Magnetization and energy dynamics in spin ladders: Evidence of diffusion in time, frequency, position, and momentum
The dynamics of magnetization and energy densities are studied in the two-leg spin- ladder. Using an efficient pure-state approach based on the concept of typicality, we calculate spatio-temporal correlation functions for large systems with up to lattice sites. In addition, two subsequent Fourier transforms from real to momentum space as well as from time to frequency domain yield the respective dynamic structure factors. Summarizing our main results, we unveil the existence of genuine diffusion both for spin and energy. In particular, this finding is based on four distinct signatures which can all be equally well detected: (i) Gaussian density profiles, (ii) time-independent diffusion coefficients, (iii) exponentially decaying density modes, and (iv) Lorentzian line shapes of the dynamic structure factor. The combination of (i) - (iv) provides a comprehensive picture of high-temperature dynamics in this archetypal nonintegrable quantum model.
Introduction. The study of low-dimensional spin systems is one of the most active fields in condensed matter physics. On the one hand, quantum spin models are of immediate relevance to describe the properties of various Mott insulators, where (quasi) one-dimensional structures like chains or ladders are realized within the bulk materials. The notion of property can be manifold in this context, including thermodynamic quantities johnston2000 (), transport characteristics such as the heat conductivity Sologubenko2000 (); Hess2001 (), as well as other dynamic features probed by, e.g., neutron scattering Lake2013 (); Mourigal2013 (), NMR Thurber2001 (), and SR Maeter2013 (), to name just a few. Particularly, developing a thorough understanding of quantum magnets, both experimentally and theoretically, is also of essential importance in order to pave the way for potential spin-based technologies in the future Wolf2001 ().
On the other hand, from a more fundamental point of view, low-dimensional spin models represent prototypical examples of interacting quantum many-body systems, allowing to study questions ranging from the foundations of statistical mechanics Cazalilla2010 () to the physics of black holes Ikhlef2012 (). In particular, long-standing questions concerning the emergence of thermodynamic behavior in isolated quantum systems have recently experienced rejuvenated attention Polkovnikov2011 (); Eisert2015 (); Gogolin2016 (); Dallesio2016 (). This upsurge of interest is not least due to the advent of controlled experiments with cold atoms and trapped ions Langen2015 (); Trotzky2012 (); Blatt2012 (), theoretical key concepts such as the eigenstate thermalization hypothesis deutsch1991 (); srednicki1994 (); rigol2005 (); Richter2018 () and the typicality of pure quantum states Gemmer2004 (); Popescu2006 (); Goldstein2006 (); Reimann2007 (), as well as the development of powerful numerical techniques schollwoeck20052011 ().
Concerning the relaxation in isolated quantum systems, an intriguing question is the role of integrability versus nonintegrability. On the one hand, integrable systems exhibit a macroscopic set of (quasi-)local conservation laws zotos1997 (); prosen2013 () which might cause anomalous thermalization vidmar2016 () as well as nondecaying currents, i.e., ballistic transport heidrichmeisner20032007 (). Nonetheless, even for paradigmatic integrable models, signatures of diffusion have been reported, e.g., for the spin- XXZ chain above the isotropic point Sirker2011 (); znidaric2011 (); steinigeweg2011_1 (); karrasch2014_2 (); Steinigeweg2017 (); Ljubotina2017 () or the Fermi-Hubbard model for strong particle-particle interactions prosen2012 (); karrasch2017 (); steinigeweg2017_2 (). On the other hand, in more realistic situations, integrability is often lifted due to various perturbations, e.g., spin-phonon coupling Chernyshev2016 (), long-range interactions Hazzard2014 (), dimerization Karrasch2013 (), as well as the presence of impurities Metavitsiadis2010 () or disorder Herbrych2013 (). Such nonintegrable systems are commonly expected to have vanishing Drude weights heidrichmeisner20032007 () and potentially exhibit diffusive transport, e.g., due to quantum chaos. However, since the dynamics of interacting systems with many degrees of freedom poses a formidable challenge to theory and numerics, signatures of clean diffusion have been found only for selected examples Michel2005 (); Monasterio2005 (); karrasch2014_2 (); Medenjak2017 (); Richter2018_2 ().
