Dynamics of Strongly Correlated Systems and Dynamical Renormalization Group

We discuss the dynamics of the central spin model with the equation of motion technique. We show that a Hierarchy of Horrelations decoupling can provide accurate results for the dynamics, in comparison with more standard decoupling methods. We also demonstrate that the use of dynamical Renormalization Group (dRG) provides a suitable analytical tool to understand the physics, and a powerful method to eliminate secular terms, which are typically present in the time evolution. In addition, this approach allows to separate purely classical non-linear contributions, from the ones due to quantum correlations.

Introduction: During the last decade, a growing interest in the understanding of dynamical quantum systems has emerged.Motivated from both, theory and experiment, a whole new area of physics is being developed, where quantum systems and their dynamics play a dominant role.While non-interacting systems are quite well understood, interacting systems can display exotic new physics such as Floquet phases [1,2], time crystals [3,4], many-body localization [5,6] and complex dynamics [7,8].While the simulation of classical complex systems with non-linearities can be already challenging (weather forecast, stock-market predictions, social behavior or swarming), an extra difficulty arises in quantum systems, due to the presence of entanglement.Furthermore, the typical appearance of secular terms introduced by the truncation of the equations, requires a careful analysis in numerical simulations.
In this letter we study the dynamics of the central spin model, and show that many-body effects and entanglement formation can be captured in simple terms.The central spin model is one of the canonical models to study the dynamics of quantum interacting systems [9].It describes a quantum spin interacting with a set of localized modes, which in many cases can reduce to 1/2 spins.It can describe molecular magnets interacting with impurities [10], flux qubits coupled to electric dipoles [11] and many other effective two-level systems interacting with localized modes.Interestingly, the dynamics in this model can be quite complex, as it is known that the localized nature of these modes makes the system nonperturbative and gives rise to a richer dynamics than the case of delocalized modes [12] (i.e., in the spin-boson model [13]).For example in some regimes, the bath dynamics is slaved to the motion of the central spin, and the memory of the bath becomes relevant [9].Here we show that the system can display very different time-scales, which can be captured numerically with a Hierarchy of Correlations decoupling.Then we demonstrate that dRG (also known as multiple-scale analysis), can capture the main features of the time evolution, and get rid of the secular terms.It is interesting that dRG is widely used in the mathematics community, but rarely used in the quantum physics community.We hope that this work shows the advantages of applying this technique to quantum models, and brings new perspectives to non-equilibrium situations.
Model: We consider the next Hamiltonian: where we have assumed that the interaction is purely longitudinal (typically due to a large crystal field anisotropy), and that the central spin and bath couple to the transverse fields B and ∆, respectively.In the limit of large interaction the central spin "sees" the longitudinal Overhauser field produced by the bath (which can be experimentally quite large, as J i does not scale as 1/ √ N which would typically be the case for delocalized modes), while each bath mode couples to a weaker field produced by the central spin.When the transverse fields are added, bath and central spin precess at different rates, and spin-flip transitions happen at certain times, due to the interaction.Different Hamiltonians with other types of couplings can also be treated using this formalism, but Eq.1 possess the minimal elements to produce interesting effects and simple analytical expressions, easy to interpret.
We calculate the Heisenberg equations of motion for the different spin components: with and Ω B (t) = (−∆, 0, J i S z (t)) being the effective magnetic fields acting on central spin and bath, respectively.Notice that the equations are non-linearly coupled due to the interaction term, and that setting ∆ = 0 freezes the bath dynamics, fixing the bath polarization P z = i I z i [14].To analyze the dynamics of both, system and bath, we focus on some specific regime.An interesting situation happens when B is large, while J i and ∆ are much smaller.In this case B corresponds to some control field applied to manipulate the qubit, while ∆/B ≪ 1 if the bath corresponds to, for example, nuclear spins which have a smaller magnetic moment.In these cases J i /B ≪ 1 as well, because the interaction between the nuclear isotopes and the qubit is typically weak.Notice that J i /B ≪ 1 does not mean weak coupling for the central spin, as for N ≫ 1, the Overhauser field can be quite large.It only implies that the back-reaction from the central spin to each individual bath spin is small.Finally, Eqs.2 and 3 couple to higher order correlators such as I z i S x,y and I x,y i S z , which in general need to be calculated as well.The simplest solution is obtained by factorization of these correlators, neglecting correlations between the central spin and the bath.This results in the well known Mean-Field (MF) equations.These equations neglect entanglement between spins, but retain the nonlinear nature of the coupling between the central spin and the bath.For numerical simulations we consider the case of homogeneous couplings J i = J, but this assumption is not crucial, as the formalism allows to consider disorder without any changes in the equations.This assumption mainly simplifies the exact diagonalization calculation, used to check the accuracy of the analytical results.
MF and dynamical Renormalization Group: Fig. 1 shows the numerical solution of the MF equations of motion (blue) for the case ∆ = 0 (dynamical spin bath), (black) the exact solution for ∆ = 0 (frozen spin bath) and (red) the exact solution for ∆ = 0 (dynamical spin bath).At short times the different solutions agree very well because for ∆, J i ≪ B, there is natural time-scale separation between system and bath dynamics.Due to the large number of bath spins, which we assume are initially polarized (the initial state for the bath can be chosen arbitrarily), the effective magnetic field Ω S is large along the longitudinal direction, and the central spin precesses with small amplitude, proportional to B/P z ≪ 1 and with frequency ω 0 = B 2 + J 2 P 2 z .At longer times the bath depolarizes, reducing the frequency of the oscillations in the central spin, until both systems become resonant and the central spin suddenly flips with an instanton-like transition.Notice that these features are already captured from the MF solution, but correlations between spins produce an additional amplitude suppression, close to the instanton transition (Fig. 1, red).Dynamical RG is a similar method to standard RG, with the peculiarity that the physical cut-off corresponds to the initial time, and the running constants to the initial conditions [15,16].The main advantages of dRG (also known as multiple-scale analysis in the literature [17,18]) is that it allows to systematically obtain longer timescales in a straightforward manner, which helps to gain some intuition about the physics behind, and that it identifies secular terms with the appearance of new timescales, which can be absorbed in the boundary conditions.Hence, even non-linear differential equations that are intrinsically divergent (and correspond to some effective low-energy model of a system) can be studied, and the time of validity of the low energy model extracted from their solutions.The first step is to identify a small dimensionless expansion parameter ǫ to expand perturbatively around it.Concretely, one needs to identify the fastest time scales and then build the corrections to longer timescales.Due to the natural time-scale separation in this problem, a good starting point is when the bath is static, which is obtained to lowest order in ǫ if Ω B (t) = ǫ (−∆, 0, J i S z (t) ).This is equivalent to assume that the effect of the central spin on the bath is small due to J i , ∆ ≪ B. In this case, to lowest order in ǫ, one has This reproduces the case ∆ = 0 in Fig. 1, and does not capture the instanton-like transition which happen at longer time-scales.If one is interested in the effect of disorder, just by defining the disorder cumulants J i , J i J j , . . . it is possible to derive an analytical expression for the disordered averaged dynamics S α 0 .However, to simplify the discussion and compare with the exact diagonalization results, all plots are particularized for the homogeneous case J i = J.
