Anomalous low-energy properties in amorphous solids and the interplay of electric and elastic interactions of tunneling two-level systems
Alexander Churkin, Shlomi Matityahu, Andrii O. Maksymov, Alexander L. Burin, Moshe Schechter
aa r X i v : . [ c ond - m a t . d i s - nn ] F e b Anomalous low-energy properties in amorphous solids and the interplay of electricand elastic interactions of tunneling two-level systems
Alexander Churkin,
1, 2
Shlomi Matityahu,
2, 3, 4
Andrii O. Maksymov, Alexander L. Burin, and Moshe Schechter Department of Software Engineering, Sami Shamoon College of Engineering, Beer-Sheva, Israel Department of Physics, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel Institute of Nanotechnology, Karlsruhe Institute of Technology, D-76344 Eggenstein-Leopoldshafen, Germany Department of Physics, NRCN, P.O. Box 9001, Beer-Sheva 84190, Israel Department of Chemistry, Tulane University, New Orleans, LA 70118, USA (Dated: February 21, 2020)Tunneling two-level systems (TLSs), generic to amorphous solids, dictate the low-energy prop-erties of amorphous solids and dominate noise and decoherence in quantum nano-devices. Theproperties of the TLSs are generally described by the phenomenological standard tunneling model.Yet, significant deviations from the predictions of this model found experimentally suggest the needfor a more precise model in describing the low-energy properties of amorphous solids. Here weshow that the temperature dependence of the sound velocity, dielectric constant, specific heat, andthermal conductivity, can be explained using an energy-dependent TLS density of states. The re-duction of the TLS density of states at low energies relates to the ratio between the strengths ofthe TLS-TLS interactions and the random potential, which is enhanced in systems with dominantelectric dipolar interactions.
I. INTRODUCTION
Understanding the low-temperature physics of disor-dered and amorphous materials has emerged as one of themost intriguing and challenging problems in condensedmatter physics [1, 2]. Below about 1 K, such systems ex-hibit physical properties that are not only qualitativelydifferent from those of crystalline solids, but also showa remarkable degree of universality [2–5]. For instance,the specific heat and thermal conductivity are approxi-mately linear and quadratic in temperature, respectively,while the internal friction Q − is nearly temperature-independent and varies slightly between different mate-rials.This behavior of amorphous solids has been primarilyinterpreted with the model of tunneling two level sys-tems (TLSs) [6, 7], suggesting the presence of atoms orgroups of atoms tunneling between two nearly degenerateconfigurations, which will be referred to as the standardtunneling model (STM). There were numerous sugges-tions targeted to describe the nature of tunneling sys-tems and their universality, including the soft-potentialmodel [8] and its further developments (see Ref. 9 andreferences therein), interaction-based models targeted toaccount for quantitative universality of TLSs [1, 10–13],glass-transition-based theory [14, 15] and models basedon the polaron effect [16, 17]. Similarly to the STM,all these theories account for the existence of TLSs atlow temperatures and the resulting thermodynamic andacoustic properties of glasses. Yet, their predictive valuelies in their deviations from the STM, which has to bechecked against experimental observations [18].Marked examples of discrepancies between experimen-tal results and theoretical predictions of the STM are thedeviations from integer powers of the temperature depen-dence of the specific heat and thermal conductivity, see below, and the anomalous temperature dependence of thesound velocity and dielectric constant. The STM predictslogarithmic temperature dependence, with a maximumfor the sound velocity and a minimum for the dielectricconstant, with a slope ratio of 1 : − . − n ( E ) ∝ E µ , with 0 . < µ < .
