Transport and Optical Conductivity in the Hubbard Model:
A High-Temperature Expansion Perspective.
We derive analytical expressions for the spectral moments of the dynamical response functions of the Hubbard model using the high-temperature series expansion. We consider generic dimension as well as the infinite- limit, arbitrary electron density , and both finite and infinite repulsion . We use moment-reconstruction methods to obtain the one-electron spectral function, the self-energy, and the optical conductivity. They are all smooth functions at high-temperature and, at large-, they are featureless with characteristic widths of order the lattice hopping parameter . In the infinite- limit we compare the series expansion results with accurate numerical renormalization group and interaction expansion quantum Monte-Carlo results. We find excellent agreement down to surprisingly low temperatures, throughout most of the bad-metal regime which applies for , the Brinkman-Rice scale. The resistivity increases linearly in at high-temperature without saturation. This results from the behaviour of the compressibility or kinetic energy, which play the role of the effective carrier number. In contrast, the scattering time (or diffusion constant) saturate at high-. We find that to a very good approximation for all , with at high temperatures. The saturation at small occurs due to a compensation between the density-dependence of the effective number of carriers and that of the scattering time. The -dependence of the resistivity displays a knee-like feature which signals a cross-over to the intermediate-temperature regime where the diffusion constant (or scattering time) start increasing with decreasing . At high-temperatures, the thermopower obeys the Heikes formula, while the Wiedemann-Franz law is violated with the Lorenz number vanishing as . The relevance of our calculations to experiments probing high-temperature transport in materials with strong electronic correlations or ultra-cold atomic gases in an optical lattice is briefly discussed.
Electronic transport is one of the most poorly understood properties of strongly-correlated electron systems. A universally observed characteristic is the absence of resistivity saturation Gunnarsson et al. (2003). In contrast to systems dominated by the electron-phonon coupling Gunnarsson et al. (2003); Calandra and Gunnarsson (2002, 2003); Werman and Berg (2016), most correlated systems are ‘bad metals’ at high temperature Emery and Kivelson (1995); Hussey et al. (2004). Their resistivity exceeds the value which corresponds in the Drude-Boltzmann picture to a mean-free path of the order of the lattice spacing (). This characteristic Mott-Ioffe-Regel (MIR) resistivity (typically of the order of 100-300 cm in oxides) is smoothly crossed at high temperature and the resistivity remains metallic-like with .
Early discussions in the context of underdoped cuprate superconductors extrapolated the bad-metal behavior at high temperatures to a low-temperature state that is an unusual metal without quasiparticles Emery and Kivelson (1995). While the situation remains controversial for cuprates Mirzaei et al. (2013); Barišić et al. (), there is ample evidence that other transition metal-oxides which are bad metals at high temperature do become good Fermi liquids at low temperatures, the best characterized case being SrRuO Tyler et al. (1998); Mackenzie and Maeno (2003).
The bad-metal behavior and its implications for optical spectroscopy and photoemission have been discussed by Deng et al. Deng et al. (2013) within the dynamical mean-field theory (DMFTGeorges et al. (1996)) (see also Refs. Merino and McKenzie, 2000; Xu et al., 2013). It was demonstrated that quasiparticle excitations disappear only at temperatures well above the low temperature Fermi liquid scale below which the resistivity is quadratic in temperature. For the large- doped Hubbard model, the scale at which the MIR value is reached was identified as the Brinkman-Rice scale, of order with the doping level counted from half-filling and the (half-)bandwidth. The asymptotic high-temperature state was not, however, fully characterized in that work (for an early study of high- transport within DMFT, see Refs. Pálsson and Kotliar, 1998; Palsson, 2001) . Vučičević et al. Vučičević et al. (2015), also working within the DMFT framework, proposed a connection between bad metallic behaviour and Mott quantum criticality and argued that the temperature at which the MIR value is reached coincides with the quantum Widom line associated with the metal-insulator transition occurring at low temperatures. Recently, incoherent transport within the bad-metal regime became the subject of renewed attention in the context of ‘holographic’ approaches to hydrodynamics and transport in quantum fluids, see e.g. Refs. Hartnoll, 2009 for reviews and Ref. Hartnoll, 2015; Pakhira and McKenzie, 2015 for a recent discussion of the incoherent regime of transport in this framework.
