A Generic Microscopic Theory for the Universality of TTLS Model Meissner-Berret Ratio in Low-Temperature Glasses
Tunneling-two-level-system (TTLS) model has successfully explained several low-temperature glass universal properties which do not exist in their crystalline counterparts. The coupling constants between longitudinal and transverse phonon strain fields and two-level-systems are denoted as and . The ratio was observed to lie between and for 18 different kinds of glasses. Such universal property cannot be explained within TTLS model. In this paper by developing a microscopic generic coupled block model, we show that the ratio is proportinal to the ratio of sound velocity . We prove that the universality of essentially comes from the mutual interaction between different glass blocks, independent of the microscopic structure and chemical compound of the amorphous materials. In the appendix we also give a detailed correction on the coefficient of non-elastic stress-stress interaction which was obtained by Joffrin and LevelutJoffrin1976 ().
It has been more than 50 years since the first experimentZeller1971 () by Zeller and Pohl showed that at ultra-low temperatures below 1K, the thermal and acoustic properties of amorphous solids (glasses) behave strikingly different from that of the crystalline counterparts. Anderson, Halperin and Varma’sAnderson1972 () group and PhillipsPhillips1987 () independently developed a microscopic phenomenological model which was later known as the tunneling-two-level-system (TTLS) model, to explain the anomalous properties of low-temperature glasses. It successfully explained several universal properties of glasses which were never observed in crystalline solids, e.g., linear heat capacity (two orders of magnitude greater than the contribution of heat capacity from conducting electrons), saturation, echoes, quadratic heat conductivity etc..
In TTLS model, people assume that there are a group of tunneling-two-level-systems randomly embedded in the glass materials. The effective Hamiltonian of low-temperature glass in TTLS model is the summation of long wavelength phonon Hamiltonian, a set of (non-elastic) two-level-system Hamiltonian and the coupling between phonon strain fields and two-level-systems. The coupling constant between longitudinal phonon and two-level-system is denoted as ; for transverse phonon, it is denoted as . The coupling constants and are adjustable parameters in TTLS model, and they were assumed to have no specific relation. However, in 1988 Meissner and BerretBerret1988 () measured 18 different kinds of glass materials below the temperature of 1K, including chemically pured materials (e.g., a-SiO), chemically mixed materials (e.g., BK7) and organic materials (e.g., PMMA). They find that the ratios between and are not arbitrary: they range from 1.44 to 1.84, and most of them are around 1.51.6. Such observation suggests that the ratios are quite universal, regardless of the chemical compounds and microscopic structures of amorphous materials. TTLS model cannot explain this universality, because the model is based on the coupling constants. Therefore, we believe that there must be a more general model to describe the universal properties of low-temperature glasses, including the universal ratio . In the rest of this paper, we use “Meissner-Berret Ratio” to stand for “the ratio between and ”.
Besides this problem, there are a number of other problems in TTLS model. First, while TTLS successfully explained several universal propeties of amorphous solids below 1K, there are more universalities which cannot be explained within TTLS model around the temperature of 10KLeggett2011 (), e.g. the plateau of thermal conductivity. Second, the TTLS model itself contains too many adjustable parameters. Experimental results could be fitted by adjusting these parameters. Third, the model lacks the consideration that as the interaction with phonon strain field, TTLS must generate a mutual RKKY-type interactionJoffrin1976 (). Taking this virtual-phonon exchange interaction into account may not only change current theoretical results, but also question the validity of TTLS model.
The purpose of this paper is to focus on the universality of Meissner-Berret ratio (the universal for various kinds of glasses) by developing a theory of coupled generic blocks. We start by expanding non-elastic part of glass Hamiltonian in orders of intrinsic and external phonon strain fields and to calculate the resonance phonon energy absorption for longitudinal and transverse external phonon fields. Within TTLS model, the resonance energy absorption per unit time is proportional to the square of coupling constants : . In our generic coupled block model, the resonance energy absorption is proportional to the imaginary part of non-elastic susceptibility . Therefore, if we want to prove the universal property of , we are actually proving the universal property of . We combine a set of single blocks together to form a super block of glass. We allow the virtual phonons to exchange with each other to set up the RKKY-type many-body interaction between different single blocks. By putting in virtual phonon exchange interaction, we set up the renormalization recursion relation of resonance phonon energy absorption between single block and super block glasses. We repeat the real space renormalization procedure from starting microscopic length scale to obtain the resonance energy absorption at experimental length scales, and try to prove the universal ratio of . Since the RKKY-type many-body interaction is independent of materials’ microscopic structure and chemical compound, we believe that the ratio at experimental length scale will be able to explain the material-independent Meissner-Berret ratio.