3. Even stronger energydependence of the DOS in a-SiO was recently extractedfrom measurements of dielectric loss using superconduct-ing lumped element resonators [31]. These findings aresupported by earlier experiments which show indirect ev-idences for energy-dependent DOS: in deviations fromSTM predicted integer values for the temperature depen-dence of the specific heat, C ∝ T α , and of the thermalconductivity, κ ∝ T − β , with α, β ≈ . − . ∝ T . [34–36], which may arisedue to dipolar interactions between the TLSs, assuminga DOS n ( E ) ∝ E µ , with µ ≈ . n ( E ) ∝ E µ with µ ≈ . /f noise in superconducting resonators at low temperatures(see, however, Ref. 43). Still, it is not clear what the ori-gin of such marked energy dependence of the TLS-DOSmay be.Here we calculate the single-particle TLS-DOS assum-ing TLS disorder energy being not much larger than theTLS-TLS interaction energy. At zero temperature wefind the TLS-DOS to be significantly reduced, and well-described by a power law, the power being approximatelythe ratio between interaction and disorder. Since thesingle-particle TLS-DOS involves the excitation energiesof single TLSs in the environment of all other TLSs, itis temperature dependent. Indeed, at finite temperaturethe pseudo-gap at low energies closes gradually.Intriguingly, we find that energy-dependent TLS-DOSaccounts well not only for the anomalous power lawsof the temperature dependence of the specific heat andthermal conductivity, but also for the anomalous tem-perature dependence of the sound velocity and dielec-tric constant. We discuss the energy dependence of theTLS-DOS within the Dipolar Gap model [29] and theTwo-TLS model [13]. Using the latter model we showthat TLS-TLS interactions not much smaller than therandom fields arise once TLS-TLS interactions are dom-inated by the electric dipolar interaction. Relation toexisting experimental results is then discussed.The paper is organized as follows: In Sec. II we in-troduce the generic model for TLSs, albeit allowing forarbitrary ratio between the typical TLS-TLS interactionsat short distances and the typical random field. We thendiscuss the relation between this model and the DipolarGap model. In Sec. III we first present (Sec. III A) the nu-merical results for the single-particle TLS-DOS for differ-ent ratios of interactions to random fields, and the result-ing temperature dependence of the thermal conductivityand specific heat (Sec. III B). We then address (Sec. III C)the anomalous temperature dependence of the sound ve-locity and dielectric constant, within the Dipolar Gapmodel, and within the model allowing for stronger TLS-TLS interactions. In Sec. IV we discuss, within the Two-TLS model, the possibility of TLS-TLS interactions en-hancement as a result of dominance of electric interac-tions over elastic interactions in amorphous solids. Wethen summarize in Sec. V. II. MODEL AND TLS-DOS
At low energies the system of interacting TLSs can bemodelled by the effective Hamiltonian [6, 7, 11, 24] H TLS = X i h i τ zi + X i ∆ ,i τ xi + 12 X i = j J ij τ zi τ zj , (1)where τ zi and τ xi are the Pauli matrices that representthe TLS at site i . The first term is the bias energy of theTLSs resulting from their interaction with static disorder.The total bias energy of TLS i is therefore ∆ i ≡ h i + P j J ij τ zj , and the total energy of a TLS is given by E = p ∆ + ∆ . Within the STM one assumes that J ij ≪ h i ,and that h i are homogeneously distributed, leading tothe ansatz P (∆ , ∆ ) = P / ∆ and density of states n = P L . Here L = ln ( ˜ E/ ∆ , min ), with ˜ E being a largeenergy of the order of the disorder energy and ∆ , min denoting the minimum tunneling amplitude of the TLSs.Generally, however, one allows energy dependence of theTLS-DOS, i.e. n ( E ) = P ( E ) L .The second term in the Hamiltonian (1) denotes TLStunneling. Whereas this term is of utmost importanceto dynamic properties, it has a small effect on the TLS-DOS, especially at energies &
10 mK relevant to most ex-periments. We therefore consider henceforth the random-field Ising Hamiltonian H = X i h i τ zi + 12 X i = j J ij τ zi τ zj , (2)with h i = h c i and J ij = c ij J / ( R ij /R + C ), where c i and c ij are normally distributed random variables withzero mean and unity variance, R ij is the distance betweenTLS i and TLS j , R is the typical distance betweennearest TLSs, J denotes typical nearest neighbor TLS-TLS interaction, C is a short distance cutoff, and h isthe typical random field. Generally, the interaction termcomprises both elastic and electric TLS-TLS interactions.Adding weak interactions to the STM, i.e. considering J /h ≪
1, and assuming T = 0, one finds the emergenceof a dipolar gap at low energies [11, 24, 29] n ( E ) = n c ˜ J n · log ( ˜ J /R E ) , (3)where c = 2 π/
3, ˜ J ≡ J R is the interaction constant,and n ≡ n ( E = J ) = 1 / ( h R ). Widespread ex-perimental evidence for the low-temperature universal-ity of acoustic properties in amorphous solids dictate,for the elastic interactions, a value of J /h = ˜ J n ≈ . − .
