Density Dynamics in Translationally Invariant Spin- Chains at High Temperatures: a Current Auto-Correlation Approach to Finite Time- and Length-Scales
Abstract
We investigate transport in several translationally invariant spin- chains in the limit of high temperatures. We concretely consider spin transport in the anisotropic Heisenberg chain, the pure Heisenberg chain within an alternating field, and energy transport in an Ising chain which is exposed to a tilted field. Our approach is essentially based on a connection between the evolution of the variance of an inhomogeneous non-equilibrium density and the current auto-correlation function at finite times. Although this relationship is not restricted to the case of diffusive transport, it allows to extract a quantitative value for the diffusion constant in that case. By means of numerically exact diagonalization we indeed observe diffusive behavior in the considered spin chains for a range of model parameters and confirm the diffusion coefficients which were obtained for these systems from non-equilibrium bath scenarios.
pacs:
05.60.Gg, 05.30.-d, 05.70.LnI Introduction
Although transport in low-dimensional quantum systems has intensively been investigated theoretically in the past years, there still is an ongoing interest in understanding the transport phenomena in such systems, including their temperature and length scale dependence X. Zotos (1999); B. N. Narozhny et al. (1998); M. Michel et al. (2003); F. Heidrich-Meisner et al. (2003); D. A. Rabson et al. (2004); F. Heidrich-Meisner et al. (2007); X. Zotos and P. Prelov̌sek (2003); J. Benz et al. (2005); R. Steinigeweg et al. (2006); L. F. Santos (2008); K. Fabricius and B. M. McCoy (1998); J. Sirker et al. (2009); T. Prosen and M. Žnidarič (2009); S. Langer et al. (2009); P. Prelovšek et al. (2004); M. Michel et al. (2008); U. Brandt and J. Stolze (1986); J. M. R. Roldan et al. (1986); K. Sakai and A. Klümper (2003); K. Saito (2003); C. Mejía-Monasterio et al. (2005); C. Mejía-Monasterio and H. Wichterich (2007); A. Karahalios et al. (2009); K. Saito et al. (2000); J. Gemmer et al. (2006); R. Steinigeweg et al. (2007); L.-A. Wu and D. Segal (2008); A. Rosch and N. Andrei (2000); R. Steinigeweg et al. (2009a); H. Castella et al. (1995); R. Steinigeweg et al. (2009b). Those works have often addressed a qualitative classification of the occurring transport types into ballistic or normal diffusive behavior and, in particular cases, the crucial mechanisms which are responsible for the emergence of diffusion have been studied. In this context the role of non-integrability and quantum chaos is frequently discussed as an at least necessary condition H. Castella et al. (1995); A. Rosch and N. Andrei (2000); C. Mejía-Monasterio et al. (2005); R. Steinigeweg et al. (2006); L. F. Santos (2008). Significant theoretical attention has been devoted to spin- chains X. Zotos (1999); B. N. Narozhny et al. (1998); M. Michel et al. (2003); F. Heidrich-Meisner et al. (2003); D. A. Rabson et al. (2004); F. Heidrich-Meisner et al. (2007); X. Zotos and P. Prelov̌sek (2003); J. Benz et al. (2005); R. Steinigeweg et al. (2006); L. F. Santos (2008); K. Fabricius and B. M. McCoy (1998); J. Sirker et al. (2009); T. Prosen and M. Žnidarič (2009); S. Langer et al. (2009); P. Prelovšek et al. (2004); M. Michel et al. (2008); U. Brandt and J. Stolze (1986); J. M. R. Roldan et al. (1986); K. Sakai and A. Klümper (2003); K. Saito (2003); C. Mejía-Monasterio et al. (2005); C. Mejía-Monasterio and H. Wichterich (2007); A. Karahalios et al. (2009), e.g. to the prominent anisotropic Heisenberg chain (XXZ model) X. Zotos (1999); B. N. Narozhny et al. (1998); M. Michel et al. (2003); F. Heidrich-Meisner et al. (2003); D. A. Rabson et al. (2004); F. Heidrich-Meisner et al. (2007); X. Zotos and P. Prelov̌sek (2003); J. Benz et al. (2005); R. Steinigeweg et al. (2006); L. F. Santos (2008); K. Fabricius and B. M. McCoy (1998); J. Sirker et al. (2009); T. Prosen and M. Žnidarič (2009); S. Langer et al. (2009); P. Prelovšek et al. (2004); M. Michel et al. (2008); U. Brandt and J. Stolze (1986); J. M. R. Roldan et al. (1986); K. Sakai and A. Klümper (2003). Most controversial appears the question whether or not the (finite temperature) transport in the pure Heisenberg chain is ballistic X. Zotos (1999); B. N. Narozhny et al. (1998); M. Michel et al. (2003); F. Heidrich-Meisner et al. (2003); D. A. Rabson et al. (2004); F. Heidrich-Meisner et al. (2007); X. Zotos and P. Prelov̌sek (2003); J. Benz et al. (2005); R. Steinigeweg et al. (2006); L. F. Santos (2008); K. Fabricius and B. M. McCoy (1998); J. Sirker et al. (2009).
