Conservation laws, integrability and transport in one-dimensional quantum systems
In integrable one-dimensional quantum systems an infinite set of local conserved quantities exists which can prevent a current from decaying completely. For cases like the spin current in the model at zero magnetic field or the charge current in the attractive Hubbard model at half filling, however, the current operator does not have overlap with any of the local conserved quantities. We show that in these situations transport at finite temperatures is dominated by a diffusive contribution with the Drude weight being either small or even zero. For the model we discuss in detail the relation between our results, the phenomenological theory of spin diffusion, and measurements of the spin-lattice relaxation rate in spin chain compounds. Furthermore, we study the Haldane-Shastry model where the current operator is also orthogonal to the set of conserved quantities associated with integrability but becomes itself conserved in the thermodynamic limit.
In classical dynamics the KAM theorem quantitatively explains what level of perturbation can be exerted on an integrable system so that quasi-periodic motion survives.Arnol’d (1978) A classical system with Hamiltonian and phase space dimension is integrable if constants of motion exist (i.e., the Poisson bracket vanishes, ) which are pairwise different, . Defining integrability for a quantum system is, however, much more complicated and no analogue of the KAM theorem is known. For any quantum system in the thermodynamic limit an infinite set of operators exist which commute with the Hamilton operator. This can be seen by considering, for example, the projection operators onto the eigenstates of the system, with . In quantum systems described by tight-binding models with short-range interactions it is therefore important to distinguish between local conserved quantities , where is a density operator acting on adjacent sites , and nonlocal conserved quantities like the projection operators mentioned above. In a field theory, a conserved operator is local if it can be written as integral of a fully local operator. Most commonly, quantum systems are called integrable if an infinite set of local conserved quantities exists which are pairwise different. This definition includes, in particular, all Bethe ansatz integrable one-dimensional quantum systems. Here the local conserved quantities can be explicitly obtained by taking consecutive derivatives of the logarithm of the appropriate quantum transfer matrix with respect to the spectral parameter.Takahashi (1999)
In recent years, many studies have been devoted to the question if integrability can stop a system from thermalizingKinoshita et al. (2006); Hofferberth et al. (2007); Rigol et al. (2008) or a current from decaying completely.Castella et al. (1995); Zotos et al. (1997); Zotos (1999); Benz et al. (2005); Rosch and Andrei (2000); Alvarez and Gros (2002a, b); Fujimoto and Kawakami (2003); Jung and Rosch (2007a); Narozhny et al. (1998); Heidrich-Meisner et al. (2003); Jung et al. (2006); Klümper and Sakai (2002); Sirker et al. (2009) That conservation laws and the transport properties of the considered system are intimately connected is obvious in linear response theory (Kubo formula), which relates the optical conductivity to the retarded equilibrium current-current Green’s function. Specializing to a lattice model with nearest-neighbor hopping, the Kubo formula reads111This formula also applies for a lattice model with longer range hopping if we replace the kinetic energy operator by the operator obtained by taking the second derivative of the Hamiltonian with respect to a magnetic flux penetrating the ring (see also Sec. III).
Here, is the system size, the temperature, the kinetic energy operator, the spatial integral of the current density operator, and the brackets denote thermal average. The real part of the optical conductivity can be written as
A nonzero Drude weight implies an infinite dc conductivity. Castella et al. [Castella et al., 1995] showed that it is possible to circumvent the direct calculation of the current-current correlation function in the Kubo formula and compute the finite temperature Drude weight using a generalization of Kohn’s formula.Kohn (1964) This formula relates the Drude weight to the curvature of the energy levels with respect to magnetic flux. The observation that integrable and nonintegrable models obey different level statistics,Berry and Tabor (1977); Bohigas et al. (1984); Poilblanc et al. (1993) as well as the calculation of the Drude weight from exact diagonalization for finite size systems,Zotos and Prelovšek (1996) led to the conjecture that anomalous transport, in the form of a finite , is a generic property of integrable models.Castella et al. (1995)
This conjecture is corroborated by the relation between the Drude weight and the long-time asymptotic behavior of the current-current correlation functionZotos et al. (1997)
In view of this relation, ballistic transport, , means that the current-current correlation function does not completely decay in time. The operators in Eq. (3) form a set of commuting conserved quantities which are orthogonal in the sense that . That conserved quantities provide a lower bound for the long-time asymptotic value of correlation functions is a general result due to MazurMazur (1969). In fact, it can be shown that the equality holds if the right-hand side of Eq. (3) includes all conserved quantities , local and non-local.Suzuki (1971)
The implications of Mazur’s inequality for transport in integrable quantum systems were pointed out by Zotos et al. [Zotos et al., 1997]. For integrable models where at least one conserved quantity has nonzero overlap with the current operator, , Mazur’s inequality implies that the Drude weight is finite at finite temperatures. This happens, for instance, for charge transport in the Hubbard model away from half-filling and for spin transport in the model at finite magnetic field. By contrast, for nonintegrable models, for which all local nontrivial conservation laws are expected to be broken, the right-hand side of Eq. (3) is expected to vanish so that . In these cases the delta function in Eq. (2) is generically presumed to be broadened into a Lorentzian Drude peak. The dc conductivity is finite and the system is said to exhibit diffusive transport.
