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Therefore, these molecules do not realize a notice- able diode effect in a steady state solid-molecule-solid configuration when constant temperatures (TH and TC) are maintained at the boundaries15. Pump-probe tran- sient spectroscopy experiments demonstrated unidirec- tional vibrational energy flow between different chemical groups (e.g., nitro and phenyl)42,43; corresponding ob- servations of steady state asymmetric heat flow through molecules are still missing44. Can harmonic systems support the diode effect? In this paper, our goal is to revisit the problem of steady state heat transfer in asymmetric harmonic junctions and make clear the conditions for the realization of a ther- mal diode effect. In our model all components are har- FIG. 1. (a) N-particle chain connecting two heat baths (mod- monic: the reservoirs, representing e.g. solids, the chain eled by Langevin thermostats), hot and cold, with bead 1 con- (molecule), and their couplings. Furthermore, we do not nected to the hot bath and bead N coupled to the cold one. In effectively include anharmonicity by making parameters this example, N = 2. (b)-(c) N-site chain made of NH + NC temperature dependent. As we had just discussed, micro- exterior beads coupled to Langevin heat baths and NI parti- cles in the central, interior zone. The imposed temperature at scopic harmonic chains that bridge two harmonic solids, the edges may be homogeneous as in (b), or inhomogeneous a heat source and a heat drain at constant temperatures as in (c), the latter potentially realizing a TGHO diode. In TH and TC, respectively, as depicted in 1(a)-(b), can- (b)-(c), we used N = 8 total number of beads. not act like a diode irrespective of structural asymmetry. However, once we modify the boundary condition as we show in Fig. 1(c) and impose thermal gradients in the II. MODEL AND METHOD: LINEAR CHAIN contact region, the junction can rectify heat due to the COUPLED TO HEAT BATHS (multi-affinity) boundary conditions, with particles di- rectly coupled to different baths. A. Model We exemplify this scenario, referred to as the temperature-gradient harmonic oscillator (TGHO) chain We focus on a 1D harmonic oscillator chain with a to- in Fig. 1(c). The hot solid is divided into several regions tal of N beads. The chain is coupled at its to edges to H H with externally controlled temperatures, T1 > T2 > two thermostats, also referred to as solids. In simulations H T3 . Similarly, the colder region may be divided into of heat transport through solid-molecule-solid junctions, domains with externally-controlled temperatures. This typically, rather than including the solids’ atoms explic- setup can be realized experimentally by controlling lo- itly, they are emulated through Langevin baths to which cal temperatures (as in trapped-ions chain in optical the first and last atoms of the molecule are attached, see 46 lattices ), or computationally, as a mean to introduce Fig. 1(a). This setup was considered in numerous com- thermal gradients in structures, the result of genuine in- putational studies, see e.g.4,15,27, and since the system as elastic scatterings. a whole is microscopically harmonic, it cannot support the diode effect. Our analysis is performed using formally-exact expres- sions for the heat current based on the quantum Langevin Let us now consider a more complex picture of a junc- equation4. Both classical and quantum harmonic diodes tion with several beads on each side (NH , NC) each are demonstrated, with quantum effects leading to an im- attached to an independent Langevin noise term. The proved performance of the TGHO diodes. Furthermore, NI interior particles are not thermostated. For exam- we describe a unique, purely-quantum TGHO diode, ple, in Fig. 1(b)-(c) we display an N = 8-bead chain which does not have a classical analogue. As for clas- where atoms 1, 2 and 3 coupled to hot baths, while beads sical diodes, we perform classical molecular dynamics 6,7, and 8 connected to colder reservoirs. We can think (MD) simulations of heat flow in anharmonic junctions to about this scenario in two different ways: We may re- demonstrate the extent of the diode effect under explicit gard all N beads as part of the molecular system, with anharmonicity in comparison to the TGHO diode. the heater and sink reservoirs (implemented via Langevin noise) acting on several edge sites. Alternatively, we can Altogether, in this work we: (i) Derive conditions picture this setup as a molecule made of the NI interior for realizing a new type of thermal diodes, the TGHO beads only (4 and 5), with the modelling of the thermal diode based on structural asymmetry and inhomogeneous reservoirs enriched: The solids are described by NH and temperature boundary conditions, (ii) identify a purely- NC physical beads, each connected to an independent quantum TGHO diode, (iii) make clear conditions for Langevin bath. In fact, this latter approach has been realizing thermal diodes in either genuine or effective har- adopted in molecular dynamics simulations of thermal monic models. conductance of nanoscale systems. It allows to engineer 3 a nontrivial phonon spectral function within a standard The steady state heat current can be evaluated inside (white noise) Langevin simulation method47. the chain by calculating heat exchange between beads, What about the temperatures imposed at the bound- or at the contact region with each bath. Using the latter aries? We consider two cases: (i) The temperature is approach, the classical (C) heat current from bath l to 4 homogeneous at the edges, TH = T1,2,3 and TC = T6,7,8. its attached bead is (kB ≡ 1, ~ ≡ 1), That is, beads 1, 2, and 3 are coupled to three inde- ∞ 2 pendent Langevin baths, but each is maintained at the C ω 2 J = γlγm dω |(G(ω))l,m| (Tl − Tm). (3) same temperature (and similarly for the cold side). This l π m Z−∞ scenario is depicted in Fig. 1(b). (ii) A temperature pro- X file is implemented at the edges: beads 1, 2, and 3 are The summation is done over every thermostat. In what coupled to Langevin baths with a thermal gradient such follows, we introduce the compact notation that the temperature of the attached baths follow the ∞ 2 ω 2 trend T1 > T2 > T3 > T4 > T5 > T6, see Fig. 1(c). It M ≡ γ γ dω |(G(ω)) | , (4) lm l m π l,m is not required that all temperatures vary; at minimum Z−∞ we require two affinities (three baths of different tem- and write down J C = M (T − T ). peratures). We refer to this scenario as the temperature l m lm l m It can be shown that Eq. (3) generalizes in the quan- gradient harmonic-oscillator chain. tum (Q) case to4 P In what follows, we show that these two cases are fun- damentally distinct. In the first setup, Fig. 1(b), a diode ∞ 3 Q ω 2 cannot J = γlγm dω |(G(ω))l,m| [nl(ω) − nm(ω)], effect show up even under structural asymmetries; l π m −∞ remember that we work with harmonic oscillators. In X Z contrast, in the second scenario, Fig. 1(c), a diode effect (5) develops in both the classical and quantum regimes when ω/Tl −1 the gradients are distinct and structural asymmetry is in- with nl(ω) = [e −1] , the Bose-Einstein distribution G troduced. Moreover, we show that in a certain setup, a function of bath l of temperature Tl. Here, (ω) is a G−1 TGHO chain can support a purely-quantum diode—with symmetric matrix. The matrix (ω) for the five-site no corresponding classical analogue. model that we simulate below is given in Appendix A. To calculate the net heat current, we separate the heat baths into two groups, NH heat sources placed to the left B. Langevin equation formalism of the interior region, and NC heat sinks at the other side. The total input heat power is

