The finite-size scaling analysis

In this section we will show how to extract ν and g_cfrom the finite-size scaling formulas, and to estimate the statistical uncertainties as well as the systematic errors. Two infer-ence frameworks, which are the frequentist inferinfer-ence and Baysian inferinfer-ence, will be used to estimate the statistical uncertainties. We will also show the consistance of these two frameworks. Finally, the chi-square test will help us to qualify the finite-size scaling model.

In Eq.(4.7), the finite-size scaling formula has been shown. Practically, we should do the Taylor expansion of the f_i(tL^1ν) in Eq.(4.7) for the fitting, such as

y = (1 + bL^−ω) [

a0+ a1(tL^ν1) + a2(tL^1ν)²+ a3(tL^1ν)³+· · ·]

, (4.8)

where ai are constants for Taylor expansion. Different order of Taylor expansion for the model will be used to extracte the ν and the g_c. In order to estimate parameters and their statistical uncertainties in Eq.(4.8), there are two methods in different inference frame-works, which will be described in the following.

In frequentist frameworks, the method of least-squared is treated as our estimator E(x, σ), which can be denoted as

E(x, σ) = arg min

where x_iare the given data having the corresponding g_iand L_i, and σ_iare the statistical er-rors of the data. In addition, y_i(θ) is the model of Eq.(4.8), where θ ={b, ω, a0, a₁, a₂,· · · }.

Since it is not possible to redo the simulations for thousand times and figure out the distri-bution of parameters from the estimator, to solve this problem, resampling technique is a more efficient way to do so. Here, the resampling technique is to generate a new data set

˜x, which has the mean value and standard deviation as original data. Then, the estimator generate parameters ˜θ = E(˜x, σ) from the resampled data set. Repeating many times of the procedure, it will form a probability distribution of parameters due to the statistical uncertainties, such as that shown in Figure.4.4. The final results of the parameters and the

Figure 4.4: The probability distribution from resampling

From the resampling technique, the parameters distribution is generated by the estimator. The model we use in this example is the same as the Eq.(4.8) with the Taylor expansion up to fifth order. The data in the figure are the ground state values of Q₁ with p = 0.5 (left) and Q₂ with p = 0.9 (right), which are the same as those in Figure.4.3. The gcand ν in the figure are the mean value of the distribution.

statistical errors are the mean and the standard deviation of the distribution, respectively.

In Bayesian frameworks which is a totally different perspective from frequentist frame-works, the posterior probability distribution will be used to evaluate the statistical uncer-tainties of parameters in Eq.(4.8). The posterior in Bayesian statistic is the conditional probability function of model’s parameters for given the data set and the model, which can be written down explicitly as

P (θ|x) = L(x|θ)π(θ)

M (x) , (4.10)

where x is assumed to be sampled from the distribution of X(θ), L(x|θ) is the likelihood, π(θ) is the prior and M (x) is the evidence. In our case, X(θ) is assumed to be independent Gaussian distributions, whose mean and standard deviation are y_i(θ) and σ_i. Therefore,

resampling posterior

(Q_i, L_min, n-th order) g_c ν g_c

(Q₁, 8, 3) 2.7380(15) 0.811(6) 2.7378(11) 0.812(4) (Q₁, 8, 4) 2.7389(16) 0.812(6) 2.7384(11) 0.812(4) (Q1, 8, 5) 2.7399(17) 0.805(6) 2.7392(12) 0.805(4) (Q₁, 12, 3) 2.7380(16) 0.804(6) 2.7377(11) 0.807(5) (Q₁, 12, 4) 2.7387(16) 0.806(6) 2.7383(12) 0.805(5) (Q1, 12, 5) 2.7398(17) 0.802(6) 2.7388(14) 0.803(5) (Q₁, 16, 3) 2.7382(15) 0.802(7) 2.7378(10) 0.805(5) (Q₁, 16, 4) 2.7386(15) 0.803(7) 2.7381(10) 0.804(5) (Q1, 16, 5) 2.7395(17) 0.804(7) 2.7385(11) 0.804(5) (Q₂, 8, 3) 2.7394(16) 0.793(6) 2.7391(11) 0.794(4) (Q₂, 8, 4) 2.7401(16) 0.793(6) 2.7396(11) 0.795(4) (Q2, 8, 5) 2.7405(16) 0.790(6) 2.7404(12) 0.791(4) (Q₂, 12, 3) 2.7396(17) 0.790(6) 2.7388(12) 0.793(5) (Q₂, 12, 4) 2.7403(17) 0.791(6) 2.7397(13) 0.793(5) (Q2, 12, 5) 2.7411(19) 0.788(6) 2.7405(14) 0.790(5) (Q₂, 16, 3) 2.7399(16) 0.790(7) 2.7390(10) 0.792(5) (Q₂, 16, 4) 2.7403(17) 0.790(7) 2.7394(11) 0.792(5) (Q2, 16, 5) 2.7410(18) 0.792(7) 2.7398(12) 0.793(5) Table 4.1: Results of g_cand ν (p = 0.5) from the resampling and posterior