To obtain the next time-scale we expand the equations of motion to first order in ǫ.Now the bath acquires dynamics and obeys the next equations: The solution contains secular terms that grow as ǫ (t − t 0 ).For example, the longitudinal component reads [23]: This indicates that the perturbative solution is valid up to a time t ∼ 1/ǫ∆, however the secular terms can be dealt with and their presence indicates the appearance of a new time-scale, whose effect can be quite large.To get rid of the secular terms one just needs to assume that the initial conditions from the lowest order solution are now functions of a new time-scale τ = ǫt such that m i (τ ).
Then as the solution must be independent of this arbitrary "cut-off" in time, its derivative must vanish, producing the next flow equations for the initial conditions (details in the Appendix): where m i and M are functions of τ .Notice that this is a highly non-linear equation, which needs to be solved simultaneously with the flow equation for M (τ ).To first order in ǫ, the equations of motion for the central spin are analogously solved and display secular terms as well.Interestingly, one now has quadratic secular terms as well, which result from the time dependence on m z i (τ ), that renormalizes the central spin frequency.The flow equations are in this case: These equations can be numerically solved or in some limits, analytically solved.Fig. 2 shows a comparison between the exact MF solution and its first order approximation using dRG, for the same parameters as those considered in Fig. 1.The comparison shows that the flow equation captures the instanton-like solutions produced by the resonance between bath and central spin.Notice the importance of secular terms in comparison with nonsecular corrections, which only produce small amplitude changes and neglect frequency shifts.
Quantum correlations: We have so far neglected the effect of quantum correlations, and from the comparison between exact diagonalization and the MF solution in Fig. 1, one can see that their main effect is to suppress the coherent oscillations as one approaches the instanton transition.This suppression is enhanced as N increases, and because it is a consequence of correlations with the bath, it can be related with decoherence, under certain assumptions [24].To include the effect of correlations we calculate the equation of motion for the spin-spin correlator: As in Eqs.2 and 3, higher order correlators appear, and one must choose a decoupling scheme.In Fig. 3 we plot the dynamics obtained using three different decoupling methods: The simplest one (MF+C1 in Fig. 3) factorizes all correlators in Eq.8.As this is the simplest possible decoupling scheme, it will be valid for very short times only, but still encodes processes neglected by the simple MF solution.A better decoupling scheme (MF+C2 in Fig. 3) consists in rewriting the 3-spin correlator in Eq.8 as , which should be a good approximation when the bath is largely polarized (away from the instanton transition).This gives a better approximation, but the amplitude suppression is not very well captured and nonphysical values for the magnetization eventually appear at long time.Finally, an alternative decoupling scheme, that provides the best results and is also simple, consists in separating correlators into its correlated and uncorrelated parts [19] I β n S α = I β n S α + I β n S α c (Hierarchy in Fig. 3).In this case one must find the equation of motion for the correlated part, which keeping only dominant terms reads: This equation of motion is obtained by assuming that correlated parts are small, at least to short time (see Appendix for details on the different decoupling schemes).
As the hierarchy of correlations decoupling provides the best numerical results, we can now find analytical approximations using dRG, free of secular terms.Importantly, this leads to flow equations for the entanglement between spins and then, demonstrate which contributions are crucial as time evolves, even for initially uncorrelated states.
To lowest order, the equations for the fastest time scales are analog to the MF equations, however to first order, correlated and uncorrelated parts couple.The flow equations for the bath are unchanged, but for the central spin are: where a αβ = i J i c αβ i and c αβ i is the initial condition for the correlated part I α i S β c .The most important change is that now M y can flow, and that assuming that correlations do not develop over time (a αβ (τ ) = 0 ∀ τ ), one recovers de MF flow equations.For the correlated parts one finds: The slow dynamics divides in two processes: The precession in the transverse field ∆, and the suppression of oscillations due to the spin-spin interaction.Fig. 4 shows a comparison between the exact dynamics, the one numerically obtained by the hierarchy of correlations decoupling and its lowest order approximation obtained from dRG.It can be seen that the first order approximation provides very good agreement, only missing a small frequency renormalization, while the full numerical solution agrees quite well with the exact solution, away from the instanton transition.Furthermore, although all initial correlations vanish, the flow equation makes c zy i (t) = 0 for all t > 0 which means that these cross-correlations are crucial and control the dynamics.
Conclusions: We have shown that it is possible to obtain good approximations for the dynamics of strongly correlated systems by numerical and analytical methods.Numerically, a Hierarchy of Correlations decoupling can capture the main features (modulation of frequency and amplitude, and the suppression of coherent oscillations), but due to the truncation, secular terms are likely to appear.For this, we have shown that dRG provides accurate analytical approximations free of secular terms.Hence, both approaches should be complementary, as they allow to study different aspects of the solutions: The full numerical solution provides better accuracy, but the control of the secular terms can be difficult to handle, and the physical understanding of the different processes playing a role in the dynamics remains obscure; on the other hand the use of dRG provides an analytical method to study the non-linear dynamics for long times, free of secular terms, and with a separation of time-scales based on the physics of the problem, disentangling the contributions from different physical processes.
For the central spin model we have considered a nontrivial regime with a wide range of time scales due to the slaved dynamics of the central spin to the bath.We have demonstrated that the instanton-like transitions, due to resonant processes between both systems, and that the suppression of the coherent oscillations, due to correlations with the bath of spins, can be captured analytically by flow equations for arbitrary values of N .These results directly apply to the understanding of the dynamics during the interaction between a qubit and the surrounding localized modes, which is important for the design of quantum computers, as it is expected to dominate T 2 at low temperatures [9,[20][21][22].calculation ǫ is set to one and the corresponding longer time-scales are 1/∆ and 1/Ji, which are still long due to their small value.
[24] The case of homogeneous couplings Ji = J has a recurrence time which can be observed in the exact diagonalization results, therefore it is not strictly correct to use the term decoherence.However, for time scales of some experiments and due to the presence of disorder, the process of suppression of coherent oscillations due to correlations between the bath and the central spin can be interpreted as a precursor of real decoherence in a realistic setup.