We write down the classical Hamiltonian and corre- NH sponding classical equations of motion (EOM); a quan- J = Jl, (6) tum description based on Heisenberg EOM directly l=1 X follows4, and it equals the total output heat current at the colder N N+1 baths. p2 1 H = i + k (x − x − a)2. (1) We now reiterate that a thermal diode effect can- 2m 2 i−1 i i−1 i=1 i i=1 not appear in harmonic chains coupled to heat baths X X at two different temperatures (single affinity setup). If Here, x0 and xN+1 are fixed, setting the boundaries. a is NH beads are coupled to heat baths at TH and sim- the equilibrium distance between nearest-neighbor sites. ilarly, NC beads are attached to reservoirs at temper- At this stage, we assume that every particle i is cou- ature TC , the net quantum heat current is given by 3 pled to an independent heat bath. This coupling is Q ω 2 J = γlγm dω |(G(ω))l,m| [nH (ω) − incorporated using the Langevin equation with a fric- l∈NH m∈NC π n (ω)]. This expression is symmetric under the exchange tion constant γ and stochastic forces ξ (t) obeying the C R i i of temperaturesP P even if long range interactions are in- fluctuation-dissipation relation associated with exchang- G ′ cluded so that (ω) is a full matrix. Thus, this setup ing energy with a heat bath, hξi(t)ξi′ (t )i = 2Tiγiδ(t − ′ cannot support a diode effect. The multi-affinity scenario t )δ ′ . In the model for the diode below, we specify the i,i is discussed in the next section. interior region (which is not thermostated) by setting its friction constants to zero. However, the TGHO effect is generic and can be discussed even when every bead is attached to a thermostat. The classical EOM for the displacements are III. TGHO DIODES

mix¨i = −ki−1(xi − xi−1 − a)+ ki(xi+1 − xi − a) In this Section, we describe the principles behind the

− γivi + ξi(t), (2) TGHO diode. We begin by exemplifying this effect in an N = 5-bead chain depicted in Fig. 2, then we generalize with vi as the velocity of the ith particle. the discussion to longer systems. As a case study, we set 4

we get (Appendix A):

2 ∞ 2 2 γ |k1k2k3 −ω + iγω + k4 + k5 | M = dωω2 14 π |det G−1|2 Z−∞
γ2 ∞ |k k k (−ω2 + iγω + k + k )|2 M = dωω2 2 3 4 0 1 . 25 π |det G−1|2 Z−∞ (10) Therefore, asymmetry in the central zone (see definitions in Fig. 2), in the form k2 6= k3 cannot lead to the re- quired asymmetry M14 6= M25, since these terms are not sensitive to the asymmetry. For the diode effect to hold, structural asymmetry must be included in the ther- mostated zones. For example, it could be introduced in FIG. 2. A thermal rectifier based on an N = 5-bead har- the form k1 = k0 6= k4 = k5. In appendix A we consider monic chain. Two beads at the boundaries are considered chains of arbitrary size NI , with NH = NC = 2 and prove part of the solids, and they directly exchange energy with that structural asymmetry must be introduced within the Langevin thermostats. (a) In the forward direction we set thermostated zones to realize a diode. T1 >T2 >T4 >T5 and calculate the total heat input J from the baths attached to sites 1 and 2. (b) In the backward di- Furthermore, in a chain of length N with NB beads in rection we interchange the temperatures such that bead 1 (2) each thermostated zone, is now attached to a thermal bath at temperature T5 (T4), NB NB and similarly for the other half. In this case we calculate the C total heat input J˜ from the hot baths, attached now to beads J = (Ti − TN+1−j )Mi,N+1−j i=1 j=1 4 and 5. X X NB NB C J˜ = (TN+1−i − Tj )Mi,N+1−j . (11) i=1 j=1 NI = 1, NH = NC = 2; beads 1 and 2 are connected to X X hot baths, beads 4 and 5 are coupled to colder reservoirs, Therefore, the central bead 3 is not thermostated. This separation is arbitrary and in practice should be based on the physical NB structure. ∆J = [(Ti − Tj ) + (TN+1−i − TN+1−j)] Mi,N+1−j We begin with the classical (C) limit, Eq. (3). The to- i=1 6 X Xj=i tal heat input in the forward (J) direction, corresponding (12) to the setup of Fig. 2(a) is Physically, the two asymmetries (structural and in the C J = (T1 − T4)M14 + (T2 − T5)M25 (7) applied thermal gradients) are achievable in molecular