The organization of this paper is as follows: in section 2 we set up the main goal of this paper, the problem of universal Meissner-Berret ratio in TTLS model. We expand the general glass Hamiltonian in orders of intrinsic and external phonon strain fields, and introduce the most important concepts of this paper, namely the non-elastic stress tensor and non-elastic susceptibility. In section 3 we set up the renormalization recursion relation for the resonance phonon energy absorption between single block and super block glasses. In section 4 we repeat the renormalization process from microscopic length scale to experimental length scale and carry out the Meissner-Berret ratio, . We prove at experimental length scale, the Meissner-Berret ratio is proportional to the sound velocity ratio . We also use the least square method to investigate the statistical significance between the theory and experiment for the data of 13 amorphous materials listed in section 4. In section 5 we give a detailed discussion on the resonance phonon energy absorption contribution from the electric dipole-dipole interaction for dielectric amorphous solids. The contribution to Meissner-Berret ratio from electric dipole-dipole interaction is renormalization irrelevant. In Appendix (A) we give a detailed correction on the coefficient of non-elastic stress-stress interaction , which was first obtained by Joffrin and LevelutJoffrin1976 ().
Ii The Set up of Meissner-Berret Ratio Problem
ii.1 The Definitions Non-elastic Hamiltonian, Stress Tensor and Susceptibility
Let us consider a block of amorphous material. Since our purpose is to discuss the universal property of “Meissner-Berret ratio” in amorphous materials below the temperature of 10K, we want to begin our investigation from the famous tunneling-two-level-system model (TTLS model)Phillips1987 (). In this model we assume that there are a group of TTLSs randomly embedded in the glass material, with the location of the -th TTLS at . With the presence of external phonon strain fields , the effective Hamiltonian of amorphous material in TTLS model is the summation of long wavelength phonon Hamiltonian , a group of tunneling-two-level-systems Hamiltonian, the couplings between two-level-systems and phonon intrinsic strain fields , and the couplings between two-level-systems and phonon external strain fields :
where the first, second, third and fourth terms stand for the long wavelength phonon Hamiltonian (we will also call it “purely elastic Hamiltonian, ”), the Hamiltonian of a group of two-level-systems, the couplings between two-level-systems and intrinsic phonon strain fields , and the couplings between two-level-systems and external phonon strain fields . The two-level-systems Hamiltonian are written in the representation of energy eigenvalue basis, with the energy splitting ; and are thr diagonal and off-diagonal matrix elements of the coupling between two-level-system and phonon strain field at location , and by definition they are no greater than 1; is the local intrinsic phonon strain field at the position of the -th two-level-system; is the product of external phonon strain field amplitude and external phonon strain field wave number ; is the frequency of external phonon strain field; are the coupling constants between two-level-systems and longitudinal/transverse phonon strain fields.
Because the external phonon strain fields are coupled to two-level-systems, if the energy splitting of a certain two-level-system, matches , then this two-level-system can resonantly absorb external phonon energy. By using Fermi golden rule, the resonance phonon energy absorption per unit time is proportional to the square of coupling constants:
where stands for the resonance phonon energy absorption for longitudinal external phonon fields, while stands for the transverse phonon energy absorption. We assume that the phonon strain is identical for longitudinal and transverse external phonon strain fields. In TTLS model, are assumed to be independent of each other. In other words, they have no specific relation.
However, in 1988, Meissner and BerretBerret1988 () measureed 18 different kinds of glass materials, including chemically pured materials (for example, amorphous SiO), chemically mixed materials (for example, BK7) and organic materials (for example, PMMA and PS). They find that the ratio between longitudinal and transverse coupling constants ranges from 1.44 to 1.84. Most of the ratios are around 1.51.6. In the rest of this paper, let us name the ratio between and as “Meissner-Berret ratio”. TTLS model cannot explain such universality, because the model is based on the coupling constants. Therefore, we would like to believe, that there must be a more general model to describe the universal properties of low-temperature glasses, including the universal Meissner-Berret ratio.
In this subsection we want to set up a multiple-level-system model from the generalization of 2-level-system model. At this stage of setting up our model, we have not applied any external phonon strain field yet. We will consider external phonon strain fields in subsection 2(C). We begin our discussion by considering a single block of glass with the length scale much greater than the atomic distance . In the subsection 2(B), we will combine a group of such single blocks to form a “super block”. We will consider the RKKY-type interaction between these single blocks, which is generated by virtual phonon exchange process. For now, we do not consider RKKY-type interaction and focus on the Hamiltonian of single block glass only.
