哈里斯準則對淬火無序二維量子自旋系統的有效性

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(1)國立臺灣師範大學理學院物理學系碩士論文 Department of Physics College of Science. National Taiwan Normal University Master’s Thesis. 哈里斯準則對淬火無序二維量子自旋系統的有效性 Validity of Harris criterion for two-dimensional quantum spin systems with quenched disorder. 彭兆宏 Jhao-Hong Peng. 指導教授：江府峻博士 Advisor: Fu-Jiun Jiang, Ph.D.. 中華民國 109 年 1 月 January, 2020.

(2) Acknowledgements 本論文是在江府峻教授的耐心指導下完成，我非常感謝江教授在許多方面對我的教誨，他要求我在分析數據要非常謹慎，並且在研究中提出犀利的觀點，使我受益良多。在研究的不同階段都會遇到不同的問題，教授總會一步一步的教導我遇到問題的解決方式，最後領導我完成這一份對我來說意義重大工作。在研究上我也要感謝譚登瑞學長與黃崚偉學長不厭其煩地與我討論，在討論的過程中讓我能從不同的角度看相同的問題，使我能夠跳脫一些主觀的想法。他們也會與我分享新的所學與看法，這都使我更有力量的完成這篇論文。我也非常感謝其他研究室的同學們，雖然研究的領域不一樣，但總會給我有用的建議，讓我的研究生生活更多采多姿。最後我要感謝我的家人一直以來對我的照顧，在研究的路上如果沒有父母親的支持，或許無法走到今天這一步。. i.

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(4) Abstract Inspired by many evidence showing that the Harris criterion could be violated in quantum phase transitions, we study the second-order quantum phase transition of a spin-1/2 antiferromagnetic Heisenberg model with a specific quenched disorder. In particular, various strengths of randomness are considered in our investigation. The studied models will undergo quantum phase transitions by tuning the dimerized-couplings which are close related to the strength of randomness. In addition, the strength of the employed randomness is controlled by a parameter p which is in the range from 0 to 1, where the clean model corresponds to p = 0. In this study, we use the stochastic series expansion with efficient loopupdate to perform the large-scale quantum Monte Carlo simulation and compute certain physical observables of the model. The critical exponent of the correlation length is evaluated from the finite-size scaling analysis with the Binder ratios as the observables. In order to estimate the statistical uncertainties in a self-consistent way, we analyze the data in the Bayesian inference framework. In the case of p = 0, we find that the critical exponent of the correlation length ν is 0.702(9) which is in reasonably good agreement with the result of O(3) universality class, and doesn’t fulfill the Harris inequality ν > 2/d, where d is the spatial dimension and is 2 in this case. Remarkably, while we find that those ν of p ≤ 0.8 do not fulfill the Harris inequality ν > 2/d, the ν associated with p = 0.9 satisfies such. This interesting phenomenon is not iii.

(5) pointed out explicitly before in the literature. Keywords: Quantum phase transition, Quenched disorder, Harris criterion, Quantum Monte Carlo. iv.

(6) Contents. Acknowledgements. i. Abstract. iii. 1. Introduction. 1. 2. Critical Phenomenon. 5. 2.1. Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 2.2. Quantum phase transition . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.3. Finite-size scaling hypothesis . . . . . . . . . . . . . . . . . . . . . . . .. 8. 2.4. Quenched disorder and Harris criterion . . . . . . . . . . . . . . . . . . .. 9. 3. 4. 5. Stochastic Series Expansion Method. 13. 3.1. Series expansion of the patition function . . . . . . . . . . . . . . . . . .. 13. 3.2. Spin- 12 AFM Heisenberg model . . . . . . . . . . . . . . . . . . . . . . .. 15. 3.3. Basic concepts of Monte Carlo sampling . . . . . . . . . . . . . . . . . .. 16. 3.4. Algorithm of stochastic series expansion . . . . . . . . . . . . . . . . . .. 19. Simulation and Data Analysis. 25. 4.1. The two-dimensional spin- 12 Heisenberg with quenched disorder . . . . .. 25. 4.2. Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26. 4.3. The finite-size scaling analysis . . . . . . . . . . . . . . . . . . . . . . .. 30. Conclusion. 37 v.

(7) Bibliography. 39. vi.

(8) List of Figures 2.1. The region of finite-size scaling . . . . . . . . . . . . . . . . . . . . . . .. 8. 3.1. The external dimension of propagated states . . . . . . . . . . . . . . . .. 21. 3.2. illustrate loop-update . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21. 3.3. example of loop-update . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. 4.1. Two different pairing patterns . . . . . . . . . . . . . . . . . . . . . . .. 26. 4.2. Testing β-doubling scheme . . . . . . . . . . . . . . . . . . . . . . . . .. 28. 4.3. The ground state value of Q1 and Q2 . . . . . . . . . . . . . . . . . . . .. 29. 4.4. The probability distribution from resampling . . . . . . . . . . . . . . . .. 31. 4.5. The probability distribution from the posterior . . . . . . . . . . . . . . .. 33. vii.

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(10) List of Tables 2.1. The critical exponents of various universality classes with different symmetry and dimension are presented in this table. . . . . . . . . . . . . . .. 6. 4.1. Results of gc and ν (p = 0.5) from the resampling and posterior . . . . . .. 32. 4.2. Results of gc and ν (p = 0.9) from the resampling and posterior . . . . . .. 34. 4.3. The final results of the ν and gc for all the considered p. . . . . . . . . . .. 35. ix.