05 [6, 7]. Below we discuss the energy-dependentTLS-DOS and its consequences within the dipolar gapmodel, and within a model of the Hamiltonian (2), tak-ing, however, J /h to be not much smaller than unity- possible reason may be domination of electric dipolarinteractions. III. RESULTSA. TLS-DOS
We now calculate the TLS-DOS within the model pre-sented by the Hamiltonian (2) with J /h = 0 . , .
3. Todemonstrate the power-law-like energy dependence of thelow-energy TLS-DOS we perform Monte Carlo simula-tions on cubic lattices of size L , with L = 8 ,
12, and pe-riodic boundary conditions are imposed. TLSs are placedrandomly in the lattice with concentration x = 0 .
5, andwe choose h = 10 K in accordance with its calculatedvalue for KBr:CN [44]. We note that the choice of latticestructure is for convenience, and the randomness of TLSpositions in the amorphous solids is retained by the ran-dom dilution and by the randomness in c i and c ij . Wefurther note that the lattice constant R denotes typicaldistance between adjacent TLSs, rather than interatomicspacing. We use a short distance cutoff equal to R (i.e. C = 1) to account for the finite size of the TLSs, butdecreasing the value of the cutoff has minimal effect onour results. (a) -1 (b) FIG. 1. (Color online) (a) Single-particle TLS-DOS at T =0 . , . L = 12 and J = 2 , J /h = 0 . , . J = 2 K, corresponding to thelimit of negligible TLS-TLS interactions. (b) Zoom in to lowenergies. Solid lines describe low-energy fits, using Eq. (4),for the curves corresponding to T = 0 . n ∝ E µ for the curves corresponding to T = 0 .
02 K.Note that the power µ decreases with increasing temperature. Simulated annealing MC simulations are performed at42 temperatures decreasing from 300 K to 0 .
02 K, andthen at 2 µ K to emulate zero temperature, and are usedto calculate the single-particle DOS, n ( T, E ). While the system does not fully equilibrate within the simulated an-nealing technique, we verify that the final state at 2 µ Kis stable against single and double spin flips. This con-stitutes the sufficient condition for the determination ofthe DOS given by the Efros-Shklovskii stability crite-rion [26, 27]. The single-particle DOS at a given tem-perature is then calculated by measuring the excitationenergies of single TLSs in a given realization, and aver-aging over 10 independent disorder realizations.In Fig. 1 we plot n ( T, E ) as a function of energyfor T = 0 .
02 K and T = 0 . J = 2 , L = 12. In the absenceof interactions, the DOS is well-described by a Gaus-sian [solid black curve in Fig. 1(a)] with width of order h = 10 K [13, 30]. The dipolar interactions producean Efros-Shklovskii type pseudo-gap for energies below ∼ J [26, 27, 29]. As T →
0, the DOS at low energiesapproaches a form well-described by power law energydependence, n ( T → , E ) ∝ E µ , with µ ≈ . − . µ depends on J , see inset of Fig. 1(b)].The dipolar gap is suppressed as the temperature in-creases, yielding a DOS which at low energies is ratherwell-approximated by the function n ( T, E ) ≈ B ( T )( T + E ) µ ( T ) / . (4)We note that the pseudo-gap closes at finite temperatureand therefore the power µ ( T ) decreases with increasingtemperature. B. Thermal conductivity and specific heat