We further define intrinsic phonon strain field at position : if denotes the displacement relative to some arbitrary reference frame of the matter at point , then strain field is defined as follows
We write down our general glass Hamiltonian as . Let us separate out from the glass general Hamiltonian , the purely elastic contribution . It can be represented by phonon creation-annihilation operators as follows:
where represents phonon polarization, i.e., longitudinal and transverse phonons.
Subtracting the purely elastic part of Hamiltonian , we name the left-over glass Hamiltonian as “the non-elastic part of glass Hamiltonian, ”. We expand the left-over Hamiltonian up to the first order expansion of long wavelength intrinsic phonon strain field. We name the coefficient of the first order expansion to be “non-elastic stress tensor ”, defined as follows:
Now let us compare Eq.(II.1) with TTLS model Hamiltonian: the zeroth order expansion of non-elastic Hamiltonian with respect to strain field , , is the generalization from two-level-system Hamiltonian to multiple-level-system Hamiltonian; non-elastic stress tensor is the multiple-level generalization of the matrix which couples to phonon strain field in TTLS model. In the rest of this paper, we denote to be the non-elastic Hamiltonian excluding the coupling between intrinsic phonon strain and non-elastic stress tensor . We denote to be the non-elastic Hamiltonian including the stress tensor –intrinsic phonon strain field couplings (see the second equation of Eq.(II.1)).
Let us denote and to be the -th eigenstate and eigenvalue of the non-elastic Hamiltonian . Such a set of eigenbasis is a generic multiple-level-system. Now we can define the most important quantity of this paper, namely the non-elastic stress-stress susceptibility (i.e., linear response function). Let us apply an external infinitesimal testing strain, . The non-elastic Hamiltonian of amorphous material, , will provide a stress response . Then we are ready to define the non-elastic stress-stress susceptibility (complex response functionAnderson1986 ())
In the rest of this paper we will always use , , and to represent non-elastic Hamiltonians , , susceptibility and stress tensor respectively, while we use , and to represent the elastic Hamiltonian, susceptibility and stress tensor, respectively. In Eq.(II.1) the stress response of non-elastic Hamiltonian, , is defined as follows: (please note from now on we use to stand for )
where is the time-dependent partition function of non-elastic Hamiltonian. With the presence of external testing strain field , the amorphous material receives a time-dependent perturbation . In the representation in which is diagonal, the perturbation has both of diagonal and off-diagonal matrix elements. The diagonal matrix elements of external perturabtion shift the energy eigenvalues: , resulting in the shifts of probability function and partition function: , . The off-diagonal matrix elements change the eigenstate wavefunctions: , where is the wavefunction in the interaction picture, and is the stress tensor operator in the interaction picture: .
We further define the space-averaged non-elastic susceptibility for a single block of glass with the volume , which will be very useful in later discussions,
It is very useful to apply the assumption in the rest of this paper, that our amorphous material is invariant under real space SO(3) rotational group. Therefore, the non-elastic susceptibility obeys the generic form of an arbitrary 4-indice isotropic quantity: , where is the non-elastic part of glass compression modulus and is the non-elastic shear modulus.
Where is the partition function, and we set . Please note that according to the definition of non-elastic susceptibility in Eq.(II.1), the imaginary part of non-elastic susceptibility is negative for positive . Because for arbitrary quantum number we always have , the definition of in Eq.(II.1) is only valid when ; when or , in Eq.(II.1) one of the delta-functions will vanish. Therefore when or , the imaginary part of non-elastic susceptibility is simplified as follows,
Eq.(II.1) is the supplemental definition of the imaginary part of non-elastic susceptibility. It is convenient to rewrite the imaginary part of non-elastic susceptibility in Eq.(II.1) into the “imaginary part of reduced non-elastic susceptibility ” as follows, for future use:
Please note, that by definition is also a negative quantity for . Again, for an arbitrary isotropic system, the imaginary part of reduced non-elastic susceptibility must satisfy the genetic isotropic form as well,
The newly-defined quantities are negative as well. Please note that we use to stand for the imaginary part of reduced non-elastic compression/shear moduli. The real part of reduced non-elastic susceptibility can be obtained by Kramers-Kronig relation from the imaginary part of it.
ii.2 Virtual Phonon Exchange Interactions
Within single-block considerations, non-elastic stress tensor and non-elastic part of glass Hamiltonian are simply generalizations from 2-level-system to multiple-level-system. There is not much difference between the multiple-level-system model and TTLS model. However, if we combine a set of single blocks together to form a “super block”, the interactions between single blocks must be taken into account. Since the non-elastic stress tensors are coupled to intrinsic phonon strain field, if we allow virtual phonons to exchange with each other, it will generate a RKKY-type many-body interaction between single blocks. This RKKY-type interaction is the product of stress tensors at different locations:
where the coefficient was first derived by Joffrin and LevelutJoffrin1976 (). We give a further correction to it in Appendix (A).