+(T1 − T5)M15 + (T2 − T4)M24. junctions by connecting a molecule to distinct solids: Different materials are characterized by different phonon properties such that the force constants at the left side Reversing the temperature profile as in Fig. 2(b), T1 ↔ would be distinct from those at the right side, leading to T5 and T2 ↔ T4, the reversed (J˜) current is the required spatial asymmetry (ii). Furthermore, given that different materials are employed at the two sides, J˜C = (T − T )M + (T − T )M (8) 5 2 14 4 1 25 it is reasonable to assume that a total imposed gradi- +(T5 − T1)M15 + (T4 − T2)M24. ent ∆T would be divided unevenly on the two bound- ary regions such that condition (i) is satisfied. (In real The sum of the opposite currents, which quantifies the materials, these gradients develop due to lattice anhar- diode effect is monicity.) Most importantly, we reiterate that imposing structural asymmetry (k 6= k in Fig. 2) while using C ˜C 2 3 ∆J ≡ J + J identical boundaries (k0 = k1 = k4 = k5) cannot result = [(T1 − T2) − (T4 − T5)] (M14 − M25). (9) in thermal rectification in our model. In Appendix B, we discuss the corresponding TGHO We can now identify the necessary conditions for realizing diode effect for harmonic chains with local trapping (pin- the diode effect, ∆J 6= 0: (i) The temperature gradients ning) potentials. We show that the TGHO diode effect should be distinct at the two boundaries, (T1−T2) 6= (T4− can develop only once pinning potentials at the two ther- T5). (ii) The setup should include a spatial asymmetry mostated regions are different–applying as well unequal such that M14 6= M25. Asymmetry should be introduced thermal gradients. This setup could correspond to a lin- in the thermostated region, as we prove next. Explicitly, ear chain of trapped ions as described in Refs.19,20. assuming the friction constants are uniform, γ1,2,4,5 = γ, We now discuss several aspects of TGHO chains: 5

(i) Absence of rectification with two affinities. If 1.4 the beads at the thermostated segments are coupled to equal-temperature baths, T = T and T = T in Fig. 2, 1 2 4 5 1.5 then ∆J = 0 irrespective of structural asymmetry imple- 1.5 1.2 mented via e.g. mass gradient, differing force constants 1 1 or couplings to the baths. 1 We emphasize that rectification does not develop in this single-affinity scenario even when the model is made 0.5 0.5 more complex, e.g. by making the statistics of the baths 0.8 quantum, including long-range (yet harmonic) interac- 0.5 1 1.5 0.5 1 1.5 tions, or by allowing the baths to couple to all beads (with different strengths). This observation emerges from the analytic structure of the Landauer heat current expres- FIG. 3. Contour plot of the rectification ratio in an N = 5- sion. particle harmonic chain with NH = NC = 2. (a) Classical (ii) Classical and quantum TGHO diodes. As case and (b) quantum calculation with T1 = 1, T2 = 0.5, T4 = we showed in Eq. (9), ∆J 6= 0 once the gradients are 0.2, T5=0.1 and γ = 1; the central bead is not coupled to a thermostat. We introduce different harmonic force constants different, (T1 −T2) 6= (T4 −T5), unless a mirror symmetry at the thermostated regions, but use k2 = k3 = 1 for the is imposed with M = M . To break the symmetry 14 25 interior part, masses are set at m = 1. between M14 and M25, the thermostated regions should be made structurally asymmetric, i.e. k1 6= k4 M M Purely-quantum TGHO diode. 14 15 (iii) In the quan- 2 0.06 2 0.02 tum limit, the temperatures in Eq. (9) appear within 0.05 the Bose-Einstein distribution functions, included in the 0.015 frequency integral. In this case, as long as at least three 0.04 affinities are applied, e.g. T1 > T2 > T4 > T5, and 1 0.03 1 0.01 even when the gradients are equal, (T1 − T2) = (T4 − T5), 0.02 0.005 thermal rectification would show up (assuming structural 0.01 asymmetry is included as required.) (iv) Self consistent reservoir method. The TGHO 1 2 1 2 system is distinct from the self consistent reservoir (SCR) M M method, which was discussed in e.g. Refs.48–53 in the con- 2 24 2 25 0.06 text of thermal rectification in quantum chains. The role 0.2 0.05 of the SCRs is to mimic anharmonicity. These fictitious 0.04 thermal baths are attached to interior beads while de- 1 0.15 1 0.03 manding zero net heat flow from the physical system to the SCRs. The temperature of the SCRs is dictated by 0.02 0.1 this condition. In contrast, in the TGHO chain the ther- 0.01 mostats are responsible for the power input and output 1 2 1 2 from the system, and their temperature is freely assigned as independent boundary conditions.