Being well-approximated with a power law DOS at lowenergies, we expect the TLS-DOS calculated from theHamiltonian (2) and plotted in Fig. 1 to account wellfor the deviations from integer power law exponents ofthe temperature dependences of the thermal conductivityand specific heat as observed in amorphous solids. Hav-ing calculated n ( T, E ), the thermal conductivity κ ( T ) isfound by calculating [5] κ ( T ) = 13 X α Z ∞ C ph ,α ( E ) v α ℓ ph ,α ( E ) dE ∝ Z ∞ E dET sinh ( E/ T ) tanh( E/ T ) n ( T, E ) , (5)where C ph ,α ( E ) = E / (cid:0) π ~ v α T sinh ( E/ T ) (cid:1) isthe Debye heat capacity for phonons at a given en-ergy E and polarization α , v α is the sound velocityand ℓ − ,α ( E ) = (cid:0) πγ α E/ρv α (cid:1) P ( T, E ) tanh( E/ T ) ∝ (cid:0) πγ α E/ρv α (cid:1) n ( T, E ) tanh( E/ T ) is the phonon inversemean free path due to interaction with resonant TLSs(i.e., TLSs with energy splitting equal to the phonon en-ergy), characterized by the coupling strength γ α , and ρ denotes the mass density. The prefactor in Eq. (5) con-tains material-dependent constants which are indepen- -2 -1 (a) -2 -1 -4 -3 -2 -1 (b) FIG. 2. (Color online) Temperature dependence of (a) Ther-mal conductivity (in arbitrary units) and (b) specific heat,obtained by Eqs. (5) and (6) with the TLS-DOS n ( T, E ) com-puted by simulated annealing MC simulations with L = 8 , J = 2 , κ ∝ T − β and C ∝ T α . Calculations correspond to arelaxed system close to equilibrium, see text. dent of temperature. To study the temperature depen-dence of the thermal conductivity we calculate the lastintegral in Eq. (5), and represent the thermal conductiv-ity in arbitrary units. Similarly, the specific heat C ( T ) isevaluated, taking the Boltzmann constant k B = 1, as [5] C ( T ) = Z ∞ n ( T, E ) E dE T cosh ( E/ T ) . (6)Figure 2 shows log-log plots of the thermal conductiv-ity and the specific heat as a function of temperature,for J = 2 , L = 8 ,
12. In all cases, the thermalconductivity and the specific heat obey a power law de-pendence, κ ∝ T − β and C ∝ T α , with α and β in therange 0 . − .
2. Note that we do not consider here theslow logarithmic time dependence of the specific heat, re-sulting from the large variance in TLS relaxation times,that can enhance the temperature dependence. Our re- -1 -2-1.5-1-0.500.511.52 X: 1.64Y: -1.749 (a) -3 -2 -1 -1.5-1-0.500.5 (b) FIG. 3. (Color online) Temperature dependence of (a) soundvelocity [23] and (b) dielectric constant of vitreous silica [22].Solid lines correspond to fits by the sum of Eqs. (7) and (8),using for ω the experimental values [90KHz in (a), 1KHzin (b)], and using the fitting parameters A [9 . . P γ /ρv [3 . · − in (a), 8 · − in(b)]. The DOS is taken from Eq. (4) using temperature inde-pendent finite µ > µ = 0 as is given by the STM. sults correspond to a given long time, as the system isout of equilibrium. C. Sound velocity and dielectric constant
Given the above mentioned long-standing discrepancybetween STM predictions and experimental results, it isof interest to study the consequences of energy-dependentTLS-DOS on the temperature dependence of the soundvelocity and dielectric response at low temperatures. Thetemperature dependence of these quantities has two con-tributions coming from the resonant and relaxation pro-cesses [5]. Considering the sound velocity, the contribu- -2 -1 -11-10.5-10-9.5-9-8.5-8-7.5-7-6.5 FIG. 4. (Color online) Temperature dependence of acous-tic velocity, derived from the TLS-DOS obtained numericallyfrom Eq. (2), for 12 different temperatures, for L = 12, J = 2(circles) and J = 3 (squares). Solid lines are fits by the sumof Eqs. (7) and (8), using Eq. (4) for the TLS-DOS with tem-perature independent µ as a fitting parameter. Discrepancybetween the power of the calculated energy-dependent DOS( µ = 0 . , .
