Nonmonotonic Plasmon Dispersion in Strongly Interacting Coulomb

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find that other long-range potentials (e.g., 1/r2) also system. In the presence of spin or valley degrees of free- manifest similar finite-momentum nonmonotonic collec- dom, the instabilities in the spin or valley collective ex- tive mode dispersion for sufficiently strong interaction. citations do not affect the plasmon as long as the charge Thus, the predicted features in this work are not due to mode is decoupled from other collective degrees of free- the particular form of the 1/r potential. Possibly, these dom. In addition, the degeneracy due to spin and valley nonmonotonic features represent the generic signatures will effectively enhance the density-density interaction of the strongly interacting 1D systems with long-range which is typically subsumed in an overall multiplicative potentials. degeneracy factor N (N = 2 for spin degenerate sys- The rest of the paper is organized as follows: After tems) in the electron polarizability function. These com- introducing the model in Sec. II, we review the basic plications affect only the plasmon dispersion quantita- derivations of the plasmon dispersion in Sec. III. Then, tively and will be discussed if needed. We do not further we show the nonmonotonic dispersion and calculate the consider effects of spin or other degrees of freedom and specific heat in Sec. IV. In Sec V, we discuss the possibil- focus on the charge density excitations (i.e. plasmons) ity of observing our predicted nonmonotonicity in can- of the 1D system (sometimes these modes are known as didate materials (e.g., silicon quantum wire and twisted “holons”). bilayer graphene at sub-degree twist angles), explain potential experimental challenges, and conclude.

III. DERIVATION OF PLASMON DISPERSION II. MODEL The 1D interacting fermion Hamiltonian given by We are interested in the plasmon oscillation of 1D Eqs. (1) and (2) can be solved by bosonization [7, 8] interacting electrons. The simplest model for this is after linearizing the single-particle dispersion around the the 1D spinless fermion along the x-axis described by Fermi energy. In particular, the bosonic mode associated Hˆ = Hˆ0 + HÎ , where with the density fluctuation is precisely the plasmon. On the other hand, the random phase approximation (RPA) ∂2 gives exactly the same energy dispersion for plasmon ex- Hˆ = dx c†(x) x µ c(x), (1) 0 −2m − citations as that obtained from bosonization [6, 7, 12]. Z We will first review the derivation of 1D plasmon disper- 1 ′ ′ ′ HÎ = dxdx ρ(x)V (x x )ρ(x ). (2) sion with both bosonization and RPA calculations to set 2 − Z a context for our predicted maxon-roton feature in the In the above expressions, c is the fermionic annihila- plasmon dispersion. tion operator, m is the electron effective mass, µ is the chemical potential, ρ(x) = c†(x)c(x), and V (x x′), − the inter-particle potential, encodes the density-density A. Linearized theory: Coulomb Luttinger liquid interaction. We are particularly interested in the unscreened Coulomb interaction, characterized by V (x) = e2/(κ√x2 + d2) where e is the electron charge, κ is the The interacting theory in one dimension can be studied dielectric constant, and d is the “transverse size” (i.e. via the standard bosonization method [3, 13]. To incor- width of the quantum wire). Note that it is important to porate the interaction, we first linearize the band in the have the transverse (i.e. normal to the x-axis) dimension vicinity of the Fermi energy. The physical fermion can be approximated by c eikF xR + e−ikF xL, where R and