FIG. 4. The elements Mij for the classical model corre- sponding to simulations in Fig. 3. Rectification arises due to IV. SIMULATIONS the asymmetry M14 =6 M25. Parameters are the same as in Fig. 3. A. Classical and Quantum TGHO diodes

54 Rectification effect can be measured in different ways, current definition see . with ∆J 6= 0, defined in Eq. (9), or based on a recti- We show that both classical and quantum calculations fication ratio, R ≡ |J/J˜|. We demonstrate the TGHO can create the diode effect. In Fig. 3(a), the rectification diode effect in Fig. 3, where we study the effect in ratio reaches up to R ≈ 1.4 in both the classical and the 5-bead system corresponding to Fig. 2. We imple- quantum cases. While the effect is not very large, it is ment spatial asymmetry by using different force con- in fact comparable to rectification ratios emerging due to stants, k0 = k1 6= k4 = k5. The current was calculated an anharmonic potential, as we discuss below in Fig. (8). as the total input heat (6) (confirmed to be identical to In Fig. 3(b) we display the behavior of the quantum the total heat dissipated to the cold baths) by numeri- TGHO diode, indicating on a somewhat stronger diode cally integrating Eqs. (3) and (5) with a fine frequency effect (bottom-right domain). grid up to a cutoff frequency larger than all other en- How can we tune the system to increase the rectifica- ergy scales For a discussion of the subtleties of the heat tion ratio? As can be seen from the analytic form of the 6 heat current for a 5-bead chain, there are four terms that play a role in the rectification ratio, M14, M25, M15 and (a) (b) M24. These contributions are displayed in Fig. 4. We 1.22 conclude that at large asymmetry (bottom-right part), 1.21 M14 should dominate—once the gradients are made large. At this region, roughly R ≈ |(T1 − T4)/(T5 − T2)|, 1.18 1.19 which is ≈ 2 in our parameters, close to the achieved maximal rectification ratio of 1.4. 1.17 Thus, a viable strategy to increase rectification is to 1.14 impose large structural asymmetry between the two ends, 1.15 as well as apply significantly-unequal thermal gradients at the left and right side. The large spatial asymmetry 5 10 5 10 15 20 results in the the dominance of a single transport path- way. Furthermore, by imposing a large gradient at the FIG. 7. Behavior of the rectification ratio with (a) NB , left side, ∆TH , and a small gradient at the right side, number of thermostated sites, and (b) NI , number of interior ∆TC , with a small temperature drop on the central re- sites. We set temperatures and gradients as T1 = 1, T2 = 0.5, gion (such that in the example used, T ∼ T ) the rec- 4 2 T4 = 0.2, T5 = 0.1 thus ∆TH = 1 − 0.5 and ∆TC = 0.2 − tification ratio of the model scales as R ∝ |∆TH /∆TC|. 0.1. In panel (a), a linear gradient is assumed within each Below (Fig. 7) we further show that in long chains, the thermostated region. In panel (b), NH = NC = 2. Other rectification effect is suppressed with NB = NH,C , but parameters are γ = 1, and force constants in the left (right) it only weakly depend on NI . We therefore suggest that thermostated region at 1 (.1); other constants are set to 1. 1 ∆TH Simulations were performed using classical expressions. R ∝ NB ∆TC .

(a) Classical (b) Quantum 5 5 B. Purely-quantum TGHO diode 4 4 1.2 3 3 1.1 1 The dependence of the classical and quantum TGHO 2 2 0.9 diode effect on the local gradients is presented in Fig 5. 1 1 As described above, the diode effect is enhanced when 0.8 0 0 e.g. the left side experiences a large thermal gradient, 0 5 0 5 while temperatures at the right side are almost identical. The classical case cannot support the diode effect when FIG. 5. Dependence of rectification on the temperature the local gradients are equal, ∆TH = ∆TC . In contrast, quantum statistics allows the diode behavior under equal differences ∆TH = T1 − T2 and ∆TC = T4 − T5 in (a) classical and (b) quantum calculations. Rectification is enhanced when gradients. This effect is illustrated in the behavior along one gradient is very large and the other small. Here, T1 = 10, the diagonal of Fig. 5(b), presented for clarity in Fig. 6. T2 = 10 − ∆TH , T4 = ∆TC , T5 = 0. The force constants are k0 = k1 = 2, k2 = k3 = 1, k4 = k5 = .1, m = 1 and γ = 1.

1.12 C. Length dependence of the TGHO diode effect 1.09

1.06 Fig. (7) displays the behavior of the rectification ra- tio as the size of the system increases. In panel (a) we increase the number of thermostated sites N while fix- 1.03 B ing the overall temperature differences ∆TH and ∆TC, assuming a linear gradient in each region. We find that 1 rectification decays as the number of thermostated sites 0 1 2 3 4 5 increases. In contrast, the rectification ratio persists and saturates as we increase the number of sites in the in- FIG. 6. Purely-quantum TGHO diode operating when the terior region, NI . This saturation is expected since in thermal gradients at the two boundaries are equal, ∆TH = harmonic chains thermal transport is ballistic. Thus, the ∆TC ; we display the diagonal of Fig. (5). Parameters are impact of the central region on the the rectification ef- T1 = 10, T2 = 10 − ∆T , T4 =∆T , T5 = 0. fect should become independent of length, NI for long enough chains. 7

