Topics in Classical and Quantum Phase Transitions

TOPICS IN CLASSICAL AND QUANTUM PHASE TRANSITIONS

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

Kohjiro Kobayashi, B.S., M.S.

*****

The Ohio State University

2008

Dissertation Committee: Approved by

Professor Trivedi Nandini, Adviser Professor David G. Stroud Adviser Professor Ralf A. Bundschuh Graduate Program in Professor Fengyuan Yang Physics ABSTRACT

Classical and quantum phase transitions for several models are considered. First, a network of parallel superconducting wires weakly coupled together by the proximity eﬀect described by the Ginzburg-Landau theory is studied by mapping onto

N-distinguishable two dimensional quantum-mechanics problem with a perpendicular imaginary magnetic ﬁeld. Using a mean-ﬁeld approximation, the critical temperature for onset of interwire phase coherence is obtained for a given coupling. Second, the thermodynamic properties of a collection of N small Josephson junctions coupled to a single-mode resonant electromagnetic cavity is considered employing these models:

(a) the model which includes all the quantum-mechanical levels of the junction, (b) the Dicke model with only the two lowest energy levels per junction, and (c) a modiﬁed

Dicke model which contains an additional XY-like coupling between the junctions. In all cases, within a mean-field approximation, for an N-independent junction-cavity coupling, there is a critical junction number N above which there is a continuous transition from incoherence to coherence with decreasing T with a non-Bose distribution for the cavity photon occupation numbers. The third problem investigated is a disorder driven quantum phase transition in a class of transition metal oxides described by a single orbital Hubbard model at half filling. This model contains the interplay of disorder and transitions between phases with differing conducting and

ii magnetic properties. The frequency and temperature dependent conductivity is cal-

culated to characterize the transport properties and the corresponding behavior of the

spin susceptibility is calculated to identify the magnetic character of the phases. The

dc conductivity vanishes at low disorder in the Mott phase as well as at high disorder

in the highly localized phase. However, rather surprisingly, it shows an enhancement

at intermediate disorder strengths in a ”metallic” regime. The spin susceptibility

shows a Curie-1/T dependence in the ”metallic” regime because of the presence of local moments.

iii To my family

iv ACKNOWLEDGMENTS

I would like to express my greatest thank to my advisor Professor Nandini Trivedi,

for her grateful advice and encouragement. It has been an honor and privilege to

work with her. Without her, my research project reported here would not have been

possible.

Another greatest thank goes to Professor David Stroud. I had leaned many important lessons from his vast understanding, knowledge, and wisdom working with him. This work would not have been possible without his guidance and patience.

I would like to extend my thank to Professor Ralf Bundschuh and Professor

Fengyuan Yang for serving in my dissertation and candidacy exam. Moreover, I would like to thank Professor Sabra J. Webber for serving in my ﬁnal oral exam.

I would also like to thank my fellow group members, all of my physics classmates and friends, and many people at the Ohio State University who are related to me for much support and help. I have truly enjoyed working with them and have learned not only physics but also many important things for my life during my studies here.

Finally, I would like to thank funding and much support from physics depart- ment at the Ohio State University and computational support provided by the Ohio

Supercomputer Center.

v VITA

March 26, 1976 ...... Born in Fukaya, Japan

1999 ...... B.S. Physics (Yokohama City Univer- sity) 2006 ...... M.S. Physics (The Ohio State Univer- sity) 2001-present ...... Graduate Teaching Assistant, The Ohio State University.

