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On the critical behavior of the XX spin-1/2 chain under correlated quenched disorder

GETELINA, J. C. A. On the critical behavior of the XX spin-1/2 chain under correlated quenched disorder. 2016. 82 p. Dissertation (Master in Science) Instituto de Física de São Carlos, Universidade de São Paulo, São Carlos, 2016. This work provides a full description of the critical behavior of the XX spin-1/2 chain under correlated quenched disorder. Previous investigations have shown that the introduction of correlation between couplings in the random XX model gives rise to a novel critical behavior, where the infinite-randomness critical point of the uncorrelated case is replaced by a family of finite-disorder critical points that depends on the disorder strength. Here it is shown that most of the critical exponents of the XX model with correlated randomness are equal to clean (without disorder) chain values and do not depend on disorder strength, except the critical dynamical exponent and the anomalous dimension. The former increases monotonically with disorder strength, whereas the results obtained for the latter are unreliable. Furthermore, the scaling relations between the critical exponents were also tested and it was found that those involving the system dimensionality, namely the hyperscaling and Fisher’s scaling relations, are not respected. Measurements of the Rényi entanglement entropy of the system at criticality have also been performed, and it is shown that the scaling behavior of the correlated-disorder case is similar to the theoretical prediction for the clean chain, displaying the same finite-size correction and a disorder-dependent effective central charge in the leading term of the scaling. Further corrections to the scaling of the entanglement entropy were also investigated, but the results are inconclusive. The model was studied via exact numerical diagonalization of the corresponding Hamiltonian.


Introduction
Most of the research in theoretical condensed matter physics is concerned about the elaboration and solution of homogeneous (clean) models, mainly because these models are easier to solve and may even have an analytical solution. However, an idealized clean system does not describe quite well how real matter is. Materials usually have impurities or defects randomly distributed throughout them, i.e. they are disordered. (1-3) The study of disordered models came into prominence recently, thanks to the advance in computer simulations and the development of new mathematical techniques, such as the renormalization group. (4-6) Clean models are now being compared to their disordered counterpart to investigate how the introduction of disorder may affect the system behavior.
One example of such model is the XX spin-1/2 chain, which consists of a onedimensional (1D) lattice of interacting spins whose x and y components are coupled. For the clean version of the XX model all those couplings are equal to the same constant, whereas for the disordered version they are realizations of a random variable with a given probability distribution. As a critical system, the XX model is fully described by its set of critical exponents. Research shows that the introduction of any amount of disorder completely changes the critical behavior of the XX model. (7) Recently, a new configuration for the disordered XX model was proposed. (8) In this special setup, called here as the XX CD model, the random couplings J i are correlated, such that the odd couplings are equal to their first right neighbor, i.e. J 2i−1 = J 2i . At first glance one could infer that this change would not affect the system properties, since from a macroscopic perspective the model is unaltered. However, it has been shown that the XX CD model behaves in a unique way: Some of its critical exponents depend on how strong the disorder is, being equal to the clean value for weak disorder and exhibiting novel critical behaviors as disorder gets stronger. (9) This unusual behavior motivates a detailed study about the XX CD model. Until now only a few properties of this system have been evaluated; a complete description of its critical behavior is still lacking.
The XX model displays a so-called quantum critical behavior, i.e. it undergoes a quantum phase transition (QPT). (10)(11)(12)(13)(14) QPTs are phase transitions driven by quantum fluctuations alone; they occur at zero temperature (T = 0). Thus, one could infer that the interest in QPTs is just theoretical. However, it has been observed experimentally systems with quantum critical behavior. (15)(16)(17) To observe this behavior one should consider sufficiently low temperatures, such that the thermal fluctuations are less significant than the quantum ones, i.e. kT ω C , where ω C is the characteristic frequency of the system.
The difference between classical and quantum phase transitions is that the latter does not allow temperature as a control parameter; all known properties of CPTs are extended to QPTs, such as the classification in first-order and continuous phase transitions, scale invariance, universality classes, etc. (For more details on CPTs see (18)(19)(20).) The similarities between quantum and classical phase transitions have been reinforced after the development of the quantum-to-classical mapping, which transforms a d-dimensional quantum critical system in a (d + 1)-dimensional classical one. (12,21) This mathematical trick consists of inserting an extra imaginary time dimension, which is represented as infinite copies of the original system. Using this technique one can, for instance, map a single quantum spin (with dimension d = 0) into a classical 1D Ising model.
Since the temperature T is not a possible control parameter for QPTs, these transitions are usually tuned by the parameters that appear in the Hamiltonian of the system. For spin chains, for example, such parameters are the couplings between spins or the external magnetic field. Therefore, to introduce disorder in a spin chain model one must make the couplings or the external field random. The former case is known as random-T C disorder (or random mass in analogy with field theory), while the latter is known as random field disorder. (22) Randomness can be further classified according to its relation with the typical time scale of measurements: If disorder is time independent it is known as quenched disorder (i.e. frozen), whereas if it depends on time it is called as annealed disorder. In general, quenched disorder is more interesting than annealed disorder, because the effects of randomness are enhanced.
The XX model under uncorrelated quenched disorder has already been fully described. (23) Conversely, for the XX CD model there are not many results available. First investigations, though, show that the introduction of correlated quenched disorder gives rise to novel critical behavior. Besides the theoretical interest in describing the XX CD model, recent investigations have shown that a similar correlated-random model describes the behavior of a polymer with high conductivity. (24)(25)(26) What follows in the next chapters is a complete description of the critical behavior of the XX CD model, by determining the full set of critical exponents and checking the so-called scaling relations that connect these exponents with each other. Furthermore, entanglement properties of the XX CD model are also studied. The results are compared to the theoretical predictions of the clean chain.
The remainder of this text is organized as follows. Chapter 2 presents the model and the method used for solving it, besides providing a discussion about the effects of disorder on the phase transition. In Chapter 3 the quantities of interest are evaluated and their corresponding critical exponents are determined. Chapter 4 shows the entanglement measurements, which is a trending topic in condensed matter physics. Concluding remarks are reported in Chapter 5.

The XX model
This chapter gives a discussion about the general properties of the XX model. In the first section the Hamiltonian is introduced, followed by an outline of the Jordan-Wigner transformation, which maps the interacting spin problem to a free fermion one. The second section presents the condition for criticality and the phases of the XX model. A derivation of the so-called Harris criterion is also presented, which determines whether weak disorder is perturbatively relevant for the critical behavior. The last section finally introduces the Hamiltonian of the XX CD model in the appropriate basis. There is also a comment about a mathematical trick that yields faster calculations.

Mapping to free fermions
The general Hamiltonian of the 1D XX spin-1/2 chain with nearest neighbor interactions is where J i > 0 are the couplings, S α i are the usual spin-1/2 operators * and L is the system size which shall be considered always even. For the clean chain the couplings are constant (i.e. J i = J), whereas for the disorder chain the couplings J i are realizations of a random variable.
The subject of this work is the so-called XX CD model, where the couplings are random and also correlated with each other. To introduce this correlation between couplings, the odd sublattice couplings (i.e. J 2i−1 ) are randomly generated and the even sublattice ones are defined as equal to their left neighbor, i.e. J 2i = J 2i−1 . Thus, the Hamiltonian of the XX CD model can be written as To solve the Hamiltonian above one can transform S ± i into fermionic creation and annihilation operators c † i and c i , respectively. (27) Such operators respect the following anti-commutation relations and are defined as This change of basis is known as the Jordan-Wigner transformation. Notice that it preserves the form of Eq. 2.3, since c † i c j = S + i S − j , except for the last term in the summation, which depends on the boundary conditions. Considering a cyclic chain, i.e. S ± L+1 = S ± 1 , and using Eq. 2.6 and Eq. 2.7 one finds Thus, the transformed Hamiltonian with periodic boundary conditions is This is the free fermion representation of the XX model. The spins are now represented by fermionic particles hopping in the lattice with a energy cost equal to the coupling. The mapping between spins and fermions is given by (2.10) Therefore, according to Eq. 2.10, if the action of the number operator c † i c i in the i-th site gives one, there is a spin up in the z-direction in that site ( The problem now is to find the eigenvalues and eigenvectors of Eq. 2.9. For the clean case (J i = J) this can be done exactly via a Fourier transform on the fermionic operators in Eq. 2.9, whereas the disordered case requires a numerical routine. The advantage of the free fermion representation for the disordered model is that it reduces significantly the size of the matrix to be diagonalized -from a 2 L × 2 L to a L × L matrix -and, therefore, requires much less computational effort.
Nevertheless, there are some interesting properties of the Hamiltonian that are regardless of the couplings nature. For instance, as it is shown in Appendix A, the spectrum of H is symmetric, i.e. there are L/2 positive and negative eigenvalues (recall that L is always even) such that λ i = −λ L+1−i , where λ i is the i-th eigenvalue and the spectrum is in ascending order (λ 1 < λ 2 < · · · < λ L ).
The ground state |GS is obtained by filling the vacuum |0 with quasiparticles whose energy is below the Fermi level, which is µ = 0. Therefore, since half of the eigenvalues are negative, the ground state is where η † i is the i-th fermionic creation operator in the momentum space (see Appendix A). The ground-state energy E 0 is equal to the sum of all negative eigenvalues, i.e.
The spin average for the other directions is also zero, because S x i and S y i are written as linear combinations of S ± i and S ± i = 0, since these operators do not preserve the number of particles. Thus, there is no magnetization in the ground state of the XX model, even though the temperature is zero. Those spin fluctuations are, thereby, caused uniquely by quantum effects.

The dimer phase transition and the effects of disorder
The seminal work that provided the solution of the XX model by mapping to free fermions also shows that the spin-spin correlation function scales as a power law. (27) The power-law scaling of the correlation function is a signature of a critical system. Thus, this result indicate that the XX model is critical. Indeed, the condition for criticality is guaranteed whenever (28) (2.12)  It is clear that this criterion is fulfilled for the clean chain and for the XX CD model, since the former has constant coupling and the latter has J 2i = J 2i−1 . Conversely, for the uncorrelated random chain one can choose a probability distribution that still satisfies Eq. 2.12.
Using Eq. 2.12 one can define the distance from criticality as (2.13) According to Eq. 2.13, the system is driven out of criticality if the values of the odd or even couplings are changed. Multiplying the even couplings by a constant a, Eq. 2.12 gives, for the clean chain and for the XX CD model, θ = ln (a) . (2.14) For a → 0 the even couplings vanish, yielding the formation of dimers in the odd sublattice. Similarly, if a 1 the interactions from the odd sublattice are negligible, which results in dimers in the even sublattice (see the phase diagram in Fig. 1.). Hence, there are two ordered phases in the phase diagram of the XX model which are called as the odd-dimer and even-dimer phases. The critical exponents can be found in either of the ordered phases for θ close to zero. Thus, expanding Eq. 2.14 for a close to one gives Here it is chosen the odd-dimer phase to make the measurements, i.e. a < 1.
With the distance from criticality defined, one can now ask about the effects of disorder under the phase transition (PT) of the XX model. The most fundamental question would be if the PT occurs at all after introducing randomness. It has been already shown that disordered systems can indeed undergo a PT, but in early studies it was believed that randomness would eventually destroy the transition. To understand this concern, let the uncorrelated random XX chain be divided in subsystems of equal size. Since the couplings are random, each subsystem could have its own distance from criticality θ and it was not known how to evaluate θ for the bulk system. This is the reason why random couplings are known as the random-T C disorder.
This same argument, however, was used later to deduce a criterion that determines whether disorder is relevant to the PT of a system, as it is shown in the following. Suppose that a d-dimensional spin chain has a bulk nonzero θ and a corresponding correlation length ξ. This chain is divided in n subsystems of size ξ d . One can calculate the average distance from criticality θ and also its standard deviation ∆θ. According to the central limit theorem, the standard deviation scales as ∆θ ∼ ξ −d 2 . Dividing this relation by θ and using that ξ ∼ θ −ν , where ν is the clean correlation length exponent, one finds For consistency of the clean criticality, ∆θ/θ → 0. This is satisfied if the exponent on the right-hand side of Eq. 2.16 is negative, since ξ → ∞ at the critical point. Hence, The expression above is known as the Harris criterion (12,29,30), and it gives a necessary but not sufficient condition for preserving the clean critical behavior of a system under the influence of disorder. When Eq. 2.17 is not fulfilled, disorder becomes relevant and a new critical behavior emerges. There are two known possibilities for this new behavior, namely the finite-disorder critical point (FDCP) and the infinite-randomness critical point (IRCP). The former changes the universality class of the clean critical point (CCP), yielding new critical exponent values. Conversely, the difference between the IRCP and the CCP is more significant: The usual power-law scaling of some quantities is replaced with an exponential scaling, i.e. the scaling becomes activated. An example of such activated scaling in the IRCP is the relation between the relaxation time and the correlation length. (12) Notice that any amount of disorder affects the PT of the XX model, since dν = 1. Indeed, it was observed that under uncorrelated disorder this system exhibit the so-called activated scaling, which is a characteristic of the IRCP. (5, 23) Another property of those random critical points is that the observables are not self-averaging at criticality. (31)(32)(33)(34) This means that the width of an observable distribution either stays constant (FDCP) or increases (IRCP) under coarse-graining, i.e. as L gets bigger. The consequence of non-self-averaging quantities is requirement of more disorder realizations as the lattice size increases.
The surprising result of the XX CD model is that the CCP is preserved for weak disorder, even though the Harris criterion is not fulfilled. The criterion, however, should be rewritten to consider the correlation between couplings. Assuming that there is a spatial correlation between the couplings J i and J j that decays with the distance r = |j − i| as a power law with exponent b > 0, Eq. 2.17 is thus rewritten as (35) min {d, b} ν > 2. (2.18) Nonetheless, for the case studied here the correlation is irrelevant to the criterion, since to obtain J 2i−1 correlated only with its right neighbor one must have b → ∞.
This contradiction between the Harris criterion and the behavior observed for the XX CD model does not indicate that the former is wrong. Rather, the criterion is not applicable to the XX CD model because of the way it is constructed. Making J 2i = J 2i−1 removes the random-T C property of the system, which was the starting point in the deduction of Eq. 2.17. Therefore, using the Harris criterion to evaluate the XX CD model is a fallacy. Furthermore, other apparent violations of the Harris criterion have already been reported. (36)

The correlated disorder case
The Hamiltonian of the XX CD model in the free fermion basis with periodic boundary conditions is This chain is depicted in Fig. 2. The couplings J i are generated by a pseudo-random numerical routine with a uniform distribution bounded between zero and one. The default distribution can be modified by taking the D-th power of the given random number, which yields a new parameterized probability density function where D is the so-called disorder strength.
Using Eq. 2.21 one can show that the ratio between the standard deviation σ J and the average J of the couplings is Since the ratio σ J / J increases monotonically with D, this definition of disorder strength is a reasonable one. Such definition, though, is not unique. One could choose, for instance, a truncated Gaussian distribution with fixed average and define disorder strength as the variation of the standard deviation. The distribution in Eq. 2.21 is preferred because it allows to investigate the effects of relatively strong disorder. Besides, it gives smooth curves for the critical exponents as a function of D, as shown in the next chapter.
For disordered systems each observable must be averaged over various disorder realizations. To obtain a reliable statistics, it is considered here at least 10 3 disorder realizations. Each one of them represents a unique Hamiltonian with a given set of couplings. The Hamiltonian is diagonalized through a LAPACK subroutine and, with the eigenvalues and eigenvectors in hand, one should be able to compute the observables. When the observable requires some algebra to find the final expression, the procedure is shown in the appendices. Some quantities of interest may take a considerable amount of time to give a reliable precision. Thus, to reduce this computation time one can diagonalize H 2 instead of H. With this trick the problem changes to diagonalizing two L/2 × L/2 matrices instead of one L × L matrix. Taking the square of Eq. 2.20 and making the following change of basis where |i represents the i-th site, gives a block diagonal matrix. The upper left block H 2 odd corresponds to fermions hopping in the odd lattice, and it is given by while the matrix for fermions hopping in the even lattice is (2.24) The (L/2 + i)-th eigenvalue of H is equal to the square root of the i-th eigenvalue of either The eigenvectors of the original Hamiltonian can also be retrieved, as it is shown in Appendix A.
However, despite the reduction of computation time, one should be considerate while calculating observables from the eigenvalues or eigenvectors of the squared Hamiltonian. As it is discussed in Appendix B, the precision of this method is only reliable for D ≤ 2.0.

Critical exponents of the XX model with correlated disorder
This chapter gives the full set of critical exponents, namely {α, β, γ, δ, η, ν}, of the XX CD model. The chapter is divided in two sections, one for the system at the critical point (a = 1) and the other for the system out of criticality (a = 1). In each section the quantities of interest are calculated for various disorder strengths D. The corresponding critical exponents are determined via linear regression and are usually shown as a function of D. Furthermore, the four scaling relations between the exponents are investigated and it is shown that two of them are only satisfied for the clean system (D = 0), while the other two still hold for D > 0.

Exponents at criticality
At criticality there is the correlation function exponent η to be determined; the other critical exponents are found exclusively out of criticality. Besides η, here are also determined the so-called critical dynamical exponent z and two thermodynamic exponents α T and γ T which are related to the specific heat and the magnetic susceptibility, respectively. The thermodynamic quantities are shown in the first part of this section, whereas the second one is dedicated to correlation function calculations.
It is important to remark, though, that in general the observables for a disordered system are averages and, therefore, have an associated error. Here, however, the error bars are not shown for simplicity, mostly because they are about the size of the points. Moreover, the errors of the critical exponents given by the fitting method are also neglected, since they may not represent the actual uncertainty. Instead, it is considered an a priori absolute error in the order of 10 −2 for every critical exponent, which is the usual precision in studies of critical behavior.

Dynamical and thermodynamic exponents
The first exponent evaluated here is the critical dynamical exponent z. This exponent establishes an interplay between the relaxation time ξ τ and the correlation length ξ by the relation ξ τ ∼ ξ z . To determine z without considering the temporal evolution of the system, one can replace the typical time scale ξ τ with the inverse of the typical energy scale, i.e. ξ τ ∼ ∆ −1 , where ∆ is the gap energy. Moreover, the finiteness of the system prevents the divergence of the correlation length at criticality. Thus, according to finite-size scaling methods (37,38), one can replace ξ by the lattice size L, which yields The gap ∆ in the equation above is defined as the energy difference between the first excited state |ψ 1 and the ground state |GS . The first excited state is built by creating the lowest-energy particle above the Fermi level, i.e.
Notice that the additional fermion changes the sign of the boundary term in the Hamiltonian. Thus, for each disorder realization one must diagonalize two matrices, one for N = L/2 and another for N = L/2 + 1. Denoting the i-th eigenvalue of the former and the latter matrices as λ i and λ i , respectively, the gap energy is, thereby, . 3 shows the plot of ∆ vs. L for various disorder strengths D. The correlated disorder case is also compared to the uncorrelated one. Notice that the curves of the former respect the power-law scaling and have a disorder varying slope, while the curves of the latter start to bend as L increases. The bending is a signature of the activated scaling, which is a known feature of the IRCP. For D = 0.1 this behavior cannot be observed because L is much smaller than the so-called crossover length ξ C ; only for L ξ C the true scaling behavior of the system is achieved. From Fig. 3 it is clear that exists a disorder-dependent crossover length for the uncorrelated chain. Indeed, previous studies show this crossover for other quantities of interest. (39,40) Conversely, it turns out that for the XX CD model there is no such crossover length, as indicated by renormalization group studies. (41) The proof of this statement, however, is beyond the scope of the present work.
The curves slope in Fig. 3 correspond to the critical dynamical exponent, which is shown as a function of disorder strength in Fig. 4 * . From this figure one can notice that * The plot of z vs. D has already been reported (9), but here a wider disorder range was considered.  the CCP is preserved up to a critical disorder D C ≈ 0.3. For stronger disorder (D > D C ) the system falls into a line of FDCPs, where the critical dynamical exponent z grows linearly with D. Moreover, one can compare the two sets of data points and observe that they approach the line z = D + 1/2 as the exponent is fitted from larger chains. This line, however, was not predicted theoretically; it is rather an observation.
Besides the critical dynamical exponent, one can also evaluate the exponents of the specific heat C and the magnetic susceptibility χ. These quantities are defined as derivatives of the free energy. Thus, one must consider finite temperature to use statistical mechanics methods and compute the partition function and, consequently, the free energy (for more details see Appendix C). After some algebra one can find the following expressions It is expected that the thermodynamic quantities above exhibit a power-law behavior with respect to the temperature T , i.e. C ∼ T α T and χ ∼ T γ T at criticality. Fig. 5 shows a plot of the specific heat vs. the temperature for the clean system and various lattice sizes. Notice that there is a region where the power-law scaling is violated. This region corresponds to temperature values that are smaller than the finite-size gap. Thus, since the gap is inversely proportional to L, the range of temperature where the power-law scaling is observed is wider for larger chains, as one can verify in Fig. 5. Furthermore, T should not get large enough to the point where the thermal fluctuations overcome the quantum ones and the quantum critical behavior is lost. Therefore, one must assure that the fitting region fulfills ∆ < T < 1.
Setting L = 2 12 and T = e −x , with x integer and x ∈ [2, 6], the specific heat and the magnetic susceptibility were computed for various disorder strengths, averaging over 10 4 disorder realizations. The respective exponents α T and γ T were fitted from the data. Fig. 6 shows those exponents plotted as a function of disorder strength D. Notice that α T and γ T are related with the critical dynamical exponent z according to the following expressions and This is in agreement with the results of the renormalization group methods introduced in Ref. (41).   Source: By the author.

Spin-spin and dimer-dimer correlation functions
The spin-spin correlation function in the α-direction is defined as where r = |j − i| is the distance between spins. It was discussed in Chapter 2 that the single-operator average vanishes, i.e. S α i = 0, which leaves only the double-operator average S α i S α j to be determined. Appendix D outlines the procedure to compute S α i S α j for the longitudinal (α = z) and transverse (α = x) components. Here the y-component is equivalent to x one, since the system is invariant under a rotation of π/2 about the z-axis.
It is known that the correlation function scales as where · · · represents the average † . Thus, since the correlation length diverges at criticality, i.e. ξ → ∞, G α (r) should scale as a power law. Fig. 7 shows the x and z-components of the spin-spin correlation function for various disorder strengths D and chain size L = 1024. The data were averaged over 10 3 disorder realizations. Taking the absolute value is necessary to show the log-log plot, since G α (r) has negative values. Notice that the power-law scaling is respected for the longitudinal and transverse components. The curves of the former, however, do not change significantly as disorder gets stronger, but for the latter the slope is clearly disorder dependent. Fig. 7 also compares the average correlation to the typical one G α typ (r), which is defined as Comparing the average and typical correlations may provide a further understanding about the random critical point. For instance, it has been observed that for a system in an FDCP the average and typical correlations display similar behaviors, whereas for the IRCP those quantities behave differently from each other. (42) Indeed, for the uncorrelated disorder case it has been shown that G α typ (r) scales exponentially instead of the power-law scaling of the average correlation. (23) Thus, the purpose of computing the typical correlation of the XX CD model is to contrast with the known results for the uncorrelated disorder case.
From the plot of Fig. 7, and within the region 1 r L, one can also determine the critical exponent a α for each of the correlation functions. Fig. 8 shows a α vs. the disorder strength D for α = x and α = z. Notice that the former is different than the latter, which is in agreement with the results for the clean chain. Conversely, for the uncorrelated   Knowing the correlation function exponent a α allows one to determine the critical exponent η. However, for the XX model in general (i.e. for the clean and random cases) one should not evaluate η from the spin-spin correlation function. The actual exponent must come from the correlation function of the operator that gives the order parameter of the model. For instance, the sum of the spin operator S i over the chain gives the magnetization. This quantity, though, is not a suitable order parameter for the XX model, since it is equal to zero in the ordered phase.
Therefore, one must define another operator that yields a valid order parameter for the XX model. Here it is chosen the so-called dimer operator M i , which is defined as The index i represents the i-th dimer of the chain. One can check for the clean chain that M i = 0 at criticality and M i = 0 in the ordered phase. The corresponding order parameter m of the dimer operator is defined as the average over the lattice of the dimer It is important to remark, though, that the choice of the order parameter for a model may not be unique; the only requirement is that it must be zero at the critical point and nonzero out of criticality. For the XX model, for example, it has also been proposed the so-called string order parameter. (42)(43)(44) With the definition of the dimer operator in Eq. 3.11, one can now compute the dimer-dimer correlation function G dd (r), which is given by (3.13) Fig. 9 shows the typical and average G dd (r) for various disorder strengths D and lattice size L = 1024. The samples were averaged over 10 4 disorder realizations. Notice that for weak disorder (D < D C ) the typical correlation coincides with the average one. As disorder increases, though, the slope of the former varies more rapidly. This can be verified in Fig. 10, which gives the plot of the exponents a dd as a function of D for the average and typical correlations.

Exponents at the dimer phase (out of criticality)
Similarly to the previous section, this one is divided in two parts. The first part is dedicated to the thermodynamic quantities, while the second one provides the study of the correlation function. In the first part the thermodynamic critical exponents, namely   {α, β, γ, δ}, are determined and the respective scaling relations are tested. The last critical exponent ν is determined in the second part, where the other two scaling relations are also investigated.

Thermodynamic exponents
Before discussing the quantities that give the remaining critical exponents, it is investigated here what happens to the gap energy ∆ when the system is driven out of criticality. It has been already reported that for clean half-integer spin chains the gap is closed (∆ = 0) at criticality and it begins to open as the system is driven out of the critical point. (45,46) The numerical results from the previous section show that the finite-size gap of the clean and correlated-random chains exhibit a power-law decay with the chain size L, i.e. ∆ ∼ L −z . Thus, at the thermodynamic limit (L → ∞) both systems are gapless. However, it is shown in the following that in the ordered (dimer) phase the XX CD model behaves differently from the clean chain; the gap is still closed out of criticality. Fig. 11 shows the gap ∆ plotted as a function of θ for various lattice sizes and two values of disorder strength. Plotting an observable for various L allows one to determine whether the displayed behavior represents the one expected at the thermodynamic limit. For instance, if the separation between curves for a fixed D diminishes as L increases one may conclude that the observed behavior is correct. This is clearly not the case for D = 1.0, where the separation between curves remains approximately constant. Thus, the gap in Fig. 11 for D = 1.0 is due to finite-size effects; for a sufficiently large chain the system is gapless. However, for D = 0.1 one cannot derive the same conclusion by analyzing Fig. 11 alone.
The reason why the gap is indeed closed even for every value of D is that the introduction of randomness provides a nonzero probability of existing a region of size l where the couplings are small. The probability of finding such region increases with lattice size L and disorder strength D. Furthermore, this region of small couplings will dominate the behavior of the system only if it is much larger than the characteristic scale of the system, i.e. l ξ. One can test this statement with the following model: the clean chain is divided in two complementary parts, A and B of sizes l and L − l, respectively. For the subsystem A the constant coupling is defined as J A = 0.1, while for the subsystem B the coupling is J B = 1. The gap energy ∆ is thus calculated for various l. disorder realizations, vs. the chain size L for three disorder strengths. Observe that J min decays with an exponent equal to D. Thus, neglecting the disorder dependent prefactor, to get a J min for D = 0.1 close to the one obtained for D = 1.0 one should consider a lattice L 10 times larger. Therefore, one can conclude that the XX CD model is also gapless in the ordered phase regardless of disorder strength; the gap in Fig. 11 is due to finite-size effects. This conclusion can be extended for every random spin-1/2 chain with a probability distribution that allows couplings close enough to zero.
After discussing what happens to the gap energy out of criticality, the focus returns to determining the critical exponents. The first one evaluated here is α, which is related to the specific heat. Differently from the previous section, the exponent is now determined by varying θ and setting T = 0. However, the problem is how one can compute specific heat without temperature, since this quantity is defined as the derivative of the internal energy with respect to T .
Using the statistical mechanics definition of C, one can make an analogy and define a specific heat C for QPTs. For instance, at T = 0 the ensemble-averaged (internal) energy corresponds to the ground-state energy E 0 . Furthermore, for the XX model the constant a is the parameter that has the same role of temperature for classical transitions. Thus, C can be defined as where it was used that dθ = da. Since the derivative in Eq. 3.14 cannot be calculated analytically, one must consider methods of numerical differentiation.   . 13 shows the plot of C as a function of θ for various disorder strengths D. The ground-state energy E 0 was computed for θ = 2 −x , with x integer and x ∈ [2,9], and averaged over 10 5 disorder realizations. The plot of E 0 vs. θ was interpolated using a cubic spline, giving 12 equally spaced points. The data from Fig. 13 correspond to the centered differentiation of the interpolated lines. Notice that C becomes smaller as disorder is increased.
Moreover, it is possible to affirm from Fig. 13 that all curves are consistent with an exponent α = 0, even though the rule of thumb for determining critical exponents was not respected. This rule states that one should consider at least three orders of magnitude for θ to obtain a reliable estimate for the exponent. One can notice, however, that the curves tend to a constant as they approach criticality. Thus, to consider smaller θ the number of disorder realizations should increase to guarantee that a given point does not have a greater E 0 than another one to its right side. If, however, the ground-state energy is calculated with a poor statistics, the numerical differentiation may provide misleading results.
The next critical exponent evaluated is γ, which describes the scaling of the susceptibility χ with respect to θ. The susceptibility can be calculated from the dimerdimer correlation function G dd (r). According to the fluctuation-dissipation theorem (18), Fig. 14 shows the susceptibility χ vs. the distance from criticality θ for several disorder strengths D. The susceptibility was calculated from G dd (r) averaged over 10 3 disorder realizations and for chain size L = 512. Notice that χ behaves similar to C ; it tends to a constant as θ gets closer to zero. Thus, one can conclude once again that the data are consistent with a critical exponent γ = 0.
The last thermodynamic quantity to be evaluated is the order parameter m, as defined in Eq. 3.12. It is important to remark that the dimer operator calculated out of criticality exhibits an unexpected behavior. Fig. 15 shows the dimer operator M i vs. its respective position i for various disorder strengths D and chain size L = 128. Notice that there are some points where M i < 0; i.e. they are at even dimer phase. This was not expected to occur for the XX CD model, since the correlation between couplings followed by the dimerization should assure that the odd sublattice had stronger couplings than the even one. Thus, M i should be positive all over the chain. Indeed, one can verify this for the clean chain.
The introduction of randomness in the chain apparently creates a collective effect that allows the existence of negative values of M i . Nonetheless, one can observe in Fig. 15 that the density of points above the dashed line is greater than below. If, however, the number of positive and negative points were equal to each other, the order parameter m would be zero and the system would be at the critical point.
For systems under random-T C disorder, such as the uncorrelated random XX model, there is a nonzero probability of finding a relatively large region that is in a different phase than the bulk system. (47)(48)(49) These are known as rare regions and they usually dominate the behavior of the phase transition. (12, 50) However, even though Fig. 15 shows dimers at opposite phases, these rare regions are fundamentally different and do not play an important role at the QPT.
From the order parameter defined in Eq. 3.12 one can determine two critical exponents, viz. β and δ. To find the former the procedure is the same as before; i.e. plotting m vs. θ. Conversely, to find the other exponent the distance from criticality is fixed at θ = 0 and the order parameter is plotted as a function of the so-called conjugate field B. The introduction of this field leads to a new Hamiltonian H , which is given by where B > 0. Thus, one just need to replace the odd and even couplings by (J i + B) and (J i − B), respectively. conjugate field for various disorder strengths and lattice size L = 512. The data were averaged over 10 5 disorder realizations. The curves are compared to a dashed line with slope equal to clean exponent values, i.e. β = 1.0 and 1/δ = 1.0. Notice that the curves tend to become parallel to the dashed line as θ decreases. Table 1 gives the fitted exponent values for each disorder strength. The fitting region corresponds to the four leftmost points. Observe that only for D = 1.5 the δ exponent is relatively apart from the clean value; the others are in good agreement. Thus, to find a slope closer to unity one should consider smaller values of B.
From the results shown in this section, one can reasonably assume that for the XX CD model the four thermodynamic critical exponents remain equal to the clean value as disorder is introduced. Furthermore, one can substitute α = γ = 0 and β = δ = 1.0 in the following scaling relations 2 − α = β (δ + 1) , (3.18) and check that they are indeed respected. However, one cannot assert from numerical results alone that the critical exponents are indeed disorder independent; other theoretical methods are required to prove the independence with respect to D. Nevertheless, it is rather unreasonable to believe that there may exists a disorder value above D = 1.5 from which the behavior of the system suddenly changes.

Spin-spin and dimer-dimer correlation functions
The last critical exponent to be determined is ν which describes the scaling of the correlation length ξ, i.e. ξ ∼ θ −ν . Thus, according to Eq. 3.9, one can find the correlation length from the linear regression of the logarithmic of the correlation function. and Fig. 18 show, respectively, the spin-spin and dimer-dimer correlation functions for several distances from criticality θ. Notice that for all correlation functions the behavior of the disordered chain with D = 1.0 (continuous lines) is similar to clean one (dot-dashed lines), which suggests that the correlation length exponent ν is disorder independent.
One can determine the critical exponent from the plot of ξ vs. θ. The correlation length can be fitted from Fig. 17 and Fig. 18 in the region where the curves are linear. However, notice that for θ = e −5 this linear behavior is not clear. This occurs when the correlation length is much larger than the lattice size (i.e. ξ L) and the system is essentially at criticality and the power-law behavior dominates. Therefore, for the considered chain size (L = 2 11  Source: By the author. x spin-spin: ν x , a x z spin-spin: ν z , a z dimer-dimer: ν dd , a dd clean(D = 0) 1.00(2), 0. can get a rough estimate of the exponent from just three points. If more data are desired one should consider larger chains, which may increase significantly the computation time required and, consequently, become unpractical. Table 2 shows the fitted exponent ν for the spin-spin and dimer-dimer correlation functions, for various disorder strengths D. Notice that ν is indeed disorder independent, within the given error, for every correlation function. Furthermore, even though it is not possible to know the exact exponent value, one can conclude that ν is close to unity for the dimer-dimer correlation function, regardless of disorder strength. For the spin-spin correlations, ν = 1 corresponds to clean chain exponent value, which can be determined exactly.
To check whether ν = 1 is a good estimate for the critical exponent one can observe if the data shown in Fig. 17 and Fig. 18 collapse into a single universal function. One can determine this function from the scaling hypothesis, which states that the correlation function G α (r) is homogeneous with respect to its parameters (18,19), i.e. λG α (r, θ) = G α λ a r, λ b θ , (3.19) where λ, a and b are constants to be determined.
The first step is to eliminate the dependency on θ, which is achieved by making It is also desired that the universal scaling Φ depends only in the parameter x = r/ξ, such that Φ (x) comprises the two limiting cases, namely the power-law (x → 0) and exponential (x → ∞) behaviors. Thus, since ξ ∼ θ −ν one can find that a/b = −ν, which gives Furthermore, using that Φ (x) ∼ x −aα and G α (r, θ) ∼ r −aα in the limit x → 0, one can find that 1/b = a α ν. Therefore, the universal scaling function is Fig. 19 and Fig. 20 show Φ (x) for the spin-spin and dimer-dimer correlations, respectively, for the clean and disordered chains with D = 0.1 and D = 0.5. The distance r was replaced with the so-called chord length r , which is defined as The chord length is the natural finite-size length of the clean critical chain. Here it is assumed that this correction is also valid for the XX CD model. Indeed, Chapter 4 shows that this correction is satisfactory for the disordered chain. The chord length is employed to minimize finite-size effects, such as the curve bending in its rightmost end (see, for example, Fig. 17 and Fig. 18). Furthermore, to guarantee that all curves end in the same point it is fixed Lθ = 16.
Notice from Fig. 19 and Fig. 20 that the collapse fails for r θ < 1, except for the longitudinal spin-spin correlation. The reason of this failure can be the neglected further corrections of the power-law scaling. For instance, to obtain a more accurate exponent one should consider (3.24) or even more corrections. This is generally true for every quantity that exhibits power-law scaling. Conversely, the z spin-spin correlation shows a good collapse because it is known that for the clean chain the scaling is exactly  Table 2.
Source: By the author.
with r odd. Thus, since only relatively small D was considered in Fig. 19, the clean chain scaling is sufficient. If, however, one considers stronger disorder the curves would start to separate from each other, since disorder introduces further corrections that become more relevant as D increases.
Nevertheless, one should observe that the points below r θ = 1 correspond to about 6% of the chain. Thus, for the great majority of the lattice, the collapse is satisfactory for all the correlation functions evaluated. Therefore, the results suggest that for the XX CD model the correlation length exponent ν is disorder independent and ν = 1 is a reliable estimate.
Finally, with an estimate of the critical exponent ν one can now check the validity  Table 2.
Source: By the author.
of the other two scaling relations, viz.
and 2 − α = (d + z) ν, (3.27) which are known as Fisher's scaling and hyperscaling relations, respectively. The original hyperscaling relation, though, does not include the critical dynamical exponent z. The substitution d → d + z is motivated from the quantum-to-classical mapping, which states that for a d-dimensional quantum critical system there is a classical (d + z)-dimensional equivalent. However, one should be concerned about this substitution, since not every quantum critical system have a classical counterpart. Nevertheless, if the critical dynamical exponent is not included the hyperscaling relation does not hold even for the clean chain.
Although replacing d with d+z is necessary to fulfill the scaling relations, the Harris criterion in Eq. 2.17 does not require such replacement and the reason is the following: The quantum-to-classical mapping introduces an imaginary time direction where the couplings are constant instead of random. This extra direction, therefore, do not contribute to the distribution of distances from criticality, which was the starting point in the deduction of the criterion.
Even though it is not explicit, Eq. 3.26 also depends on the critical dynamical exponent z, because of the relation between η and a dd which is given by  Moreover, it is clear that the hyperscaling relation in Eq. 3.27 also fails for D > D C . The breakdown of the hyperscaling relation is a known characteristic of systems that are above the so-called upper critical dimension d + c . For instance, the mean-field Ising model (which is above d + c ) has exponents α = 0 and ν = 1/2, thereby violating Eq. 3.27. Recently, though, it has been proposed that the hyperscaling relation for systems with d ≥ d + c should be rewritten as (51) 2 − α = νd q , (3.29) where q = d/d + c . Notice that for Eq. 3.29 the substitution d → d + z is not important, since d cancels out. However, for the XX model the upper critical dimension is d + c = 3 and, consequently, Eq. 3.29 does not hold. Similarly, it has also been suggested a different Fisher's scaling relation for finite-size systems with d ≥ d + c (52), but since Eq. 3.29 fails as well this assumption of the emergence of a system with d ≥ d + c seems rather implausible. Another explanation that one may deliver is that Eq. 3.27 is indeed respected, but the order parameter chosen here does not provide the actual critical exponents of the system. Therefore, assuming that Eq. 3.27 holds and noticing that α and z are independent of the chosen order parameter, the actual correlation length exponent ν should give Since for strong disorder z ∝ D, ν → 0 as D → ∞, which is an unlikely scenario.

Entanglement properties of the XX model with correlated disorder
This chapter provides calculations of the so-called Rényi entanglement entropy for the XX CD model. The first section shows that the clean chain scaling function also holds for the XX CD model (this result is published in (41)). Furthermore, the central charge is determined and plotted as a function of disorder strength. Higher-order corrections in the scaling of the entanglement entropy are considered in the second section.

Scaling function and the central charge
Entanglement measurements used to be a exclusive topic of quantum information studies. Recently, however, such measurements became popular in condensed matter physics (53,54), mainly because it provides an insight on long-range correlations. The most used quantity for measuring entanglement in many-body systems is the so-called von Neumann entropy S vN . Dividing the system in two complementary parts A and B of sizes x and L − x, respectively, S vN is thus given by where ρ A = T r B ρ is the reduced density matrix, i.e. the partial trace of the density matrix ρ over the subsystem B.
For every 1D clean critical system it is known that the leading scaling behavior of the entanglement entropy is logarithmic (55,56), i.e.
where c is the so-called central charge. Entanglement measurements became so widespread because of their relation with the central charge, which is used for classifying the system. For instance, it is known that the XX model belongs to the universality class with c = 1.
Recently, it has been found that the XX chain with uncorrelated disorder also exhibits a logarithmic scaling for the entanglement entropy, similar to Eq  corrections can be found for the clean chain using conformal field theory (CFT) methods. Conformal invariance is the combination of rotational, translational and scale invariance. CFT approaches are best employed for two-dimensional (2D) classical critical points, or equivalently 1D quantum critical points.
Using CFT methods it has been found that finite-size effects diminish as one replaces x in Eq. 4.2 with the so-called chord length (58,59) The chord length is the line that connects two points of a circle, as Fig. 22 shows. Such scaling function, though, is not valid for random systems, since the introduction of disorder breaks conformal invariance. For instance, a conformally invariant model should have a dynamical exponent z = 1. However, for the XX CD model Fig. 4 shows that z = 1 above D C , while for the uncorrelated case z → ∞.
Nevertheless, one can search for a scaling function for random chains by expanding f (x) as a Fourier series, regarding periodic boundary conditions and symmetric properties. (60) For instance, it has been found that a good approximation for the scaling function of the uncorrelated random XX chain is with b 1 ≈ 0.153. (61) Conversely, for the XX CD model it is shown in the following that a satisfactory scaling function is the clean system chord length, i.e. Eq. 4.3, even for D > D C where the necessary condition z = 1 is not obeyed.
Before showing the results for the scaling function of the XX CD model, it is outlined the procedure to compute the entanglement entropy. Here it is investigated the so-called Rényi n-entropies, which are defined as Notice that in the limit n → 1 Eq. 4.5 retrieves the von Neumann entropy (Eq. 4.1).
To calculate the entanglement entropy, one must first define the so-called correlation matrix where the expectation values c † i c j are calculated from the eigenvectors of the Hamiltonian at criticality, as shown in Appendix A. Since the chain is periodic, the subsystem size x ranges from one to L/2. For each value of x one must diagonalize Eq. 4.6 and with the respective eigenvalues λ i calculate the entanglement entropy, which is given by (62,63) and for the von Neumann entropy (n = 1). Notice that, besides averaging over the number of disorder realizations, one can also average the entanglement entropy over the chain by varying the index i in Eq. 4.6.
From CFT methods it has been shown that the Rényi entanglement entropy of the clean chain scales as (58,59) S n (x) = c 6 1 + 1 n ln [f (x)] + a 0 + subleading terms. (4.9) Cancelling out the constant a 0 by subtracting S n (L/2) from S n (x) and neglecting the subleading terms one can find (4.10) Thus, one can use Eq. 4.10 to determine the scaling function. Notice, however, that an input value for the central charge is required. This value is not known a priori for the XX CD model, but one can consider a self-consistent approach. Such approach consists of fitting the central charge from the entanglement entropy calculations by assuming that the CFT scaling is valid. Then, using the fitted central charge as input in Eq. 4.10, one can show that the data is in good agreement with the proposed scaling. After determining the scaling function of S n (x) for the XX CD model, one can now find the central charge for various disorder strengths. Fig. 24 shows S n (x) for various n and two chain sizes L. It was also considered two values of disorder strength, namely D = 1.0 (top panel) and D = 2.0 (bottom panel). The data were averaged over 10 3 disorder realizations. Notice that all curves are parallel to the dashed lines, which give an estimate of the central charge. The value of c is also increasing as disorder gets stronger, contrasting with the uncorrelated disorder case where the effective central charge is universal. (57) Moreover, observe that an oscillatory behavior appears for n = 1 and the amplitude of the oscillations increases with n. These oscillations are due to higher-order corrections in the scaling of S n (x), which are considered in the next section.