27 for J = 2 ,
3, respectively, see Fig. 1) and fit( µ = 0 . , . J = 2 ,
3, respectively) is attributed tothe temperature dependence of the power µ not taken intoaccount in the fit using Eq. (4). tion of the resonant process is of the form δv res v = − L γ ρv Z ∞ n ( T, E ) dEE tanh (cid:18) E T (cid:19) , (7)where v and γ are characteristic values for the velocityand for the interaction constant. For the relaxation pro-cess one has δv rel v = − L γ ρv Z ∞ n ( T, E ) dE T cosh ( E/ T ) × Z √ − x dxx
11 + ω [ Ax E coth( E/ T )] , (8)such that δv/v = ( δv res + δv rel ) /v . Here ω is the probingfrequency and A ≡ ω/T , where T is a crossover temper-ature of the order of the temperature at which the soundvelocity obtains a maximum value [5]. The correspond-ing expressions for the dielectric constant ǫ are obtainedby substituting γ / ( ρv ) → p / (4 πǫǫ ), where p is theTLS dipole moment and ǫ is the vacuum permittivity.Fitting of experimental data for the temperature de-pendence of the sound velocity and dielectric constant re-quires a numerical calculation of the TLS-DOS n ( T, E )at many values of the ratio J /h , which is a compli-cated task. We therefore take first a simpler approachand consider the dependence of the TLS-DOS on energyand temperature as given in Eq. (4), allowing the power µ to serve as a free fitting parameter independent of tem-perature. In Fig. 3 we show fits to typical experimental data for sound velocity and dielectric response in amor-phous solids at low temperatures. These data, displayingthe usually found ratio of 1 : − T , are wellfit by the sum of Eqs. (7) and (8), using the DOS n ( T, E )of Eq. (4) with a rather small power µ ≈ . − .
07. Sucha small power is consistent with the dipolar gap theoryprediction for the energy dependence of the TLS-DOS,given by Eq. (3) [29].To analyze the quality of the above fitting, we nowcompare it with the predictions of the full numericalsimulation, using the above results for the numericallycalculated n ( T, E ) for the Hamilonitan 2 with ratios J /h = 0 . , .
3. In Fig. 4 we plot the sound velocitycalculated as the sum of Eqs. (7) and (8), using the nu-merically calculated DOS for twelve temperatures belowand above the temperature corresponding to the maxi-mum in sound velocity. We find the ratio between thelogarithmic slopes below and above the crossover tem-perature to be roughly 1 : − J /h = 0 .
2, and evena steeper descent beyond the crossover temperature for J /h = 0 .
3. We then find a rather good fit of the nu-merical data using Eq. (4) with a fixed (temperature-independent) power µ for the DOS in Eqs. (7) and (8).We note that the temperature-independent values ob-tained ( µ = 0 . , . J /h = 0 . , .
3, respectively)are intermediate between the powers µ describing thenumerically simulated TLS-DOS at T = 0 .
02 K ( µ =0 . , .
27 for J /h = 0 . , .
3, respectively) and the nu-merically simulated TLS-DOS at T = 0 . µ = 0 . , . J /h = 0 . , .
3, respectively, see Fig. 1(b)].Based on this analysis, one observes that the constantvalue of µ used to fit the numerical data in Fig. 4 under-estimates the exponent of the energy dependence of theTLS-DOS at T = 0, and thus the value of J /h . Accord-ingly, we expect that the T = 0 exponent describing theenergy dependence of the TLS-DOS of the experimentalsystem in Fig. 3 will be larger than the temperature-independent exponent µ ≈ . − .
07 obtained in Fig. 3using the approximate form of Eq. (4). Our numericalresults therefore suggest that values of J /h larger thanthose consistent with the dipolar gap theory of the STMmay be needed to account for the temperature depen-dence of the sound velocity at low temperatures. As thedielectric constant differs from the sound velocity onlyby an overall prefactor, our results and conclusions abovehold also for the anomalous temperature dependence ofthe dielectric constant. IV. ELECTRIC DIPOLAR TLS-TLSINTERACTIONS
In Sec. III B we have discussed the effect of the inter-actions being not much smaller than the random field onthe energy dependence of the TLS-DOS at low energies,
Interaction betweenNN general defects TLS disorder energy Interaction between NN TLSsTwo-TLS model ∼ γ / ( ρv R ) ∼ T g ≈ − h ∼ γ s γ w / ( ρv R ) ≈
10 K J ∼ γ / ( ρv R ) ≈ . − . ∼ γ / ( ρv R ) ∼ T g ≈ − h ∼ γ s γ w / ( ρv R ) ≈