d of the 1D system in the definition of the Coulomb in- ≈ teraction in order to avoid the well-known singularity of L denote the long-wavelength fermionic fields near the the 1D Coulomb coupling [6]. For any given system the right and left Fermi points respectively in the 1D nonin- ˆ precise value of d (of the order of the transverse width teracting Fermi surface. H0 given by Eq. (1) becomes of the 1D system) can be calculated microscopically [11]. The Fourier transform of the potential V (x) is given by ˆ † † 2 H0 vF dx R ( i∂xR) L ( i∂xL) , (3) ˜ 2e → − − − V (q)= κ K0( q d) where K0 is the zero-th order modi- Z fied Bessel function| | of the second kind. The momentum transfer due to this potential is relative to the inverse where vF = 2µ/m is the Fermi velocity with µ the width of the quantum wire (1/d), which determines the chemical potential equal to Fermi energy at zero tempera- momentum scale of the nonmonotonic maxon-roton be- ture. Since wep are interested in only the long-wavelength havior of the 1D plasmon dispersion to be described be- fluctuation (i.e. momenta smaller than kF , the Fermi low. momentum), the density operator in Eq. (2) can be ex- The spinless model given by Eqs (1) and (2) describes pressed as ρ ρ = R†R+L†L (where we ignore the 2k ≈ 0 F the charge degrees of freedom in an interacting 1D sys- and 4kF components) considering only small momenta. tem, which is of interest in the current work since plas- With these approximations, the interacting fermions can mons are the collective charge density excitations of the be mapped to a noninteracting bosonic model. This 3 bosonized Hamiltonian is v Hˆ = dx F (∂ φ)2 + (∂ θ)2 2π x x Z h i 1 ′ 1 ′ 1 ′ + dxdx ∂ θ(x) V (x x ) ∂ ′ θ(x ) 2 π x − π x Z (4) dq v(q) 1 = K(q)q2φ˜( q)φ˜(q)+ q2θ˜( q)θ˜(q) , 2π 2π − K(q) − Z (5) where φ and θ are the phase-like and plasmonic (phonon- like) bosonic fields respectively, v(q) is the scale- FIG. 1: Scale-dependent Luttinger parameter and Luttinger dependent velocity, and K(q) is the scale-dependent Lut- exponent. We plot the Luttinger parameter K(q) with dif- 2 tinger parameter (not to be confused with K0 in the ferent α = 2e . Inset: The Luttinger exponent, γ(q) = πvF κ 1D Coulomb potential, which is a modified Bessel func- [K(q) + 1/K(q) − 2] /8, is plotted. Both the K(q) and γ(q) tion). We can obtain the scale-dependent velocity and approach to the noninteracting limit for a sufficiently large Luttinger parameter by solving v(q)K(q) = vF and |q|d. Yellow line: α = 10; blue line: α = 50; green line: V˜ (q) α = 100; brown line: α = 250; red line: α = 500. Note v(q)/K(q)=1+ π . Straightforwardly, that the momentum is relative to 1/d rather than the Fermi wavevector kF . V˜ (q) v(q)=vF 1+ , (6) s vF π 1 states in the twisted bilayer graphene) since the curva- K(q)= . (7) ˜ ture is basically absent. 1+ V (q) vF π Conventionally, for a quadratic single-particle disper- q Since the Luttinger parameter depends on the mo- sion, only the long-wavelength limit [i.e. Eq. (8)] is con- mentum, we might be interested in the scale- sidered in the literature discussing 1D plasmons and Lut- dependent Luttinger exponent as defined by γ(q) = tinger liquid holons. Going beyond the long-wavelength [K(q)+1/K(q) 2] /8 0. γ(q) = 0 suggests K(q)=1 limit where terms higher than leading order in momenta (non-interacting− limit).≥ The scale-dependent K(q) and are kept in the theory, we need to worry about the finite γ(q) are plotted in Fig. 1. band curvature (e.g., nonlinear Luttinger liquid theory The equation of motion of the plasmonic boson θ can [14]). In the next section, we turn to a complementary be obtained by integrating over the phase boson φ in the approach that incorporates the band curvature exactly. partition function. The plasmonic dispersion [8] is