However, in many cases of interest the known local conserved quantities associated with integrability are orthogonal to the current operator because of their symmetry properties. This happens, for instance in the model at zero magnetic field . In terms of spin-1/2 operators the model reads
Here is the number of sites, parametrizes an exchange anisotropy, and is the exchange constant which we set equal to in the following. One can show that all local conserved quantities of the model are even under the transformation , , whereas the current operator is odd.Zotos et al. (1997) In these cases the integrability-transport connection has remained a conjecture. Nonetheless, several works have presented support for a finite Drude weight in models where the current operator has no overlap with the local conserved quantities, at least in some parameter regimes. The list of methods employed include exact diagonalization (ED),Narozhny et al. (1998); Heidrich-Meisner et al. (2003); Jung and Rosch (2007a); Rigol and Shastry (2008) Quantum Monte Carlo (QMC)Kirchner et al. (1999); Alvarez and Gros (2002b); Heidarian and Sorella (2007); Grossjohann and Brenig (2010) and Bethe ansatz (BA).Fujimoto and Kawakami (1998); Zotos (1999); Benz et al. (2005)
In particular for the model, ED results for chains of lengths up to sitesHeidrich-Meisner et al. (2003) suggest that at high temperatures the Drude weight extrapolates to a finite value in the thermodynamic limit for values of exchange anisotropy in the critical regime, including the Heisenberg point. The Drude weight appears to vanish for large values of anisotropy in the gapped regime, but the minimum value of anisotropy for which it vanishes cannot be determined precisely. Although no conclusions can be drawn about the low temperature regime, the claim is that if the Drude weight is finite at high temperatures it must also be finite and presumably even larger at low temperatures. The weakness of this method is that it assumes that the finite size scaling of , which is not known analytically and is only obtained numerically for , can be extrapolated to the thermodynamic limit. This is not necessarily true for strongly interacting models where even at high temperatures there may be a large length scale above which the behavior of dynamical properties changes qualitatively.
In the QMC approach for the model in Refs. [Alvarez and Gros, 2002a, b], the Drude weight was obtained by analytic continuation of the conductivity function , which is a function of the Matsubara frequencies , to real frequencies. This method uses a fitting function to try to extract a decay rate which broadens the Drude peak if the Drude weight vanishes, but it clearly fails to find decay rates that are smaller than the separation between Matsubara frequencies, . In fact, the application of this method has even led to the conclusion that the Drude weight is finite for some gapless systems that are not integrable.Kirchner et al. (1999); Heidarian and Sorella (2007) This is hard to believe, considering that Eq. (3) requires the existence of nontrivial conservation laws which have a finite overlap with the current operator in the thermodynamic limit in order for the Drude weight to be finite. An attempt was made to explain the Drude weight for nonintegrable models described by a Luttinger liquid fixed point based on conformal field theoryFujimoto and Kawakami (2003), but this analysis neglects irrelevant interactions that lead to current decay and render the conductivity finite.Rosch and Andrei (2000)
Although the Drude weight for the model has been calculated exactly by BA at using Kohn’s formula Shastry and Sutherland (1990), the calculation of by BA is hindered by the need to resort to approximations in the treatment of the excited states. The BA calculation by ZotosZotos (1999) follows an ansatz proposed by Fujimoto and KawakamiFujimoto and Kawakami (1998) that employs the thermodynamic Bethe ansatz (TBA), which relies on the string hypothesis for bound states of magnons.Takahashi (1999) This approach predicts that the Drude weight is finite and decreases monotonically with for the model in the critical regime at zero magnetic field, except at the Heisenberg point, where vanishes for all finite temperatures. Benz et al. [Benz et al., 2005] criticized the TBA result and pointed out that it violates exact relations for at high temperatures. These authors presented an alternative BA calculation of the Drude weight based on the spinon and anti-spinon particle basis and predicted a different temperature dependence than the TBA result. In particular, the Drude weight is found to be finite for the Heisenberg model at zero magnetic field. Actually, for values of anisotropy near the isotropic point this approach predicts that increases with at low temperatures. Like the TBA result, the result based on spinons and anti-spinons violates exact relations at high temperatures. A consistent calculation of the Drude weight by applying the BA to the finite temperature Kohn formula is therefore still an unresolved issue.