In this context, we study the dynamics of magnetization and energy in the spin- ladder, using an efficient pure-state approach based on the concept of typicality Gemmer2004 (); Popescu2006 (); Goldstein2006 (); Reimann2007 (); Hams2000 (); iitaka2003 (); sugiura2013 (); elsayed2013 (); monnai2014 (); steinigeweg2014 (). The Hamiltonian with periodic boundary conditions reads
where are spin- operators on lattice site , denotes the coupling along the legs (and sets the energy scale throughout this Letter), and is the interaction on the rungs. While for , decouples into two separate chains and is integrable in terms of the Bethe ansatz kluemper2002 (), this integrability is broken for any . Numerous works Zotos2004 (); Jung2006 (); Langer2009 (); znidaric2013_1 (); Karrasch2015 (); Steinigeweg2016 () have explored the dynamics of the spin ladder (1), also including various modifications such as four-spin terms Nishimoto2009 (), Kitaev-type couplings Metavitsiadis2017 (), as well as XX-ladder systems znidaric2013_2 (); steinigeweg2014_2 (). However, while these studies often discuss either spin or energy transport only, they also mostly focus exclusively on the dynamics of densities or currents, either in time or in frequency.
In this Letter, we do not focus on a particular quantity and representation and provide a comprehensive picture of high-temperature dynamics in the spin- ladder. As a main result, we unveil the existence of genuine diffusion both for magnetization and energy. In particular, this result is based on the combination of four distinct signatures: (i) Gaussian density profiles, (ii) time-independent diffusion coefficients, (iii) exponentially decaying density modes, and (iv) Lorentzian line shapes of the dynamic structure factor. We present these signatures for large systems with up to lattice sites.
Setup. We study the dynamics of time-dependent expectation values
where the time argument has to be understood as , and the operator denotes the local densities of magnetization or energy (cf. Fig. 1). Furthermore, the (unnormalized) pure initial state is prepared as
where the complex coefficients are randomly drawn from a Gaussian distribution with zero mean (Haar measure bartsch2009 ()) and the denote a set of orthonormal basis states (e.g. the Ising basis) of the full Hilbert space with dimension . Moreover, the operator in Eq. (3) is essentially equivalent to except for a constant offset which renders the eigenvalues of nonnegative (see Supp. Mat. Supplemental ()). Exploiting the concept of dynamical typicality bartsch2009 (); Reimann2018 (), as well as , the expectation value can be connected to an equilibrium correlation function at formally infinite temperature Supplemental (),
where translational invariance has been exploited, and with discrete momenta and . Furthermore, a subsequent Fourier transform from time to frequency domain eventually yields the dynamic structure factor ,
with the finite frequency resolution .
It is further instructive to establish a relation between density dynamics and current-current correlation functions. To this end, let us introduce the time-dependent diffusion coefficient Steinigeweg2009 ()
where denotes the isothermal susceptibility Suscep () and the spin- or energy-current operators follow from a lattice continuity equation Supplemental (). To proceed, we note that the states realize an initial density profile which exhibits a peak at InitDelta (). This initial peak will gradually broaden with time and its spatial variance for is given by
Note that a diffusive process requires . such that karrasch2014_2 (), above the mean free time.
Method. We rely on the typicality relation (4) in order to calculate spatio-temporal correlation functions for spin and energy. This approach turns out to be powerful, since the action of on the pure state can be efficiently evaluated without the diagonalization of , e.g., by means of a Trotter decomposition deReadt2006 (); Supplemental (), or other approaches Dobrovitski2003 (); Weisse2006 (); Varma2017 (). In practice, this fact enables us to treat ladders with up to rungs, i.e., lattice sites in total. If not stated otherwise, we always take into account the full Hilbert space (e.g. for ).
Furthermore, let us stress that the numerical costs of the Fourier transforms (5) and (6) are practically negligible. Therefore, we essentially obtain all information on the dynamics of either magnetization or energy from the time evolution of the single pure state and the measurement of local operators . Note that in the case of energy, is not diagonal in our working basis and the preparation of by means of the square root projection in Eq. (3) can become cumbersome Supplemental ().