V˜ (q) 2e2 B. Random phase approximation ωp(q)= vF q 1+ = vF q 1+ K0( q d). | |s vF π | |s πvF κ | | (8) In addition to the bosonization approach, we study the problem using the RPA. In fact, the 1D plasmon dis- Now, we express the above equation in dimensionless persion was originally computed using the RPA a long units as follows: time ago [6], and much theoretical work has been done on 1D plasmon dispersion using the RPA in the context ω˜ (˜q)=˜q 1+ αK (˜q), (9) p 0 of studying collective modes in semiconductor quantum 2 d p 2e wires [12, 15, 16]. In contrast to the band linearization whereω ˜p(˜q) = ωp(q) ,q ˜ = q d, and α = . vF πvF κ approximation in bosonization, we keep the full quadratic The long-wavelength property is| distinct| from the con- single-particle dispersion and treat the interaction eﬀect ventional (short-range) Luttinger liquids. Forq ˜ 1, ≪ via the summation of the inﬁnite series of the “bubble” ω˜p(˜q) q˜ ln(2/q˜). Forq ˜ 1,ω ˜p(˜q) q˜ the same diagrams. We emphasize that the 1D RPA plasmon mode as the≈ noninteracting dispersion.≫ For fermions≈ with ad- p reproduces the exact bosonization long-wavelength re- ditional degrees of freedom (e.g., spin and valley), the sult, but the RPA enables us to go beyond the long- dimensionless interacting parameter α will be enhanced wavelength linearized limit. by the degeneracy factor N. (We take N = 1 above; The standard dynamic dielectric function in RPA is for spinful fermions, N = 2). Another important remark given by is that the Luttinger liquid approach can apply beyond the strict long-wavelength limit in the 1D Dirac system ǫ(ω, q)=1 V˜ (q)Π (ω, q), (10) (e.g., edges of topological insulators and 1D domain-wall − 0 4 where the irreducible polarization function [6]

m ω2 ω2 Π (ω, q)= ln − − , (11) 0 2π q ω2 ω2 | | − + 2 q vF 2 and ω± = vF q = vF q q . The plasmon | |± 2m | |± 2kF corresponds to the zero of the dielectric function ǫ(ω, q) [given by Eq. (10)]. The plasmon dispersion in RPA is then

1 A(q)ω2 ω2 2 ω (q)= + − − , (12) p A(q) 1 − FIG. 2: Plasmon dispersion in the Coulomb Luttinger liq- 2|q|π uids. The plasmon dispersion given by Eq. (9) with diﬀer- where A(q) = exp ˜ . Equation (12) is consis- 2 kF V (q)/vF 2e dωp(q) ent α = . Inset: The plasmon velocity, vp ≡ , tent with the resultsh in Ref.i [6] when setting the spin πvF κ dq degeneracy factor N = 1 (spinless case). In the limit is plotted. The zeros of vp identify the maxon and roton wavevectors. Yellow line: α = 10; blue line: α = 50; green q /k 1, we expand ω (q) to O(q2/k2 ) and recover | | F ≪ p F line: α = 100; brown line: α = 250; red line: α = 500. The the Luttinger liquid result [8] given by Eq. (8). The plas- black dashed line (α = 0) corresponds to a noninteracting 1D mon dispersion in the large q d limit [A(q) 1] is pre- | | ≫ linearly dispersing fermion and sets the upper bound of the cisely the upper bound of the particle-hole continuum. particle-hole continuum. The plasmon dispersion develops a Note that the long-wavelength limit in the formal many- roton like minimum when α > 33 (estimated numerically). body theory corresponds to q k whereas the 1D plas- Note that the momentum is relative to 1/d rather than the ≪ F mon dispersion depends also on an independent dimen- Fermi wavevector kF . sionless parameter qd in addition to the parameter q/kF . Thus, the 1D plasmon dispersion depends on two independent length parameters: d and 1/n, where n is the 1D electron density in the system deﬁning kF = πn/N.