the likelihood can be easily written down as

L(x|θ) =∏

In the Bayesian statistic, the prior is the knowledge before analyzing the data. In this sence, we use the flat prior, which is given by−10 < b < 10, 0 < ω < 5 and 0.5 < ν < 1.5 and zero otherwise. Because the M (x) in Eq.(4.10) is not the function of θ, it can be ignored.

To make sure that the prior we choose would not affect the results for g_cand ν, we shift the range of the prior to be narrower and see no significant change of g_cand ν.

However, it is not easy to construct the distribution of the posterior due to lots of di-mension in the parameter space. Conventionaly, Markov Chain Monte Carlo (MCMC), which is a sampling technique in continuous parameter space, can solve this problem ef-ficiently [7]. We show the result in Figure.4.5. The final result of the parameters and the statistical errors are the parameters which maximum a posterior (MAP) and the standard deviation of the posterior distribution, respectively.

Figure 4.5: The probability distribution from the posterior

In Bayesian frameworks, the posterior probability distribution show the statis-tical uncertainties of model’s parameters. The model we use in this example is the Eq.(4.8) with the Taylor expansion up to fifth order. Also, the data is the ground state values of Q1with p = 0.5 (left) and Q2 with p = 0.9 (right), which is the same as that in Figure.4.3. The gcand ν in the figure are the mean value of the distribution.

The statistical uncertainties of parameters in the finite-size scaling model has been evaluated in the frequentist frameworks and the Bayesian frameworks. Also, we would like to present the fitness between the data and the model. Here, we select the parameters set which has the maximum posterior and calculate the χ² value. The comparison of the two results which come from the resampling and the posterior is shown in Table.4.1 and Table.4.2. For g_cand ν, these two methods have the same results in statistical perspective.

From Table.4.1 and Table.4.2, it is easy to see that the ν obtained from carrying out finite-size scaling using the observable Q1 differ slightly from that of Q2. However, both the results from Q1and Q2behave the same when p increases. To extract the final results, we average the ν over all Q₁, Q₂ and different order of the Taylor expension. With these steps of analyzing the data, we can conclude that when p increases, ν increases accordingly from (around) 0.7 to (around) 1.06. Therefore, our results lead to different conculsion with that of J.T. Chayes et al., which claims that the ν should satisfy the Harris criterion when quenched disorder is presented. Our conclusion differs from that of Dao-Xin Yao et al. as