Higher-order corrections of the scaling
The previous section shows that the finite-size correction of the clean chain, viz. the chord length (Eq. 4.3), is also in good agreement with the XX CD model. This result instigates an extension of the comparison with the clean chain by considering the subleading terms of the scaling. Up to first order, the CFT scaling of the Rényi entanglement entropy (4.11) Therefore, in this section the data of S n (x) for the XX CD model are fitted to Eq. 4.11. For every disorder strength and chain size, the entanglement entropy is averaged over 10 3 disorder realizations. The four parameters, namely a 0 , c, a 2 and φ, are shown as a function of n, which varies from one to ten with a step of 0.1. To determine the parameters, the usual least-squares method is accompanied by a numerical procedure, as explained in Appendix E, because of the nonlinear parameter φ. Fig. 26 shows the parameters a 0 and a 2 vs. n (panels (a) and (b), respectively) for various disorder strengths D. These parameters are not compared to clean chain. It is important, though, to observe that in panel (b) the parameter a 2 goes to zero as n → 1 for each of the considered value of D. This result is in agreement with the clean chain behavior, where a 2 = 0 for n = 1. Fig. 27 and Fig. 28 show the central charge c as a function of the parameter n for three disorder strengths, namely D = 0.1, D = 1.0 and D = 2.0. It is expected that the central charge is constant with respect to n. However, from Fig. 27 and Fig. 28 one can clearly observe that c varies with n. An explanation for such unexpected behavior are the finite-size effects. Thus, to test this hypothesis, the central charge is fitted for various chain sizes L. As the lattice size gets larger, one should observe the curves becoming more horizontal, which is indeed verified for every disorder strength. It is also expected that the curves approach each other with increasing L, which is not observed in Fig. 28. Another issue is that higher-order corrections should be considered as n increases. (66) However, this would make the fitting procedure even more difficult. In addition, the introduction of disorder may also require further corrections that cannot be predicted by CFT.
Lastly, Fig. 29 * shows the exponent φ vs. the parameter n for disorder strength D = 0.1 and D = 1.0. For D = 0.1, notice that the curves become more horizontal as the chain size L is increased and they stay relatively close to the clean chain value. The region n < 2 is not reliable, since as n → 1 the prefactor a 2 gets closer to zero and the subleading correction is irrelevant. One can also observe that the exponent φ changes with disorder, as for D = 1.0 it remains approximately close to three. The data for D=2.0 is not shown because the results were not satisfactory.
For completeness, Fig. 30 compares the data of S n (x) with the corresponding best fitting curve (dashed lines), which was obtained using Eq. 4.11. The inset gives a closer look for n = 3 in the region 1 x L. Notice that the fitted curves are in good agreement with the numerical data. As the parameter n increases, though, the difference between them becomes more significant. One can explain such discrepancy with the neglected higher-order corrections, which are required for greater n.  Source: By the author.

Conclusion
The complete set of critical exponents of the XX CD model, namely {α, β, γ, δ, η, ν}, has been determined. It has been found that most of them are universal, i.e. disorder independent, and equal to the clean chain exponent. The only exception is the anomalous dimension η. Besides the usual set of critical exponents, the so-called critical dynamical exponent z was also determined. It has been shown that z increases monotonically with disorder strength D and is equal to the clean value for D ≤ D C ≈ 0.3. Thus, one may conclude that for weak disorder (i.e. below D C ) the clean critical point is preserved. As disorder gets stronger, though, the system falls into a family of finite-disorder critical points, each of them characterized by a unique z.
However, this disorder-dependent dynamical exponent results in unreliable values for the anomalous dimension η. For instance, for sufficiently strong disorder, η < 0 which is not acceptable. Such negative values of η are obtained from the premise that, for quantum critical systems, one should replace d with d + z. As another consequence of this substitution, two scaling relations, viz. hyperscaling and Fisher's scaling relations, do not hold for disorder strength above D C . The other two scaling relations that do not involve dimensionality are respected even for D > D C .
There is not much literature available about the validation of the hyperscaling and Fisher's scaling relations for random quantum phase transitions. Thus, it would be interesting to test such relations for other models. However, a more appropriate procedure would be to derive those relations from the beginning for quantum critical points and see where the introduction of disorder may interfere. As shown here, just replacing d for d + z is not sufficient to validate the scaling relations. In addition, the failure of two scaling relations may indicate that there are more than two independent critical exponents.
The observed numerical results strongly suggest that the critical exponents are disorder independent (except for η and z). However, from numerical studies alone it is not possible to prove this independence. This study should be accompanied with theoretical approaches, such as the renormalization group. It is not known, though, how to extract all quantities of interest from the renormalization group of finite-disorder critical points; this method is best employed for infinite-randomness critical points, such as the XX model with uncorrelated disorder.
Alongside the usual study of the critical behavior, it has also been performed calculations of the Rényi entanglement entropy. Surprisingly, it has been found that the leading scaling behavior in Eq. 4.9 and the finite-size correction of the clean system (namely the chord length) are in good agreement for the XX CD model as well. This result is intriguing because the entanglement entropy scaling and the chord length are derived from conformal field theory methods and a necessary condition for conformal invariance is that z = 1, which does not occur for D > D C . Thus, this result suggests that the logarithmic scaling in Eq. 4.9 and the chord length may be derived from a more general assumption than conformal invariance, which may also include random systems.
Furthermore, it has been found that the corresponding central charge c increases monotonically with disorder strength D, contrasting with the uncorrelated disorder case where the effective central charge is universal. It has also been shown the central charge as a function of the parameter n of the Rényi entanglement entropy. It was expected that c and the exponent φ (introduced in the first-order correction) were universal, i.e. independent of n. However, this independence was not verified, mostly because of finite-size effects. Nevertheless, one could obtain better results if chains with open boundaries were considered instead. (67) Moreover, to achieve a deeper understanding of the entanglement properties of the XX CD model one should consider a study of the real-space entanglement spectrum. (68) The next step is to investigate other systems under correlated disorder and check if the same behavior occurs. Indeed, the correlated disorder XXZ spin-1/2 chain has been investigated and it turns out that it behaves similarly to the XX CD model. (41) However, renormalization group calculations show that this behavior for the XXZ model is due to finite-size effects; for sufficiently large lattices the uncorrelated case is retrieved. Conversely, for the XX CD model there is no such crossover length, although it has not been proven here.
This appendix outlines the procedure to write the Hamiltonian in its diagonal basis. This is a necessary step to evaluate expectation values such as c † i c j , which are related to some of the quantities of interest, e.g. the correlation function. Some matrix properties that simplify calculations are also presented.