Integer-spin Heisenberg lattice models are not integrable, but their low energy properties are often studied in the framework of the continuum O(3) nonlinear sigma model, which is integrable. Interestingly, BA calculations for the nonlinear sigma model predict a finite that is exponentially small at temperatures below the energy gap.Fujimoto (1999); Konik (2003) In fact, it has been arguedKonik (2003) that the finite Drude weight is due to at least one nonlocal conserved quantity of the quantum model which is known explicitlyLüscher (1978) and has overlap with the current operator. This result is not consistent with the semiclassical results by Damle and SachdevDamle and Sachdev (1998, 2005), which predict diffusive behavior for gapped spin chains at low temperatures. No nonlocal conserved quantities that overlap with the current operator are known explicitly for the integrable model. For finite chains such quantities can be constructed explicitly, however, in a numerical study of chains with no definite conclusions could be made whether or not the overlap remains finite in the thermodynamic limit.Jung and Rosch (2007a)
The debate about the role of integrability in transport properties of integrable models is connected with the question of diffusion in spin chains. The term spin diffusion first appeared in the context of the phenomenological theory,Bloembergen (1949); de Gennes (1958); Steiner et al. (1976) where it refers to a characteristic form of long-time decay of the spin correlation function. In the phenomenological theory, the case where the total magnetization in the direction of quantization, , is conserved is considered. It is then said that spin diffusion occurs if the Fourier transform of the spin-spin correlation function for small wavevector decays with time as , where is the diffusion constant. Provided that the behavior at small dominates, this implies that the Fourier transform decays as . This theory was formulated to explain results of inelastic neutron scattering experiments in three-dimensional ferromagnets at high temperatures. The assumptions of the theory are usually motivated by the picture that at high temperatures the spin modes are described by independent Gaussian fluctuations. The decay of the correlation function can then be interpreted as a random walk of the magnetization through the lattice as in classical diffusion.
It is important to note that this definition of diffusion is not obviously related to that of diffusive transport given earlier, since the two definitions refer to two different correlation functions. In the phenomenological theory, transport is found to be diffusive because the local magnetization obeys the diffusion equation and the dc conductivity is therefore finite. However, more generally diffusion in the autocorrelation function for the density of the conserved quantity does not exclude the possibility of ballistic transport, understood as a nonzero long-time value for the current-current correlation function.
The applicability of the phenomenological theory of diffusion to spin chains described by the integrable model is of course questionable. Even at high temperatures, the spin dynamics is likely to be constrained by the nontrivial conservation laws and the assumption of independent modes is not expected to hold. In the one case where the long-time behavior of the autocorrelation function can be calculated exactly, namely the XX model, which is equivalent to free spinless fermions, diffusion does not occur since for large , as opposed to expected for diffusion in one dimension. For decades, a great deal of effort has been made to compute the autocorrelation function for the model with general values of anisotropy, particularly for the Heisenberg model.Carboni and Richards (1968); Böhm et al. (1994a, b); Fabricius et al. (1997); Fabricius and McCoy (1998); Starykh et al. (1997); Sirker (2006) While ED is always limited to small systems and short times (out to in units of inverse exchange constant for sitesFabricius and McCoy (1998)), QMCStarykh et al. (1997) is plagued by the analytic continuation and cannot resolve singularities associated with the long-time behavior. Although the more recently developed density matrix renormalization group (DMRG) method applied to the transfer matrixSirker and Klümper (2005) works directly in the thermodynamic limit, it is also restricted to intermediate times and has not detected a diffusive contribution at low temperatures.Sirker (2006) Nonetheless, these works have concluded in favor of the existence of diffusion for the Heisenberg model at high temperatures.
Although diffusion was originally proposed to describe spin dynamics at high temperatures, the paradigm has been used to interpret nuclear magnetic resonance (NMR) experiments that measure the spin-lattice relaxation rate of spin chains at low temperatures.Boucher et al. (1976); Takigawa et al. (1996a); Thurber et al. (2001); Kikuchi et al. (2001) Strictly speaking, linear response theory expresses in terms of the Fourier transform of the transverse spin-spin correlation function . The small magnetic field applied in NMR experiments breaks the rotational spin invariance of Heisenberg chains from SU(2) down to U(1) and the total are not conserved. However, if the experiments are in the regime of temperatures small compared to the exchange constant, but large compared to the nuclear or electronic Larmor frequencies then the transverse correlation function can be traded for the longitudinal one calculated for the electronic Larmor frequency. We will discuss this point in more detail in Section II.7. Up to -dependent form factors that stem from the spatial dependence of the hyperfine couplings, is then proportional to the dynamical autocorrelation , with equal to the Larmor frequency of an electron in a magnetic field . If diffusion is present, with given as in the phenomenological theory, behaves as in dimensions. In particular, in the one-dimensional case diverges at low frequencies as . This type of behavior has been observed for gapped spin chainsTakigawa et al. (1996b) and gapless spin chains with large half-integer .Boucher et al. (1976) More surprisingly, spin diffusion has also been observed in chain compounds by NMRThurber et al. (2001); Kikuchi et al. (2001) and by muon spin resonance.Pratt et al. (2006) It is important to note that in the NMR experiment of Ref. [Thurber et al., 2001], the -dependence of the form factor suppresses the contribution from modes in the autocorrelation function, so that the signal is completely dominated by modes.
The observation of spin diffusion in Heisenberg chains is puzzling from the point of view of the integrability-transport conjecture. Since the experimental diffusion constant was found to be fairly large,Thurber et al. (2001) it becomes important to determine whether the diffusion constant is mainly determined by integrability-breaking interactions present in the real system or by umklapp processes already contained in the integrable Heisenberg model.
Recently, we have shown using a field theory approach that the long-time behavior of the autocorrelation function and the transport properties are directly related and can be obtained from the same retarded Green’s function at low temperatures.Sirker et al. (2009) The analytical results were supported by DMRG calculations for the time-dependent current-current correlation function as well as by a comparison with the NMR experiment on SrCuO.Thurber et al. (2001) We also argued that ballistic transport can be reconciled with diffusion in the autocorrelation function because ballistic channels of propagation can coexist with diffusive ones. This can be made precise with the help of the memory matrix approach,Rosch and Andrei (2000) which allows one to incorporate known conservation laws in the low energy effective theory. However, ballistic and diffusive channels compete for spectral weight of the spin-spin correlation function. Our field theoretical results for the chain at – valid at finite temperatures small compared to the exchange energy – are in very good agreement with the diffusive response measured experimentallyThurber et al. (2001) and with time-dependent DMRG results if we assume that the Drude weight vanishes completely. Although a small Drude weight at finite temperatures cannot be excluded, a combination of the numerical data with the memory matrix approach implies that it has to be smaller than the values obtained in the BA calculation by Klümper et al. [Benz et al., 2005] and by QMC calculationsAlvarez and Gros (2002a, b). The results in Ref. [Sirker et al., 2009] were further supported by a recent QMC study.Grossjohann and Brenig (2010) In the latter work the problems arising from analytical continuation of numerical data were circumvented by comparing with the field theory resultSirker et al. (2009) transformed to imaginary times.
The purpose of this paper is to provide details of the calculations for the chain in Ref. [Sirker et al., 2009]. Furthermore, we present an extension of these methods to charge transport in the attractive Hubbard model as well as a discussion of the transport properties of the Haldane-Shastry chain. Our paper is organized as follows: In Sec. II we study the spin current in the spin- Heisenberg chain. We discuss the relation between the current-current and the spin-spin correlation function at low temperatures, explain in detail how our results relate to previous BA and QMC calculations, and discuss consequences for electron spin resonance and the finite-temperature broadening of the dynamic spin structure factor. In Sec. III, we discuss spin transport in the Haldane-Shastry model and point out that the current operator is a nonlocal conserved quantity in the thermodynamic limit. In Sec. IV we show that many of the results we obtained for the spin current in the Heisenberg model also directly apply to the charge current in the attractive Hubbard model. Finally, we give a summary and some conclusions in Sec. V.
Ii The spin current in the spin- model
The spin-current density operator is defined from the continuity equation for the density of the globally conserved spin component
which for the model yields
For , the summed current operator has a finite overlap with the local conserved quantities of the model. The simplest nontrivial conserved quantity is
Here is the energy current operator as obtained from the continuity equation for the Hamiltonian density for Zotos et al. (1997); Klümper and Sakai (2002). It can be verified that is conserved in the strong sense that . The thermal conductivity therefore only has a Drude part which can be calculated exactly by BA.Klümper and Sakai (2002) According to Mazur’s inequality, Eq. (3), the overlap of with provides a lower bound for the Drude weight of the spin conductivity
The advantage of this formula is that – contrary to Eq. (1) – it does not require the calculation of dynamical correlation functions and is thus much more accessible by standard techniques. The evaluation of (8) becomes particularly simple in the limit leading toZotos et al. (1997)
where is the magnetization. At low temperatures, on the other hand, standard bosonization techniques can be applied. Furthermore, BA can be used to evaluate (8) for all temperatures as will be shown in Sec. II.1.
All other local conserved quantities can be obtained either recursively by applying the so-called boost operatorZotos et al. (1997) or by taking higher order derivatives of the quantum transfer matrix of the Hamiltonian with respect to the spectral parameter. The local operators obtained this way act on more and more adjacent sites but they are all even under particle-hole transformations. As a result, the Mazur bound for the Drude weight, Eq. (3), vanishes for .
ii.1 Low energy effective model and the Mazur bound
Here, is the standard Luttinger model and and are the leading irrelevant perturbations due to Umklapp scattering and band curvature, respectively. The bosonic field and its conjugate momentum obey the canonical commutation relation . The long-wavelength () fluctuation part of the spin density is related to the bosonic field by . The spin velocity and the Luttinger parameter are known exactly from Bethe ansatz
In this notation, at the free fermion point () and at the isotropic point (). The amplitudes , , and are also known exactlyLukyanov (1998)
In the gapless phase, the Mazur bound can be calculated in the low-temperature regime using the field theory representations of and . In the continuum limit, the continuity equation becomes
Using the bosonized form for the spin density, we obtain for the effective model (II.1)
In the following we neglect the corrections to the current operator due to band curvature terms and calculate the Mazur bound for the Luttinger model using the current operator
and the energy current operator
At zero field, the overlap vanishes because and have opposite signatures under the particle-hole transformation . A small magnetic field term in Eq. (4), on the other hand, can be absorbed by shifting the bosonic field (here we set )
In this case, the conserved quantity becomes Rosch and Andrei (2000)
We calculate the equal time correlations within the Luttinger model and find
We note that in the limit the Mazur bound obtained from the overlap with saturates the exact zero temperature Drude weight Shastry and Sutherland (1990).
One can also use the Bethe ansatz to calculate the Mazur bound in Eq. (8) exactly. To do so we computed the equal time correlations using a numerical solution of the nonlinear integral equations obtained within the Bethe ansatz formalism of Refs. [Sakai and Klümper, 2005; Bortz and Göhmann, 2005]. In Fig. 1, the numerical Bethe ansatz solution for small values of magnetization is compared to the field theory formula (18).
Remarkably, the free boson result in Eq. (18) fits well the behavior of the exact Mazur bound out to temperatures of order .
ii.2 Retarded spin-spin correlation function
We will now proceed with the field theory calculations in the following way. We first assume that the Drude weight is not affected by unknown nonlocal conserved quantities. In this case we are left with the following picture based on the Mazur bound and including all conserved quantities related to integrability: For finite magnetic field the Drude weight is a continuous function of temperature. At zero field, however, the Drude weight is only finite at but drops abruptly to zero for arbitrarily small temperatures. Within the effective field theory such a possible broadening of the delta-function peak at finite temperatures has to be related to inelastic scattering between the bosons which can relax the momentum. Such a process is described by the Umklapp term in Eq. (II.1) which we have ignored so far. We will now include this term as well as the band curvature terms in a lowest order perturbative calculation. We want to stress that such an approach does not know about conservation laws which could protect a part of the current from decaying. The parameter-free result derived here is expected to be correct if the Drude weight is indeed zero and allows to check the validity of the assumption by comparing with experimental and numerical results. Importantly, we can also systematically study how this result is modified if such conservation laws do exist after all. This case will be considered in section II.3.
We are interested in the retarded spin-spin correlation function , which can be obtained from the Matsubara correlation function
by the analytic continuation . We will show that in the low-temperature limit this correlation function determines both the decay of the current-current correlation function as well as the spin-lattice relaxation rate . In the low-energy limit, we follow Ref. [Oshikawa and Affleck, 1997] and relate the long-wavelength part of the retarded spin-spin correlation function to the boson propagator
We calculate the self-energy by perturbation theory to second order in and first order in . We first focus on the half-filling case (); the finite field case is discussed at the end of this sub-section.
The contribution from Umklapp scattering reads
where Schulz (1986)
where is the beta function. For , we need a cutoff in the integral in Eq. (23). However, the imaginary part of does not depend on the cutoff scheme used Schulz (1986). The expansion of Eq. (23) for yields both a real and an imaginary part for . The calculation of is also standard. In contrast to , the result for is purely real, as band curvature terms do not contribute to the decay rate. The end result is
In the anisotropic case, , the parameters are given by
Here and ( and ) are the parts stemming from the band curvature (umklapp) terms, respectively. In Eq. (II.2) we have used the following functions
with being the Digamma function.
At the isotropic point, , Umklapp scattering becomes marginal and can be taken into account by replacing the Luttinger parameter by a running coupling constant, . In this case we find
Following Lukyanov Lukyanov (1998), the running coupling constant is determined by the equation
where is the Euler constant. We remark that a similar calculation was attempted in Ref. [Fujimoto and Kawakami, 2003], but there the imaginary part of the self-energy was neglected.
This calculation can be extended to finite magnetic field. Shifting the field as in Eq. (16), the Umklapp term in Eq. (10) becomes . As long as , it is reasonable to keep this oscillating term in the effective Hamiltonian. However, in a renormalization group treatment, after we lower our momentum cut-off below , it should be dropped. Our formula for the self-energy, in Eq. (21) is thus modified to
where is given by Eqs. (II.2) and (23) as before. As a consequence, the relaxation rate for will now be given by (up to logarithmic corrections in the isotropic case). In the next section we show that the retarded current-current correlation function can be obtained at low energies using the calculated self-energy. We will also discuss what the shortcomings of the self-energy approach are and show that these shortcomings can be addressed by taking conservation laws into account explicitly.
ii.3 Decay of the current-current correlation function
The time-dependent current-current correlation function can be written as
where is the retarded current-current correlation function. The latter appears in the Kubo formula for the optical conductivity (1) with and the current operator given by . One can easily show that
The Kubo formula (1) can therefore also be written as
allowing us to use the results for the boson-boson Green’s function from the previous section. At zero temperature the irrelevant operators in (II.1) can be ignored and the Drude weight of the model can be obtained using the free boson propagator
in agreement with Bethe ansatz.Shastry and Sutherland (1990)
with the real part being given by
For we find, in particular, a Lorentzian
As expected, the self-energy approach predicts zero Drude weight whenever is nonzero. This result is inconsistent with Mazur’s inequality if there exist conservation laws which protect the Drude weight. We know that this is the case for any arbitrarily small magnetic field , while our calculations in sub-section IIB gave a non-zero at non-zero field. Whether or not such a conservation law exists also for is an open question. We will try to tackle this problem by first studying how the self-energy result is modified by a conservation law, followed by a comparison with numerical and experimental results.