IV. STRONGLY INTERACTING COULOMB LUTTINGER LIQUID

Most of the low-energy properties of the Coulomb Lut- tinger liquids have been studied systematically [7–9]. In particular, the ground state develops a Wigner-crystal- like quasi long-range order with slowly spatially decaying (slower than any power-law) 4kF oscillations regardless of the interaction strength [8]. Therefore, the Coulomb FIG. 3: Plasmon dispersion of 1D Coulomb interacting 2 Luttinger liquids cannot in general be characterized by fermions with RPA. We plot Eq. (12) with diﬀerent α = 2e . πvF κ a single exponent as in the short-range case, and the Yellow line: α = 10; blue line: α = 50; green line: α = 100; Luttinger parameter is now scale-dependent [9]. Here, brown line: α = 250; red line: α = 500. For |q|d > 6, we show that “strongly” interacting Coulomb Luttinger the plasmon dispersion merges with the upper bound of the liquids can develop a nonmonotonic plasmon dispersion. particle-hole continuum. We have set kF d = 5 for all of the Concomitantly, the speciﬁc heat in the strongly interact- curves. Note that the momentum is relative to 1/d rather ing regime develops a low-temperature suppression aris- than the Fermi wavevector kF . ing directly from the plasmon nonmonotonicity.

and/or by changing the substrate to modify the dielec- A. Non-monotonic plasmonic dispersion tric background. Since vF = πn/(Nm), one can enhance α simply by decreasing the 1D carrier density through The plasmon dispersion derived in the previous section external gating. has been verified in a number of experiments [4, 17, 18]. A natural question arises: can a larger α, correspond- We can simply take Eq. (9) and insert the material- ing to a stronger interacting system, give a qualitatively different plasmon dispersion? To answer the above ques- dependent interaction parameter α, which depends only 2 on the background dielectric constant (κ) and the bare tion, we examine Eq. (9) by varying α = 2e . In Fig. 2, πvF κ Fermi velocity. In principle, therefore, α can be tuned the calculated plasmon dispersion develops an enhanced by changing the electron density (e.g., through gating) peak and a local minimum for a sufficiently large α > αc, 5

FIG. 5: Diagram of qualitatively distinct plasmon dispersion regimes using RPA calculations. The black dots represent the interaction threshold (αc) for realizing nonmonotonic plasmon dispersion given by Eq. (12). For α > αc (orange shaded regime), the maxon-roton feature is manifest. The value of αc decreases when kF d increases. The value of αc using RPA saturates at ∼ 33 (the αc obtained from the Luttinger liquid dispersion) for kF d ≫ 1.

tors are around 2/d (i.e. where the plasmon dispersion nonmonotonicity occurs). Note that the nonmonotonicity disappears for α < αc, explaining why earlier work missed the maxon-roton structure in the plasmon dispersion since the regime of large α (small κ and small vF ) was never studied before in the context of 1D plasmon dispersion theories. An important question is the stability of this predicted dispersion in ﬁnite-wavevector Coulomb Luttinger liquid calculations, especially the robustness against the ﬁnite curvature. To answer this, we check Eq. (12) which follows from the full quadratic band using the full RPA theory. In Figs. 3 and 4, we show that the nonmonotonic plasmon dispersion persists but with a value of αc

FIG. 4: Plasmon dispersion using RPA with varying kF d. We strongly depending on kF d. In Fig. 5, we show that αc plot Eq. (12) with various α [(a) 50, (b) 100, and (c) 200] and monotonically decreases as a function of kF d and satu- different values of k d. Yellow dashed line: k d = 0.5; blue rates at 33 [the same αc as obtained from the Luttinger F F ∼ dashed line: kF d = 1; green dashed line: kF d = 2; brown liquid dispersion in Eq. (9)] for k d 1. We conclude F ≫ dashed line: kF d = 5; red dashed line: kF d = 50. Note that the nonmonotonic 1D plasmon dispersion does sur- that the momentum is relative to 1/d rather than the Fermi vive in the presence of the band curvature effects, and wavevector kF . both the standard bosonized Luttinger liquid theory and the full RPA give the maxon-roton plasmon dispersion at finite momenta (q 2/d) provided the Coulomb coupling exceeds a critical∼ value. As presented in Figs. 4 and 5, where α is the critical value of the dimensionless in- c a larger value of k d indeed enhances the maxon-roton teraction parameter for producing this nonmonotonicity. F feature in the full RPA plasmon dispersion. Thus, the The nonmonotonic dispersion is reminiscent of the fa- maxon-roton feature, arising in the strongly interacting mous roton-maxon dispersion in liquid helium. Large system, is not an artifact of the linearized Luttinger liq- α corresponds to either a