The matrix representation of H in the old basis is
where M is the matrix in Eq. 2.20 and c † (c) is a row (column) vector with the L fermionic operators, i.e.
It is desired to rewrite the Hamiltonian in the diagonal basis, i.e.
where η † and η are the diagonal fermionic operators and where λ i is the i-th eigenvalue of M with λ 1 < λ 2 < · · · < λ L .
The change of basis above is called a similarity transformation, in which the eigenvalues of the matrix are unaltered. This transformation can be represented in general as where V (V −1 ) is the (inverse) transformation matrix to be determined. For hermitian matrices, such as M , one can show that V is the matrix with the eigenvectors v of M stored column wise, i.e., To show that the matrix above satisfies the transformation notice that Since the eigenvectors matrix V is unitary (or orthogonal, since v ∈ R), V −1 = V T and where V T is the transpose of V and I is the identity matrix. Thus, which is indeed the matrix D.
Using the transformation in Eq. A.5 and comparing Eq. A.1 with Eq. A.3 one finds that c † M c = η † V T M V η. Therefore, the i-th component of c † and c is, respectively, The expectation value c † i c j is evaluated on the ground state |GS of the system. The ground state is constructed by creating fermions up to the Fermi level µ = 0, i.e.
where |0 is the vacuum. Hence, the expectation value is The operator η m will first act on the ground state and annihilate a preexisting fermion. Therefore m < L/2 + 1, which can be represented as a step function θ L 2 − m . The second operator η † k will create a fermion in a vacancy. However, if k = m the right eigenstate will be different than the left one, which will result c † i c j = 0. Thus, a Kronecker delta (δ k,m ) is also needed, yielding Notice that the interchange between i and j do not change the result, i.e. c † i c j = c † i c j † .
As it turns out, the eigenvectors matrix V has a property that results in c † i c j = 0 whenever i + j is even. One can check from V that v i,,j = (−1) i−j v i,L−j+1 . (A.14) Moreover, the orthonormality condition of eigenvectors requires that k=1 v i,k v j,k = 1/2. For i = j the right side of the equation is zero, which means that one of the multiplying terms on the left side must be zero as well. If i + j is even the term within brackets equals two, which implies that the summation is zero, i.e. c † i c j = 0 for i + j even. Thus it is only necessary to evaluate c † i c j when i = j and their sum gives an odd number.
Besides using this property one can further reduce computation time by diagonalizing the squared Hamiltonian H 2 instead of H, as discussed in Chapter 2. Let V odd and V even be the eigenvector matrices of H 2 odd and H 2 even , respectively. The procedure to retrieve the eigenvector matrix V of H from the two block matrices is straightforward: Insert the entries of V odd and V even in the odd and even lines of V , respectively; i.e. v 2i−1 = v odd i / √ 2 and v 2i = v even i / √ 2. The entries must be divided by √ 2 because the program gives normalized eigenvectors. Furthermore, the given eigenvectors are stored column wise in crescent order. Thus, the combination of the first column of V odd and V even corresponds to either the L/2-th or L/2 + 1-th column of V . To check whether the i-th eigenvalue is positive or negative one can compute λ i = v i Hv i . Using Eq. A.13 one can obtain right after the opposite eigenvector.

APPENDIX C -Expressions for the specific heat and the magnetic susceptibility
This appendix shows how to derive expressions for the specific heat and magnetic susceptibility using the statistical mechanics approach.
From the canonical ensemble formalism, the specific heat C is defined as (19) where f is the free energy density and T is the temperature. The definition of the magnetic susceptibility χ is similar to the specific heat, except that the derivative is with respect to an external magnetic field B, i.e.
where the derivative is taken in the limit B → 0.
The introduction of the field B changes the Hamiltonian of the system, since one must also considered the interaction of the spins with the field. Thus, the new Hamiltonian is defined as H = H + H B , where is the term that comprises the spin-field interactions, considering that B is constant and points in the z-direction.
To compute the free energy it is easier to write the new Hamiltonian H in its diagonal basis, as shown in Appendix A for H. To write H B in the same basis one can use Eq. 2.10 to map S z i to the fermionic operators c † and c. Furthermore, according to Eq. A.9 and Eq. A.10, one can find that Thus, since the eigenvectors are orthogonal (i.e. L i=1 v ij v ik = δ j,k ), L i=1 c † i c i = L i=1 η † i η i . Using the notation n i = η † i η i , H in the diagonal basis is, thereby, The next step is to calculate the so-called partition function Z, which is defined as where the set {n j } represents the j-th microstate of the system. Substituting Eq. C.5 in Eq. C.6 gives The sum of exponents can be replaced with a product of exponentials. This product commutes with the sum over all microstates. Since the value of n i is either one or zero, the expression for the partition function becomes The free energy F is given in terms of the partition function as F = −T ln (Z) , (C.9) Thus, substituting Eq. C.8 in the equation above gives (C.10) Noticing that the sum over all eigenvalues is zero and that the two terms with B cancel out, one can divide Eq. C.10 by L and find the free energy density (C.11) Therefore, with Eq. C.11 the calculation of the specific heat and the magnetic susceptibility is straightforward. The first derivative with respect to the temperature is using B = 0. The second derivative is since the spectrum is symmetric and cosh (x) = cosh (−x). Similarly, one can take the second derivative of Eq. C.11 and find that For simplicity, it is considered the following substitution: 2i − 1 → i, 2i → j, 2i + 1 → k, 2j − 1 → l, 2j → m, 2j + 1 → n.
Notice that the distance between pairs of spin operators is always one. Thus, according to Eq. D.11 one can write Similarly, for the y-direction one can find that Therefore, G dd (r) can be written as Using Eq. D.2 and that B i A j = 0 for |j − i| even, one can simplify the expression above until it reads Retrieving the original indexes and using Eq. D. 13 gives (D.21) entropy scaling with subleading terms This appendix explains the fitting procedure of the entanglement entropy with a subleading correction to the scaling (Eq. 4.11).
The least-squares method consists in determining the set of parameters a that minimize the following quantity Taking the gradient of Q with respect to a and setting it equal to zero gives the following system of equations

(E.8)
Notice that all the equations above are nonlinear with respect to a 3 and linear with respect to the other parameters. The linear parameters can be calculated exactly, after determining