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(12) Chapter 1 Introduction Studying phase transitions is an important topic for both the theoretical and the experimental physicists. Many interesting phenomena like self-similarity, scaling behavior and universality appear at the critical point of a second-order phase transition [3] [11]. In the realistic materials, most of the systems have certain kinds of disorder. Therefore, how the disorder affects the phase transitions has become a serious problem [6] [13] [14] [15] [20]. For classical phase transitions, there is the Harris criterion stating that for all the disorder systems the inequality ν > 2/d must be satisfied, where ν is the exponent of the correlation length and d is the dimension of the systems. [4] [8] However, many evidences show that the phase transitions at zero temperature, where the quantum fluactuation dominate, don’t fulfill the Harris criterion necessarily. [16] [24] To investigate how the disorders affect the phase transition in a quantum system, we start from a clean model that doesn’t follow the inequality of Harris criterion. Then a disorder is added into the clean system. By increasing the strength of randomness, we probe the critical exponent of the correlation length from finite-size scaling hypothesis (νF ) of the system with different strength of randomness. There are three scenarios people found for νF when the disorder is presented in the clean system, which doesn’t satisfy the Harris inequality. The first scenario expected for classical systems is that the νF will have a dramatic change which leads to νF > 2/d, hence satisfies the Harris criterion. The second scenario is observed in Ref.[16][24], and the calculated results of the νF remain the same as that of the clean model. The third scenario is that the infinite randomness 1.

(13) fixed point is emerged due to the quenched disorder [18], which can be observed in the ground state of 1D model [6]. To test the Harris criterion, we consider the two-dimensional dimerized spin-1/2 antiferromagnetic (AFM) Heisenberg model on the square lattice. If one increases the strength of certain couplings, the AFM ordering will vanish and the singlet-pair state will emerge. The quantum phase transition between these two phases is a second-order phase transition, whose critical behavior is described by the O(3) universality class in 3-dimension. Here we want to mention that a quantum system with d-dimension at zero temperature can be mapped to a classical thermal dynamic system with (d + 1)-dimension. The exponent of the correlation length is about 0.7112, which doesn’t satisfy the inequality of the Harris criterion for two-dimensional quantum system. To introduce the disorder into this model, we make the dimerized couplings as random variables with a certain distribution, which has a parameter p measuring the randomness. Here p has the range from 0 to 1. We would like to investigate how νF behaves when p is varied. To study the quantum phase transition of the considered spin-1/2 Heisenberg model, we carry out large scale quantum Monte Carlo simulations using the stochastic series expension (SSE) with very efficient loop update algorithm [22]. The quantum phase transition occurs at the zero temperature of the system. Therefore, to find the true ground states, the β-doubling scheme plays an important role [20]. We simulate the system with different strength of the randomness, and extract the νF with the finite-size scaling method. Surprisingly, the νF gradually increases from 0.702 to 1.04, which does not follows any scenario mentioned above. This thesis is organized as follows. In chapter 2, we mention some basic concepts of the critical phenomena, derive the finite-size scaling ansatz phenomenologically, and talk about the argument and the concept of Harris criterion. In chapter 3, we show how the stochastic series expansion method works. Also, we review the basic idea of the Monte Carlo sampling technique. In chapter 4, we describe the details of the model we used and present the finite-size scaling analysis in both the frequentist and the Bayesian frameworks. Finally, in chapter 5, we conclude this work and discuss some possible future 2.

(14) works for further investigating the new universality class which is emerged by the employed disorder considered in this study.. 3.

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(16) Chapter 2 Critical Phenomenon 2.1. Basic concepts. The critical phenomena occur when a system undergoes the second-order phase transition. Most of the cases, the second-order phase transition corresponds to the spontaneous symmetry breaking in the classical statistical field theory [11]. For example, the transition of a magnet from the paramagnetic phase to ferromagnetic phase breaks the O(3) symmetry of the Hamiltonian without applying the external field. For the system around the critical point, where symmetry breaking just occurs, many interesting phenomena emerge, which are so called the critical phenomena. In the following, we will introduce these critical phenomena. For a system at the critical point, people found that many physical quantities diverge with the power law. The divergence of the correlation length is the key point in the critical phenomenon, which make the system have properties of scaling invariant and selfsimilarity. The correlation length is the characteristic length of the correlation function for local observables. For a magnet, the order parameter is the local magnetization m(x), whose correlation function is denoted as ⟨m(x)m(x′ )⟩, where ⟨· · · ⟩ is the ensemble average. The behavior of the correlation function is exponential decay when |x − x′ | is large, which can be written as. ⟨m(x)m(x′ )⟩ ≈ e− 5. |x−x′ | ξ. ,. (2.1).

(17) model. Heisenberg XY Ising Ising. sym. dim. O(3) D = 3 O(2) D = 3 Z2 D = 2 Z2 D = 3. α γ -0.112(18) 1.389(14) −0.0146(8) 1.3177(5) 0 7/4 0.11008(1) 1.237075(10). ν η 0.704(6) 0.027(2) 0.67155(27) 0.0380(4) 1 1/4 0.629971(4) 0.036298(2). Table 2.1: The critical exponents of various universality classes with different symmetry and dimension are presented in this table.. where ξ is the correlation length. If the temperature T of the system approaches the critical temperature Tc , the correlation length diverge as. ξ ∝ |t|−ν , where t =. T −Tc Tc. (2.2). and ν is the critical exponent of the correlation length. Similarly, the. susceptibility χ and the specific heat cv diverge as the correlation length with the critical exponents γ and α, respectively. Because both the susceptibility and specific heat are the response function, their divergence indicate the thermal fluctuations in the macroscopic world. Furthermore, the divergence of the correlation length makes the system have the property of scaling invariance, whose correlation function should be a homogeneous function such as ⟨m(0)m(x)⟩ ∝ |x|−(d−2+η) ,. (2.3). where η is the anomalous dimension. Interestingly, not only the correlation length can be the homogeneous function but most of the physical observables can be, such as the free energy. If we derive the specific heat and susceptibilit from free energy, we would find the critical exponents follow some identities, such as α = 2 − dν,. (2.4). γ = ν(2 − η), which are called the hyper-scaling relations. Another interesting phenomenon is the universality, which is saying that some of the 6.

(18) physical quantities, such as critical exponents, don’t depend on the microscopic details. Theoretically, the universality can be described by the fixed point of the renormalization group theory, which only depends on the dimension and the symmetry of the system. Here, we list some critical exponents, which correspond to different universality classes, in Table.2.1 [2] [9] [12].. 2.2. Quantum phase transition. In the previous section, we know that thermal fluctuation plays an important role in the phase transition. However, people still found evidences of phase transition when the system is at zero temperature, where the thermal fluctuation vanish and the quantum fluctuation dominates. The phase transition which takes place at zero temperature is called quantum phase transition. Unlike the classical phase transition, the quantum phase transition occurs from changing the external field or increasing the pressure. Usually, a quantum thermal system, which has the spatial dimension d, can be mapped to a classical thermal system with (d+1)-dimension. The additional dimension of the classical system and the temperature of the classical system is mapped from the inverse temperature and the parameters in the Hamiltonian which can be the strength of the external field or coupling constant of the quantum system. Therefore, the second-order quantum phase transition with d spatial dimension has the critical behavior corresponding to the universality class with (d + 1)-dimension. The system we are interested in is the 2-dimensional quantum Heisenberg model with antiferromagnetic order (AFM). Tuning certain couplings from small to large, the ground state of the system will change from the AFM ordering state to the singlet-pair state. Also, between the AFM ordering state and singlet-pair state, the system encounters the secondorder quantum phase transition with O(3) and D = 3 universality class. More details of the studied system will be described in section 4.1. 7.

(19) a. L À ξ. L ≈|t|−ν. t =0. b. ξ À L. 1/L Figure 2.1: The region of finite-size scaling The crossover behavior of the observable Q in two different region due to the finite-size effect. (a) The region which has L ≫ ξ is irrelevant to finite-size effect, where Q ≈ ξ κ/ν . (b) The region which has ξ ≫ L, where Q ≈ Lκ/ν .. 2.3. Finite-size scaling hypothesis. In this work, we are interested in critical exponents for a system which undergoes a secondorder phase transition. To extract critical exponents for the system with a finite size, the finite-size scaling hypothesis plays an important role. The competition between correlation length and the linear system size can be studied in two different cases. The first situation is that when the system size L ≫ ξ, where ξ is the correlation length in the infinite system size. In this first case, the finite size effect is irrelevant and the observable Q diverge with critical exponent κ. We can derive the critical behavior of Q from correlation length such as. Q ∝ ξ κ/ν .. (2.5). From Eq.2.5, we can understand that the ξ is an charateristic length scale for the Q. The second case is that ξ ≫ L, where the finite size effect is dominant. Therefore, L now is the characteristic length scale instead, where ξ in the Eq.(2.5) can be replace by L. Q ∝ Lκ/ν . 8. (2.6).

(20) The finite-size scaling hypothesis assumes that Q in the second case can be expressed as a universal function and a scaling function such that ( ) Q(t, L) = Lκ/ν g˜ L/ξ , where the ξ is written as a function of t with t =. T −Tc . Tc. (2.7) In Eq.(2.7), the ξ can be replace. by t−ν and after that we rearrange the function to arrive at. Q(t, L) = Lκ/ν g(tL1/ν ).. (2.8). Here, while Eq.(2.8) is derived phenomenologically, it can be proved from the renormalization group theory. Finally, the correction due to small system size is written as [1]. Q(t, L) = Lκ/ν (1 + bL−ω )g(tL1/ν ),. (2.9). which will be used in the following sections.. 2.4. Quenched disorder and Harris criterion. The quenched disorder is a certain kind of disorder that is time-independent. For example, the spins sitting on a lattice with random couplings is a quenched disorder system. Because the translational symmetry vanishes in the quenched disorder system, it is hard to solve the system analytically. Therefore, there are some questions could be asked as follows. Could a phase transition be observed in a quenched disorder system? If so, could the phase transition in the quenched disorder system be second-order? What is the critical behavior of the quenched disorder system if it has a second-order phase transition? Harris proposes the Harris criterion, which said that the phase transition in a quenched disorder system has a stable critical point only if ν > 2/d, in 1974. To understand the Harris criterion, let us consider a bunch of quenched disorder systems with finite system size L, and each of the quenched disorder systems is statistically independent. Theoretically, the critical point appears when L goes to ∞. However, practically, we can do the 9.

(21) extrapolation of finite-size system to locate the divergent point Tc∗ . Collecting all Tc∗ from each system, the ∆Tc (L) is the standard deviation of the collected Tc∗ . Assuming that the |Tc∗ − Tc | is proportional to the average value of the coupling strength, where Tc is the critical point for the infinite system size, we will have the relationship between ∆Tc and L as. ∆Tc (L) ∝ (Ld )− 2 . 1. (2.10). Let’s consider a system with infinite system size. To have the stable critical point for the system, the phase transition from one phase to another phase should simultaneously occur for different region in the system. Otherwise, the phase coexitance will appear and the sharp transition will be destroy. In this case, the reasonable size of the subsystem is ξ, where ξ is the correlation length. Therefore, the inequality

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(25) ∆Tc (ξ) ≪

(26) T − T c

(27). (2.11). should satisfy, for phase transition occuring simultaneously for each subsystem. From Eq.(2.10) and (2.2), the Eq.(2.11) can be rewritten as. aξ − 2 ≪ bξ − ν , d. 1. (2.12). where the a and the b are constants. Because the ξ diverges as |T − Tc |−ν , the inequality is satisfied for T → Tc only if. 2 ν> , d. (2.13). which is as known as the Harris criterion. For quantum phase transitions, things become more complicated than the classical phase transitions. As we mentioned above, a quantum system at zero temperature can be mapped to a classical system with (d + 1)-dimension. However, only the disorder on the spatial dimension is statistically independent. The dimension in the Harris criterion is the 10.

(28) same as the spatial dimension for the quantum phase transition, but this scenario still needs to be confirmed. To summarize, testing the validity of Harris criterion for quantum phase transitions is an important subject.. 11.