We now present our numerical results. In particular, we will focus on the isotropic case (see Supp. Mat. for Supplemental ()). Starting with dynamics in time and real space, Figs. 2 (a) and (b) show the density profiles of magnetization and energy for large systems with . One can clearly observe the initial peak at (or almost peak InitDelta ()) which broadens for times . Moreover, on the (short) time scales depicted, does not reach the boundaries of the system, i.e., trivial finite-size effects do not occur.
For a more detailed discussion, Figs. 3 (a) and (b) show cuts of at fixed times . For these times, one finds that the data are well described by Gaussians over several orders of magnitude,
While these Gaussians already suggest diffusion both for magnetization and energy, it only is an sufficient criterion if scales as as well. Consequently, Figs. 3 (c) and (d) show the widths of the density profiles obtained from Eq. (8) (symbols) in comparison with the respective quantities and (lines), calculated from the current autocorrelations, cf. Eqs. (7) and (9). Generally, one observes an excellent agreement between density and current dynamics. Moreover, after a linear increase at short times, eventually saturates at a constant plateau , and correspondingly . Thus, based on our numerical analysis in time and real space, we unveil the existence of diffusive transport in the spin- ladder both for magnetization as well as energy. This is a first main result of this Letter.
Next, let us also study density dynamics in momentum space, where the lattice diffusion equation decouples into separate Fourier modes Supplemental (). Since the results for magnetization (Fig. 4) and energy (Fig. 5) are qualitatively very similar, we here restrict ourselves to a detailed discussion of the former. In Fig. 4 (a), the density modes are shown for various momenta in a semilogarithmic plot. On the one hand, for large , exhibits pronounced oscillations and essentially decays on a time scale . On the other hand, for the two smallest wave numbers and , we find clean exponential relaxation
where we have introduced the abbreviation for sufficiently small , and can be extracted from the constant plateau in Fig. 3 (c).
Going from time to frequency domain, Fig. 4 (b) shows the corresponding dynamic structure factors for . One observes that the data can be accurately described by Lorentzians of the form,
Note that, while we display the time data in Fig. 4 (a) only up to intermediate time scales, the Fourier transform (6) is routinely performed for a much longer cut-off time (here ) in order to achieve a high frequency resolution. The exponential relaxation [Fig. 4 (a)] and the Lorentzian line shapes [Fig. 4 (b)] clearly confirm our earlier observations in the context of Fig. 3, i.e., the occurrence of genuine spin diffusion in the spin- ladder. This is another main result of the present work.
In addition to the long-wavelength limit, Fig. 4 (c) shows at momentum . For this momentum, one finds that is practically -independent up to and exhibits a constant plateau. It is instructive to compare this result to the dynamics of the one-dimensional XY model. Since the XY chain is equivalent to a model of free fermions, is known exactly Fabricius1997 () and reads (for , , and )
Here, is the Bessel function of first kind (and order zero), and exhibits a square-root divergence at , cf. Fig. 4 (c). Following an approach introduced in Ref. Knippschild2018 (), the dynamics is interpreted as being generated by an integro-differential equation comprising a memory kernel . Moreover, the rung couplings and terms of the spin ladder (1) are effectively treated as a perturbation to the bare XY model, giving rise to an exponential damping of this memory kernel (for small perturbations), (see Supp. Mat. for details Supplemental ()). As shown in Fig. 4 (c), the effective dynamics generated by this (heuristic) approach reproduces the structure factor of the spin ladder remarkably well, even though the perturbation is not small. Thus, while clear signatures of diffusion can be found for [cf. Figs. 4 (a) and (b)], the relaxation of the density modes for can be qualitatively understood as the (damped) dynamics of free fermions, e.g., since the wavelength is smaller than the mean-free path. Note that similar behavior has been found also for XXZ chains Fabricius1997 (); Herbrych2012 ().
Eventually, Fig. 4 (d) shows the dynamic structure factor for all momenta . We find that exhibits a broad continuum in the center of the Brillouin zone. Moreover, for small momenta , one can clearly identify the diffusion peaks discussed above.