FIELDS OF STUDY

Major Field: Physics

vi TABLE OF CONTENTS

Page

Abstract...... ii

Dedication...... iv

Acknowledgments...... v

Vita ...... vi

ListofTables...... x

ListofFigures ...... xi

Chapters:

1. Introduction...... 1

2. Brief Review of Classical and Quantum Phase Transitions ...... 5

2.1 ModeloftheJJA ...... 5 2.2 Classical Thermal Phase Transition ...... 9 2.3 QuantumPhaseTransition ...... 11

3. Theory of Fluctuations in a Network of Parallel Superconducting Wires . 16

3.1 Introduction ...... 16 3.2 Formalism ...... 18 3.2.1 Mapping to a quantum mechanics problem for interacting superconducting wires when B~ is perpendicular to the wires 18 3.2.2 Probability distribution of the order parameter ...... 21 3.2.3 Phase only model and mean-ﬁeld approximation ...... 23

vii 3.3 ResultsandDiscussion ...... 25 3.3.1 Nomagneticﬁeld...... 25 3.3.2 Perpendicular magnetic ﬁeld ...... 28 3.4 Summary ...... 32

4. Statistical Mechanics of Small Josephson Junctions Coupled to a Single- ModeResonantCavity ...... 34

4.1 Introduction ...... 35 4.2 Many Josephson Junctions Interacting with a Single-mode Cavity 37 4.2.1 ModelHamiltonian...... 37 4.2.2 Mean-fieldapproximation...... 39 4.3 Dicke Model and Generalized Dicke Model ...... 43 4.3.1 ModelHamiltonians ...... 43 4.3.2 Statistical mechanics of Dicke model using Glauber coherent stateexpansion...... 45 4.3.3 Statistical mechanics of modified Dicke model using a mean- fieldapproximation...... 48 4.4 NumericalResults ...... 51 4.5 ThermodynamicLimit ...... 60 4.6 Discussion ...... 62

5. Conductivity and Spin Susceptibility for the Disordered 2D Hubbard Model 65

5.1 Introduction ...... 65 5.2 Model...... 68 5.3 Method ...... 70 5.4 Review for Numerical Simulation of the 2D Disordered Hubbard Model...... 70 5.5 Results ...... 74 5.5.1 Frequency dependent conductivity at T =0...... 75 5.5.2 DCconductivity ...... 78 5.5.3 Frequency dependence of the conductivity for ﬁnite temperature ...... 80 5.5.4 Sumrulefortheconductivity ...... 82 5.5.5 Uniform static spin susceptibility ...... 84 5.6 Conclusion ...... 86

Appendices:

viii A. Comparison between Analytical Solutions and Numerical Results from the inhomogeneous Hartree-Fock Method in Several Limiting Cases of 1D DisorderedHubbardModel ...... 88

A.1 Tight-binding model, (U = 0 and Vi =0)...... 88 A.2 The disordered Hubbard model with t =0...... 89 A.3 Non-disordered case at a half ﬁlling of the Hubbard model with t =0 91 A.4 Hubbard model at half ﬁlling (No disorder) ...... 92 A.4.1 Totalenergy ...... 92 A.4.2 LocalMoment ...... 92 A.4.3 GapfromMFT...... 93

B. Formalism of the Expectation Values on the Numerical Simulation . . . 96

B.1 Spin degree of a freedom: Magnetic Properties ...... 96 B.1.1 Localmoments ...... 96 B.1.2 Spin-spin correlation function ...... 97 B.1.3 Magneticstructurefactor ...... 98 B.1.4 Spin susceptibility for ω =0...... 98 B.1.5 Spin susceptibility for ω =0...... 100 6 C. Probability Distribution of the Exchange Coupling Constant,J...... 102

C.1 The probability distribution of coupling constants based on the per- turbationtheory ...... 104 C.2 The probability distribution of coupling constants based on exact diagonalization ...... 105 C.2.1 Case of two nearest neighbor sites ...... 105 C.2.2 Case:(b) ...... 107 C.2.3 Case:(c) ...... 108 C.2.4 Case:(d) ...... 109 C.2.5 Case:(e) ...... 110