10 K J ∼ p / (4 πǫǫ R ) ≈ − τ )-TLSs at low energies, dominating low-temperature physics; and the interactions between( τ )-TLSs. (i) (top row) as derived by the Two-TLS model with dominant elastic interactions [13], (ii) (bottom row) as presentedhere for the Two-TLS model with strong electric dipolar interactions. Note the small TLS disorder energies in comparison tothe value of 300 − and consequently on the anomalous power laws of thetemperature dependence of the specific heat and thermalconductivity. In Sec. III C we have shown that such a J /h ratio, larger than dictated by experimental resultsfor the elastic interactions [1], may be needed to explainthe temperature dependence of the sound velocity andthe dielectric constant at low temperatures. In this sec-tion we discuss what may be a cause for an enlargedratio of interaction strength to random field strength,and specifically the consequences of TLSs having largerelectric dipolar interaction compared to their phonon-mediated interaction.Amorphous solids show quantitative universality intheir low-temperature acoustic properties. This univer-sality suggests a small and universal value for the quan-titatively universal product P ˜ J , translating to a smalland universal ratio between the elastic interaction andthe random field. The emergence of a larger ratio of J /h in the presence of dominant electric dipolar in-teractions is naturally obtained within the theoreticalframework of the Two-TLS model [13]. We thus beginwith a presentation of the main features of the Two-TLSmodel relevant to our discussion. A more detailed dis-cussion of the model is deferred to Appendix A.First considering only elastic interactions, the Two-TLS model divides TLSs into two groups, with bimodaldistribution of their interaction strengths with the strain,denoted by γ w for the weakly interacting τ -TLSs, whichcorrespond to the abundant TLSs at low energies, and by γ s for the other defects, where g ≡ γ w /γ s ≈ .
02 [13, 44–48]. The Hamiltonian 2 is then derived as the low-energyeffective Hamiltonian of the system (see Ref. [13] and alsoApp. A), with h i ≈ c i γ w γ s ρv R ; J ij ≈ c ij γ ρv R ij . (9)Here, R is the typical distance between nearest two-leveldefects, R ij denotes the distance between τ -TLS i and τ -TLS j , and the parameters c i , c ij ∼ O (1) can be regardedas normally distributed random variables [45].The form of the Hamiltonian (2) is equivalent to thatof the STM Hamiltonian, albeit within the Two-TLSmodel one can derive the typical magnitude of the in-teractions between the weakly interacting TLSs, as wellas the typical magnitude of the random field. Since typical disorder energy at nearest neighbor distance is ≈ γ / ( ρv R ) ∼ T g , where T g ≈ − τ -TLS, which is g times smaller, isgiven by h ≈
10 K [13, 30, 44, 47], and that TLS-TLSinteractions at nearest neighbor distance have a typicalvalue of J ≈ gh ≈ . ≪ h [13, 30, 47].Consider now the electric dipolar interaction, J ij ≈ c ij p πǫǫ R ij . (10) The above characteristics of the TLSs within the Two-TLS model, including the relative smallness of the ran-dom fields, and the extreme smallness of the elastic TLS-TLS interaction, allow for the possibility of electric dipoleinteractions to dominate over the elastic TLS-TLS inter-actions, and to be not much smaller than the typical biasenergies of the TLSs, i.e. J /h . . Experimentally, the ratio between the electric and elas-tic interactions can be deduced from combined measure-ments of dielectric loss and acoustic loss on the samematerial. Such measurements were carried out for BK7and coverglass [49], indicating somewhat larger electricthan elastic TLS-TLS interactions (see detailed analysisin Appendix B). Typical values of γ w ≈ − p ≈ − O and Si N [59]), suggest that larger values of J electric0 /J elastic0 =( p ρv ) / (4 πγ ǫǫ ) may be expected.In table I we summarize the typical energy scales of (a)interactions between general defects, which is of the or-der of the glass transition, and: (b) disorder energy and(c) typical TLS-TLS interaction energy - for the abun-dant ( τ -)TLSs at low temperatures. All energy scalesare denoted within the Two-TLS model with: (i) domi-nant elastic TLS-TLS interactions, and (ii) strong electricdipolar TLS-TLS interactions.The relation suggested above between the magnitudeof the power of the energy dependence of the TLS-DOSand the dominance of the electric dipolar interaction overthe elastic interaction can be further examined by con-sidering disordered lattices, where the TLS concentration -1 -1.5-1-0.500.511.522.53 (a) -1 (b) FIG. 5. (Color online) Temperature dependence of acousticvelocity. Points denote experimentally obtained values, takenat ω = 90KHz [23]. Solid lines correspond to fits by the sumof Eqs. (7) and (8), using Eq. (4) for the DOS with finite µ > µ = 0. (a) For (SrF ) − x (LaF ) x . Hereas x is enhanced so does the strain, and consequently theratio of dipolar interaction to elastic interaction is decreased.Reduction of the power µ is in agreement with theory. Param-eters used for best fits are P γ /ρv = 1 . · − , . · − and A = 7 . . x = 0 . , .