strong interaction (i.e. small uid theory as it exists in the full RPA theory too except κ) or a small Fermi velocity (low carrier density). To that in the full RPA, the nonmonotonicity depends both identify α , we compute the derivative of the plasmon c on the dimensionless Coulomb coupling strength (α) and dispersion with varying α. The condition of getting zero the dimensionless Fermi momentum (kF d). plasmon velocity [vp(q)= dωp(q)/dq = 0] corresponds to The maxon (peak) and roton (local minimum) α q˜K (˜q) K (˜q) = 1. We find that α 33 numer- 2 1 − 0 c ≈ wavevectors depend on the width of the quantum wire icallyh using the linearizedi Luttinger liquid theory [i.e. (d) instead of the Fermi wavevector. With a wide quan- Eq. (9)]. In addition, the maxon and roton wavevec- tum wire (i.e. large d), the maxon and roton wavevectors 6 can be much smaller than the Fermi wavevector. This suggests that the nonmonotonicity in the plasmon dispersion comes from the “low-energy” properties of the Coulomb Luttinger liquids since it can occur already for q k . Mathematically, the long-wavelength 1D plas- ≪ F mon dispersion ωp(q) vF q ln[2/(qd)] already gives the nonmonotonic dispersion,≈ clearly showing that the maxon-roton feature is inherentp in the long-range nature of the Coulomb coupling which is inherently dependent on the cutoff scale “d.” Moreover, the nonmonotonic part of the plasmon dispersion (especially the maxon peak) is always well above the 1D particle-hole continuum and is undamped by particle-hole pair creations. We therefore expect to observe a sharp essentially delta-function-like FIG. 6: Specific heat of the Coulomb Luttinger liquids with spectral peak in the dynamical structure factor corre- various values of α. The low-temperature suppression in the sponding to the maxon-roton dispersion feature which specific heat depends on the value of α and therefore can should clearly show up in inelastic scattering experiments identify the regime realizing nonmonotonic plasmon disper- such as light scattering and phtoemission spectroscopies. sion. Black dashed line: α = 0; yellow solid line: α = 10; An important remark is that the scale-dependent Lut- blue solid line: α = 50; green solid line: α = 100; brown solid tinger parameter K(q) is always monotonically increas- line: α = 250; red solid line: α = 500. ing to 1 with increasing q (see Fig. 1 with no nonmonotonicity whatsoever). This implies that the nonmonotonic plasmon dispersion does not change the well- curves roughly follow a linear temperature dependence, known zero-temperature Wigner-crystal-like phase of the the prefactor gets smaller for larger α. We find that the 1D Coulomb systems [8, 19]. Nevertheless, we might ex- low-temperature specific heat is significantly reduced for pect that the nonmonotonicity of the plasmon dispersion α αc. Such a reduction of the low-temperature spe- ≫ contributes to the finite-temperature properties where cific heat is caused by the small density of states of the higher energy excitations may contribute. In the next low-energy plasmon for large α where the maxon-roton section, we turn to the specific heat which provides a dispersion feature arises. It is important to note that the way to identify the strongly interacting regime (the same low-temperature specific heat cannot be described by an regime of α that realizes the nonmonotonic plasmon dis- exponential Arrhenius scaling but appears to be close to persion). linear in temperature. This is consistent with the gapless nature of the plasmon in one dimension. However, the precise low-temperature form is not known analytically B. Specific heat except that we find that it is significantly suppressed for strong interaction. This strong suppression for large α is When the plasmon oscillation develops a nonmono- directly connected with the nonmonotonic maxon-roton tonic dispersion, the density of states of the plasmon di- plasmon dispersion. verges at the maxon and roton energies. In addition, the The results in this section are only on the charge sector dispersion ωp(q 0) q ln[2/(qd)] implies a vanishing contribution to the specific heat. There may be contribu- low-energy density→ of∼ states. These singular features will tions to the specific heat from spinons and other excita- be reflected in the thermodynamicp quantities like specific tions outside the charge sector. Nevertheless, we expect a heat which we discuss in this section. large suppression of the low-temperature specific heat for The specific heat can be computed straightforwardly strongly interacting Coulomb Luttinger liquids as shown based on the plasmon dispersion in Eq. (9). The expres- in Fig. 6 since the contribution from plasmons will indeed sion for the specific heat is be significantly suppressed at strong coupling.