Even though there certainly is a large variety of different methods for the investigation of transport in quantum systems, we concentrate here on two of the main approaches in detail which are also most relevant in the context of the present work. The first approach may be classified as a direct one, since it is rather close to an experimental measurement setup: Within the theory of open quantum systems H.-P. Breuer and F. Petruccione (2007) the model of interest is coupled locally, e.g. at both ends of a spin chain, to reservoirs of the transported quantity, e.g. at different temperature or chemical potential K. Saito et al. (2000); K. Saito (2003); M. Michel et al. (2003); C. Mejía-Monasterio et al. (2005); C. Mejía-Monasterio and H. Wichterich (2007); H. Wichterich et al. (2007); M. Michel et al. (2008); T. Prosen and M. Žnidarič (2009); R. Steinigeweg et al. (2009b). Due to the coupling to reservoirs, a stationary non-equilibrium state eventually results for which all relevant expectation values such as the current and the spatial density profile of the transport quantity can be evaluated. Here, a vanishing profile corresponds to ballistic behavior, whereas normal diffusive dynamics is associated with a strictly linear profile, according to Fourier’s law, respectively Fick’s law, see, e.g. Ref. T. Prosen and M. Žnidarič, 2009. In the latter case a finite conductivity is simply given by the ratio of the current and the spatial density gradient of the transported quantity. This conductivity can be also understood in terms of a diffusion coefficient, since transport is driven by a gradient within the considered model and not by an external force R. Steinigeweg et al. (2009c). One may therefore directly compare with the diffusion constant of a corresponding closed scenario, where transport is not induced by the coupling to baths but by an initially inhomogeneous non-equilibrium density R. Steinigeweg et al. (2009c, b).
In order to simulate the influence of the baths the Liouville-von Neumann equation for the coherent evolution of the density matrix is routinely extended by an incoherent damping term, see, e.g. Ref. H. Wichterich et al., 2007. The derivation of such a dynamical equation from a microscopic bath model is highly non-trivial and involves a combination of various subtle approximation schemes, e.g. improper approximations may eventually lead to a mathematically correct but physically irrelevant quantum master equation (QME)H. Wichterich et al. (2007). One often intends to derive a proper QME of the Lindblad form Lindblad (1976), because its special structure allows to apply the numerically efficient method of stochastic unraveling J. Dalibard et al. (1992); K. Mølmer and Y. Castin (1996), e.g. spin- chains with about sites become numerically tractable M. Michel et al. (2008). However, a recently suggested matrix product operator Ansatz has been shown to significantly increase this number of available sites up to several dozens T. Prosen and M. Žnidarič (2009), i.e. finite size effects are drastically reduced and the extracted conductivity may be interpreted as a pure bulk property of the model, if it does not depend crucially on the concrete form and strength of the bath coupling, of course R. Steinigeweg et al. (2009b).
Another approach for the investigation of transport in quantum systems is the Green-Kubo formula (KF) which was originally derived for electrical conductance by the use of linear response theory R. Kubo et al. (1991); Mahan (2000). In that case the (frequency-dependent) electrical field is an external force which perturbs the system and the resulting current of charge through the system is the response to this external perturbation. The KF as such gives the (frequency-dependent) conductivity as the linear response coefficient in terms of a current auto-correlation function. The same approach is also used in the context of gradient-driven transport phenomena, e.g. for transport of energy or heat. In that case the current is driven by a much more complicated mechanism J. M. Luttinger (1964); R. Zwanzig (1965); F. Bonetto et al. (2000); S. Lepri et al. (2003); P. L. Garrido et al. (2001); J. Gemmer et al. (2006); L.-A. Wu and D. Segal (2008), especially since it can not be treated as a perturbation to the system. Nevertheless, simply by the replacement of the electrical current, e.g. by the energy current J. M. Luttinger (1964), the KF is used for gradient-driven transport, too. But a rigorous justification of this replacement remains still a conceptual problem, see Ref. J. Gemmer et al., 2006 or the overview paper R. Zwanzig, 1965 (and the comprehensive literature which is cited therein).