Appendix A: Dynamical RG for the Mean-Field Equations
The Mean-Field (MF) equations of motion for each component of the central spin and bath are: where we have introduced the parameter ǫ to organize the different powers of perturbations.To lowest order the equations of motion reduce to: and ∂ t I i 0 = 0 for the bath.The solutions are easily obtained by direct integration: where and M is the initial condition for the central spin.This solution describes a spin precessing in an effective magnetic field, combination of the external one B and the Overhouser field produced by the static bath α z .
The first order corrections for the bath are obtained from the next equations: whose solution is easily found to be: These solutions display small amplitude oscillatory corrections with the same frequency as the central spin, and secular terms, where for example I z i (t) decreases linearly with time, with a rate ǫ∆.The appearance of secular terms due to both ∆ and J i indicate that two new time-scales at t ∼ ∆ −1 , J −1 i appear, and to obtain a solution including them, we need to absorb these terms by deriving their corresponding flow equation.This leads to: For the central spin dynamics we have the next equation of motion to first order: reason is that we have kept certain correlated parts, but eliminated others without a physical criteria, introducing unphysical non-linear terms.If we consider a more complex decoupling, where two-spin correlations are maintained, we can just separate the three-spin correlators as The next equation is then obtained for the two-spin function: This gives the numerical results labeled as MF+C2 in the main text.The number of coupled equations is now larger, and there is back-reaction included between the different two-point correlators.Nevertheless the numerical solution is still at large disagreement with the exact solution, even producing unphysical values.Finally, an alternative decoupling is obtained by separating the n-point functions into correlated and uncorrelated parts.Then, one must determine the equation of motion for the correlated part , which is given by: To capture the suppression of the oscillations observed from exact diagonalization it is enough to consider a simple approximation of the previous equation.As correlated terms are expected to be small at short times, and ∆, J i ≪ B, we keep only the next leading terms: The first line corresponds to the fastest time-scale, coming from the central spin dynamics, while the second line is proportional to J i and suppresses the amplitude of the oscillations as the spins precess away from the longitudinal axis.For the three relevant components one has: One important new aspect introduced by the correlated parts is the appearance of quadratic terms I z n 2 , which even for large disorder, where the bath would be unpolarized on average (i.e., when i I z i ≃ 0), are non-vanishing.Their dynamics are easily obtained from the equations of motion, and for the homogeneous case with J i = J they simplify to: which we have approximated to lowest order, by factorizing I θ n S z in the second line.These set of equations is solved numerically to obtain the spin dynamics used in the main text labeled as Hierarchy.

Appendix C: Flow equations for the Hierarchy of correlations
To obtain the flow equations that encode the effect of amplitude renormalization using dRG we can consider different approximations to Eqs.B3.The simplest one consists in solving the equations for the fastest time-scale only, i.e., for terms that oscillate with frequency ω 0 .For the bath and central spin one gets the same solutions as for the MF case to lowest order (as we have assumed that correlations is a correction over the MF solution), while for the correlations (which do not yet couple to the uncorrelated parts): where α β S α c 0 = i J i I β i S α c 0 .The solution is similar to the one for the central spin, with different boundary conditions that describe the initial correlations between system and bath.
To first order the equations are now: where we have neglected the correlated terms because they are small in comparison with the others.Similarly for the central spin, to first order in ǫ one has: and for the correlations: The appearance of secular terms in the different solutions is what gives rise to the new flow equations for the system: Then one just needs to define the disorder correlators for the disordered case, or directly solve for the ordered case.Fig. 6 shows a comparison between the exact numerical simulation and approximation using the Hierarchy of Correlations decoupling at long times.It is clear that although the slow decay is not captured, because it corresponds to a higher order correction in the flow equations, the agreement is much better than the MF solution and than all the other approximations previously tried.Finally, Fig. 7 shows the solutions from the flow equations for the cross-correlator between central spin and bath, and the magnetization components M x,z (t).It is interesting to see that the periodicity of the flow equations is quite large (of the order of t ∼ 220 in units of B −1 ) and that the cross-correlator initial condition a zy (t) has large corrections, even when the system starts in an uncorrelated state (corrections of the order of ∼ B rather than J i or ∆, which is what one would find in absence of renormalization).This means that central spin and bath become highly correlated and these correlations are needed for the correct description of the dynamics at late times.

Figure 1 : 2 and
Figure 1: Comparison between the exact dynamics for ∆/B = 0 (black) and ∆/B = 0.03 (red), for the case of homogeneous couplings, N = 100 and J/B = 0.05.The dynamics of the bath spins, produced by ∆ = 0, leads to instantonlike transitions in the central spin at longer time-scales, when N ≫ 1. (Blue) Numerical solution of the MF equations.When ∆/B ≪ 1 the short time dynamics is identical to the case ∆ = 0, with a static bath.Initial condition S z (0) = 1 2

Figure 2 : 2 and
Figure 2: Comparison between (black) exact MF solution and (orange) the approximate one from the flow equations.We also plot in red the slow variable dynamics Mz (τ ), obtained by solving the flow equations.Parameters: ∆/B = 0.03, J/B = 0.05 and N = 100 for the initial condition S z (0) = 1 2

Figure 3 :
Figure 3: Comparison for the dynamics of the longitudinal magnetization S z (t) between different decoupling schemes and the exact diagonalization result.(blue) The MF+C1 decoupling, which neglects correlations, cannot capture the instanton transition.(orange) The MF+C2 includes two-spin correlations, but still gives a bad agreement with numerics.(black) The hierarchical decoupling qualitatively captures the suppression, as well as the instanton transition.Parameters: ∆/B = 0.03, J/B = 0.05 and N = 100 for the initial condition S z (0) = 1 2 , Pz (0) = N/2 and vanishing all initial correlations.

Figure 4 :
Figure 4: Comparison between the exact dynamics (green), the numerical solution using the hierarchy correlations (black) and the lowest order dRG (blue).The red dashed line shows the slow dynamics of the boundary condition Mz (τ ), which plays the role of the envelope function for the faster oscillations.

Figure 5 :
Figure 5: Comparison between the central spin dynamics between the dRG result to first order (blue) and the exact diagonalization one (yellow).The red dashed line shows the function Mz (t), which modulates the fast coherent oscillations of the central spin.Parameters: ∆/B = 0.03, J/B = 0.05 and N = 100 for the initial condition S z (0) = 1 2 and Pz (0) = N/2.

Figure 6 :
Figure 6: Comparison between the exact Hierarchy of correlations (black), its lowest order approximation using dRG (red) and the exact dynamics for the central spin (green), at long times.Parameters: ∆/B = 0.03, J/B = 0.05 and N = 100 for the initial condition S z (0) = 1 2 and Pz (0) = N/2.

Figure 7 :
Figure 7: Numerical solution of the flow equations for the running of the different initial conditions.(Red, solid line) Mz (t)still displays instanton-like behavior, but the profile is modified due to correlations with respect to the uncorrelated case (see Fig.5, red-dashed line).(Blue, dashed line) Mx (t) also shows large corrections but it is always negative, which is expected due to the positive value chosen for the interaction parameter.(Green, dot-dashed line) azy (t) displays complicated oscillations, correlated with the bath polarization and the central spin magnetization, but importantly, acquires large values that can largely modify the MF dynamics.