Finally, let us stress again that the results for energy dynamics in Fig. 5 are qualitatively very similar and confirm the existence of diffusive energy transport as well Zotos2004 (); Karrasch2015 (); Steinigeweg2016 (). Moreover, signatures of energy diffusion can also be observed in the presence of an additional magnetic field, where magnetothermal corrections can become relevant (see Supp. Mat. Supplemental ()).
Conclusion. To summarize, we have studied spin and energy dynamics in the spin- ladder for large systems with up to lattice sites. Our state-of-the-art numerical simulations have unveiled the existence of genuine diffusion both for spin and energy. In particular, this finding is based on four distinct signatures which have all been equally well detected: (i) Gaussian density profiles, (ii) time-independent diffusion coefficients, (iii) exponentially decaying density modes, and (iv) Lorentzian line shapes of the dynamic structure factor. Combining (i) - (iv), this Letter provides a comprehensive picture of high-temperature dynamics in the spin- ladder. Promising directions of research include, e.g., the application of the pure-state approach to a larger class of condensed matter systems and a wider range of temperatures Rousochatzakis2018 ().
Acknowledgments. This work has been funded by the Deutsche Forschungsgemeinschaft (DFG) - GE 1657/3-1; STE 2243/3-1 - within the DFG Research Unit FOR 2692. Additionally, we gratefully acknowledge the computing time, granted by the “JARA-HPC Vergabegremium” and provided on the “JARA-HPC Partition” part of the supercomputer “JUWELS” at Forschungszentrum Jülich. J.H. has been supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials Sciences and Engineering Division.
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Appendix A Supplemental Material
a.1 Dynamics in the presence of a magnetic field
As a natural extension of our considerations in the main part of this Letter, let us now also discuss the effect of an additional uniform magnetic field in the direction, i.e., the new local energy reads
where is defined according to Eq. (1). Concerning the current operators, the spin current is given by Eq. (S3) (independent of ) while the energy current in Eqs. (S4) and (S5) experiences a magnetothermal correction and the heat current reads
Analogous to our discussion in the context of Fig. 5, we depict in Fig. S1 the energy structure factors and in time as well as frequency domain for various momenta . Generally, we find that the presence of a finite magnetic field does not qualitatively change the behavior of and . Again, one can observe an exponential decay for the two smallest wave numbers and correspondingly a Lorentzian line shape in frequency space at these momenta. Compared to the results with shown in Fig. 5, one might even argue that those signatures of diffusion are slightly improved due to the finite magnetic field.
a.2 Density dynamics for additional parameters
While we focused on the isotropic case in the main part, let us now also discuss the dynamics of magnetization for other ratios .
To this end, Fig. S2 shows the real-space magnetization density profiles at fixed times for the interchain couplings (). Generally, we find that the profiles are very similar to each other for all and shown here. In particular, all profiles are convincingly described by Gaussian fits over several orders of magnitude.
Moreover, Fig. S3 shows a comparison of the respective structure factors and . On the one hand, for the smallest wave number , we find a clean exponential decay of , and quite remarkably, the rate of this decay in almost identical for all shown here, cf. Fig. S3 (a). This fact is also reflected in the Lorentzian shape of for this momentum [Fig. S3 (b)], which essentially coincides for all strengths of interchain couplings. On the other hand, for wave number , we find that for shows some deviations from an exponential, i.e., the hydrodynamic regime becomes smaller for smaller , which can be explained by the increased mean free path. All in all, however, the results presented in Fig. S2 and S3 suggest that the occurrence of spin diffusion is not restricted to but extends to a much wider parameter range. For a calculation of heat conductivities in a wide range of , see Ref. Steinigeweg2016S ().
a.3 Current operators and autocorrelations
In the main part of this paper, we mainly focused on the dynamics of local densities (with the exception of in Fig. 3). In this section, let us discuss the dynamics of spin and energy currents in more detail.
To begin with, the spin current is defined as Karrasch2015S ()
In order to calculate current-current correlation functions by means of a typicality-based approach directly (at finite or infinite temperature), we use the two auxilary pure states steinigeweg2014S ()
where again for . Of course, by replacing the current operator in Eqs. (S7) and (S8), it is straightforward to generalize the above approach in order to calculate dynamic correlation functions also for other operators.
While we already introduced the time-dependent diffusion coefficient for spin and energy transport in Eq. (7), the respective ac-conductivities at finite frequency are given by the Fourier transform of the current autocorrelations,
Note that the spin conductivity from Eq. (S9) must not be confused with the spatial variance introduced in Eq. (8). Omitting the possibility of finite Drude weights, the corresponding dc-conductivities then follow in the limit and can be connected to the diffusion constant via an Einstein relation , cf. Eq. (7). Let us note that, since the Fourier transforms in Eqs. (S9) and (S10) can in practice be only evaluated up to a finite cutoff time , the frequency resolution of is finite as well [see also Eq. (6) in the main text].
In Figs. S4 (a) and (b), the current autocorrelations at are shown for spin and energy transport, respectively. While in Fig. S4 (a) we show data for smaller systems with rungs only, the energy current in Fig. S4 (b) is calculated for systems with . Note however that in the latter case, we restrict ourselves to the symmetry subspaces with momentum as the current is known to be essentially independent of for such system sizes. [This crystal momentum should not be confused with the wave number below Eq. (5)]. In all cases shown here, we observe that decays to approximately zero, consistent with the absence of ballistic transport in a nonintegrable system. In the case , our results are additionally compared to data digitized from Ref. Karrasch2015S () obtained by a time-dependent density matrix renormalization group (tDMRG) approach. Generally, one finds a convincing agreement between both methods, i.e., our data for is free of significant finite-size effects and represents the thermodynamic limit. In Fig. S4 (c), spin and energy autocorrelations are depicted for the finite temperature . Also in this case, we observe that the pure-state method accurately reproduces the tDMRG data. Thus, typical pure states yield an efficient approach to correlation functions at finite temperatures as well.
In Figs. S5 (a) and (b) the respective ac-conductivities and at are shown, i.e., the Fourier transforms of the data in Figs. S4 (a) and (b). Particularly, we compare data with two different frequency resolutions and , i.e., a rather short and significantly longer cutoff time in Eqs. (S9) and (S10). In all cases, we observe a well-behaved dc-conductivity which does not (significantly) depend on , except for . Moreover, our data is again in good agreement with existing data obtained by tDMRG Karrasch2015S () and by the microcanonical Lanczos method Zotos2004S ().
a.4 Initial states and typicality relations
The purpose of this section is to derive Eqs. (4), (5), and (6) from the main part in this paper. In this context, we also show how the concept of typicality yields an efficient technique to compute high-temperature correlation functions.
We start with an equilibrium correlation function at formally infinite temperature ,
where is the smallest eigenvalue of . We realize that Eq. (S12) can be further simplified if either or . Focusing on these cases, one finds
i.e., the correlation functions and are equivalent (or in case of , only differ by a time-independent constant). This is a first important observation. Continuing with the expression given in Eq. (S11) and exploiting the cyclic invariance of the trace then also yields
where the square root operation has to be understood in a representation where is diagonal. Details on this aspect can be found below. Using the concept of quantum typicality, the trace in Eq. (S14) can be replaced by a scalar product with a single pure state which is randomly drawn according to the unitary invariant Haar measure,
where we have used the definition of our initial state from Eq. (3) and interpreted the time dependence as a property of the states and not of the operators. In summary, Eqs. (S15) to (S17) in combination with Eq. (S13) then yield the typicality relation in Eq. (4) from the main part of this Letter,
Concerning the evaluation of the square root the following comments are in order. On the one hand, in the case of this procedure is rather simple since is naturally diagonal in the Ising basis, which is routinely used as our working basis. On the other hand, in the case of the local energy, is not diagonal immediately. While this situation usually requires diagonalization, a complete diagonalization of can still be avoided since is a local operator acting non-trivially only on a small part of the Hilbert space. Thus, although the preparation of becomes more demanding in the case of , it certainly remains feasible and yields a powerful numerical approach as well. If one still wants to refrain from such square-root constructions, it is of course also possible to use two auxiliary pure states instead of just one, analogous to the calculation of current autocorrelations in Eqs. (S6) to (S8). It should be noted however, that the approach presented in this paper, using just a single pure state, will generally be more favorable concerning memory requirements and runtime (even if the initial preparation of is more costly).