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(30) Chapter 3 Stochastic Series Expansion Method The stochastic series expansion (SSE) with efficient operator-loop updating, which was developed around the year of 2000, play an important role in large scale quantum Monte Carlo simulations. In this chapter, we will review the series expansion representation and the idea of operator-loop updating. Also, a brief introduction regarding the Monte Carlo sampling with Markov processes will be presented. Before we proceed, we would like to acknowlege that the cartoon figures associated with SSE shown in the following sections are produced based on those available at [19]. In particular, the details of SSE with efficient operator-loop updating are described in [21][22].. 3.1. Series expansion of the patition function. To illustrate the SSE algorithm, let’s begin with the derivation of putting partition function into a series expansion form. For a given Hamiltonian H, the partition function is as following. Z = Tre−βH =. ∑ α. 13. ⟨α|e−βH |α⟩,. (3.1).

(31) where, the sum over α is sum over all the othogonal basis. Expanding the exponential function in the partition function into Taylor series, Eq.(3.1) can be written as. Z=. L ∑∑ (−β)n α. n!. n=0. ⟨α|H n |α⟩.. (3.2). Instead of carrying out the summation to the infinite power, it is truncated at a certain finite number L. An important trick is to collect all the Taylor series into the operator sequence SL . SL is a sequence with L components, which is the combination of I and H, where I is the identity operator. For example, SL = {I, H, I, I, · · · }. Let M be defined by ∏. M=. Si .. (3.3). Si ∈SL. Then, the partition function can be derived as. Z=. ∑ ∑ (−β)n 1 ⟨α|M |α⟩, n! CLn α S. (3.4). L. where n is the number of H in the SL . In Eq.(3.4) the denominator CnL is to reweight the operator sequence due to the over counting. To determine how large the L can make the Taylor series converge, we can evaluate the internal energy of the system. ∂lnZ ∂β 1 ∑ ∑ n (−β)n 1 =− ( ) ⟨α|M |α⟩. Z α S β n! CLn. E=−. (3.5). L. Easily speaking,. E=−. ⟨n⟩ , β. (3.6). where ⟨n⟩ is the expectation value for the number of nonidentity operator in the operator sequences. Therefore, to make the Taylor series converge L must be larger than βE. 14.

(32) 3.2. Spin- 12 AFM Heisenberg model. We move to a more specific model, which is the spin- 12 AFM Heisenberg model. The Hamiltonian of the spin- 12 AFM Heisenberg model can be written as. H=. ∑. Jij ⃗Si · ⃗Sj ,. (3.7). ⟨ij⟩. where ⟨ij⟩ is the nearest neighbor pair of spin indexes and ⃗S is the spin- 21 operator. In order to make the ground state be AFM ordering, the coupling constant Jij should be positive. Also, the system Hamiltonian can be divided into local Hamiltonian and labeled by the bond index b. Hb = Jb⃗Si(b) · ⃗Sj(b) ,. (3.8). where b ∈ {1, 2, 3, · · · , Nb } and Nb is the total bond number. To make the problem easily, we define three operators as following.. H0,b = I/Jb . H1,b =. 1 − 2Szi(b) · Szj(b) . 2. (3.9). − − + H2,b = (S+ i(b) · Sj(b) + Si(b) · Sj(b) ).. These three operators are named as identity operator, diagonal operator and off-diagonal operator, respectively. By applying the diagonal and off-diagonal operators on the local states, one finds that ⟨↑↓ |H1,b | ↑↓⟩ = 1. ⟨↓↑ |H1,b | ↓↑⟩ = 1.. (3.10). ⟨↑↓ |H2,b | ↓↑⟩ = 1. ⟨↓↑ |H2,b | ↑↓⟩ = 1. In Eq.(3.10), the | ↑↓⟩ and | ↓↑⟩ are the states of two spin in the representation of magnetization along z direction. The others combination of states, which are not listed in 15.

(33) Eq.(3.10), are zero. This implies that the eigenstates of magnetization in the z-component are good choices for |α⟩. Also, Eq.(3.8) can be derived as. Hb = −. Jb 1 (H1,b − H2,b + ), 2 2. (3.11). where the − J4b can be seen as a overall shifting of the energy in the system and it will not affect the physical properties considered here. For spin- 12 AFM Heisenberg model, Eq.(3.4) can finally be derived as L ∑ ∑ (−1)n2 β n 1 ∏ ⟨α| Z= Jb(p) Ha(p),b(p) |α⟩, 2n n! CnL α S p=1. (3.12). L. where SL is the sequence which collects the configurations of Ha,b . From Eq.(3.11), it is easy to know that n2 is the number of off-disgonal operator in SL . Note that for the p-th component of the sequence, it can be denoted as. SL (p) = [a(p), b(p)].. (3.13). Conveniently, the propagated state can be defined as ∏p−1 |α(p)⟩ =. ∥. ∏k=1 p−1 k=1. Ha(k),b(k) |α⟩ Ha(k),b(k) |α⟩ ∥. ,. (3.14). where ∥ · ∥ measures the length of a vector.. 3.3. Basic concepts of Monte Carlo sampling. In this section, we will introduce the basic concepts of Monte Carlo sampling. Starting from the original definition of magnetization, which can be written as. ⟨Mz ⟩ =. 1 Tr(Mz e−βH ). Z. (3.15). In Eq.(3.15), H and Mz are 2N × 2N matrix, where N is the number of total spins in the system. Clearly, it can not be solved in general due to the huge amount of degree of 16.