D. Comparison to an anharmonic diode 1.18 To appreciate the magnitude of the rectification ef- fect in the TGHO chain, we present in Fig. (8) the diode behavior emerging when anharmonic interactions 1.12 are explicitly added to the chain. We use the Frenkel- 10-3 Kontorova (FK) potential that was used in many 12 demonstrations of nonlinear thermal devices, e.g.,22,55,56, adding onsite potentials to Eq. (1), 1.06 8 4 2π 0 1 2 V (x)= VR/L cos x . (13) a 0 0.5 1 1.5 2   Specifically, for the five-site chain, we encode asymme- try in the force constants and in the local potentials, VL FIG. 8. Rectification ratio in the Frenkel-Kontorova anhar- vs. VR. Unlike the harmonic case, which is analytically monic chain. The setup is analogous to Fig. (1)(a) with the solvable, to treat anharmonic interactions we turn to nu- leftmost and rightmost particles thermostated. Rectification merical molecular dynamics simulations. The Langevin is achieved by adding different anharmonic onsite FK poten- equations of motion are integrated with the Br¨unger- tials to the left (first two beads) and right (last three beads) V Brooks-Karplus method; simulations were preformed by sides of the five-bead chain. Here, at the right side, R = 1 and k = 1 while at the other half of the chain VL varies and propagating the dynamics long enough to reach a steady k = 0.1. Other parameters are TH = 1, TC = .1, γ = 1. The state, then finding the heat current by averaging the lo- inset presents the currents in the forward (J) and reversed cal currents between adjacent beads. Here we compute (J˜) directions. heat current as the net power exchanged between central C k2 beads, hJ i = 2 h(v2 + v3)(x3 − x2 − a)i. We then aver- age over time and over realizations of the noise. Technical to a diode model that was based on an anharmonic force 54 details were discussed in Ref. . Results are presented in field. Fig. (8). Note that in the FK calculation, we resort to Recent studies used harmonic junctions with a single the standard modelling with a single thermal affinity, TH affinity (as in Figs. 1(a)-(b)) to realize a diode effect19,20; at the left thermostat and TC at the right side. Fur- this was achieved by making parameters such as fric- thermore, only the leftmost (bead 1) and rightmost (N) tion coefficients temperature dependent, γ1(T ), γN (T ). beads are thermostated, We refer to such models as effective harmonic-oscillator Comparing Fig. (8) to e.g. Fig. (3), we note that rec- diodes. In this case, going back for simplicity to the tification in the anharmonic FK model is comparable to classical limit, Eq. (3), the net heat current is given values received in the TGHO diode. Thus, while the by J ∝ (TH − TC)γ1(TH )γN (TC )M1N (TH ,TC), where rectification ratio demonstrated with the TGHO chain we extracted the friction coefficients from the defini- model is not impressive, is is similar to what one would tions of M1N in Eq. (4). Assuming e.g. a linear de- achieve using similar parameters in the FK anharmonic pendence of friction coefficients with the temperature of chain, a central model for diodes examined in the liter- the attached bath, γ1,N (TH ) = γ1,N + λ(TH − TC ), and ature. The FK model has been optimized to show large γ1,N (TC)= γ1,N − λ(TH − TC), with λ as the slope, one rectification ratio22; similarly, it is interesting to explore obtains a diode effect, means for enhancing the TGHO diode effect. 2 ∆J ∝ λ(TH − TC ) (γN − γ1)M1N (TH ,TC), (14) where for simplicity we assumed that the friction coef- V. DISCUSSION AND SUMMARY ficients have a small effect on the the Green’s function G(ω). The diode effect ∆J 6= 0 relies on two conditions: We described a new type of a thermal diode, which is (i) structural asymmetry in the form here of γ1 6= γN , and constructed in a purely-harmonic system when attached (ii) hidden-effective interactions λ 6= 0, making parame- to multiple thermostats, thus imposing at least two affini- ters temperature dependent. Notably, this effective har- ties. The TGHO diode operates when two conditions are monic oscillator diode scales quadratically with the tem- 2 met: The thermostated regions are (i) structurally asym- perature difference, ∆J ∝ (TH − TC ) . This quadratic metric with respect to each other and (ii) placed under scaling is the fingerprint of a hidden anharmonicity, illus- unequal thermal gradients. We further proved the onset trating a nonlinear phenomena. In contrast, the TGHO of a purely-quantum TGHO diode, which exists when diode is a linear effect, characterized by the linear scal- the reservoirs (of different temperatures) are placed un- ing of the net heat current with local temperature biases, der equal gradients. We analyzed the dependence of the ∆J ∝ ∆T , see Eq. (9). TGHO diode effect on chain length and the applied tem- Purely harmonic junctions connecting heat baths at perature gradient and further compared its performance two different temperatures cannot rectify heat. Our 8 study shows that one may achieve a diode behavior in and the hospitality of the Department of Chemistry at harmonic setups by using compound boundary condi- the University of Toronto. BKA and DS thank the Shas- tions that enforce local thermalization on several sites. tri Indo-Canadian Institute for providing financial sup- Realizing a TGHO thermal diode with a large rectifi- port for this research work in the form of a Shastri Insti- cation ratio remains a challenge. Future work will be tutional Collaborative Research Grant (SICRG). focused on testing the impact of long range interactions on the TGHO diode with the goal to enhance its perfor- mance. APPENDIX A: TGHO DIODE WITH ASYMMETRIC INTERPARTICLE COUPLINGS ACKNOWLEDGMENTS We show that rectification appears only when asym- DS acknowledges the NSERC discovery grant and the metry is encoded such that the thermostated regions are Canada Chair Program. BKA gratefully acknowledges distinct. the start-up funding from IISER Pune, the MATRICS For the five-site model with equal friction constants, grant MTR/2020/000472 from SERB, Govt. of India, G−1(ω)=

