iv1kt −iv2kt −iv3kt † aˆk (t ) = b11e αˆ k (t ) + b12e βˆk (t ) + b13e γˆ− (t ), autocorrelators, the function in the exponent is given by k (B4) 2 M(x1t1; x2t2 ) = φˆ(x1, t1)φˆ(x2, t2 ) − φˆ (x1, t1) . (B2) Next, using the definition of bosonic field in terms of creation into the above cross correlator, we obtain the following simple and annihilation operators of bosons from Eq. (1), the cross combination of the sum of three terms: correlator of bosonic fields takes the following form (zero modes are omitted): = 2 I + 2 I M(x1t1; x2t2 ) b11 1(x1t1; x2t2 ) b12 2(x1t1; x2t2 ) 2 (2π ) † ikx1−ik x2 2 φˆ(x , t )φˆ(x , t ) = [ aˆ (t )â (t ) e + b I3(x1t1; x2t2 ), (B5) 1 1 2 2 kkL2 k 1 k 2 13 kk>0

† − + + ikx1 ik x2 . aˆk (t1)ˆak (t2 ) e ] (B3) where

∞ −ηk e − − − − − + − I (x t ; x t ) = dk [eik(x1 x2 ) iv1k(t1 t2 ) − 1] 1 + f (v1 )(k) + [e ik(x1 x2 ) iv1k(t1 t2 ) − 1] f (v1 )(k), 1 1 1 2 2 k B B 0 ∞ −ηk e − − − − − + − I (x t ; x t ) = dk [eik(x1 x2 ) iv2k(t1 t2 ) − 1] 1 + f (v2 )(k) + [e ik(x1 x2 ) iv2k(t1 t2 ) − 1] f (v2 )(k), (B6) 2 1 1 2 2 k B B 0 ∞ −ηk e − + − − − − − I = ik(x1 x2 ) iv3k(t1 t2 ) − (v3 ) + ik(x1 x2 ) iv3k(t1 t2 ) − + (v3 ) . 3(x1t1; x2t2 ) dk [e 1] fB (k) [e 1][1 fB (k)] 0 k

Here we introduced the new notations v1 = λ1vs, v2 = λ2vs, v3 =−λ3vs, where λi, i = 1, 2, 3, are eigenvalues of the dynamical β − (vi ) = vik − 1 β = , , matrix; fB (e 1) is a bosonic distribution function, is the inverse temperature, and i 1 2 3. At zero temperature we have ∞ e−ηk η ik(x1−x2 )−iv1k(t1−t2 ) I1(x1t1; x2t2 ) = dk [e − 1] = ln , 0 k η − i[x1 − x2 − v1(t1 − t2 )] ∞ e−ηk η ik(x1−x2 )−iv2k(t1−t2 ) I2(x1t1; x2t2 ) = dk [e − 1] = ln , (B7) 0 k η − i[x1 − x2 − v2(t1 − t2 )] ∞ e−ηk η −ik(x1−x2 )−iv3k(t1−t2 ) I3(x1t1; x2t2 ) = dk [e − 1] = ln . 0 k η + i[x1 − x2 + v3(t1 − t2 )] Substituting Eq. (B7) into Eq. (B5) and then into Eq. (B1), we ﬁnally get the expression for the two-point correlation function in the form b2 b2 b2 iη 11 iη 12 −iη 13 G(x1, t1; x2, t2 ) ∝ . (B8) x1 − x2 − v1(t1 − t2 ) + iη x1 − x2 − v2(t1 − t2 ) + iη x1 − x2 + v3(t1 − t2 ) − iη

APPENDIX C: THERMAL HALL CONDUCTANCE can be expressed as [31] Here we calculate the thermal Hall conductance of the dk 1 π 2 k2 = (vi ) = B 2. chiral edge channel coupled to acoustic phonons. According nQi i(k) f (k) T (C2) 2π B v 6 h to Eqs. (15) and (16) of THE main text there are three modes i which propagate with velocities vi, i = 1, 2, 3, in the χi = The thermal Hall conductance is evaluated as the derivative of ±1 direction. The dispersion relation is given by Eq. (13) the current with respect to temperature [31], namely, of the main text and has the linear form i(k) = vik.At ∂ π 2 2 temperature T , each mode is characterized by the equilibrium IQ kB (v ) / − KH = = χQ T, (C3) bosonic distribution function f i (k) = (evik T − 1) 1. Thus, ∂T 3 h B the thermal current is given by [31] χ = χ = where the prefactor Q i i 1 is the difference be- 3 tween the number of upstream and downstream modes. There- 2 = χ , fore, the thermal KH and electrical GH = e /h Hall conduc- IQ ivinQi (C1) i=1 tances are related by the Wiedemann-Franz law as in the case of free electrons. Namely, KH /GH = TL0, where L0 = 3 2 where the energy density of mode i (restoring natural units) (π /3)(kB/e) is a free-electron Lorenz number.

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