(34) freedom. However, not all the degree of freedom are important. This will be more clearly in the representation of series expansion. By deriving Eq.(3.15) from Eq.(3.12), the magnetization can be obtained from the sum over all configurations α and SL , as L ∏ 1 ∑ ∑ (−1)n2 β n Mz [α] ⟨Mz ⟩ = ⟨α| Jb(p) Ha(p),b(p) |α⟩. Z α S 2n n! CnL p=1. (3.16). L. In Eq.(3.16), |α⟩ is chosen to be the eigenstate of the magnetization in z direction. Also, the Mz [α] is the shorthand of the ⟨α|Mz |α⟩. To simplify the problem, we can collect all other things inside the summation, such that ∑ ⟨Mz ⟩ =. Mz [α]W[α, SL ] , α,SL W[α, SL ]. ∑. α,SL. (3.17). where. W[α, SL ] =. L ∏ β n (L − n)! Jb(p) Ha(p),b(p) |α⟩. ⟨α| 2n L! p=1. (3.18). However, the degree of freedom of α and SL are still very large. The summation over all possible configurations is not practical. Here comes the idea of important sampling in Monte Carlo techniques. Instead of considering all the configurations, we sample each of them with probability pµ , where µ is one of the configurations in the system. In the representation of the series expansion, µ = [α, SL ]. If we choose that. pµ ∝ W µ ,. (3.19). the magnetization in Eq.(3.17) is just the expectation value during the sampling, which can be denoted as. ⟨Mz ⟩ = ⟨Mz;µ ⟩µ .. (3.20). To sample configuration µ with probability pµ , the Markov process is the kernel of the Monte Carlo techniques. Markov processes can be seen as the ”random walk” in the 17.

(35) configuration space. The transition probability P (µ → ν) from configuration µ to ν only depends on the current state µ and the target state ν. For using Markov processes to do the sampling, the conditions of ergodicity and detail balance play an important role. The condition of ergodicity is saying that Markov processes can travel over all the state µ for pµ ̸= 0. Otherwise, Eq.(3.19) should never hold. If the system has ergodicity, we can imagine that the the system will go to the equilibrium state for a large enough number of Markov processes steps. For the system in the equilibrium state, the probability of finding states µ and ν should follow the detail balance condition, such that. qµ P (µ → ν) = qν P (ν → µ).. (3.21). In Eq.(3.21), qµ and qν are the probability of finding states µ and ν, respectively, when system is in equilibrium. It should be distinguished from the probability pµ , which we expect to be the probability for sampling the state µ. To make qµ = pµ and qν = pν , we should find a good transition probability that satisfy. P (µ → ν) pν = . P (ν → µ) pµ. (3.22). During the Markov processes, there are many candidates of ν from the current state µ. The procedure of Markov processes is the following. First, drop a dice to determine which candidate we should pick. Then, drop a dice again to determine if we should go to the candidate state or stay at the current state. Therefore, the transition probability can be separate into selection probability g(µ → ν) and acceptance probability A(µ → ν), such as. P (µ → ν) = g(µ → ν)A(µ → ν).. (3.23). From Eq.(3.22) from Eq.(3.23), the detail balance condition will becomes. g(µ → ν)A(µ → ν) pν = . g(µ → ν)A(ν → µ) pµ 18. (3.24).

(36) Most of the time, the selection probability is fixed. Therefore, we only have the degree of freedom for choosing the acceptance probability. Here, we would like to introduce the Metropolis choice for the acceptance probability, which can be written as [ p g(ν → µ) ] ν A(µ → ν) = min 1, . pµ g(µ → ν). (3.25). It is easy to show that this choice would satisfy the detail balance condition.. 3.4. Algorithm of stochastic series expansion. In this section, we will introduce how to combine the series expansion with Monte Carlo sampling in an efficient way, which is called the stochastic series expansion (SSE) [22]. The whole procedure of SSE can be separated into five parts. They are the diagonal operators updating, the off-diagonal operators updating, the spin flipping, adjusting the cut-off, and measuring observables. Diagonal operators update- Given [α, SL ] with non-zero weight, there is a possibility to replace the indentity operator in the operator sequence by the diagonal operator at bth bond. This procedure can be denoted as SL (p) = [0, 0] → SL (p) = [1, b]. For inserting the diagonal operator at bth bond, one should check that H1,b |α(p)⟩ ̸= 0. From Eq.(3.10), this mean that σi(b) (p) ̸= σj(b) (p), where σi (p) ∈ {1, −1} is the spin that the state with on the ith site in the pth propagated state. Also, there is a possibility to remove diagonal operator at bth bond, which can be denoted as SL (p) = [1, b] → SL (p) = [0, 0]. The acceptance probability A of inserting and removing the diagonal operator can be derived from Eq.(3.25) as [ βJb Nb ] A([0, 0] → [1, b]) = min 1, , 2(L − n) [ 2(L − n + 1) ] . A([1, b] → [0, 0]) = min 1, βJb Nb 19. (3.26).

(37) In Eq.(3.26), the selection probability for inserting or removing a diagonal operator are g([0, 0] → [1, b]) = 1/Nb or. (3.27). g([1, b] → [0, 0]) = 1, respectively. For inserting a diagonal operator, there are Nb choices. However, there is only one way to remove a diagonal operator. Finally, to complete a diagonal operator updating, we should do the same thing from p = 1 to p = L for the operator sequence. Off-diagonal operators update- For the off-diagonal operator updating, we can not do the same thing as that done in the diagonal operator. The main problem is that the offdiagonal operator will flip the spin during the propagating state. To avoid the zero-weight, there is an additional condition must statisfy, which can be written as. |α(L)⟩ = |α(0)⟩.. (3.28). Therefore, we need an algorithm, which can generate off-diagonal operators and doesn’t violate above condition. Fortunately, the operators loop-update method can solve this problem in an efficient way. To illustrate the operators loop-update, we imagine the propagate states as an additional dimension τ , such as that shown in Figure.3.1. Then, we pick up any leg on an operator and draw a line along the τ direction until it hits another operator. When the line hits the new operator, there are four ways to hit and get out the new operator as demonstrated in Figure.3.2 and keep going along the τ direction. Finally, the line will form a closed loop as that depicted in Figure.3.3a, which is called operator-loop. After constructing operator-loop, we need to determine if the loop can be flipped or not with equal probability. If the loop is flipped, the diagonal operator will change to the off-diagonal operator and vice versa. However, the operator will not change if it is traversed by a loop twice. Figure.3.3 shows an example of flipping the loop. When all the possible operator-loop being constructed and flipped (or not being flipped), the operator loop-update part is finished. Spin flipping- Spin flipping is relative simple part in the SSE algorithm. One can easily 20.