2 −ω + iγω + k0 + k1 −k1 2 −k1 −ω + iγω + k1 + k2 −k2 2  −k2 −ω + k2 + k3 −k3  2  −k3 −ω + iγω + k3 + k4 −k4   2   −k4 −ω + iγω + k4 + k5       with zero elsewhere. As discussed in the main text, in our setup, NH = NC = 2, NI = 1; two beads are thermalized at the boundaries and a single bead at the centre is not directly coupled to heat baths. The diode effect for this system can be quantified by Eq. (9), and it is controlled by the asymmetry between M14 and M25. We provide now explicit expressions for these terms, as defined in Eq. (4). det(C14) −1 First, G14 = det(G−1) , where C14 is the minor of G , missing the row 1 and column 4, 2 −k1 −ω + iγω + k1 + k2 −k2 2 −k2 −ω + k2 + k3 C14 = (A1)  −k3 −k4  2  −ω + iγω + k4 + k5 so   2 det(C14)= −k1k2k3(−ω + iγω + k4 + k5). (A2) Similarly, 2 −ω + iγω + k0 + k1 −k1 2 −k2 −ω + k2 + k3 −k3 C25 = 2 (A3)  −k3 −ω + iγω + k3 + k4  −k4  so   2 det(C25)= −k2k3k4(−ω + iγω + k0 + k1). (A4)

In order to obtain det(C14) 6= det(C25) we need to introduce an asymmetry, for example, setting k0 6= k5 or k1 6= k4. The parameters of the interior (unthermalized) region, k2 and k3, play no role determining whether or not there will be rectification. Nevertheless, they can control the magnitude of the effect. Explicitly, 2 ∞ 2 2 γ |k1k2k3 −ω + iγω + k4 + k5 | M = dωω2 (A5) 14 π |det G−1|2 Z−∞
The denominator is a degree 20 polynomial of ω. Its exact value depends on all the system parameters. Other contributions to the current are given in terms of

det(C15)= k1k2k3k4, (A6) 2 2 det(C24)= k2k3(−ω + iγω + k0 + k1)(−ω + iγω + k4 + k5). 9

Longer chains have the analogous property that force constants between beads not connected to heat baths play no role in rectification: Asymmetry must appear between the sections directly thermalized by baths. More precisely, in an N-bead chain with NH = NC = 2, so that beads 1, 2; N − 1, and N are connected to thermostats we get

N−2

det(C1(N))= k1kN−1 −ki (A7) i=2 ! Y N−2 2 det(C1(N−1))= −k1 −ki (−ω + iγω + kN−1 + kN ) i=2 ! Y N−2 2 det(C2(N))= −kN−1 −ki (−ω + iγω + k0 + k1) i=2 ! Y N−2 2 2 det(C2(N−1))= −ki (−ω + iγω + k0 + k1)(−ω + iγω + kN−1 + kN ) i=2 ! Y Rectification appears when det(C1(N−1)) 6= det(C2(N)), thus k1 6= kN−1 and/or k0 6= kN ; asymmetry in the central region force constants, k2,...,kN−2 is not sufficient to enact rectification.

APPENDIX B: TGHO DIODE WITH interparticle potentials are assumed identical. For the ASYMMETRIC ONSITE POTENTIALS five-particle chain with harmonic onsite potentials, the inverse Green’s matrix has form In this Appendix we include asymmetry by introduc- ing local trapping potentials with force constant k˜; the

2 −ω + iγω +2k + k˜1 −k −k −ω2 + iγω +2k + k˜ −k  2  −1 −k −ω2 +2k + k˜ −k G (ω)= 3  −k −ω2 + iγω +2k + k˜ −k   4   2 ˜   −k −ω + iγω +2k + k5       We again set NH = NC = 2 and NI = 1; two beads are thermalized at each boundary, while the single bead at the centre (particle 3) is not thermalized. The diode effect for this system can be quantified by Eq. (9), and it is controlled by the asymmetry between M14 and M25. We provide now explicit expressions for these terms to analyze the required source of asymmetry. The elements det(Cij ) take the form

4 det(C15)= k , (B1) 3 2 det(C14)= −k (−ω + iγω +2k + k˜5), 3 2 det(C25)= −k (−ω + iγω +2k + k˜1), 2 2 2 det(C24)= k (−ω + iγω +2k + k˜1)(−ω + iγω +2k + k˜5).

In this case, rectification can show up once k˜1 6= k˜5 resulting in M14 6= M25. As in the case of asymmetric interparticle forces, this holds for chains of any size. For an N-bead chain with NH = NC = 2, N−1 det(C1(N))= k , (B2) N−2 2 ˜ det(C1(N−1))= −k (−ω + iγω +2k + kN ), N−2 2 ˜ det(C2(N))= −k (−ω + iγω +2k + k1), N−3 2 ˜ 2 ˜ det(C2(N−1))= k (−ω + iγω +2k + k1)(−ω + iγω +2k + kN ).