In this article, we address these issues, and more generally the behaviour of transport and response functions, from a high-temperature perspective. We setup a general formalism for the high-temperature expansion of single-particle Green’s functions and two-particle response functions and apply this formalism to the Hubbard model. The presented formalism allows for the analytical evaluation of moments of these correlation functions and allows us to make general statements on the behaviour of transport and optical conductivity in the high-temperature bad-metal regime, which shed light on transport mechanisms in this regime. High-temperature series for thermodynamic properties of the Hubbard model have been developed and applied by several authors Plischke (1974); Kubo and Tada (1983, 1984); Pan and Wang (1991); Bartkowiak and Chao (1992); Putikka et al. (1998); Scarola et al. (2009); De Leo et al. (2011), but remarkably little previous work has been devoted to high-temperature series for dynamical response functions and transportPairault et al. (2000). In Ref. (Khatami et al., 2014), the high-temperature series was applied to high orders to the single-particle spectral function of the two-dimensional infinite- Hubbard model. In Ref. (Khatami et al., 2013), the results were compared favorably to the “Extremely Correlated Fermi Liquid” (ECFL) theory Shastry (2011); Hansen and Shastry (2013); Shastry and Perepelitsky (2016) for the tJ model. This method may also have applications to understanding the conductivity in models that display Many-body Localization Basko et al. (2006); Pal and Huse (2010).
In the limit of infinite dimensions, we are able to obtain quantitative results, using moment-reconstruction methods, for the resistivity, thermal transport coefficients and frequency-dependence of the optical conductivity of the Hubbard model. These results are successfully compared to solutions of the DMFT equations using the numerical renormalization-group method.
This article presents the formalism and its applications in details. In order to facilitate its reading, we provide in Sec. II an overview of its organization and of the main results.
Ii Overview of main results and outline
ii.1 General formalism for high-temperature expansion of dynamic correlations.
This article is based on a general formalism for expanding correlation functions as a series in inverse powers of temperature. The general formula for the spectral density associated with the correlation function of a two-particle operator reads, when is dimensionless:
In this expression, is an energy scale which can be conveniently chosen to be the hopping amplitude from the non-interacting part of the Hamiltonian. Unless specified otherwise we set . The ’s are dimensionless functions of momentum and normalized frequency. For the Hubbard model, they depend solely on the density , while for the finite- model they also depend on the dimensionless coupling . The full frequency dependence of the functions cannot be derived in general from a high-temperature series approach, reflecting the fact that the low-frequency long-time regime is not directly accessible in this framework. Instead, we derive in Sec. III a general formula for the moments of these functions, namely . Note that being a bosonic correlator, is an even function of frequency, and therefore odd moments are zero. For cases where enough moments can be calculated, we will attempt to approximately reconstruct the frequency-dependence of .
The expression of the moments is obtained at a fixed value of the electron density, which is defined per site as ( at half-filling). The chemical potential has a high-temperature expansion:
Since at high temperatures,
has a finite high- limit. The dominant term is given by the atomic limit. For it reads
while for finite it is
A similar high- expansion can be performed for the single-particle Green’s function and self-energy. The expansion applies for in the vicinity of , and has the general form:
In this expression and are the spectral functions associated with the Green’s function and self-energy, respectively. In Sec. III we derive a general expression for the moments , from which the moments of can also be obtained. The spectral functions and have a non-trivial frequency-dependence at , given by and . Hence, the lower and upper Hubbard bands in the single-particle spectrum have a non-trivial shape and a finite width in this limit and do not simply reduce to the atomic limit, as previously pointed out by Palsson and Kotliar Palsson (2001); Pálsson and Kotliar (1998).
Using these general formulas, we have derived explicit analytical expressions for several moments of the current-current correlation function (conductivity) , and . In the limit of infinite dimensions we managed to derive a larger number of moments, allowing for approximate reconstruction of some of the dynamical correlation functions and comparison to numerical solutions of the DMFT equations (Sec. VIII). A summary of the moments calculated is given in Table 1.
ii.2 High-temperature transport and optical conductivity for : general results.
In Sec. V we show, using an inspection of the general formalism and some simplifications applying at , that the high-temperature expansion for the optical conductivity takes in this case the following form:
In this expression, can be taken to be of order , with a lattice spacing (corresponding in a quasi two-dimensional system to a sheet resistance of one quantum per plaquette). In the proximity of a Mott insulator (i.e., for small doping), it becomes equivalent to the Mott-Ioffe-Regel (MIR) value of the conductivity. In the rest of the paper, we work in a system of units in which , i.e. we normalize the conductivity to .
This expression calls for the following remarks:
In the limit the general expansion (1) simplifies and only odd powers of remain.
The optical conductivity in the high- incoherent regime at is a smooth featureless function of frequency involving only the scale .
The functions are dimensionless and depend only on density . We provide in Sec. V.2 analytical expressions of the zeroth moments of and , as well as the second moment of , on a -dimensional cubic lattice. Due to the -sum rule Maldague (1977); Bari et al. (1970); Sadakata and Hanamura (1973); Jaklič and Prelovšek (2000); Shastry (2006) the zeroth moments are simply related to the corresponding high- expansion of the kinetic energy Calandra and Gunnarsson (2003).
The functions have non-singular behavior (i.e. do not vanish) in the limit of a half-filled band. Expression (7) is written in a way that captures the dominant singularities in as the Mott insulator at is approached. In this limit, the conductivity vanishes, as expected.
In the (empty band) limit, we find that the zeroth moment of the optical conductivity vanishes linearly in , while the higher order moments vanish quadratically in . This has consequences for dc-transport that will be summarized below.
Hence, the dc-resistivity has the general expansion for :
where , and the ’s are dimensionless coefficients depending on density, but which are non-singular in the limit, and in the case of in the limit as well. Based on these general expressions, we can draw the following physical conclusions:
The resistivity at high- has a linear dependence on temperature, with a slope that diverges as the Mott insulator is approached, .
In contrast, the slope of the -linear dc resistivity reaches a finite value as (i.e. at a fixed , the resistivity saturates in the low-density limit). We show that this suprising result can be interpreted as a compensation between the density-dependence of the effective number of carriers and that of the scattering time. Furthermore, we find that to a very good approximation (see Figs. 9 and 11).
thus the saturated resistivity sets the overall size of the diverging resistivity. This is the case not only in the asymptotic limit ( being the bare bandwidth), but even in the experimentally relevant temperature range .
The first deviations from linearity, as is reduced, occur for , of order . At that scale, the resistivity is of order . Hence, close to the Mott insulator, the first deviations from linearity occur at a scale where the resistivity is still much larger than the Mott-Ioffe-Regel limit, i.e. well into the ‘bad-metal’ regime.
This is consistent with the observation made in Ref. (Deng et al., 2013), that the scale at which the MIR limit is reached is the Brinkman-Rice scale .
These properties of the resistivity can be rationalized by noting that the dominant -dependence at high temperature is entirely controlled by that of the effective carrier number. To see this, we can use either of the following expressions for the conductivity:
In the first expression, is the electronic compressibility and the diffusion constant defined from Fick’s law . Combining the latter with yields the above expression for , which expresses the Einstein relation Kokalj (2015); Hartnoll (2015). The second expression, closer in spirit to the standard analysis of the conductivity Kohn (1964); Shastry and Sutherland (1990) and the Drude formula for the complex conductivity , relies on the fact that the integrated spectral weight of the optical conductivity is proportional to the kinetic energy (f-sum rule Maldague (1977); Bari et al. (1970); Sadakata and Hanamura (1973); Jaklič and Prelovšek (2000); Shastry (2006)). Note that for , all the spectral weight is contained in an energy range of order the bandwidth. Hence, the (absolute value of) the kinetic energy can be interpreted as setting the effective number of carriers , which vanishes in both the low-density and the Mott insulating limits. In this view, the transport scattering time can be defined as:
The compressibility and kinetic energy have, up to prefactors, the same high- expansion:
Hence, the -linear behaviour of the resistivity at high- is directly related to the vanishing of the compressibilityKokalj (2015), or of the kinetic energy (effective carrier number) as , as pointed out by Calandra and Gunnarsson Calandra and Gunnarsson (2003). In contrast, the diffusion constant and the transport scattering rate both reach a finite limit as . It is in that sense that one can talk of ‘saturation’ at high temperatureKokalj (2015). Furthermore, the divergence of the slope of the -linear resistivity upon approaching the Mott insulator at is also captured by the vanishing of or . Note that, in contrast, in the low-density limit both the diffusion constant and the scattering time diverge as in order to insure that this slope approaches a finite value. Putting things together, we find that , where at high temperatures. This formula is surprising and works for all densities between and . This is a characteristic of the high-temperature (bad-metal) regime .