Bibliography ...... 115

ix LIST OF TABLES

Table Page

3.1 Correspondence on the mapping between Q.M. and S.C. (in each case, the left-hand variable corresponds to the parameters on the quantum mechanics problem and the right hand variable corresponds to the parameters on the superconductor wires) ...... 21

x LIST OF FIGURES

Figure Page

1.1 Comparison between a quantum phase transition and a classical phase transition using a phase diagram of square arrays (solid squares) and triangular arrays (solid triangles) obtained from experimental data in zero magnetic ﬁeld, f = 0 (from Ref. [3]), where the vertical axis is temperature and horizontal axis, x is a coupling ratio. The solid line and the dotted line are guides to the eye. The resistance per junction by applying a small transport current is measured to characterize these phases. The resistance of the superconducting phase is zero, that of 3 the insulating phase is a ﬁnite value with R > 10 R0 where R0 is the normal state resistance, and that of the resistive/conductive phase is a value with R 103R . The quantum phase transition corresponds to ≤ 0 the red line on the x-axis at zero temperature and the classical phase transition corresponds to the blue line or green line, which are parallel tothey-axis...... 3

2.1 A model of single Josephson junction connected by the voltage.... 6

2.2 A model of 2D square Josephson junction arrays...... 9

2.3 Phase states as a function of a temperature on classical XY model. . 10

2.4 Phase states as a function of K = U on Josephson junction arrays EJ model...... q 13

2.5 Experimental data of the resistance per junction as a function of temperature for six diﬀerent coupling strengths. The dashed horizontal line is 16.4kΩ, which corresponds to the universal conductivity that is independent of the microscopic details of the system. (From Ref. [3]). 13

xi 2.6 A phase diagram of square arrays (solid squares) and triangular arrays (solid triangles) obtained from experimental data in zero magnetic ﬁeld, f = 0, where the vertical axis is temperature and horizontal axis, x is a coupling ratio. The solid line and the dotted line are guides to the eye. The resistance per junction by applying a small transport current is measured to characterize these phases. The resistance of the superconducting phase is zero, that of the insulating phase is a 3 ﬁnite value with R > 10 R0 where R0 is the normal state resistance, and that of the resistive/conductive phase is a value with R 103R . ≤ 0 (fromRef.[3])...... 14

3.1 Temperature dependence of the phase-ordering for inﬁnite long wires. 27

0 3.2 Phase diagram of tc = Tc/Tc as a function of α for several values of length of the wires, 100ξ , 1000ξ , 2000ξ , 5000ξ , and where ξ = 42A.˚ 29 0 0 0 0 ∞ 0

0 3.3 Phase diagram of tc = Tc/Tc as a function of α for several values of magnetic ﬁeld strength, f = 0.2, f = 0.4, f = 0.6, and f = 0.8 for inﬁnitylengthwires...... 31

3.4 Temperature dependence of the critical ﬁeld strength, fc for α = 0.2, 0.4, 0.6, 0.8,and1forinﬁnitylongwires...... 32

4.1 Coherence order parameter λ(N, t), plotted as a function of the number of junctions N, forn ¯ =0 and 0.5 at values of the scaled temperature t =0and0.12, as indicated in the legend, using J =0.2U, ~ω =0.15U, and g =0.1...... 52

4.2 Coherence order parameter λ(¯n, t), at several values of the scaled temperature t = 0, 0.12, 0.14 and 0.16, as indicated in the legend, using J =0.2U, ~ω =0.15U, g =0.1, and N =110...... 53

4.3 Temperature dependence of coherence order parameter λ(t) forn ¯ = 0, 0.1, 0.2, 0.3, 0.4 and 0.5, using J = 0.2U, ~ω = 0.15U, g = 0.1, and N =110...... 54

4.4 Average Cooper pair number diﬀerence n at t = k T/U =0, 0.12, 0.16, 0.2, h i B and 0.24. In all cases, J = 0.2U, ~ω = 0.15U, g = 0.1, and N = 110 where tc(¯n =0)=0.124 and tc(¯n =0.5)=0.181...... 55

xii 4.5 Photon number distribution P (n) for the Josephson junction model at various temperatures: t = 0, 0.12, 0.13, and 0.14. In all cases, J = 0.2U, ~ω =0.15U, g =0.1, N = 110, andn ¯ = 0. For these parameters, tc = 0.131. P (n) represents the probability that there are exactly n photonsinthecavitymode...... 57