It is possible to accommodate the existence of nontrivial conservation laws by resorting to the memory matrix formalism of Ref. [Rosch and Andrei, 2000]. This approach starts from the Kubo formula for a conductivity matrix which includes not only the current operator , but also “slow modes” () which have a finite overlap with . The idea is that if is a slowly decaying function of time, the projection of into governs the long-time behavior of the current-current correlation function and consequently dominates the low-frequency transport. The overlap between and the slow modes is captured by the off-diagonal elements of the conductivity matrix. In practice, only a small number of conserved quantities is included in the set of slow modes, but the approach can be systematically improved since adding more conserved quantities increases the lower bound for the conductivity Jung and Rosch (2007b). It is convenient to invert the Kubo formula for the conductivity matrix using the projection operator method Jung and Rosch (2007b). We introduce the scalar product between two operators and in the Liouville space
Here, is the Liouville superoperator defined by . For simplicity, we assumed that all the slow modes have the same signature under time-reversal symmetry. The conductivity matrix can be written as
The conductivity in Eq. (1) is the component of . The susceptibility matrix is defined as
We denote by
where is the memory matrix given by
It can be shown that if there is an exact conservation law (local or nonlocal) involving one of the slow modes, the memory matrix has a vanishing eigenvalue, which then implies a finite Drude weight.
In the following we apply the memory matrix formalism to calculate the conductivity for the low-energy effective model (II.1) at , allowing for the existence of a single conserved quantity , . The conductivity matrix is then two-dimensional. We choose and so that is diagonal. At low temperatures , we can use the current operator in Eq. (II.1); to first order in , we obtain
where is defined in Eq. (II.2) and we have used the fact that due to the vanishing of the superfluid density we have . Likewise, can be calculated within the low energy effective model once a conserved quantity has been identified. The remaining approximation is in the calculation of the memory matrix to second order in Umklapp. This is analogous to the calculation of the self-energy in Eq. (21). Using the conservation law, we can write
For finite magnetic field, a conserved quantity is given by (see Eq. (17)) and thus in this case. Equating Eq. (36) for and (48), we find that the memory matrix approach is equivalent to adopting the self-energy
As expected, the memory matrix result reduces to the self-energy result for . The difference between the self-energy and the memory matrix result is of higher order in the Umklapp interaction. Therefore, the conservation law is not manifested in the lowest-order calculation of the self-energy. Although Eq. (50) is not correct beyond , it suggests that the memory matrix approach corresponds to a partial resummation of an infinite family of Feynman diagrams which changes the behavior of in the limit from to .
where and . The first term on the right hand side of Eq. (II.3) can be associated with the ballistic channel and the second one with the diffusive channel. The calculation is valid also in the finite field case if with as already discussed below Eq. (48). This means that we obtain a correction of the relaxation rate in the memory matrix formalism which is exactly of the same order as obtained previously from the self-energy approach in this limit (see Eq. (II.2)). However, now we see that the weight is transferred accordingly from the diffusive into a ballistic channel, a fact which is missed in the self-energy approach.
Substituting Eq. (II.3) into Eq. (II.3) we can now evaluate the integral. From the second term in (II.3) we obtain an integrand which has poles at frequencies and with . Since the contributions of the latter poles can be ignored at times leading to
with . We can further expand the denominator in powers of . The first order contribution is real and given by
The imaginary part of the correlation function is obtained in second order in the expansion and is thus suppressed by an additional power of . From Eq. (53) we see that in the limit the current-current correlation function approaches the value
consistent with the Mazur bound for the Drude weight. For intermediate times , we obtain the linear decay
independent of if . Therefore a small Drude weight cannot be detected in this intermediate time range.
ii.4 Comparison with Bethe ansatz and numerical data for the current-current correlation function
In the previous section we have derived a result for the optical conductivity, Eq. (II.3), and for the real part of the current-current correlation function, Eq. (53), at low temperatures by a memory-matrix formalism. In this approach we have explicitly taken into account the possibility of a nonlocal conservation law - leading to a nonzero Drude weight at finite temperatures. By comparing our results with the Bethe ansatz calculationsZotos (1999); Benz et al. (2005) and numerical calculations of we will show that a diffusive channel for transport does indeed exist. Furthermore, we will try to obtain a rough bound for how large and therefore the Drude weight can possibly be.