In Sec. IX we address the high- expansion of the thermal conductivity and thermoelectric Seebeck coefficient. The latter reaches at high- the value given by Heikes formula. We show furthermore that the Wiedemann-Franz law does not apply at high-temperature, and that the Lorenz number vanishes as .
ii.3 High-temperature transport and optical conductivity for in large dimensions, and DMFT
In Sec. VII, we consider the infinite- Hubbard model in the limit of large dimensions. In this limit, the self-energy becomes momentum-independent:
where the half-bandwidth is kept finite. The functions are shown to be even (odd) in for even (odd). In addition, all moments of w.r.t to vanish linearly in as . However, only the odd moments vanish linearly in as . In the finite-dimensional case, all moments continue to vanish linearly in as , while none of the moments vanish as .
In this limit, we are able to calculate a larger number of moments (see table 1) for both the optical conductivity and the self-energy. This allows us to reconstruct explicitly these correlation functions using two complementary reconstruction methods, the maximum entropy method (MEM) and Mori’s relaxation function approach for the optical conductivity. The latter can also be obtained from the reconstructed self-energy, using the fact that vertex corrections vanish in infinite dimensions Khurana (1990) so that only a bubble graph involving the convolution of two one-particle Green’s functions has to be evaluated (see Eq.(96)).
Furthermore, we obtained full numerical calculations of real-frequency correlation functions (self-energy and optical conductivity) by solving the DMFT equations using a numerical renormalization-group algorithm (NRG), and compare the results to the analytical high- expansion in Sec.VIII. In order to reach this goal, we had to take special care in adapting the current NRG codes, using a very narrow kernel for the broadening of raw spectral data into a continuous spectral function on the real-frequency axis. This leads to severe underbroadening of spectral function and oscillatory artifacts, however intergrated quantities such as optical and dc conductivity converge with the decreasing kernel width to the exact high-temperature results.
An example of such a comparison for the optical conductivity is displayed in Fig. 1. The shape of the optical conductivity displayed there confirms the qualitative points made above, and the agreement between the DMFT-NRG results and the high- expansion is seen to be excellent.
Using moment reconstruction methods, we were also able to calculate both the leading and sub-leading coefficients , , and in the high-temperature expansion Eq. (8) of the dc-resistivity. The resulting high- approximation to the resistivity is compared to the DMFT-NRG results in Fig. 2, for several densities. Remarkably, the series reproduces the NRG curves for , i.e. essentially throughout the bad-metal regime, well below its a priori range of applicability. The NRG curves confirm our finding that the slope of the resistivity in the linear high- regime saturates as and diverges like as . In Sec.VIII.3, we furthermore present a physical interpretation of the ‘knee-like’ feature displayed by the NRG resistivity curves at lower temperature, and previously noted in e.g. Ref. (Deng et al., 2013) (see also Jaklič and Prelovšek (1994, 1995)). We show that above the knee, the resistivity is mostly controlled by the temperature dependence of the effective carrier number (as given by the compressibility or kinetic energy according to Eq. (13)) while below this scale it is mostly controlled by the -dependence of the scattering rate or diffusion constant. See Sec. VIII.3 and Fig. 17 therein, as well as the concluding section, for a discussion of the transport mechanism in the different regimes.
In the very high-temperature regime, the DMFT results are most reliably obtained using the interaction expansion continuous-time Quantum Monte-Carlo Gull et al. (2011) rather than the NRG. This is also the regime in which the series works the best. On the level of the imaginary-time Green’s function, we find excellent agreement between the QMC results and the series. Performing the analytic continuation to obtain the self-energy on the real frequency axis is somewhat problematic using the QMC data. Nonetheless, we find good agreement between the two methods even for this quantity (see sec. VIII.2).
ii.4 High-temperature transport and optical conductivity for finite
For the finite- Hubbard model, the spectral function has two peaks in the high-temperature limit. In the parameter range with finite , the lower and upper Hubbard bands can be individually expanded in the form of Eq. (6):
The moments are themselves a series in the expansion parameter . The coefficients of this series depend on the density and . Upon setting , the lower Hubbard band becomes equal to the spectral function of the infinite- Hubbard model.
In sec. VI.1, we compute the zeroth through second moments of the lower and upper Hubbard bands in the case of a -dimensional hypercubic lattice, for arbitrary filling . In sec. VI.2, we use the results of sec. VI.1 along with the “bubble” formula, to address the temperature-dependence of the dc resistivity. Finally, in Sec. X, we address the question to what extent the DMFT becomes exact in the high-temperature limit and in what sense is the physics ‘local’ at high temperature. We find that while the thermodynamic potential becomes exact, the same can only be said for the zeroth moment of the local self-energy. Moreover, the self-energy has non-local contributions which survive the high- limit, but are not captured by DMFT. DMFT becomes accurate however when both a high-temperature and a high-frequency expansion are performed.