vF 2 ∞ 2 ω˜p(˜q) vF 1 [˜ωp(˜q)] e Td CV = dq˜ , (13) v 2 T d πd 0 F ω˜ (˜q) V. CONCLUSION Z e Td p 1 − h i whereω ˜p(q) is the dimensionless plasmon dispersion in We have theoretically demonstrated that a Coulomb Eq. (9). The specific heat results given by Eq. (13) are Luttinger liquid can manifest a finite-momentum non- plotted in Fig. 6. In the high-temperature limit, the monotonic 1D plasmon dispersion in the strongly inter- πT specific heat CV , which is consistent with what acting regime, preserving still the low-energy charge col- ≈ 3vF is expected for the noninteracting 1D spinless fermions lective mode behavior [i.e. ωp(q 0) q ln[2/(qd)]] [3]. In the inset of Fig. 6, the low-temperature spe- at long wavelengths. We emphasize→ that∼ our results are cific heat shows a suppression which depends strongly model-independent: both Luttinger liquid theoryp (for lin- on α. Although all of the low-temperature specific heat earized energy dispersion) and the RPA (for parabolic 7 energy dispersion) give the nonmonotonic plasmon dis- collective mode will manifest a charge density oscillation persion at finite wavenumbers. We also show that the reflecting the total charge density, and the plasmon would specific heat of the Coulomb Luttinger liquids depends remain 1D character as long as the inter-level transitions strongly on the interaction strength (α). In particular, are suppressed. A complete theory including the mul- the magnitude of the low-temperature specific heat sup- tilevel situation is well outside the scope of the current pression can identify the strongly interacting regime, the work and should follow the formalism of Refs. [16, 20, 21]. same regime realizing the nonmonotonic plasmon disper- Our current work only points out that in the strongly in- sion. teracting (i.e. α 1) situation, the charge collective ≫ Where to look for the nonmonotonicity in the 1D plas- bosonic mode (i.e. 1D plasmon) in a Coulomb Luttinger mon dispersion is an important question, which we can liquid develops a nonmonotonic energy-momentum dispersion. The actual experimental manifestation of the answer only partially. One needs a large α 1/(vF κ) ∼ physics discussed in this work may be more likely in a (since α > αc is necessary) and a large d (since q 1/d is necessary). These two conditions are in principle∼ in- lattice system (rather than in a doped system) where dependent since the experimental 1D system has four in- the Fermi velocity and the Fermi momentum may not be dependent physical parameters: n (carrier density) and directly connected with the Fermi velocity arising from m (effective mass) determining the Fermi velocity, κ (the band dispersion (and can be very small in flat bands) background dielectric constant) determining the strength and the Fermi momentum arises from the band filling, thus enabling both α 1 and k d> 1 conditions to be of Coulomb coupling, and d (the transverse width) de- ≫ F termining the cut off for the Coulomb divergence. We simultaneously satisfied. The complications stated above explain why the plasmon nonmonotonicity has not been emphasize that, as long as q kF (which is allowed ≪ observed in literature. The ideal system manifesting the since kF and d are independent physical parameters), our theory (both RPA and Luttinger liquid theory) are nonmonotonicity is a 1D system where the bare band dis- essentially exact, and hence, our predictions are inde- persion is linear to very large momenta (i.e. very small pendent of our approximation schemes. The ideal 1D band curvature) so that the complication arising from system should have very low n (and not too small an m) kF d> 1 is simply not relevant. In this sense, graphene- related systems (carbon nanotubes, graphene nanorib- so that vF