where is the unit vector of , and runs over cartesian coordinates. In the rest of this paper we will call Eq.(13) non-elastic stress-stress interaction. In the rest of this paper we always use the approximation to replace by for the pair of the -th and -th blocks, when denotes the center of the -th block, and that is the uniform stress tensor of the -th block. Also, from now on we use to denote the intrinsic phonon strain field located at the -th block. By combining identical single blocks to form a super block, the Hamiltonian without the presence of external phonon strain fields is written as
From now on we apply the most important assumption, that these space-averaged stress tensors are diagonal in spacial coordinates: , which means for the stress tensors at different locations, they lose spacial correlations.
ii.3 Glass Non-elastic Hamiltonian with the Presence of External Strain field
Next, let us consider the glass Hamiltonian with the presence of external weak strain field as a perturbation. Please note that we have already used to denote the intrinsic strain field, in this section we use to stand for the external strain field. We further use to represent the external phonon strain field at the -th single block of glass. It seems that the Hamiltonian Eq.(16) simply adds a stress-strain coupling term . However, more questions arise with the appearance of external phonon strain field.
First of all these non-elastic stress tensors might be modified. A familiar example is that external phonon strain field can modify electric dipole moments by changing relative positions of positive-negative charge pairs (to the leading order of strain): where are cartesian coordinates, and is the external phonon field. In principle we need to obtain the modification of stress tensors, to the leading order in to calcuate the resonance phonon energy absorption correction. However, we only know the qualitative behavior of in orders of external phonon strain field is . Therefore, we are only able to estimate the length scale dependence of the resonance phonon energy absorption correction from the contribution of . We will give the qualitative result of this correction in section 3, Eq.(III). In section 4 we will show that this correction of resonance phonon energy absorption is renormalization irrelevant at experimental length scales via scaling analysis.
There is a second problem arising from the presence of external phonon strain field: the relative positions of glass single blocks can be changed, resulting in the modification of non-elastic stress-stress interaction coefficient . To the first order expansion in the external phonon strain field, the modification of is given as follows,
where , , , , is the angle between and , and is the unit vector of . Finally the glass super block Hamiltonian with the presence of external phonon strain field is given by
Iii Resonance Phonon Energy Absorption of Super Block Glass
According to the previous discussions, in TTLS model the resonance phonon energy absorption per unit time is proportional to the coupling constant squared: . In this section we want to use our generic coupled block model (i.e., the Hamiltonian in Eq.(II.3)) to calculate the same quantity.
First of all, let us consider the -th single-block glass with the dimension , with the unperturbed non-elastic Hamiltonian and time-dependent perturbation , so the total Hamiltonian of single block glass is . We denote and to be the -th eigenstate and eigenvalue of unperturbed Hamiltonian . Thus the single-block resonance phonon energy absorption rate is , where is the single block interaction picture wavefunction, and and are single block interaction picture operators. For an arbitrary operator , the single block interaction picture operator is . The resonance phonon energy absorption per unit time of single-block glass is therefore given by
where is the product between external phonon strain field amplitude and phonon field wave number; is the frequency of external phonon field. In Eq.(III), according to the negativity of , the single block energy absorption rate is positive. According to the argument by D. C. Vural and A. J. LeggettLeggett2011 (), and the experiment by R. O. Pohl, X. Liu and E. ThompsonPohl2002 (), we assume that below the temperature of 10K and within a certain range of frequency , the imaginary part of reduced non-elastic susceptibilities are independent of frequency . In section 4, Eq.(29) we will discuss the order of magnitude of in details. Therefore, in our model, within single block considerations, the resonance phonon energy absorption is proportional to the imaginary part of reduced non-elastic susceptibility: . If we want to prove the universal Meissner-Berret ratio, we are actually required to prove the universal property of in our model.
Let us combine glass non-interacting single blocks together to form a super block glass with the dimension . Such a group of non-interacting single blocks has the non-elastic part of glass Hamiltonian , eigenstates and eigenvalues , where and stand for the -th eigenstate and eigenvalue for the -th single block Hamiltonian . The partition function of these non-interacting single blocks is . We combine them to form a super block, and turn on non-elastic stress-stress interaction . We assume is relatively weak compared to , so it can be treated as a perturbation. Let us denote and to be the -th eigenstate and eigenvalue of super block static Hamiltonian ( ). Their relations with and are given as follows,
Finally, we turn on the time-dependent perturbation induced by external phonon: