Nonequilibrium dynamical renormalization group: Dynamical crossover from weak to infinite randomness in the transverse-field Ising chain

Nonequilibrium dynamical renormalization group: Dynamical crossover from weak to infinite randomness in the transverse-field Ising chain

Markus Heyl Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, 6020 Innsbruck, Austria Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria    Matthias Vojta Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
July 4, 2019

In this work we formulate the nonequilibrium dynamical renormalization group (ndRG). The ndRG represents a general renormalization-group scheme for the analytical description of the real-time dynamics of complex quantum many-body systems. In particular, the ndRG incorporates time as an additional scale which turns out to be important for the description of the long-time dynamics. It can be applied to both translational invariant and disordered systems. As a concrete application we study the real-time dynamics after a quench between two quantum critical points of different universality classes. We achieve this by switching on weak disorder in a one-dimensional transverse-field Ising model initially prepared at its clean quantum critical point. By comparing to numerically exact simulations for large systems we show that the ndRG is capable of analytically capturing the full crossover from weak to infinite randomness. We analytically study signatures of localization in both real space and Fock space.


I Introduction

In equilibrium, renormalization group (RG) approaches constitute one of the central concepts for the theoretical description and understanding of many-body systems. One major challenge in the field of nonequilibrium physics Polkovnikov et al. (2011) is the development of appropriate out-of-equilibrium generalizations. Recently, RG techniques have been developed Kehrein (2006); Berges et al. (2008); Mitra (2012); Mathey and Polkovnikov (2010); Chiocchetta et al. (2015) that transfer the idea of scale separation to the out-of-equilibrium dynamics of homogeneous quantum many-body systems. For spin systems in the presence of strong disorder, RG techniques have been formulated,Vosk and Altman (2013, 2014); Pekker et al. (2014) applicable to initial states with weak entanglement, that extend concepts of the strong-disorder RG Dasgupta and Ma (1980); Bhatt and Lee (1982); Fisher (1994) to the nonequilibrium dynamical regime. Although these RG methods constitute a key step towards the analytical description of the nonequilibrium dynamics in complex systems, controlling the long-time properties is still a major challenge.

In this work we present a novel nonequilibrium dynamical renormalization group (ndRG) technique for the analytical description of the quantum real-time evolution in complex systems. The ndRG provides a general iterative RG prescription for the full time-evolution operator without the need of diagonalizing the complete Hamiltonian. The ndRG is applicable to both homogeneous as well as disordered quantum many-body systems. Importantly, the ndRG incorporates time as an additional scale which turns out to be important for the description of the long-time dynamics. The main idea behind the ndRG is to separate resonant from off-resonant processes on the basis of the energy–time uncertainty relation:


that expresses a fundamental limit onto the law of conservation of energy within a scattering process monitored over a time span (Ref. Landau and Lifshitz, 1991). In the asymptotic long-time regime energy-conserving processes dominate the dynamics as apparent, for example, in Boltzmann equations which describe the final asymptotic relaxation to thermal states in homogeneous systems. For long but finite times , however, energy fluctuations are possible such that not only precisely energy-conserving processes contribute but also all those with energy transfer as dictated by the energy–time uncertainty relation. Therefore, we will classify all processes with as “resonant” in the following. The ndRG takes advantage of this limited energy resolution by isolating resonant processes on the basis of a general factorization property of the time-evolution operator. This allows to treat the dynamics of these resonant processes, although nonperturbative in nature, analytically.

We demonstrate the capabilities of the ndRG by applying it to quantum quenches in the disordered transverse-field Ising chain. We study the system’s critical dynamics by quenching the system from its clean to its infinite-randomness critical point. We achieve this by preparing the system in the ground state of its homogeneous critical point and then studying its dynamics in presence of weak disorder. We characterize the resulting localization dynamics in both real as well as Fock space. Within the recently developed concept on many-body localization Altshuler et al. (1997); Basko et al. (2006); Nandkishore and Huse (2015); Altman and Vosk (2015), it is particularly interesting that Fock-space localization has been related to the fundamental question of quantum ergodicity Altshuler et al. (1997) and therefore to thermalization of quantum many-body systems Polkovnikov et al. (2011). We show that the ndRG is capable of describing the dynamics also in cases when the initially weak perturbation flows to strong coupling.

The remainder of this article is organized as follows. In Sec. II we introduce the ndRG. First, we outline the general idea and present the resulting ndRG equations in Sec. II.1 that can be applied directly to any model system of interest. The derivation of the ndRG is shown in detail in Sec. II.2. In Sec. III we then apply the ndRG to the disordered transverse-field Ising chain.

Ii ndRG – nonequilibrium dynamical renormalization group

The ndRG, formulated below, is an iterative coarse-graining procedure. It is designed to provide analytical access to the full time-evolution operator of complicated many-body problems without the need of diagonalizing the complete Hamiltonian. As its main goal it isolates resonant from offresonant processes as dictated by the energy–time uncertainty relation, see Eq. (1). The offresonant processes are eliminated based on scale separation analogous to RG procedures in equilibrium. The resonant processes, however, that are nonperturbative in nature, cannot be eliminated in this way and, if relevant, can drive the system to strong coupling. Based on a general decoupling mechanism for time-evolution operators we show that the dynamics of the resonant processes, although nonperturbative, can still be accessible analytically on all time scales. This is possible even though the system might flow to strong coupling as we will demonstrate for the disordered transverse-field Ising chain in Sec. III.

ii.1 The ndRG recipe

Consider a system whose Hamiltonian


at a given ultraviolet (UV) cutoff can be decomposed into an exactly solvable part and a weak perturbation , with an associated time-evolution operator


Notice that could either be chosen as a momentum or energy cutoff. Given that the UV cutoff takes a value we now aim at reducing it to by eliminating contributions of involving particles with momenta in the shell and thereby generating a renormalized effective theory for the remaining modes. Operators with a subscript , e.g., , denote the renormalized ones after the RG step from to has been made while those with a subscript refer to the initial operators before the RG step is performed.

Importantly, we do not aim at integrating out all contributions of modes in this shell, but only those which are offresonant according to Eq. (1). The resonant processes remaining in the Hamiltonian are finally dealt with on the basis of a general factorization property of time-evolution operators. This is the main feature of the ndRG and distinguishes it from other RG schemes. The ndRG eliminates the offresonant contributions from the time-evolution operator by constructing explicitly a unitary transformation yielding


with containing the dynamics of the remaining degrees of freedom. For a given Hamiltonian, the ndRG scheme works according to the following prescription which is straightforward to implement for a given model system:

  1. Identify the off-resonant processes and decompose:

  2. Determine the generator of the unitary transformation via


    Here, with the free time-evolution operator.

  3. Obtain the renormalized Hamiltonian:

  4. Iterating the above steps by successively lowering the cutoff one obtains at the end of the ndRG transformation the following representation of the full time-evolution operator


    with denoting the renormalized exactly solvable part and the remaining resonant contributions. The unitary transformation is a -ordered exponential:


    with the initial and the final UV cutoff. denotes -ordering analogous to common time ordering. The dynamics of this apparently complicated problem can be solved on the basis of a factorization property of the time-evolution operator for resonant processes:


These equations are valid up to second order in the perturbation strength, the extension to higher orders is straightforward. In what follows, we will present the derivation of the ndRG procedure. For readers interested in the application of the ndRG for a concrete model system, it is possible to directly consult Sec. III where the ndRG is applied to the disordered transverse-field Ising model.

ii.2 Derivation of the ndRG

As anticipated before, consider a Hamiltonian that can be separated into an exactly solvable part and a weak perturbation at a UV cutoff . In order to derive the ndRG procedure, let us first turn to an interaction picture with respect to the free Hamiltonian at the desired final cutoff after the RG step has been performed. Of course, is not known a priori but has to be determined self-consistently in the end which is straightforward to implement. In the interaction picture the time-evolution operator is:




with and denotes the usual time-ordering prescription. In the following, time-dependent operators will always refer to being time evolved in the interaction picture, i.e., .

ii.2.1 Disentangling theorem

The interaction picture representation of the time-evolution operator in Eq. (11) is still exact, for the complicated models of interest, however, the time-ordered exponential in cannot be evaluated easily. The goal of the RG procedure is not to find an approximate solution to as a whole. Instead, we aim at an iterative sequence of transformations by successively integrating out high-energy degrees of freedom. For that purpose we use a disentangling theorem for time-ordered exponentials:van Kampen (1974)






While the operator will be chosen such to eliminate the desired processes, the operator will determine the renormalized Hamiltonian after the RG step. The precise choice of the operator for the purpose of the ndRG will be given below.

The aim of the ndRG is to find a representation of of the following form:


for some suitable antihermitian operator such that is unitary. Before going into the details of how to derive this identity we first would like to illustrate its main consequences. Using this identity, we obtain


which gives, because is independent of time:


by defining


Bearing in mind that, using Eqs. (11,13), the time-evolution operator has been factorized according to:


one obtains


which then yields


Now one can switch back from the interaction to the original picture such that:


which is the desired identity provided is chosen such that becomes the renormalized Hamiltonian after eliminating the contribution , see Eq. (7). In the following, we now show how this can be achieved.

ii.2.2 Magnus expansion

The crucial point is that in Eq. (14) can be evaluated approximately within a controlled expansion. The main complications in computing arise from the time ordering prescription that makes it difficult, and for interesting problems impossible, to evaluate exactly. Most importantly, the operator turns out to proportional to the strength of the weak perturbation such that a Magnus expansion is applicable:


which transforms the time-ordered exponential into a conventional exponential on the expense of an infinite series. Importantly, this expansion is controlled by a small parameter which is the perturbation strength. Here, the Magnus expansion is shown up to second order which is sufficient for the targeted accuracy. In case of a larger desired precision, higher orders of the Magnus expansion have to be included.

ii.2.3 Generator of the ndRG transformation

In order to transform into the desired form we choose:


with given by the simple integral


Inserting this into the Magnus expansion, see Eq. (24), one obtains taking into account all contributions up to second order in the perturbation strength


which, using the Baker-Campell-Hausdorff formula, is equivalent to the desired expression:


again taking into account all contributions up to second order in the perturbation strength.

ii.2.4 Renormalized Hamiltonian

Having established the generator of the ndRG, it remains now to determine the renormalized Hamiltonian after the RG step. For that purpose we use that the generator of the transformation is proportional to the perturbation strength such that the following expansion for Eq. (19) is applicable:


Using the choice for in Eq. (25) this gives


taking into account all terms up to second order accuracy. According to Eq. (23), the renormalized Hamiltonian is then given by:


Importantly, however, the last contribution can be neglected. This operator contains again contributions of modes in the shell as it was for , the strength of , however, is now of second order. Eliminating this contribution in the same way as , will then only contribute beyond second order in the perturbation strength in the renormalized Hamiltonian and is therefore beyond the desired accuracy. Concluding, the final renormalized Hamiltonian taking into account all contributions up to second order in the perturbation strength reads:


which is the desired result presented already in Eq. (7).

ii.3 Factorization of the time-evolution operator

At the end of the ndRG procedure, one ends up with a Hamiltonian


By construction we have not eliminated all processes of the perturbation , but kept the resonant contributions that still have to be accounted for. This seemingly complicated problem, however, can be simplified substantially because the processes in are resonant as the renormalized time-evolution operator approximately factorizes:


This property can be seen in the following way. In the interaction picture the renormalized time-evolution operator obeys . As only contains resonant processes with we have that is approximately constant in time leading directly to the factorization in Eq. (34).

In typical problems the complexity of originates from the noncommutativity of and while their individual properties are much easier to determine. The major advantage of Eq. (34) is the separation of the resonant processes of the perturbation from the dynamics of the (renormalized) unperturbed system whose individual time evolution can be determined much easier as will be demonstrated for the Ising model with disorder below.

ii.4 Discussion

Summarizing, in this section we have introduced the ndRG. In Sec. II.1 we have presented the ndRG recipe that can be straightforwardly implemented for any model system of interest. In Sec. II.2 a detailed derivation of the resulting ndRG equations has been given.

Up to now we have not specified when to stop the ndRG transformation. Of course, the ndRG always stops when there are no remaining off-resonant modes as it will occur for the random transverse-field Ising chain, see Sec. III. Importantly, we can take advantage of the additional scale time appearing in the ndRG. Specifically, consider a mode of energy . As the time-evolution operator only contains the product we have that for times this mode is essentially inert. In other words, all modes with energies are frozen out and can be dealt with on a purely perturbative basis using time-dependent perturbation theory. Therefore, we can stop the ndRG transformation when the energy of the modes at the UV cutoff reaches and we obtain a time-dependent final UV cutoff .

This observation has important consequences. In particular, consider a system with a strong-coupling divergence where we leave the region of validity of the ndRG when reducing the UV cutoff too much. Utilizing that for not too large times the ndRG transformation can be stopped at a UV cutoff which is large enough such that the strong-coupling divergence is not yet effective it is still possible to describe the dynamics of the system. In other words, the dynamics on not too long times is still accessible on the basis of the ndRG although the system might flow to strong coupling in the asymptotic long-time limit.

We would like to emphasize that the ndRG can be generalized also to other temporal dependencies of the Hamiltonian beyond the quantum quench considered here. This is possible because time itself constitutes an essential element of this RG by construction as it is utilized explicitly for the resonance condition, for example. In this context, it might be particularly interesting to study crossovers to the adiabatic limit by considering ramps instead of quenches where universality such as Kibble-Zurek scaling Kibble (1976); Zurek (1985); Polkovnikov et al. (2011) can be observed. Moreover, the ndRG also inherits the potential to extend RG ideas to periodically driven systems. This is of particular interest in view of the recently discovered energy-localization transitions D’Alessio and Polkovnikov (2013) which represent a novel class of nonequilibrium phase transitions in complex quantum many-body systems.

Iii Quantum quenches in the disordered transverse-field Ising chain

In the previous section we have introduced the ndRG. It is the aim of the following analysis to apply the ndRG to a paradigmatic model system, the one-dimensional transverse-field Ising chain.Sachdev (2011) The transverse-field Ising chain can be solved exactly also in the presence of disorder. Therefore, it is ideally suited to demonstrate the capabilities of the ndRG by comparing to exact numerical simulations. In particular, we aim at demonstrating that the ndRG is capable to describe both the dynamics on intermediate time scales as well as in the long-time limit which is a challenging task.

In noninteracting one-dimensional quantum systems the effect of arbitrarily weak randomness is substantial: all single-particle eigenstates become localized.Abrahams et al. (1979) As a consequence, particles separated over distances larger than the localization length cannot exchange information and are therefore essentially unentangled. The localization dynamics for unentangled initial states in one-dimensional systems show universal behavior Bardarson et al. (2012); Serbyn et al. (2013) that can be attributed to a dynamical renormalization-group fixed point Vosk and Altman (2013, 2014) where localization not only happens in real but also in many-body Hilbert space.Altshuler et al. (1997); Basko et al. (2006) But how is information propagating within a disordered landscape when the system is highly entangled initially? In the remainder of this article we aim at studying this question exemplarily for the disordered transverse-field Ising chain.

iii.1 Model and setup

To study the localization dynamics out of quantum correlated states we consider a one-dimensional Ising model with transverse-field disorder:


In equilibrium, the homogeneous system with shows a quantum phase transition at separating a ferromagnetic () from a paramagnetic () phase.Sachdev (2011) According to the Harris criterion Harris (1974) the quantum critical point is unstable against weak disorder, and it has been shown that the system flows to an infinite-randomness fixed point instead.Fisher (1994)

In this paper, we develop a dynamical theory for this flow from weak to infinite randomness in nonequilibrium real-time evolution where progressing time itself drives this crossover. The quantum correlated state is initialized by preparing the system in the ground state of the clean critical model at . The localization dynamics is generated by switching on weak disorder suddenly inducing nonequilibrium real-time evolution that is formally solved by


The distribution for the random fields is chosen such that with the disorder average. Thus the ground state of the system is located right at the infinite-randomness critical point.Fisher (1994); Pfeuty (1979)

Contrary to typical condensed-matter systems where disorder is ubiquitous, systems of cold atoms in optical lattices are clean, and disorder has to be imposed, for example, by laser speckle patterns Bloch et al. (2008) providing an ideal candidate for the implementation of the anticipated nonequilibrium protocol. Moreover, the model in Eq. (35) can also be simulated within circuit QED Viehmann et al. (2013a) where disorder is also tunable.Viehmann et al. (2013b)

Numerically, this model can be solved exactly for large systems.Lieb et al. (1961) In this work we will present exact results for systems up to lattice sites. Due to the broken translational invariance in presence of disorder, however, this model is challenging for analytical methods. In equilibrium, this system has been solved exactly in the vicinity of its critical point both in the weak disorder limit McKenzie (1996) and in the vicinity of the infinite-randomness critical point.Fisher (1994) Out of equilibrium, the dynamics in the vicinity of the infinite-randomness critical point for weakly entangled initial states has been studied analytically recently.Vosk and Altman (2014) In the present work, we will address a complementary viewpoint – the localization dynamics for strongly entangled initial states at weak disorder. In particular, applying the ndRG to this model system will serve as a benchmark for gauging the capabilities of the methodology introduced in this work.

The transverse-field Ising model in Eq. (35) can be diagonalized exactly by a mapping to a free fermionic theory using a Jordan-Wigner transformation:Lieb et al. (1961)


with fermionic annihilation operators at site . For the numerical implementation we parametrize


with drawn from uncorrelated uniform distributions, i.e., , yielding . As emphasized before, this ensures that the system is located right at the infinite-randomness critical point.Pfeuty (1979); Fisher (1994) In the analytical treatment we use the parametrization


The strength of the disorder we characterize via the variance which in the weak-disorder limit becomes . It is important to note that this parametrization, which is a consequence of the condition , yields


Therefore, a larger homogeneous critical transverse field is required to destroy the ferromagnetic order as opposed to the case without disorder where . This is a consequence of the fluctuating local fields which can locally decrease the homogeneous field when favoring magnetic order.

iii.2 Summary of results

Based on the ndRG and corroborated by extensive numerical simulations we study the dynamics in the disordered transverse-field Ising chain induced by a quantum quench from the homogeneous to the infinite-randomness critical point. We investigate the resulting localization dynamics from two perspectives, namely through localization in Fock as well as real space. Here, we summarize our main findings, whose derivation and more detailed analysis can be found in Sec. III.4.

Unlike in classical systems a general understanding of ergodicity for quantum many-body systems has not yet been achieved.Polkovnikov et al. (2011) Recently, however, a framework addressing this fundamental problem has been proposed, many-body localization,Altshuler et al. (1997); Basko et al. (2006); Nandkishore and Huse (2015); Altman and Vosk (2015) where the transition from ergodic to nonergodic is associated with an Anderson localization transition in Fock space.Altshuler et al. (1997) In this work, we quantify the localization in Fock space by studying the temporal deviation of the system from its initial Fock state via the Loschmidt echo


Due to its large-deviation scaling  Silva (2008); Gambassi and Silva (2012); Heyl et al. (2013) with an intensive function we will consider in the following. We find that on intermediate time scales the rate function shows a very slow temporal logarithmic growth:


for with denoting the average over disorder realizations. An increasing implies an increasing deviation of the time-evolved state from the initial Fock state. The slow logarithmic growth of the Loschmidt rate function we interpret as an indicator of Fock space localization: the system only very slowly departs from its initial condition.

In the asymptotic long-time limit, the rate function approaches a constant value


We find that is related to the Fidelity with the ground state of the final Hamiltonian:


by calculating numerically exactly for large systems. Therefore, the asymptotic long-time behavior of the Loschmidt echo, containing, in principle, information about the full many-body spectrum, is only determined by equilibrium ground-state properties which we attribute to localization in Fock space.

Importantly, Fock-space localization and localization in real space are typically connected.Altshuler et al. (1997); Basko et al. (2006) While in homogeneous systems local correlations decay in time this is not the case in localized systems.Anderson (1958) Signatures for retaining local memory are contained in the long-time behavior of the autocorrelation function Anderson (1958); Iyer et al. (2013)


Here, denotes the cumulant and the average with respect to the initial state.

Using the ndRG, we show below that the decay of the local memory on intermediate time scales is algebraic:


for . Remarkably, the presence of disorder induces a static time-independent contribution (up to a phase factor) signaling a nondecaying local memory with a weight given by the strength of the random potential. Therefore, the system shows localization complementing the observed Fock-space localization in terms of the Loschmidt echo above for real space. We confirm the preservation of local memory using numerically exact simulations of the dynamics for the asymptotic long-time limit where we find that that .

iii.3 ndRG for the disorder transverse-field Ising chain

Having summarized the main results obtained in this work it is now the aim to show in detail their derivation on the basis of the ndRG. First, we outline the exact analytical diagonalization of the homogeneous system and then we represent the weak disorder perturbation in this basis. In Sec. III.3.1 we construct the generator of the ndRG and summarize the resulting ndRG scaling equations for the spectrum and the disorder strength. In Secs. III.3.2 and III.3.3 we show how these scaling equations are determined within the ndRG. A detailed treatment of the resonant process will be given in Sec. III.3.4. The derivations of the results for the observables, already summarized above, will be presented in Sec. III.4.

According to the ndRG recipe given in Sec. II.1 we first decompose the Ising Hamiltonian in Eq. (51) into an exactly solvable part and a perturbation which in the weak-disorder limit is:


The homogeneous part can be diagonalized explicitly using Fourier transformation and a subsequent Bogoliubov rotation: Sachdev (2011)


with . Then, the Hamiltonian becomes


In the present case where , see Eq. (40), the Bogoliubov angles can be determined asymptotically which yields


with . In this basis the disorder contribution to the Hamiltonian reads




with .

iii.3.1 ndRG generator and scaling equations

Applying the ndRG scheme to the Ising model in Eq. (51), the disorder amplitude takes the role of the perturbation strength. At a given UV cutoff one obtains for the generator of the unitary transformation, see Eq. (6):


with the restricted sum defined as


Here, we use the notation that capital energies and denote the final renormalized ones after the RG step has been performed. Additionally, we have introduced the scale that is supposed to distinguish between resonant and off-resonant processes , i.e., , according to the energy–time uncertainty relation in Eq. (1). When physical quantities have been calculated we replace by in the end with a nonuniversal constant. By keeping instead of we are able to identify whether some properties depend on the nonuniversal details of the RG cutoff and have therefore to be interpreted with care.

Using Eq. (7) from the ndRG recipe, we obtain at the critical point the following RG equations for the disorder strength and the low-energy spectrum :


which we derive in detail below in Sec. III.3.2 and Sec. III.3.3. Before, however, we aim at discussing shortly their main consequences.

Let us first discuss the renormalization of the spectrum for not too small UV cutoffs . To recapitulate, for , the initial spectrum is approximately linear with whereas for we have that is constant, see Eqs. (49) and (40). According to Eq. (55), the linear spectrum of the modes is modified due to the RG only such that its velocity obtains weak perturbative corrections for . Here, we have neglected the flow of the disorder strength . As we will analyze below, in this regime of the UV cutoff, the disorder strength only acquires weak perturbative corrections which will only contribute beyond second order in the scaling equation for the spectrum. Similarly, for , the scaling equation (55) can also be solved analytically. This yields , with the Lambert-W function and is consequently a monotonously decreasing function of . Therefore, the initial ndRG flow for the spectrum only leads to perturbative corrections at which we will neglect in the following analysis.

When where for all , however, we have to stop the ndRG transformation because the remaining modes are now all resonant. Thus, according to the ndRG recipe given in Sec. II.1 there are now no off-resonant processes that can be integrated out perturbatively and the UV cutoff where the ndRG comes to an end is given by


Importantly, however, the dynamics of the system is still accessible when stopping the ndRG transformation at this cutoff . The reason for that is discussed in Sec. III.3.4.

Notice that although the RG equation for the spectrum does not contain any randomness, this does not mean that the randomness is fully gone. In fact, it is hidden in the unitary transformation connecting the extended states of the clean with the localized wavefunctions of the disordered system.

As opposed to the spectrum, the disorder strength increases during the ndRG. According to Eq. (55) we find that


displaying a strong-coupling divergence for . Before approaching this strong-coupling divergence the ndRG, however, stops, see Eq. (56). Most importantly, this leaves us within the regime of validity of the current weak-disorder treatment.

In the following, we now aim at showing the derivation of the scaling equations in Eq. (55).

iii.3.2 Spectrum

We start by determining the renormalized spectrum. Based on Eq. (7) of the ndRG recipe, in Eq. (53) generates the following RG equation for the energies:


As (but not ) one can replace by on the right-hand side of the above equations. In the following we show how to evaluate the sums appearing in the scaling equations in Eq. (58). For illustration we take


which is a sum of over the random variables . As is the case in Wilson RG schemes the width of the momentum shell is small but still large enough to host an extensive number of states. Then, in the equation above, we sum over a large number of random variables. Thus, to leading order we can replace the sum over the random variables by the sum over its mean:


The corrections involve random variables with an associated probability distribution function that shows that their typical magnitude scales as . The mean is zero as their sign fluctuates. The variance, however, is finite and scales . Comparing this to the variance of the amplitudes and that are of the order it is clear that the corrections above will only contribute to third order and can therefore be neglected within the current accuracy of second order. Using Eq. (52) we have that yielding for the sum:


As up to corrections and all remaining contributions are now smooth functions of we can then evaluate this expression exactly:


Here, we have used , and . Inserting this result into Eq. (58) and using an analogous analysis for the sums over then yields the scaling equation for the following scaling equation for the slow modes :


Using the result for the Bogoliubov angles in Eq. (50) we see that we have to distinguish two different cases. For where for and , we have that for the low-energy modes :


When on the other hand, according to Eq. (50) such that:


When neglecting the weak renormalization of the high-energy modes in the linear regime, i.e. , then this gives Eq. (55). Additionally, we have to determine the renormalization of the eliminated fast modes whose energies have to be chosen according to the self-consistency equation:


In the following, we will neglect the influence of this renormalization of the eliminated modes . First of all, the shift of compared to is perturbative in the disorder strength . But most importantly, the eliminated modes are now completely decoupled from the remaining ones and therefore they do not influence the further RG transformation. Being only interested in the leading behavior as a function of , we can therefore neglect this final renormalization and replace by in the following.

iii.3.3 Disorder strength

Now, we derive the scaling equation, see Eq. (55), for the disorder strength. For that purpose, it is suitable to analyze the superpositions


rather than the amplitudes