is small along with a rather small κ so that the Coulomb coupling is strong. In addition, d should bons, twisted bilayer graphene, etc.) may turn out to not be too small or large, so that the characteristic wave be the ideal place to look for the maxon-roton physics number, q 1/d, where the nonmonotonicity occurs is predicted in our work. ∼ neither too large nor too small. We estimate 1D quan- Interestingly, the recent experiments in the twisted bi- tum wires made of silicon with a low carrier density to be layer graphene [22–26] may provide a new platform for the possible system for the observation of our predicted investigating the plasmon nonmonotonicity. When the maxon-roton feature, but other 1D systems should be twist angle is smaller than 1 degree, the 1D domain-wall tunable to the interesting regime since the number of in- structure (which separates the effectively AB and BA dependent tunable parameters (n, m, d, κ) is twice the stacking regions) is manifest [24–27]. It is important to number of conditions (α > αc and q> 1/d) necessary for emphasize that the 1D domain-wall states, which can be the predicted physics to occur. well described by the 1D Dirac fermion with linearized One problem to worry about is the constraint kF d> 1 dispersion [i.e. Eq. (3)], are qualitatively different from which also seems to be crucial for the manifestation of the the conventional k2 dispersing band. Thus, the plasmon maxon-roton nonmonotonicity in k2 dispersing fermion dispersion from bosonization [i.e. Eq. (8) which is ig- bands [i.e. Eq. (1)]. In Fig. 4, we show the RPA cal- norant of kF ] captures the plasmon in such 1D domain- culated 1D plasmon dispersion for α = 50, 100, 200 and wall states. The domain-wall states contain four counter- kF d = 0.5, 1, 2, 5, 50. As is clear from these plots, propagating helical pairs, arising as the boundary sep- the nonmonotonicity is favored when kF d > 1 as well arating quantum valley Hall state in the AB (winding as α 1. The interaction threshold α (the minimum number ν = 1) and BA (ν = 1) staking registries ≫ c − value of α realizing nonmonotonicity) versus kF d is plot- [27, 28]. The width of the domain-wall states (i.e. d) ted in Fig. 5. We show that α decreases as a func- is estimated to 2 3nm (based on lattice relaxed calcu- c − tion of kF d and saturates at the linearized Luttinger liq- lations in [29]). The nonmonotonic plasmon dispersion uid value (i.e. α 33) for k d 1. Unfortunately, may be observed in the twisted bilayer graphene close to c ≈ F ≫ kF d > 1 and α 1 are somewhat mutually exclusive the second or the third magic angles [30, 31] because the in doped parabolic≫ 1D electron systems such as semicon- domain-wall state velocity in the miniband is significantly ductor quantum wires since v = k /m, and α 1/v . quenched (i.e. large enhancement in α). Although the F F ∼ F Thus, a small vF (to make α large) typically implies a characteristic wave number for nonmonotonicity is con- small k as well which makes it difficult to satisfy the siderably large ( 1/d), the long-wavelength part (i.e. F ∼ condition kF d> 1. In addition, kF d> 1 condition is in smaller than 1/d) of plasmon dispersion is boosted up conflict with the system being in the strictly 1D limit as by a factor √α (with α 1), enabling a large energy the higher quantized levels in the system may become rel- separation between plasmon≫ dispersion and the particle- evant. Even in such a situation, however, the main charge hole continuum. Such a system with a large energy mis- 8