(38) Figure 3.1: The external dimension of propagated states The propagated states can be imagined as an external dimension for the system. The x direction represent the spatial direction and the τ direction represent the temporal direction. Black points and white points are down states and up states for spins, respectively. Also black and white blocks are off-diagonal operators and diagonal operators, respectively.. Figure 3.2: illustrate loop-update There are four possible ways for the loop comes in and goes out. The red arrow shows the direction when the loop encounters the operator.. 21.

(39) (a). (b) Figure 3.3: example of loop-update. If the loop is flipped, the operator, which is traversed by the loop once, will flipped from balck to white or from white to black. Also, all spins, which traversed by loop, will flip too. Therefore, we should flip the spin on |α⟩ if the loop crosses the boundary.. 22.

(40) see that if the spin is not traversed by any loop, it can be free to be spin-up-state or spindown-state without violating the condition in Eq.(3.28). Because the spin-up-state and the spin-down-state have the same weight, it can be each with the same probability. Adjusting the cut-off - In the calculations, one has to make sure that the cut-off L is larger than the number of nonidentity operators nb in the operator sequences SL . In practice, one usually keeps the ratio between L and nb to be a fixed number, like 1.3, in the simulations. This is typically done in the thermalization by enlarging or shortening SL after each complete Monte Carlo update (defined later). One can carry out such a procedure for each complete Monte Carlo update after the thermalization as well. Measuring Observable- After the thermalization, one Monte Carlo sweep consists of diagonal operator updating, off-diagonal operator updating, and spin flipping. Because the SSE algorithm is a cluster updating algorithm, the correlation between each Monte Carlo sweep is small. Therefore, it is efficient to do the measurements for each Monte Carlo sweep. Here we would like to introduce how to measure the staggered magnetization in the SSE algorithm. The staggered magnetization is the order parameter for the system in AFM order. In two dimensions, it can be written as. ⟨mzs ⟩. N 1 ∑ (−1)xi +yi Szi ⟩, =⟨ N i=1. (3.29). where xi and yi are the label for the i-th spin in the x direction and in the y direction. Because |α⟩ is the eigenstate of the staggered magnetization, it is easy to do the measurement by average each mzs [α] during the Monte Carlo sampling, such as. ⟨mzs ⟩. N 1 ∑ (−1)xi +yi Siz [α]⟩[α,SL ] . =⟨ N i=1. (3.30). For each propagated state |α(p)⟩, it has the same weight as the original state |α⟩ because of the periodic boundary condition in τ direction. Therefore, the staggered magnetization 23.

(41) can be more accurate by average over all the propagated states, which can be denoted as. ⟨mzs ⟩. L N 1∑ 1 ∑ =⟨ (−1)xi +yi Siz [α(p)]⟩[α,SL ] . L p=1 N i=1. (3.31). Similar and useful observables are ⟨(mzs )2 ⟩ and ⟨(mzs )4 ⟩, which can be derived in the same way. The second moment and the fourth moment of the staggered magnetization can be used to distinguish between the order and disorder phases. In the next chapter, we will have more detail in analyzing the Monte Carlo data.. 24.

(42) Chapter 4 Simulation and Data Analysis. 4.1. The two-dimensional spin- 12 Heisenberg with quenched disorder. For (two-dimension) square lattice with nearest neighbor interaction, the ground state of the spin- 12 Heisenberg model with uniform interaction J is in the AFM ordering. However, if certain pairs of spins has a much larger interaction J ′ and are arrange in some special patterns, the ground state for the system may attempt to be the state consists of singletpairs, which destroys the AFM ordering. People found that this system encounters the second order phase transition by tuning the interaction ratio g =. J′ . J. In our works, spins. pair up in two ways, one is the herringbone-dimer and the other is the plaquette-dimer, such as shown in Figure.4.1. The Hamiltonian for the model can be written as. H=. ∑. J ⃗Si · ⃗Sj +. ∑. Ji′′ j ′ ⃗Si′ · ⃗Sj ′ ,. (4.1). ⟨i′ j ′ ⟩. ⟨ij⟩. where the ⟨ij⟩ collects all spin indexes connected by thin line and the ⟨i′ j ′ ⟩ collects all spin indexes connected by thick line in Figure.4.1. The quenched disorder is placeded into 25.

(43) Figure 4.1: Two different pairing patterns This figure show two different types of pairing pattern. The left one is pairing in the herringbone pattern. The right one is pairing in the plaquette pattern.. Ji′′ j ′ , where. Ji′′ j ′ = J + J(g − 1)(1 ± p).. (4.2). In Eq.(4.2), the minus sign and the plus sign is in the same probability and p ∈ [0, 1] is the measurement of the randomness. For p = 0, then J ′ = gJ and we go back to the clean model. For the clean herringbone-dimer model, the critical point is founded at gc ∼ 2.4982 with the critical exponent of the correlation length ν = 0.7112 [10]. Also, for the clean plaquette-dimer model, the critical point is founded at gc = 1.8230 with the critical exponent of the correlation length ν = 0.7114 [23]. We can easily see that the gc is not universal but the ν is universal and follow the O(3) universality class. Also, in order to check the Harris criterion, we will probe into the critical exponents of the correlation length for systems with quenched disorder in the following sections.. 4.2. Observables. To extract the critical exponent and the location of the critical point, we should apply the finite-size scaling on relevant observables. Generally, we treat the location of the critical 26.