Therefore, it is the asymmetry k˜1 6= k˜N that is responsi- ble for the diode effect. Inspecting this form, we expect 10 that the TGHO chain with an asymmetry in the inter- particle force constants would support larger rectification ratios than the case with pinning potentials.

∗ [email protected] efficient was varied with temperature, thus allowing the oc- 1 I. V. Rubtsov and A. L. Burin, Ballistic and diffusive vi- currence of a diode effect in an effective harmonic junction. brational energy transport in molecules, J. Chem. Phys. To eliminate confusion, we highlight that here all param- Perspective 150, 020901 (2018). eters are temperature independent, and that the model is 2 Thermal and thermoelectric properties of molecular junc- microscopically harmonic. tions, K. Wang, E. Meyhofer, and P. Reddy, Adv. Func. 18 E. Pereira, Requisite ingredients for thermal rectification, Mater. 30, 1904534 (2020). Phys. Rev. E 96, 012114 (2017). 3 D. Segal and B. K. Agarwalla, Vibrational heat transport 19 M. A. Simon, S. Martinez-Garaot, M. Pons, and J. G. in molecular junctions, Ann. Rev. Phys. Chem. 67, 185 Muga, Asymmetric heat transport in ion crystals, Phys. (2020). Rev. E 100, 032109 (2019). 4 A. Dhar, Heat transport in low dimensional systems, Adv. 20 M. A. Sim´on, A. Ala˜na, M. Pons, A. Ruiz-Garc´ıa, and J. Phys. 57, 457 (2008). G. Muga, Heat rectification with a minimal model of two 5 D. M. Leitner, Quantum ergodicity and energy flow in harmonic oscillators, Phys. Rev. E 103, 012134 (2021). molecules, Adv. Phys. 64, 445 (2015). 21 M. Terraneo, M. Peyrard, and G. Casati, Controlling the 6 E. Fermi, J. Pasta and S. Ulam, Studies of nonlinear prob- energy flow in nonlinear lattices: A model for a thermal lems, Los Alamos Scientific Laboratory report LA-1940 rectifier, Phys. Rev. Lett. 88, 094302 (2002). (1955). 22 B. Li, L. Wang, and G. Casati, Thermal Diode: Rectifica- 7 T. Dauxois, Fermi, Pasta, Ulam, and a mysterious lady, tion of Heat Flux, Phys. Rev. Lett. 93, 184301 (2004). Physics Today 61, 55 (2008). 23 N. Li, J. Ren, L. Wang, G. Zhang, P. H¨anggi, and B. 8 J. C. Kl¨ockner and F. Pauly, Variability of the ther- Li, Colloquium: Phononics: Manipulating heat flow with mal conductance of gold-alkane-gold single-molecule junc- electronic analogs and beyond, Rev. Mod. Phys. 84, 1045 tions studied using ab-initio and molecular dynamics ap- (2012). proaches, arXiv:1910.02443. 24 M. Y. Wong, C. Y. Tso, T. C. Ho, and H. H. Lee, A 9 M. D. Losego, M. E. Grady, N. R. Sottos, D. G. Cahill, and review of state of the art thermal diodes and their potential P. V. Braun, Effects of chemical bonding on heat transport applications Int.l J. Heat and Mass Transfer 164, 120607 across interfaces, Nat. Mater. 11, 502 (2012). (2021). 10 T. Meier, F. Menges, P. Nirmalraj, H. H¨olscher, H. Riel, 25 G. Benenti, G. Casati, C. Mej´ıa-Monasterio, and M. and B. Gotsmann Length-dependent thermal transport Peyrard, From Thermal Rectifiers to Thermoelectric De- along molecular chains, Phys. Rev. Lett. 113, 060801 vices. In: Lepri S. (eds) Thermal Transport in Low Dimen- (2014). sions. Lecture Notes in Physics, vol 921. Springer, Cham. 11 S. Majumdar, J. A. Sierra-Suarez, S. N. Schiffres, W.- 2016. L. Ong, C. F. Higgs, A. J. H. McGaughey, and J. A. 26 B. K. Agarwalla and D. Segal, Energy current and its Malen, Vibrational mismatch of metal leads controls ther- statistics in the nonequilibrium spin-boson model: Ma- mal conductance of self-assembled monolayer junctions, jorana fermion representation, New J. Phys. 19, 043030 Nano Lett. 15, 2985 (2015). (2017). 12 N. Mosso, H. Sadeghi, A. Gemma, S. Sangtarash, U. 27 D. Segal and A. Nitzan, Spin-boson thermal rectifier, Phys. Drechsler, C. Lambert, and B. Gotsmann, Thermal trans- Rev. Lett. 94, 034301 (2005). port through single-molecule junctions, Nano Lett. 19, 28 E. Pereira, Thermal rectification in classical and quantum 7614 (2019). systems: Searching for efficient thermal diodes, Europhys. 13 L. Cui, S. Hur, Z. A. Akbar, J. C. Kl¨ockner, W. Jeong, F. Lett. 126, 14001 (2019). Pauly, S. Y. Jang, P. Reddy, and E. Meyhofer, Thermal 29 D. M. Leitner, Thermal boundary conductance and ther- conductance of single-molecule junctions, Nature 572, 628 mal rectification in molecules J. Phys. Chem. B 117, 12820 (2019). (2013). 14 Z. Rieder, J. L. Lebowitz, and E. Lieb, Properties of a 30 K. M. Reid, H. D. Pandey, and D. M. Leitner, Elastic Harmonic Crystal in a Stationary Nonequilibrium State, and inelastic contributions to thermal transport between J. Math. Phys. 8, 1073 (1967). chemical groups and thermal rectification in molecules, J. 15 D. Segal, A. Nitzan, and P. H¨anggi, Thermal conductance Phys. Chem. C 123, 6526 (2019). through molecular wires, J. Chem. Phys. 119, 6840 (2003). 31 S. Chen, E. Pereira, and G. Casati, Ingredients for an effi- 16 L. G. C. Rego and G. Kirczenow, Quantized thermal con- cient thermal diode, EPL 111, 30004 (2015). ductance of dielectric quantum wires, Phys. Rev. Lett. 81, 32 S. Chen, D. Donadio, G. Benenti, and G. Casati, Effi- 232 (1998). cient thermal diode with ballistic spacer Phys. Rev. E 97, 17 A thermal diode effect can be realized in “effective” 030101 (2018). harmonic systems—once anharmonicity is introduced at 33 T. Alexander, High-heat-flux rectification due to a local- a mean field level, by making parameters temperature ized thermal diode, Phys. Rev. E 101, 62122 (2020). dependent18–20. In these type of models there is a cross- 34 S. You, D. Xiong, and J. Wang, Thermal rectification graining or mean-field assumption making parameters tem- in the thermodynamic limit, Phys. Rev. E 101, 012125 perature dependent. In Ref.20 for example the friction co- (2020). 11