32 respectively.(b) For [(BaF ) . (SrF ) . ] − x (LaF ) x . Here the strain, andthus the ratio of electric to elastic interactions, are inde-pendent of x . Independence of the power µ on x is inagreement with theory. Parameters used for best fits are P γ /ρv = 2 . · − , . · − and A = 3 . . x = 0 . , .
07 respectively. can be varied [60]. Two protocols exist for the varia-tion of the concentration of TLSs [23, 61]. In the first,TLSs are the sole defects in the lattice. In this case,increasing TLS concentration increases the strain in thesystem, and thus the coupling of τ -TLSs to the phononfield [13, 45]. In the second, TLS concentration is variedin a mixed lattice, in which strain is large already in theabsence of TLSs. Thus, in the second protocol elastic and electric dipole interactions strengthen equally withincreased TLS concentration, as a result of the reducedtypical distance between TLSs. However, in the first pro-tocol, in addition to the reduced distance between TLSs,the increased strain with TLS concentration results in adecreased ratio of the electric dipolar to elastic TLS-TLSinteractions. This is reflected in the temperature depen-dence of the sound velocity as fitted with the DOS ofEq. (4), with a smaller power µ as TLS concentration isincreased [Fig. 5(a)]. However, in the mixed crystal plot-ted in Fig. 5(b) strain is large and independent of TLSconcentration, leading to a similar and small value of µ at TLS concentrations of 4% and 7%. V. SUMMARY
We have considered TLSs in amorphous solids forwhich their mutual interactions are not much smallerthan the randomness in their bias energies. Such a sce-nario emerges naturally within the Two-TLS model, pro-vided that electric interactions dominate over elastic in-teractions. Data for BK7 and coverglass [49] attest forsomewhat larger electric dipolar than elastic interactionsin these materials, and typical parameters for amorphoussolids used in superconducting resonators suggest thatstronger dominance of the electric dipolar interactionsmay be expected. Experiments in disordered lattices fur-ther show correspondence between the relative strengthof the electric dipolar interactions and the temperaturedependence of the sound velocity in agreement with ourtheory.Our results suggest a microscopic origin for the powerlaw dependence of the single-particle TLS-DOS at lowenergies, as found experimentally, and for the result-ing anomalous exponents of the low-temperature thermalconductivity and specific heat and the temperature de-pendence of the acoustic velocity and dielectric constantat low-temperatures.Energy-dependent TLS-DOS, albeit weaker, is ob-tained also within the dipolar gap theory of the STM[29]. A comprehensive study of the relation between theacoustic and dielectric responses in various amorphoussolids could attest for the abundance of systems in whichdipolar interactions dominate over elastic interactions,and for the relevance of the dipolar gap theory and theTwo-TLS model in describing the low-energy propertiesof amorphous solids at low temperatures.
ACKNOWLEDGMENTS
SM acknowledges support from the Minerva founda-tion. AB and AM acknowledge partial support by theNational Science Foundation (CHE-1462075). AB ac-knowledges the support of LINK Program of NSF andLouisiana Board of Regents and MPIPKS (Dresden, Ger-many) Visitor Program in 2018 and 2019. MS acknowl-edges support from the Israel Science Foundation (GrantNo. 821/14 and Grant No. 2300/19).