(44) point and the critical exponent of the correlation length as free parameters during the fitting in the finite-size scaling ansatz, i.e. Eq.(2.9). One of the useful observables is the spinstiffness ρs for system in the ordering phase, which satisfies. ρs ∝ (−t)ν(d+z−2) , where t =. g−gc , d is the spatial dimension and z gc. (4.3). is the dynamical exponent. Combining the. Eq.(4.3) and the Eq.(2.9), the finite-size scaling formula of spin-stiffness can be derived as. ρs Lz = (1 + b0 L−ω )f (tL ν ), 1. (4.4). where ω is the confluent exponents. Although z = 1 for clean model, z may not be 1 when the quenched disorder is presented. In this way, we also need to determine z and make the problem become complicated. Another way to extract ν, without knowing z, is using the Binder ratios as the observables. The first Binder ratio Q1 and the second Binder ratio Q2 can be written as 2. Q1 =. ⟨|mzs |⟩. (4.5). ⟨(mzs )2 ⟩. and 2. Q2 =. ⟨(mzs )2 ⟩. ⟨(mzs )4 ⟩. ,. (4.6). where mzs is the staggered magnetization which is defined in Eq.(3.29) and ⟨·⟩ is the disorder average. The finite-size scaling formulas of Q1 and Q2 can be derived as Q1 = (1 + b1 L−ω1 )f1 (tL ν ), 1. (4.7) −ω2. Q2 = (1 + b2 L. 1 ν. )f2 (tL ).. In Eq.(4.7), z is absent, which can help us determine the numerical value of ν without the effort of calculating z first. 27.

(45) 0.70. 0.44. 0.69. 0.42. 0.67. Q2. Q1. 0.68. 0.38. p=0.1, L=48. g=2.50 p=0.2, L=48. g=2.53 p=0.3, L=48. g=2.57 p=0.4, L=48. g=2.65 p=0.5, L=48. g=2.73 p=0.6, L=48. g=2.90 p=0.9, L=48. g=4.50. 0.66 0.65 0.64 100. 101. beta. 0.40 p=0.1, L=48. g=2.50 p=0.2, L=48. g=2.53 p=0.3, L=48. g=2.57 p=0.4, L=48. g=2.65 p=0.5, L=48. g=2.73 p=0.6, L=48. g=2.90 p=0.9, L=48. g=4.50. 0.36 0.34. 102. 100. 101. beta. 102. Figure 4.2: Testing β-doubling scheme The trial simulations are for testing the convergence in β which allow us to extract the ground state value of the Q1 and Q2 . For each p and g, several hundred randomness realizations are used for disorder average. And, for each realization, we use β-doubling scheme to speed up reaching the equilibrium and obtaining the ground state properties. The linear size of the system is 48, which is the largest system size in this works. Also, g is selected to be around the critical point of the corresponding p.. To measure the ground state values of Q1 and Q2 from the SSE algorithm, two things should be checked. The first one is to check if the Monte Carlo simulations go to equilibrium and the detailed balance condition is satisfied. Therefore, before measuring the observables during the Monte Carlo sampling, the thermalization procudure is needed, which make the system approach thermal equilibrium. The second one is if the temperature is low enough to have the same properties as the ground state. An efficient way called β-doubling scheme can check these two conditions at the same time [20]. Here, we do some trial simulations and found several hundred Monte Carlo sweeps is a suitable number for each individual thermalization during the β-doubling procedure. Also, Q1 and Q2 have the convergent results for system with β = 512, see Figure.4.2 for the detail. Also, we apply the jackknife method to estimate the statistical error from disorder average.. To examine if the strength of the disorder affects ν, we collect the ground state values of the Q1 and Q2 for p in 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 0.95 around their critical points with different linear size as shown in Figure.4.3. 28.

(46) 0.696 0.694 0.692. 0.440. 0.690. 0.425 2.46. 2.47. 2.48. 2.49. 2.50 g. 2.51. 2.52. 2.53. 2.54. 0.70 0.69 0.68. 2.49. 2.50 g. 2.51. 2.52. 2.53. 0.42 0.40. 0.66. 0.38. 2.54. L=12 L=16 L=20 L=24 L=28 L=32 L=36 L=40 L=44 L=48. 0.44. 0.36. 0.65 2.4. 2.5. 2.6. g. 2.7. 2.8. 2.9. 0.71. 2.4. L=12 L=16 L=20 L=24 L=28 L=32 L=36 L=40 L=44 L=48. 0.69 0.68 0.67. 0.36 g. 5.0. 5.5. 2.7. 2.8. 2.9. L=12 L=16 L=20 L=24 L=28 L=32 L=36 L=40 L=44 L=48. 0.40. 0.65 4.5. g. 0.42. 0.38. 4.0. 2.6. 0.44. 0.66. 3.5. 2.5. 0.46. Q2. 0.70. Q1. 2.48. 0.46. 0.67. 0.64. 2.47. 0.48. Q2. 0.71. 2.46. 0.50. L=12 L=16 L=20 L=24 L=28 L=32 L=36 L=40 L=44 L=48. 0.72. Q1. 0.435 0.430. 0.688 0.686. L=12 L=16 L=20 L=24 L=28 L=32 L=36 L=40 L=44 L=48. 0.445. Q2. 0.698. Q1. 0.450. L=12 L=16 L=20 L=24 L=28 L=32 L=36 L=40 L=44 L=48. 0.700. 6.0. 3.5. 4.0. 4.5. g. 5.0. 5.5. Figure 4.3: The ground state value of Q1 and Q2 Collect the ground state values of the Q1 (left) and Q2 (right) around the critical points and doing the disorder average over several hundred realizations. From top to bottom, they are p = 0.0, p = 0.5 and p = 0.9, respectively. For each p, there are several different L and g.. 29. 6.0.

(47) 4.3. The finite-size scaling analysis. In this section we will show how to extract ν and gc from the finite-size scaling formulas, and to estimate the statistical uncertainties as well as the systematic errors. Two inference frameworks, which are the frequentist inference and Baysian inference, 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 1. the Taylor expansion of the fi (tL ν ) in Eq.(4.7) for the fitting, such as [ ] 1 1 1 y = (1 + bL−ω ) a0 + a1 (tL ν ) + a2 (tL ν )2 + a3 (tL ν )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 gc . In order to estimate parameters and their statistical uncertainties in Eq.(4.8), there are two methods in different inference frameworks, 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. ∑ ( yi (θ) − xi )2. θ. i. σi. ,. (4.9). where xi are the given data having the corresponding gi and Li , and σi are the statistical errors of the data. In addition, yi (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 distribution 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 30.