However, the KF is nowadays a standard method for the investigation of transport in spin chains B. N. Narozhny et al. (1998); X. Zotos (1999); X. Zotos and P. Prelov̌sek (2003); F. Heidrich-Meisner et al. (2003); K. Saito (2003); D. A. Rabson et al. (2004); P. Prelovšek et al. (2004); J. Benz et al. (2005); F. Heidrich-Meisner et al. (2007); A. Karahalios et al. (2009), not least due to its direct computability, once a finite piece of the considered system has been exactly diagonalized. As far as numerics is concerned, exact diagonalization is restricted to spin-1/2 chains with at most 24 sites. But another difficulty arises, if the KF is evaluated on the basis of a finite piece: One often distinguishes between ballistic and normal diffusive transport by the notion of the Drude weight, essentially the conductivity’s singular contribution at zero-frequency, see the review X. Zotos and P. Prelov̌sek, 2003, for example. Whenever it is finite, the long-time behavior is expected to be ballistic, and whenever it vanishes, the dc-conductivity as the zero-frequency limit of the conductivity’s regular non-singular part determines the long-time behavior. Since for any finite system the conductivity exclusively consists of delta peaks at different frequencies, there apparently are only singular contributions and it therefore is a difficult question how to extract and extrapolate the dc-conductivity from a finite system J. Gemmer et al. (2006). Moreover, the quantitative comparison of the resulting dc-conductivity (due to a possibly hypothetic external force) with a diffusion constant (from non-equilibrium bath scenarios) is hardly possible without the detailed knowledge about an Einstein relation between both quantities.
In the present paper we do not intend to further discuss the above mentioned conceptual and methodological problems which may come along with the KF. Instead we will present a different but in a sense related approach which particularly is not concerned with the most of those problems. To this end we will firstly introduce a “typical” inhomogeneous non-equilibrium density in Sec. II and then connect the evolution of the variance of this density to a current auto-correlation function at finite times R. Steinigeweg et al. (2009c). Remarkably, in the special limit of infinitely long times this connection will be shown to yield a generalized Einstein relation which relates the diffusion constant to the dc-conductivity, i.e. as evaluated by the KF. Moreover, we will demonstrate that the great advantage of the connection is given by its direct applicability at finite times and for finite systems as well. In this context we will suggest another concept for the analysis of data which is available from current auto-correlations functions.
By the use of the suggested concept we will investigate in Secs. III-V transport in several translationally invariant spin-1/2 chains in the special limit of high temperatures. We will concretely consider spin transport in the anisotropic Heisenberg chain, the pure Heisenberg chain within an alternating field, and energy transport in an Ising chain which is exposed to a tilted field. By means of numerically exact diagonalization we indeed observe strong indications for diffusive behavior in the considered spin chains for a range of model parameters and, what is more, we are able to quantitatively confirm the diffusion constants which were found for these systems from non-equilibrium bath scenarios in Refs. C. Mejía-Monasterio et al., 2005; C. Mejía-Monasterio and H. Wichterich, 2007; M. Michel et al., 2008; T. Prosen and M. Žnidarič, 2009. Finally, we will close in Sec. VI with a summary and a conclusion.
Ii Connection between Variance and Current Auto-Correlation Function
In this Sec. we are going to introduce our approach to density-driven transport in translationally invariant quantum systems. To this end the next Sec. II.1 firstly presents the pertinent definitions and exclusively describes the general theory which eventually yields the basic Eq. (13) for the time-dependent diffusion constant . In the following Sec. II.2 we then motivate to evaluate for finite times and particularly illustrate the concept which will be used for the concrete spin chains in the subsequent Secs. Thus, the reader which is not primarily interested in the theoretical details may directly continue with Sec. II.2.
ii.1 Diffusion Constant
In the present paper we will investigate translationally invariant, one-dimensional quantum spin systems which are described by a respective Hamiltonian . In those quantum systems we will consider an overall conserved transport quantity , i.e. . This transport quantity and the Hamiltonian as well are decomposable into formally identical addends , respectively corresponding to different positions, i.e.
(1) |
Thus, is a local density of the transported quantity . Note that the may be defined on the positions of the , in between, or both. The above decomposition is further done in such a way that Heisenberg’s equation of motion for the local densities reads
(2) |
where and represent those local addends of the Hamiltonian which are located directly on the l.h.s., respectively r.h.s. of . This apparently implies a kind of locality. However, such a description can always be at least approximately enforced, if only interactions are reasonably short-ranged. For all quantum systems in the following Secs. III-V the description will be even exact, because interactions between nearest-neighbors are taken into account solely. As routinely done, the comparison with a continuity equation suggests the definition of a local current according to the scheme in Eq. (2), see Ref. J. Gemmer et al., 2006, for example. This definition is consistent, if
(3) |
where the latter holds, if is globally preserved. The total current is given by
(4) |
Once the above decomposition has been established, we can define a certain class of initial states which corresponds to some inhomogeneous, non-equilibrium density. To those ends let
(5) |
denote the deviation of the local densities from their equilibrium average , where is any stationary equilibrium state, i.e. . Then the initial state reads
(6) |
with some realization for the numbers . The factor is concretely given by
(7) |
and therefore quantifies the equilibrium fluctuations of the transported quantity .
For the special initial state we now consider the actual expectation values
(8) |
It follows that
(9) |
i.e. the sum of the numbers in Eq. (6) determines the sum of the actual expectation values .
Of particular interest is the spatial variance of the . It is given by
(10) |
If the dynamics of the was indeed generated by a discrete diffusion equation of the form
(11) |
then the evolution of this variance would read
(12) |
as long as the vanish at the ends of a chain (open boundary conditions) or are reasonably concentrated at a sector of a ring (closed boundary conditions). Even though Eq. (11) implies Eq. (12), the inverse direction generally is not true, of course.
However, in Ref. R. Steinigeweg et al., 2009a a connection of the form (12) has recently been found directly from Eq. (10), i.e. by firstly applying Heisenberg’s equation of motion to Eq. (10) and subsequently manipulating the resulting equations. The time-dependent diffusion constant reads
(13) |
and is essentially given in terms of a time-integral over the current auto-correlation function
(14) |
i.e. in the limit of high temperatures (). This limit will be considered throughout this work.
Strictly speaking, the above diffusion constant is restricted to the special initial state in Eq. (6), of course. Nevertheless, due to its concrete form, this initial state represents an ensemble average w.r.t. to typicality S. Goldstein et al. (2006); S. Popescu et al. (2006); P. Reimann (2007) or, more precisely, the dynamical typicality of quantum expectation values C. Bartsch and J. Gemmer (2009). Thus, the overwhelming majority of all possible initial states with the same as is also expected to approximately yield the corresponding to , if the dimension of the relevant Hilbert space is sufficiently large, see especially Ref. C. Bartsch and J. Gemmer, 2009. The latter largeness is certainly fulfilled for all practical purposes. Or, in other words, the concrete curves for and thus for only slightly “fluctuate” around the curve of the ensemble average. In fact, for the single-particle quantum system in Refs. M. Michel et al., 2005; J. Gemmer et al., 2006; R. Steinigeweg et al., 2007; R. Steinigeweg et al., 2009a it has been demonstrated that Eqs. (12)-(14) correctly describe the dynamics for all pure initial states which are not created explicitly in order to violate these equations, see Ref. M. Michel et al., 2005; J. Gemmer et al., 2006.
Eventually, let us relate the non-perturbative result for the diffusion constant to the standard result of linear response theory for the dc-conductivity R. Kubo et al. (1991); Mahan (2000). In the limit of high temperatures this conductivity may be written as
(15) |
where is the inverse temperature. The comparison of Eq. (15) with Eq. (13) yields
(16) |
i.e. it leads to an Einstein relation. It is well-known that the integral in Eq. (15) will diverge, whenever , the Fourier transform of , has a finite contribution at , see Ref. X. Zotos and P. Prelov̌sek, 2003, for example. As routinely done, we may hence define a Drude weight , e.g.
(17) |
ii.2 Finite Time and Length Scales
If the total current is strictly preserved, i.e. , the diffusion constant is completely governed by the Drude weight , i.e. for all . Consequently, transport is purely ballistic at each time, respectively length scale. Even if the total current does not represent a strictly conserved quantity, a non-zero Drude weight directly implies that the diffusion constant is still approximately given by the straight line in the limit . However, at some finite time or length scale the diffusion constant may nevertheless appear to be almost constant, i.e. , as expected for diffusive behavior. Such a behavior requires that the Drude weight is not relevantly large (on the corresponding scale), the finiteness of by itself is not crucial in this context. In fact, it is well-known that the classification of transport types for a given quantum system generally is a concept which crucially depends on the considered time or length scale, see Refs. J. Gemmer et al., 2006; R. Steinigeweg et al., 2007; R. Steinigeweg et al., 2009a, for example.
The above line of reasoning becomes also relevant in those situations where some model parameter induces a transition from a finite towards a zero Drude weight , e.g. where vanishes above some critical value of , say . Then at this critical value a sharp transition from ballistic towards non-ballistic transport is to be expected at the infinite time scale. However, such a transition does not necessarily appear at some finite time, respectively length scale. Even if there was a transition, a sharp one would require at the critical value a sudden jump of the Drude weight from a finite and relevantly large number (on the corresponding scale) to zero.
We hence do not concentrate merely on Drude weights and their finiteness, although those are also discussed, of course. Instead of that we focus on the diffusion constant , as defined in Eq. (13), at finite times and for finite systems as well. According to Eq. (13), we can evaluate the underlying current auto-correlation function directly in the time domain, i.e. , instead of the frequency domain, i.e. . This way we are not concerned with the problems which may arise due to the fact that is a highly non-smooth function for a finite system. We particularly exploit that the -curve does not change at sufficiently short time scales any more, when the size of a system becomes large enough. Thus, at those short time scales, interesting signatures of the infinitely large system may already be extractable for a system with an accessible size. In the following Secs. III-V this concept will be demonstrated in full detail.
Iii Ising Chain within a Tilted Field
In the present Section we will study a first example of a translationally invariant spin system. This system is an Ising chain which is exposed to a, say, tilted magnetic field . Concretely, its Hamiltonian reads () C. Mejía-Monasterio et al. (2005); C. Mejía-Monasterio and H. Wichterich (2007); T. Prosen and M. Žnidarič (2009)
(18) |
where and denote the -component, respectively -component of the total vector . Here, one might think of some magnetic field which originally was in line with the -direction and has been rotated about the -axis with the angle .
In the Hamiltonian (18) the operators () are the standard Pauli matrices (corresponding to site ); denotes the coupling strength; and represents the total number of sites. Throughout this work we use periodic boundary conditions, i.e. .
Due to the presence of the tilted magnetic field, the only trivial symmetries are translational invariance and mirror symmetry C. Mejía-Monasterio et al. (2005) such that the whole Hilbert space is decomposable into decoupled subspaces with similar dimensions. For the present model the component of the total spin in -direction (and any other direction) is non-preserved. Therefore magnetization (or spin) is not a suitable transport quantity here.
However, this model is appropriate to investigate the transport of energy, i.e. it allows to study the density dynamics of the local quantities , see Eq. (18). Note that these quantities contain contributions from both the Ising interaction and the Zeeman energy. The respective energy current is given by C. Mejía-Monasterio et al. (2005); C. Mejía-Monasterio and H. Wichterich (2007); T. Prosen and M. Žnidarič (2009)
(19) |
and the factor , as defined in Eq. (7), reads
(20) |
If the component is identical to zero (), the energy current (19) is strictly preserved. Furthermore, the Hamiltonian (18) is well-known to be integrable. But it can become quantum chaotic for , e.g. for the special set of parameters , , () C. Mejía-Monasterio et al. (2005); C. Mejía-Monasterio and H. Wichterich (2007); T. Prosen and M. Žnidarič (2009). Recently, for exactly this parameter set, a strong evidence for diffusive behavior has been found from non-equilibrium bath scenarios C. Mejía-Monasterio et al. (2005); C. Mejía-Monasterio and H. Wichterich (2007); T. Prosen and M. Žnidarič (2009).
Fig. 1 (top) shows the diffusion constant , as given by Eq. (13), for the above set of parameters. We observe that increases within the correlation time of the underlying current auto-correlation function but, already on this time scale, essentially the Drude weight governs the overall shape of the curve, at least for small lengths . But when is increased, the Drude weight rapidly becomes smaller, i.e decreases faster than a power-law, see Fig. 2 (squares). In particular there is no need to assume a finite and relevant value for in the thermodynamic limit . Already for the Drude weight is such small that the tendency of the diffusion constant to gradually develop towards a horizontal line becomes visible. Nevertheless, only from those lengths which are available from numerically exact diagonalization a definite conclusion may still be vague, particularly conclusions about the time after which possibly remains constant and about the constant value , too.
Fortunately, additional data is available in literature: In Ref. C. Mejía-Monasterio et al., 2005 the current auto-correlation function has been evaluated also by the use of an approximative numerical integrator. Based on this data the time evolution of the diffusion constant can be extracted not (b) for the length , see Fig. 1 (top inset). And in fact, for times which are larger than we observe that takes on a constant value . The latter value further is in excellent agreement with the conductivities from Ref. C. Mejía-Monasterio et al., 2005 and from Ref. T. Prosen and M. Žnidarič, 2009 not (a), as obtained therein for the same set of parameters from non-equilibrium bath scenarios. (In these works is denoted by .)
When the -component is decreased from down to , diffusive behavior eventually breaks down towards ballistic transport, because the current becomes strictly preserved for . Unfortunately, it is hardly possible to give a critical value for this transition, simply due to the limited system sizes which are accessible by the use of exact diagonalization. However, such a value may be very unsharp for a continuous transition, i.e. if the Drude weight changes smoothly and does not jump suddenly from zero (or a finite, irrelevantly small number) to a finite, relevantly large number, cf. Sec. II.
Contrary, when the -component is increased from , the indications of diffusive behavior become much more pronounced for the accessible system sizes, cf. Fig. 1 (bottom). Respective diffusion constants may therefore be suggested for this parameter regime.
Iv Anisotropic Heisenberg Chain
In this Section we will investigate transport in the anisotropic Heisenberg chain (or XXZ model) as another and certainly more interesting example of a translationally invariant spin system. It is described by a Hamiltonian of the form () X. Zotos and P. Prelov̌sek (2003); F. Heidrich-Meisner et al. (2007); M. Michel et al. (2008); T. Prosen and M. Žnidarič (2009)
(21) |
with the anisotropy parameter . Independent from the concrete choice of , the Hamiltonian (21) is integrable in terms of the Bethe Ansatz, see Ref. X. Zotos, 1999, for example.
In the presence of an external (uniform) magnetic field one may add to the Hamiltonian (21) a Zeeman term of the form . However, because the following investigation (at infinite temperature) will not depend on the concrete choice of , we set for simplicity.
As well-known, the Hamiltonian (21) is invariant under rotations about the z-axis, i.e. it commutates with . As a consequence can be diagonalized within decoupled subspaces with dimensions over , where is the quantum number w.r.t. . Moreover, due to the translational invariance as well as the mirror symmetry of (21), the problem can be reduced further by a factor .
Since represents a strictly conserved quantity, this model allows to investigate the density dynamics of the local quantities , i.e. magnetization transport (or spin transport). The respective magnetization current is given by X. Zotos and P. Prelov̌sek (2003); F. Heidrich-Meisner et al. (2007); M. Michel et al. (2008); T. Prosen and M. Žnidarič (2009)
(22) |
and the factor , as defined in (7), takes on the fixed value .
Similarly, an energy current can be also introduced via the relation . But, since its commutator with the Hamiltonian (21) vanishes exactly, this current is strictly conserved such that energy transport is purely ballistic at each time- and length-scale (and for all finite temperatures, see Ref. K. Sakai and A. Klümper, 2003).
Except for the special case of (XY model) the magnetization current is non-preserved. On that account it could be in principle possible to find diffusive behavior for non-zero anisotropies. Nevertheless, in the literature there is strong evidence that such a behavior is restricted to the regime only B. N. Narozhny et al. (1998); K. Fabricius and B. M. McCoy (1998); X. Zotos (1999); X. Zotos and P. Prelov̌sek (2003); F. Heidrich-Meisner et al. (2003); D. A. Rabson et al. (2004); J. Benz et al. (2005); F. Heidrich-Meisner et al. (2007); M. Michel et al. (2008); S. Langer et al. (2009); T. Prosen and M. Žnidarič (2009), most promising appear those which are close to M. Michel et al. (2008); S. Langer et al. (2009); T. Prosen and M. Žnidarič (2009). Those expectations are also due to results on the Drude weight X. Zotos (1999); J. Benz et al. (2005); F. Heidrich-Meisner et al. (2003), e.g. Bethe Ansatz approaches suggest that is finite for the regime and zero for the regime X. Zotos (1999); J. Benz et al. (2005). Still controversial is the special case of , where may be already zero X. Zotos (1999) or not J. Benz et al. (2005).
For completeness, we also show numerical results for the Drude weight in Fig. 5, although respective data can be found already in Ref. F. Heidrich-Meisner et al., 2003, also obtained by the use of exact diagonalization and additionally extrapolated to . This data can be transferred directly to the present investigation, if the concrete values for the Drude weight are multiplied by a factor . Further results on Drude weights from exact diagonalization can be found in, e.g. Refs. B. N. Narozhny et al., 1998; D. A. Rabson et al., 2004; F. Heidrich-Meisner et al., 2007.
iv.1 Anisotropies
Fig. 3 shows the diffusion constant , as defined in Eq. (13), for the anisotropy parameter . For the Drude weight determines the overall shape of the -curve, i.e. on all time scales is very close to the straight line . There is only an insignificant deviation from this line at short times scales below the correlation time of the underlying current auto-correlation function, see Fig. 3 (inset). For , i.e. when the length of the chain is more than doubled, the -curve remains practically the same. On that account it appears to be justified to assume a similar curve in the thermodynamic limit . This assumption is consistent with the extrapolation of the Drude weight in Ref. F. Heidrich-Meisner et al., 2003, see Fig. 5 (squares), too. We may therefore suggest purely ballistic behavior for the case .
Note that Sirker et al. have recently presented results in Ref. J. Sirker et al., 2009 which point towards a coexistence of diffusive and ballistic dynamics, where the ballistic distribution is small due to a small Drude weight. Even though we reproduce that the Drude weight is large for and does not depend significantly on , we can not exclude the possibility that the Drude weight eventually becomes small for , of course.
iv.2 Anisotropies
In Fig. 4 (middle) we display the diffusion constant for the anisotropy . We observe that increases at short time scales, i.e. within the correlation time of the underlying current auto-correlation function , and then remains approximately constant for an interval at intermediate time scales, i.e. a “plateau” is formed at those times. Finally, a renewed increase takes place on long time scales which is completely governed by the Drude weight, or by the zero-frequency distribution of . Note that the plateau can not be seen, if zero- and finite-frequency parts are treated separately from each other. It is a feature which arises from a combination of both contributions.
The above plateau remarkably is already visible for . Moreover, its “height” does not change with , while its “width” seems to increase gradually, see Fig. 4 (middle inset). The latter increase particularly appears to be plausible, because the Drude weight is commonly expected to vanish in the thermodynamic limit , as already outlined above. We hence make the educated guess that the plateau of will be continued to, say, arbitrary long times, when only becomes sufficiently large, e.g. . Then the height directly determines the concrete value of the diffusion constant, of course. And indeed, the latter value for also is in excellent agreement with the conductivity in Ref. T. Prosen and M. Žnidarič, 2009 not (a), as found therein from a non-equilibrium bath scenario. Although a respective figure is not shown for , such an agreement is additionally obtained with the results in Ref. M. Michel et al., 2008, namely and . (In these works is denoted by .) Those quantitative agreements support the correctness of our guess and the emergence of diffusive transport appears to be verified for those which are close to .
Independent from the above arguments regarding the long-time extrapolation for , we should mention that for the -curves have already converged for those times which are shorter than , see Fig. 4 (middle inset). Due to Eq. (12), this time corresponds to a length scale on the order of about sites. Thus, at least for the dynamics should be known.
For larger anisotropy parameters we find a similar situation: Again there is a clearly developing plateau at intermediate time scales, see Fig. 4 (bottom) for the case . For the accessible system sizes, however, the final plateau height does not seem to be reached yet, i.e., can not be read off with the same accuracy, as done before for . For comparison we hence indicate in Fig. 4 (bottom inset) the quantity with the dc-conductivity according to Ref. P. Prelovšek et al., 2004, obtained therein from an analysis on the basis of the standard Green-Kubo formula. The good agreement with this result is noticeable.
iv.3 Anisotropy
As already mentioned above, it is still controversial, whether or not the Drude weight is finite in the limit , when becomes exactly . However, let us for the moment assume that the Drude weight is indeed finite in that limit. Then at the infinite time scale the diffusion constant is completely governed by this non-zero Drude weight and increases linearly. At finite time scales, however, may still appear to be almost constant and feature a plateau like the one for the case . In principle such a plateau can be very wide, if only its height is large in comparison with the Drude weight such that the finiteness of is by itself not crucial in this context, cf. Sec. II.
We hence show in Fig. 4 (top) the diffusion constant for the special case . The initial short-time increase directly passes into the final long-time increase, i.e in between a horizontal line is not observable for the accessible system sizes. Nevertheless, we can not exclude the possibility that a plateau with a height on the order of will eventually develop, when the system size is further increased, cf. Fig. 4 (top inset). We may thus compare this value for with the extrapolated Drude weight in Ref. F. Heidrich-Meisner et al., 2003 not (c), for example. (The latter extrapolated is about of the long-time slope for , cf. Figs. 4 and 5.) By the use of the rough estimation we directly obtain that an interval with can not be wider than some time scale , respectively length scale . Thus, if the above extrapolation for the Drude weight was indeed correct, diffusion would be restricted to very few sites only. Contrary, if the Drude weight was zero (or a finite, irrelevantly small number), diffusion could occur for a possible arbitrary number of sites, of course.
V Heisenberg Chain within An Alternating Field
In this Section we are going to concentrate exclusively on the pure Heisenberg chain, i.e. , and investigate another modification of the Hamiltonian (21) which may lead to diffusive transport. Such a modification certainly is the incorporation of disorder, i.e. coupling constants or field strengths which vary randomly from one site to another Y. Avishai et al. (2002); Santos (2004); R. Steinigeweg et al. (2006); L. F. Santos (2008); A. Karahalios et al. (2009). But the investigation of disorder is numerically rather challenging and conceptually more subtle due to the localization phenomenon, even for a non-interacting single-particle system Anderson (1958); R. Steinigeweg et al. (2009a). Instead we will consider another scenario without disorder, where the Heisenberg chain is exposed to a strictly alternating field. This scenario is concretely described by the Hamiltonian (21) and an additional Zeeman term of the form M. Michel et al. (2008); T. Prosen and M. Žnidarič (2009)
(23) |
where is the deviation from the mean . As already done before, we may set for simplicity.
The presence of the Zeeman term (23) does not affect the commutation of and , i.e. the Hamiltonian can be diagonalized within decoupled -subspaces, too. As long as is even, there also is translational invariance (w.r.t two sites) such that the problem can be reduced further by a factor . (There is no mirror symmetry for even .)
Since still represents a strictly conserved quantity, the present model allows to investigate the transport of magnetization w.r.t. the field parameter . Note that the respective current and factor are identical to those in Sec. IV. When is increased from zero and becomes comparative with the coupling strength , the model undergoes a transition to quantum chaos T. Prosen and M. Žnidarič (2009), i.e. one may assume that the latter transition already is a first pointer towards the onset of diffusion. Even though it is entirely independent from those assumptions, we start our investigation with .
For the curve for the diffusion constant has changed from Fig. 4 (top) into Fig. 6 (top). In particular the initial short-time increase does not directly pass into the final long-time increase any more. Instead there is an oscillation in between such that, at the first view, the situation appears to be much more complicated than a simple horizontal line. However, for the -curve has already converged until the end of this oscillation, see Fig. 6 (top inset). Exactly at this position the curve seems to gradually develop a plateau with a height on the order of . The plateau becomes visible, since the Drude weight rapidly decreases with , i.e. it is about one order of magnitude smaller for than for , cf. Fig. 7 (circles). Note that the Drude weight does not fulfill a simple -dependence and decays faster than a power-law. Thus, there is no need to suppose a finite and relevant Drude weight in the limit .