Finally, let us also comment on the derivation of Eqs. (5) and (6) from the main part of this paper. Referring to Eq. (4), we realize that a cut through the density profile at fixed lattice site is equivalent to the correlation function . It is now instructive to perform the following calculation
where we have exploited translational invariance in order to compress the original double sum. Thus, we find that the intermediate structure factor can be easily obtained by a lattice Fourier transform of the real-space correlations. Furthermore, this momentum-space correlation function can also be transferred to what is usually referred to as the dynamic structure factor by another Fourier transform from time to frequency domain,
where the finite cutoff time yields a frequency resolution . Thus, starting from the correlations it is straightforward to also obtain correlation functions in momentum and frequency domain, which makes our pure-state method a rather powerful numerical approach.
a.5 XY model and damped memory kernel
The one-dimensional XY model is described by the Hamiltonian
The couping constant is set to unity in the following. By means of a Jordan-Wigner transformation, can be mapped onto a model of free spinless fermions. Furthermore, a subsequent Fourier transform (assuming periodic boundary conditions) yields the quadratic form
where , , , and () creates (annihilates) a spinless fermion at lattice site . Furthermore, the eigenenergies are given by
Since the XY model is equivalent to noninteracting lattice fermions, the dynamic structure factor in the thermodynamic limit is known exactly for arbitrary temperature and momentum Fabricius1997S (). In the case and , it reads [cf. Eq. (13)]
where is the Heaviside step function und is the Bessel function of first kind and order zero.
where is a memory kernel. Eq. (S29) establishes a direct correspondence between and and can be evaluated in both directions. Thus, given the original dynamics , the respective memory kernel can be calculated, e.g., numerically. In fact, given the expression in Eq. (S28), can be even obtained analytically and reads,
As a next step, the bare XY model is compared to the spin ladder (1). The additional rung couplings as well as the terms are treated as a perturbation which (heuristically) leads to an exponential damping of the original memory kernel,
This new memory kernel is then used to numerically evaluate Eq. (S29) in order to obtain the new (damped) dynamics .
a.6 Diffusion in lattice models
In this section, let us discuss in more detail how to detect diffusion in lattice models. To begin with, a process is called diffusive if it fulfills the lattice diffusion equation
with the time-independent diffusion constant . In the case of an initial peak profile at , a specific solution for the time dependence of is given by Richter2018_2S ()
where is the modified Bessel function of the first kind and denotes the homogeneous background. (Note that in this paper we have since .) In case of sufficiently large and long , i.e., if the discrete lattice momenta become dense, this lattice solution can be well approximated by a Gaussian. Such Gaussians have been observed in Figs. 3 (a) and (b). Specifically, the spatial variance of these Gaussians is then also given by , i.e., .
Given some general density distribution, it is in some cases instructive to study the dynamics in momentum space as well. In this context, a Fourier transform of Eq. (S32) yields
Apparently the different Fourier modes are completely decoupled and their exponentially decaying solutions have been already given in Eq. (11).
a.7 Trotter decomposition
Let us briefly give some details on the time evolution of pure quantum states by means of a Trotter decomposition. To begin with, we note that the time-dependent Schrödinger equation
is formally solved by
with , where we have set . While the exact evaluation of Eq. (S36) requires the diagonalization of , we here approximate the time-evolution operator by means of a Trotter product formula. Specifically, a second-order approximation of is given by
where . This approximation is then bounded by
where is a positive constant. In practice, the Hamiltonian is decomposed into the , , and components of the spin operators, i.e., . Since the computational basis states are eigenstates of the operators, the representation is diagonal by construction and only changes the input state by altering the phase of each of the basis vectors. Using an efficient basis rotation into the eigenstates of the or operators, the operators and can act as as well deReadt2006S ().
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