magic angle) are still openly debated [30, 31, 33]. Never- theless, the nonmonotonic plasmon dispersion predicted in this work does not rely on the magic angle fundamen- tally as long as the domain-wall velocity is small enough and the 1D approximation applies. Theoretically, we can ask if a different interaction potential (but still long-ranged) also realizes the nonmonotonic dispersion. The answer is in the affirmative. The same analysis as in the current work for other long- range potentials manifests similar nonmonotonicity in the collective mode dispersion in the strong coupling regime, showing our finding of the maxon-roton feature to be universal for 1D long-range interacting systems. For example, we check a power-law potential V (x) = 2 2 n Cn/(√x + d ) for n > 0 and find nonmonotonic collective mode dispersion for a sufficiently large Cn. We plot the plasmon dispersion for n = 2 in Fig. 7(a). For n = 2, the model can be viewed as the continuum ver- sion of the Haldane-Shastry model [34, 35] which can be solved by the Bethe ansatz. In addition, we find that a 1/[d2 sinh2(x/d)+ d2] potential, a non-power-law potential related to the Calogero-Sutherland model [36, 37], also can develop a nonmonotonic dispersion as plotted in FIG. 7: Plasmon dispersion of 1D fermions with dif- Fig. 7(b). The existence of the nonmonotonic dispersion ferent long-range potentials. (a) Inverse square potential, is indeed not particular to the 1/r Coulomb interaction 2 2 V (x) = C2/(x + d ). The dispersion is given by ωpd/vF = and should exist in other systems with long-range interactions (e.g., trapped ions [38]). It is possible that the q˜ 1+ C˜ q˜1/2K (˜q), where the dimensionless strength C˜ = q 2 1/2 2 nonmonotonic maxon-roton charge collective mode dis- C2√2π , K is the modified Bessel function of the second πvF d 1/2 persion is indeed a universal feature of all strongly inter- kind, andq ˜ = |q|d. The plasmon dispersion develops a maxon- acting 1D models with long-range potentials of any type ˜ roton feature when C2 > 33 (estimated numerically). (b) although it is desirable to establish this speculation using 2 2 V (x) = B2/ d sinh(x/d)+ d . The dispersion is given by exact techniques such as the Bethe ansatz. ˜ 1/2 ωpd/vF =q ˜q1+ B2q˜ csch(˜qπ/2), where the dimension- B2π less strength B˜2 = . The plasmon dispersion develops πvF d a maxon-roton feature when B˜2 > 21 (estimated numeri- Acknowledgments cally). Black dashed line: C2 = 0(B2 = 0); yellow line: C2 =10 (B2 = 10); blue line: C2 =50 (B2 = 50); green line: C2 = 100 (B2 = 100); brown line: C2 = 250 (B2 = 250); red We thank Shiang Fang, An´ıbal Iucci, Rahul Nandk- line: C2 = 500 (B2 = 500). All the curves are calculated via ishore, and Micheal Schecter for useful discussions. In Luttinger liquid approximation. particular, we thank Rahul Nandkishore for his insight- ful inputs in the early stage of this work. This work is supported by the Laboratory for Physical Sciences (Y- match, resulting in an undamped plasmon dispersion, is .Z.C. and S.D.S.) and by the Army Research Office under very similar to the recent study on the surface plasmon in Grant Number W911NF-17-1-0482 (Y-Z.C.). The views twisted bilayer graphene at a magic angle [32] although it and conclusions contained in this document are those of approaches the problem from the two-dimensional limit the authors and should not be interpreted as represent- rather than from the 1D limit (this work). We fur- ing the official policies, either expressed or implied, of the ther predict that the specific heat has a strong low- Army Research Office or the U.S. Government. The U.S. temperature suppression in the same regime manifest- Government is authorized to reproduce and distribute ing the plasmon dispersion nonmonotonicity. The precise reprints for Government purposes notwithstanding any values of the magic angles lower than 1 degree (the first copyright notation herein.

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