Iii Behavior of 2-particle correlation functions in the high-T limit.
iii.1 Lehmann Representation
Consider the two-particle correlator
where , and , , or . These are the particle, current, and spin operators respectively, and represents a spatial direction. We also consider the Fourier transform of , defined as
Here, . We can also write in terms of its spectral representation
where is the spectral function corresponding to .
We show quite generally, independent of the specific form of the Hamiltonian, that has the following form when the temperature is the highest energy scale in the problem:
Here, the are even functions of the frequencyKadanoff and Martin (1963), independent of the temperature, and hence this is an expansion in . can be any unit of energy, and is most conveniently taken to be a characteristic energy unit of the Hamiltonian. For example, it can be taken to be the hopping. If , , while in the case that , or , .
In the Lehmann representation, is written as Fetter and Walecka (2003)
Here, is the partition function, and and are indices that run over all of the eigenstates. We assume the system to have inversion symmetry, and therefore . We now expand the term in the brackets on the RHS of Eq. (20) in powers of , the inverse temperature. After some simplifications, this yields
iii.2 Short-time Expansion
In this section, we derive a closed form expression for the moments of the functions appearing in Eq. (19). We begin by writing the Hamiltonian in the following way
Here, is independent of the temperature, and the only part of the Hamiltonian which depends on the temperature is the chemical potential . Also, we will assume that conserves particle-number, therefore . In order to be able to control the density through the chemical potential, it must be the case that as , is proportional to . Therefore, we can write as a series in the inverse-temperature
where , and . Here, the partition function has been eliminated, and only “connected” diagrams have been retained in the expectation values. The meaning of connected diagram in this context is that we keep only terms which are intensive, i.e. do not scale with the size of the lattice (see Ref. (Perepelitsky, 2013)). Expanding the time-dependent exponentials, we obtain
The RHS of Eq. (27) can now be expanded in powers of . This expansion, performed in Appendix A, will allow us to derive a closed-form expression for the moments of . First, we derive some additional useful formulae.
Eq. (28) is the high-frequency expansion of Eq. (18), while Eq. (29) is Eq. (18) evaluated at . Plugging Eq. (19) into Eqs. (28) and (29) and comparing with Eq. (117) yields the moments of . For even,
while for odd, .
Iv Behavior of 1-particle correlation functions in the high-T limit.
iv.1 Lehmann Representation
Consider the single-particle Green’s function
Its Fourier transform, defined as where , can be written in terms of the Dyson(-Mori) self-energy :
Here, is obtained from the high-frequency limit of the Green’s function, while is some unspecified dispersion. In the case of the finite- Hubbard model , and (Dyson self-energy), where is the dispersion of the lattice. In the case of the infinite- Hubbard model and (Dyson-Mori self-energy). Both the and can be written in terms of their respective spectral densities:
We will show that when is the highest energy scale in the problem, and can be expanded in the following series in :
Note that in order to achieve this expansion, the frequency must be re-centered around . The new frequency variable measures the displacement of from .
The Lehmann representation for the spectral density is Fetter and Walecka (2003)
We now define , where and are dimensionless. Plugging these in, we find that
Expanding all exponentials in powers of , we find that has the form of Eq. (35). For example,
iv.2 Short-time expansion
Just as was the case for , can be expanded in power of . This expansion is derived starting from Eq. (43) in Appendix B. Its final form is given in Eq. (LABEL:Gk3). Making the substitution in Eq. (33), performing a high-frequency expansion, plugging in Eq. (35), and comparing with Eq. (LABEL:Gk3) yields the moments of the :
V Infinite- Hubbard model
The infinite- Hubbard model Hamiltonian is
The Hubbard operator projects the state onto the state , where and can be any of the three allowed states ,, or . acts only on the site . We can write this Hamiltonian as , where is the hopping term in the Hamiltonian. In the notation of sec.(III), , and . A special feature of this model is that
where for odd, and for even. This is due to the fact that an odd number of hops cannot return the system back to its initial configuration. Moreover, the two identical operators must contribute an even number of hops. Furthermore, the real space expectation value
where is the Manhattan distance between site and site . This is the case since the number of hops required to get from to is equal to the separation between them, while the number of remaining hops must be even for the system to return to its initial configuration.
v.1 General form of the conductivity
To discuss the conductivity, we choose , in which case . Applying Eq. (48) with yields
We will see below that the moments of are each proportional to exactly one power of . Explicitly pulling the factor out of the functions