Remarkably, the above suggested for agrees well with the conductivity from Ref. M. Michel et al., 2008, as found therein from a non-equilibrium bath scenario. The deviation from this conductivity is on the order of solely. However, the small deviation may be explained as follows: In Ref. M. Michel et al., 2008 the conductivity is evaluated by the use of the Lindblad equation, i.e., it is extracted from the steady state of a finite chain which is at both ends weakly coupled to baths at different temperatures. This weak bath-coupling is essential for the validity of the Lindblad approach. But, for a finite chain, a too weak bath-coupling may also yield a value which is lower than the correct value, say, , see Ref. R. Steinigeweg et al., 2009b. Since in Ref. M. Michel et al., 2008 the independence of the conductivity from the bath-coupling strength is not analyzed in detail, we suppose that the above deviation results from a slightly too weak coupling of chain and reservoirs.
For larger field parameters suggestions are less reliable, since the initial oscillations of the -curves become much more pronounced, as already visible in Fig. 6 (bottom). Even though there are oscillations, still remains strictly positive, i.e. those oscillations probably are no pointer towards insulating behavior.
Note that definite conclusions can not be made for the parameter regime , since the curves for continuously change from Fig. 4 (top) into Fig. 6 (top), i.e. one basically is concerned with the problem that the Drude weight governs the overall shape of the curve, at least for accessible system sizes.
Vi Summary and Conclusion
In the present paper we have investigated transport in several translationally invariant spin- chains in the special limit of high temperatures. We have concretely considered spin transport in the anisotropic Heisenberg chain, in the pure Heisenberg chain within an alternating field, and energy transport in an Ising chain which is exposed to a tilted field.
To this end we have firstly reviewed on a recently derived connection between the evolution of the variance of some “typical” inhomogeneous non-equilibrium density and the current auto-correlation function at finite times R. Steinigeweg et al. (2009c). In the limit of infinitely long times this connection was shown to yield a generalized Einstein relation which relates the diffusion constant (in the absence of any external force) to the dc-conductivity (as the linear response coefficient in the presence of an external force, i.e. as evaluated from the standard Green-Kubo formula R. Kubo et al. (1991); Mahan (2000)). However, we have additionally demonstrated that the great advantage of the above connection is given by its direct applicability at finite times and for finite systems, e.g. at short times interesting signatures of an infinitely large system may be extractable for a system with an accessible size.
By means of numerically exact diagonalization we have indeed observed strong indications for diffusive behavior in the considered spin chains for a range of model parameters. Moreover, the suggested diffusion constants have been found to be in quantitative agreement with recent results on diffusion coefficients which were obtained for the same spin chains from numerically involved investigations of non-equilibrium bath scenarios in Refs. C. Mejía-Monasterio et al., 2005; C. Mejía-Monasterio and H. Wichterich, 2007; M. Michel et al., 2008; T. Prosen and M. Žnidarič, 2009.
Amongst all those and our findings at high temperatures the emergence of diffusive transport of magnetization appears to be verified in the anisotropic Heisenberg chain with the anisotropy parameter M. Michel et al. (2008); T. Prosen and M. Žnidarič (2009), despite the integrability of the model. It is known that the onset of quantum chaos is not a sufficient condition for diffusion R. Steinigeweg et al. (2006); L. F. Santos (2008) but the latter result suggests also that non-integrability is not necessary at all.
Unfortunately, it is still an open question whether spin transport in the pure Heisenberg chain is diffusive or ballistic. Simply by the use of numerically exact diagonalization we were not able to reach those system sizes which are required for any definite conclusion on that question.
Therefore the next step certainly is the application of approximate methods in order to obtain the current auto-correlation function at finite times, either numerical ones, e.g. Suzuki-Trotter decompositions H. F. Trotter (1959); M. Suzuki (1990), or analytical ones, e.g. projection operator techniques S. Nakajima (1958); R. Zwanzig (1960); H.-P. Breuer and F. Petruccione (2007); R. Steinigeweg et al. (2009a) or moment methods U. Brandt and J. Stolze (1986); J. M. R. Roldan et al. (1986). Approximative methods will also be in indispensable, when the investigation is extended from one-dimensional spin chains to, e.g. more-dimensional spin lattices.
Apart from the above methodic details, a physically interesting question is the dependence of transport on temperature, of course. Since the connection between the evolution of the variance and the current auto-correlation function is not restricted to infinite temperature, further investigations may be done in this direction, too. However, we expect that for low temperatures the convergence of the diffusion constant at finite times is much slower, when the system size is increased. Thus, for low temperatures conclusions on the basis of a finite system may be less reliable.
Acknowledgements.
We sincerely thank H. Wichterich, M. Michel and F. Heidrich-Meisner for fruitful discussions. We further gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft.References
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