(48) 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 gc and 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 frameworks, the posterior probability distribution will be used to evaluate the statistical uncertainties 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 yi (θ) and σi . Therefore, 31.

(49) resampling (Qi , Lmin , n-th order) (Q1 , 8, 3) (Q1 , 8, 4) (Q1 , 8, 5) (Q1 , 12, 3) (Q1 , 12, 4) (Q1 , 12, 5) (Q1 , 16, 3) (Q1 , 16, 4) (Q1 , 16, 5) (Q2 , 8, 3) (Q2 , 8, 4) (Q2 , 8, 5) (Q2 , 12, 3) (Q2 , 12, 4) (Q2 , 12, 5) (Q2 , 16, 3) (Q2 , 16, 4) (Q2 , 16, 5). gc 2.7380(15) 2.7389(16) 2.7399(17) 2.7380(16) 2.7387(16) 2.7398(17) 2.7382(15) 2.7386(15) 2.7395(17) 2.7394(16) 2.7401(16) 2.7405(16) 2.7396(17) 2.7403(17) 2.7411(19) 2.7399(16) 2.7403(17) 2.7410(18). posterior. ν 0.811(6) 0.812(6) 0.805(6) 0.804(6) 0.806(6) 0.802(6) 0.802(7) 0.803(7) 0.804(7) 0.793(6) 0.793(6) 0.790(6) 0.790(6) 0.791(6) 0.788(6) 0.790(7) 0.790(7) 0.792(7). gc 2.7378(11) 2.7384(11) 2.7392(12) 2.7377(11) 2.7383(12) 2.7388(14) 2.7378(10) 2.7381(10) 2.7385(11) 2.7391(11) 2.7396(11) 2.7404(12) 2.7388(12) 2.7397(13) 2.7405(14) 2.7390(10) 2.7394(11) 2.7398(12). 0.812(4) 0.812(4) 0.805(4) 0.807(5) 0.805(5) 0.803(5) 0.805(5) 0.804(5) 0.804(5) 0.794(4) 0.795(4) 0.791(4) 0.793(5) 0.793(5) 0.790(5) 0.792(5) 0.792(5) 0.793(5). Table 4.1: Results of gc and ν (p = 0.5) from the resampling and posterior. the likelihood can be easily written down as. L(x|θ) =. ∏ i. [ ( y (θ) − x )2 ] 1 i i √ exp − . σi 2πσi. (4.11). 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 gc and ν, we shift the range of the prior to be narrower and see no significant change of gc and ν. However, it is not easy to construct the distribution of the posterior due to lots of dimension in the parameter space. Conventionaly, Markov Chain Monte Carlo (MCMC), which is a sampling technique in continuous parameter space, can solve this problem efficiently [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. 32.

(50) Figure 4.5: The probability distribution from the posterior In Bayesian frameworks, the posterior probability distribution show the statistical 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 Q1 with p = 0.5 (left) and Q2 with p = 0.9 (right), which is the same as that in Figure.4.3. The gc and 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 gc and 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 Q1 and Q2 behave 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 33.

(51) resampling (Qi , Lmin , n-th order) (Q1 , 8, 3) (Q1 , 8, 4) (Q1 , 8, 5) (Q1 , 12, 3) (Q1 , 12, 4) (Q1 , 12, 5) (Q1 , 16, 3) (Q1 , 16, 4) (Q1 , 16, 5) (Q2 , 8, 3) (Q2 , 8, 4) (Q2 , 8, 5) (Q2 , 12, 3) (Q2 , 12, 4) (Q2 , 12, 5) (Q2 , 16, 3) (Q2 , 16, 4) (Q2 , 16, 5). gc 4.884(30) 4.873(31) 4.869(31) 4.861(35) 4.854(35) 4.858(36) 4.842(35) 4.851(36) 4.854(38) 4.892(37) 4.876(38) 4.873(39) 4.907(46) 4.887(45) 4.890(47) 4.894(42) 4.891(44) 4.890(46). posterior. ν 1.132(19) 1.134(19) 1.129(19) 1.084(20) 1.083(20) 1.083(20) 1.059(21) 1.055(20) 1.055(20) 1.086(18) 1.088(18) 1.085(18) 1.062(19) 1.059(18) 1.059(19) 1.046(19) 1.039(19) 1.038(19). gc 4.883(25) 4.868(23) 4.870(22) 4.858(25) 4.858(26) 4.869(35) 4.838(29) 4.849(29) 4.861(21) 4.900(26) 4.883(28) 4.885(27) 4.907(31) 4.893(32) 4.898(31) 4.897(33) 4.902(33) 4.899(37). ν 1.127(14) 1.128(14) 1.121(14) 1.084(14) 1.082(15) 1.088(16) 1.061(15) 1.054(16) 1.061(16) 1.081(12) 1.083(13) 1.082(13) 1.062(13) 1.057(14) 1.059(14) 1.047(14) 1.041(14) 1.038(14). Table 4.2: Results of gc and 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 determinaions of gc and 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 gc for 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 deviations from the bootstrapping method times Np , where Np 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 because an alternative method of analysis than that of [17] is used in obtaining the outcomes demonstrating in Table. 4.3.. 34.

(52) p 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95. ν 0.702(9) 0.702(6) 0.724(6) 0.745(10) 0.776(11) 0.804(12) 0.841(13) 0.890(15) 0.940(19) 1.04(4) 1.06(5). gc 2.4981(2) 2.5056(2) 2.5308(4) 2.5732(7) 2.6383(11) 2.7397(13) 2.8939(20) 3.1466(38) 3.605(15) 4.796(76) 6.552(41). Table 4.3: The final results of the ν and gc for all the considered p.. 35.

(53) 36.

(54) 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. 37.

(55) 38.

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