Appendix A: Two-TLS model
While Hamiltonian (2) is standard in the theory ofinteracting TLSs [24], we now derive it as a low-energyeffective Hamiltonian of the Two-TLS model [13]. Thisallows the determination of the energy scales of the TLS-TLS interactions and of the random fields, and the ratiobetween the two energy scales for both cases of elasticdominated and electric dominated TLS-TLS interactions.The Two-TLS model was microscopically derived [13]and thoroughly validated [44–48] for disordered latticeswhich share the same universal low temperature phe-nomena with amorphous solids [2]. Here we assume thegeneralized validity of the Two-TLS model in describingTLSs in amorphous solids. Such a view point is sup-ported by: (i) TLS-dictated low-energy properties showthe same universal phenomena in disodered lattices andin amorphous solids [2], and careful experimental worksuggests that universality in both groups of systems isof the same origin [60, 62, 63] (ii) The Two-TLS modelderives the smallness and universality of phonon attenu-ation as is given by the universal dimensionless “tunnel-ing strength” [2, 6, 7] for both disordered lattices andamorphous solids (iii) the Two-TLS model was founduseful in explaining TLS pure dephasing and nonlinearabsorption in superconducting qubit and microresonatorcircuits, not accounted for by the STM [64–66].First considering only elastic interactions, the Two-TLS model divides TLSs into two groups, with bimodaldistribution of their interaction strengths with the strain,leading to the TLS-phonon interaction Hamiltonian [13] H int = − X i X α,β (cid:0) η i δ αβ + γ i w ,αβ τ zi + γ i s ,αβ S zi (cid:1) ε iαβ . (A1)Here, ε iαβ is the strain tensor at TLS i , and the sum over α and β runs over the three Cartesian coordinates x , y and z . τ -TLSs possess an intrinsically small interactionwith the strain, and as a result weak TLS-TLS interac-tions [13, 44–48]. Such TLSs constitute the predominantdegrees of freedom at low energies [13, 30, 44]; their stateis described by the Ising variable τ zi ( τ zi = ±
1) and itsweak coupling to the strain is given by the tensor γ i w ,αβ . S -TLSs are described by the Ising variables S zi ( S zi = ± | γ i s ,αβ | ∼ γ s , where g ≡ γ w /γ s ≈ .
02 [13, 44–48].The first term of Eq. (A1) describes a volume energy dueto the strain field which is independent on the orientationof defects, where usually η i . γ s .The density of states (DOS) of S -TLSs strongly dimin-ishes at low energies [13, 30, 44], and at low temperatures,for most purposes, these TLSs can be treated as frozenvariables having no dynamics. They then contribute an additional term to the energy of the same order of mag-nitude ( ∝ γ s ) as the volume term. By integrating out thephonon amplitudes, at lowest order perturbation theory,one obtains the Hamiltonian (2) as the low-energy effec-tive Hamiltonian of the system [13], with typical randomfields and interactions as are given in Eq. (9) in the maintext. Appendix B: Magnitude of the elastic and electricinteractions
The parameters c i , c ij ∼ O (1) in Eq. (9) contain theangular dependence of the random field and the interac-tion. Generally, the c ij parameters have a complicateddependence on the relative orientation and position ofthe TLSs, and can be regarded as normally distributedrandom variables [45]. Under this assumption one can es-tablish a connection between the interaction strength ex-pressed as the dimensionless parameter P h| ˜ J , el |i for theelastic interaction and P h| ˜ J , dip |i for the electric dipolarinteraction, assuming that only terms with the transversesound velocity v t are significant for the elastic interaction.Here ˜ J , el and ˜ J , dip are interaction constants for elasticand electric interactions, respectively.Then for the elastic interaction one can express the in-ternal friction as Q − = π P γ / ( ρv t ), where γ is the av-erage squared of the off-diagonal component of the TLS-strain interaction constant tensor, and the logarithmicslope of the temperature dependence of the sound ve-locity in the resonant regime as C el = P γ / ( ρv t ) [49].The average absolute value of TLS-TLS interactionconstant[12, 45] has been evaluated numerically assum-ing independent Gaussian distributions of elastic tensorcomponents, similarly to Ref. 67, and it can be expressedas P h| ˜ J , el |i ≈ . Q − = 1 . C el . Similar analysisfor the electric dipolar interaction yields P h| ˜ J , dip |i ≈ .
36 tan δ = 0 . C dip , where C dip is the slope of the log-arithmic temperature dependence of the dielectric con-stant. Consequently, the ratio of the two averaged inter-action constants r ≡ h| ˜ J , dip |i / h| ˜ J , el |i can be expressedas r = 0 .
36 tan δ/ (1 . Q − ) = 0 . C dip / . C el . Usingthe available experimental data for loss tangent and in-ternal friction we find for SiO that elastic interactionsdominate ( C el = 0 . − . · − [68], Q = 3 . · − [69],yields r = 0 . C dip =1 . · − , C el = 3 . · − yields r = 1 .
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