35 S.-A. Wu and D. Segal, Sufficient conditions for ther- port in trapped ion chains, New J. Phys. 16, 063062 (2016). mal rectification in hybrid quantum structures, Phys. Rev. 47 I. Sharony, R. Chen, and A. Nitzan, Stochastic simulation Lett. 102, 095503 (2009). of nonequilibrium heat conduction in extended molecule 36 L.-A. Wu, C. X. Yu, and D. Segal, Nonlinear quantum junctions, J. Chem. Phys. 153, 144113 (2020). heat transfer in hybrid structures: Sufficient conditions for 48 F. Bonetto, J. L. Lebowitz, and J. Lukkarinen, Fourier’s thermal rectification, Phys. Rev. E 80, 041103 (2009). Law for a harmonic crystal with self-consistent stochastic 37 S. H. S. Silva, G. T. Landi, R. C. Drumond, and E. Pereira, reservoirs, J. Stat. Phys. 116, 783 (2004). Heat rectification on the XX chain, Phys. Rev. E 102, 49 E. Pereira and H. C. F. Lemos, Symmetry properties of 062146 (202). heat conduction in inhomogeneous materials, Phys. Rev. 38 V. Balachandran, G. Benenti, E. Pereira, G. Casati, and E 78, 031108 (2008). D. Poletti, Heat current rectification in segmented XXZ 50 D. Segal, Absence of thermal rectification in asymmetric chains, Phys. Rev. E 99, 032136 (2019). harmonic chains with self consistent reservoirs: An exact 39 V. Balachandran, S. R. Clark, J. Goold, and D. Poletti, analysis, Phys. Rev. E 79, 012103 (2009). Energy current rectification and mobility edges, Phys. Rev. 51 M. Bandyopadhyay and D. Segal, Quantum heat transfer Lett. 123, 020603 (2019). in harmonic chains with self consistent reservoirs: Exact 40 J. Senior, A. Gubaydullin, B. Karimi, J. T. Peltonen, J. numerical simulations, Phys. Rev. E 84, 011151 (2011). Ankerhold, and J. P. Pekola, Heat rectification via a su- 52 E. Pereira, H. C. F. Lemos, and R. R. Avila,´ Ingredients perconducting artificial atom, Comm. Phys. 3, 40 (2020). of thermal rectification: The case of classical and quantum 41 A. Iorio, E. Strambini, G. Haack, M. Campisi, and F. Gi- self-consistent harmonic chains of oscillators, Phys. Rev. E azotto, Photonic heat rectification in a coupled qubits sys- 84, 061135 (2011). tem, arXiv:2101.11936. 53 R. Moghaddasi Fereidani and D. Segal, Phononic heat 42 B. C. Pein, Y. Sun, and D. D. Dlott, Unidirectional vibra- transport in molecular junctions: quantum effects and vi- tional energy flow in nitrobenzene, J . Phys. Chem. A 117, brational mismatch, J. Chem. Phys. 150, 024105 (2019). 6066 (2013). 54 N. Kalantar, B. K. Agarwalla and D, Segal, On the def- 43 B. C. Pein, Y. Sun, and D. D. Dlott, Controlling vibra- initions and simulations of vibrational heat transport in tional energy flow in liquid Alkylbenzenes, J. Phys. Chem. nanojunctions, J. Chem. Phys. 153, 174101 (2020). B 117, 10898 (2013). 55 B. Li, L. Wang, and G. Casati, Negative differential ther- 44 Nanoscale solid-state thermal rectifiers were realized e.g. mal resistance and thermal transistor, App. Phys. Lett. 45 in Ref. based on mass-graded carbon and boron nitride 88, 143501 (2006). nanotubes. 56 L. Wang and B. Li, Thermal logic gates: computation with 45 C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl, phonons, Phys. Rev. Lett. 99, 177208 (2007). Solid-state thermal rectifier, Science 314, 1121 (2006). 46 M. Ramm, T. Pruttivarasin, and H. H¨affner, Energy trans-