resampling posterior

(Q_i, L_min, n-th order) g_c ν g_c ν

(Q₁, 8, 3) 4.884(30) 1.132(19) 4.883(25) 1.127(14) (Q₁, 8, 4) 4.873(31) 1.134(19) 4.868(23) 1.128(14) (Q1, 8, 5) 4.869(31) 1.129(19) 4.870(22) 1.121(14) (Q₁, 12, 3) 4.861(35) 1.084(20) 4.858(25) 1.084(14) (Q₁, 12, 4) 4.854(35) 1.083(20) 4.858(26) 1.082(15) (Q1, 12, 5) 4.858(36) 1.083(20) 4.869(35) 1.088(16) (Q₁, 16, 3) 4.842(35) 1.059(21) 4.838(29) 1.061(15) (Q₁, 16, 4) 4.851(36) 1.055(20) 4.849(29) 1.054(16) (Q1, 16, 5) 4.854(38) 1.055(20) 4.861(21) 1.061(16) (Q₂, 8, 3) 4.892(37) 1.086(18) 4.900(26) 1.081(12) (Q₂, 8, 4) 4.876(38) 1.088(18) 4.883(28) 1.083(13) (Q2, 8, 5) 4.873(39) 1.085(18) 4.885(27) 1.082(13) (Q₂, 12, 3) 4.907(46) 1.062(19) 4.907(31) 1.062(13) (Q₂, 12, 4) 4.887(45) 1.059(18) 4.893(32) 1.057(14) (Q2, 12, 5) 4.890(47) 1.059(19) 4.898(31) 1.059(14) (Q₂, 16, 3) 4.894(42) 1.046(19) 4.897(33) 1.047(14) (Q₂, 16, 4) 4.891(44) 1.039(19) 4.902(33) 1.041(14) (Q2, 16, 5) 4.890(46) 1.038(19) 4.899(37) 1.038(14) Table 4.2: Results of g_cand ν (p = 0.9) from the resampling and posterior

well, which claims that the ν remains the same as that of clean model when the quenched disorder is considered. [4] [24]

Finally, since Tables 4.1 and 4.2 are obtained by considering most of the available data in the analysis, to understand the impact from using different range of g on the determi-naions of g_cand ν, we have also carried out the Bayesian analysis for data sets obtained by considering various range of g. The final results of ν and gcfor p = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 0.95 are shown in Table. 4.3 which are produced by weighted mean for all different results considering above. Also, the errors are the standard devi-ations from the bootstrapping method times N_p, where N_p for each p is the number of outcomes used in the bootstrap resampling procedure. The results shown in Table. 4.3 are slightly different from that of [17] (However, they are statistically consistent). This is be-cause an alternative method of analysis than that of [17] is used in obtaining the outcomes demonstrating in Table. 4.3.

p ν g_c

0.0 0.702(9) 2.4981(2) 0.1 0.702(6) 2.5056(2) 0.2 0.724(6) 2.5308(4) 0.3 0.745(10) 2.5732(7) 0.4 0.776(11) 2.6383(11) 0.5 0.804(12) 2.7397(13) 0.6 0.841(13) 2.8939(20) 0.7 0.890(15) 3.1466(38) 0.8 0.940(19) 3.605(15) 0.9 1.04(4) 4.796(76) 0.95 1.06(5) 6.552(41)

Table 4.3: The final results of the ν and gcfor all the considered p.

Chapter 5 Conclusion

Inspired from the evidence of the quantum phase transition violating the Harris criterion presented in Refs. [16], [24], we study a spin-1/2 antiferromagnetic Heisenberg model with the quenched disorder firstly employed in Ref. [16]. In particular, we investigate how the strength of randomness affects the considered quantum phase transitions due to randomness. In our calculations, we find that for 0.3 ≤ p ≤ 0.8 the ν obtained are different from the clean model (Which is ν = 0.7112(5) theoretically), but still do not satisfy the Harris criterion. However, when p = 0.9 the determined ν = 1.04(4) fulfills the Harris criterion. For increasing the randomness, ν changes gradually from 0.702(9) to 1.04(4). Before this work, no similar results for the quantum phase transitions are pointed out explicitly. Also, there is no any theoretical predictions for our results. Therefore, more numerical evidence and new theory are needed to clarify and explain the scenario observed here.

In our future works, we would like to investigate the quantum critical region for the same model. For two-dimensional AFM Heisenberg model in the quantum ciritical region, there are universal behaviors for the correlation length, uniform susceptibility, staggered susceptibility and specific heat [5]. The relations for those observables mentioned above correspond to the universal constant X, Ω, Ξ and Ψ, respectively. Therefore, we can study how the strength of randomness affects these universal constants, which can also be the evidence for new universality class emerging when the disorder is presented.

Part of the outcomes presented in this thesis appear in [17] as well.

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