A finite Drude weight at finite temperatures has been obtained in two independent Bethe ansatz calculationsZotos (1999); Benz et al. (2005), however, the obtained temperature dependence is rather different. In both works the finite temperature Kohn formulaCastella et al. (1995), which relates the Drude weight and the curvature of energy levels with respect to a twist in the boundary conditions, is used. While Zotos [Zotos, 1999] uses a TBA approach based on magnons and their bound states, Klümper et al. [Benz et al., 2005] use an approach based on a spinon and anti-spinon particle basis. Both approaches have been shown to violate exact relations at high temperatures and are therefore not exact solutions of the problem. The reason is that within the Bethe ansatz both approaches use assumptions which have been shown to work in the thermodynamic limit for the partition function. This, however, does not seem to be the case for the curvature of energy levels relevant for the Drude weight. Nevertheless, this does not exclude that these results become asymptotically exact at low temperatures and arguments for such a scenario have been given in Ref. [Benz et al., 2005].
Doing so we obtain excellent agreement. From this we draw two conclusions: First, this BA calculation predicts that even at finite temperatures the transport is purely ballistic. Second, the temperature dependent parameter which we obtained from field theory in first order in band curvature and second order in Umklapp scattering is consistent with this BA approach. We note that the case () is particularly interesting because here the contributions from band curvature and from Umklapp scattering in Eq. (II.2) both yield a contribution with diverging prefactors. These divergencies cancel leading to a Drude weight
with where is the Euler constant and the Riemann zeta function. As is also the case for other thermodynamic quantitiesSirker and Bortz (2006) we see that the coinciding scaling dimensions of the terms stemming from Umklapp scattering and band curvature lead to a term at this special point. For () band curvature gives the dominant temperature dependence while Umklapp scattering dominates for (). For () a term, which arises in 4th order perturbation theory in Umklapp scattering and which is not included in our calculations, becomes more important than the term from band curvature. Therefore the agreement for in Fig. 2 is not quite as good as for the other values. For , Umklapp scattering becomes marginal leading to a logarithmic temperature dependence of the Drude weight (56) with as given in (II.2).
To see whether or not such a large Drude weight as predicted by Klümper et al. is possible and to discuss the second BA approach by Zotos we now turn to a numerical calculation of . A dynamical correlation function at finite temperatures can be obtained by using a density matrix renormalization group algorithm applied to transfer matrices (TMRG).Sirker and Klümper (2005); Sirker (2006); Sirker et al. (2009) This algorithm uses a Trotter-Suzuki decomposition to map the 1D quantum model onto a 2D classical model. For the classical model a so-called quantum transfer matrix can be defined which evolves along the spatial direction and allows one to perform the thermodynamic limit exactly. In order to calculate dynamical quantities, a complex quantum transfer matrix is considered with one part representing the thermal density matrix and the other part the unitary time evolution operator. By extending the transfer matrix one can either lower the temperature (imaginary time) or increase the real time interval for the correlation function. The calculation of the current-current correlation function is particularly complicated because it involves the summation of all the local time-dependent correlations
In order to obtain converged results, the two-point correlations for distances up to have to be summed up. This is not possible using exact diagonalization which is restricted to considerably smaller system sizes. A good check is obtained by considering the free fermion case . Here can be calculated exactly and is non-trivial, however, commutes with the Hamiltonian leading to (58) being a constant.
We now discuss the same parameter set and as used above in more detail. In Fig. 4 the real and imaginary parts of , obtained in a TMRG calculation with a states per block kept, are shown. Because the imaginary part is very small, an extrapolation in the Trotter parameter was necessary restricting the calculations to smaller times than for the real part.
In the limit , the real part would directly yield the Drude weight (possibly zero). As already discussed, the BA result by Klümper et al. corresponds to purely ballistic transport, . From Fig. 4 we see that this is not consistent with the numerical data. The BA calculation by Zotos, on the other hand, predicts a Drude weight which requires . While formula (52) with seems to fit the numerical results best, we would need to be able to simulate slightly longer times (a factor of should be sufficient) to clearly distinguish between and . Most importantly, however, the numerical data clearly demonstrate that the decay rate is nonzero.
In addition to the zero field case, we have also calculated at relatively large magnetic fields and various temperatures. As shown in Fig. 5 we find that in such cases appears to converge to a finite value within fairly short times.