4.6 Comparison between the predictions of the mean-field approximation, the Dicke model, and the modified Dicke model for the critical temperature [part (a)] and the average photon number, a†a (t) for N = 70 h i [part (b)], atn ¯ = 0.5. For the Josephson junction model, we use the parameters J = 0.2U, ~ω = 0.15U, and g = 0.1; these lead to m m Nc (0) = 64.0 and tc = 0.0681. The corresponding critical numbers Dicke Dicke for the Dicke model are Nc (0) = 37.1 and tc = 0.168. For the modified Dicke model, we show plots with Ω~ω/ ξ 2 =0.2, 0.4, 0.6 and | | 0.8...... 59

5.1 Systematicphasediagramofcuprate...... 67

5.2 The above figure (a) is density of states as a function of E = ǫ µ where − µ is the chemical potential and the below figure (b) is spectral gap from the density of states (blue line) and the AF order parameter (red line) as a function of disorder, V. U =4t at half-filling and the system size is 32 32. The data is averaged over 10 disorder realizations. (From × Ref.[70])...... 72

5.3 Plots of the density, ψ (~r) 2 at the Fermi energy. The density is σ | nσ | high at the dark gray area. U = 4t at half-ﬁlling and the system size P is 32 32. The data is averaged over 20 disorder realizations. (From × Ref.[70])...... 73

5.4 Plots of the density, ψ (~r) 2 at the Fermi energy. The density is σ | n | high at the dark gray area. U = 4t at half ﬁlling and the system size P is 32 32. The data is averaged over 20 disorder realizations. (From × Ref.[70])...... 74

xiii 5.5 The frequency dependence of the conductivity, σ(ω), for Vd =1, 2, 3, 4, and 5. The system size is 32 32, U = 4t, and the data is averaged × over 20 disorder realizations. The low frequency region of σ(ω) for Vd = 3t is obtained by a third order polynomial ﬁt of the current- current correlation function, Λ = ωσ. For Vd = 1t there is clearly a ﬁnite Mott gap at the low frequency. Thus, σ(ω) is suppressed at low frequency and there is a peak in the frequency when ω is around the Mott gap. With increasing disorder the spectral weight above the Mott gap is transfered to lower spectral region and the peak of σ(ω) moves to the low frequency. At V 2.2t, the gap in the conductivity d ≃ closes because energy scale of disorder is comparable with the gap. Near Vd = 3t, the conductivity near ω = 0 is most enhanced and the frequency where the conductivity has the maximum value is lowest. For Vd = 4t and Vd =5t, the dc conductivity, σ(0), is zero due to the Anderson insulating regime. In Region 2, the peak location of σ(ω) moves to the low frequency as the Mott gap closes but with increasing disorder the peak location again moves to the high frequency in Region 3 76

5.6 (a) The gap in Re[σ(ω)] as a function of disorder, Vd. (b) ωpeak as a function of disorder. The system size is 32 32, U =4t, and the data × is averaged over 20 disorder realizations. ωpeak is obtained by ﬁtting Re[σ(ω)] in Fig. 5.5 near the peak using Lorentzian. The gap in the conductivity clearly closes at V 2.2t. ω in Region 2 is lower than d ≃ peak thatinRegion1and3...... 77

xiv 5.7 Temperature dependence of the dc conductivity, σdc(ω), for Vd =1, 2, 3, and 4. The system size is 16 16, U = 4t, and the data is averaged × over 20 disorder realizations. The dc conductivity at low T for Vd =3t is obtained by a third order polynomial ﬁt of the current-current correlation function, Λ = ωσ. At low T , the dc conductivity in Region 1 is zero because of the Mott gap, the system with Vd =3t is more conducting than Vd = 1t due the close of the Mott gap, and with increasing disorder, σdc is suppressed because of an Anderson localization of electrons. On the other hand, at high T , σdc in Region 1 is higher than the others because energy scale of temperature is comparable with the Mott gap and as disorder increases, σdc decreases due to the increase of transport scattering of electrons and disorder. It is demonstrated that disorder can drive a metallic phase in the system. At low temperature, the dc conductivity in Region 1 and 2 decreases as lowering temperature, signifying an insulating phase. However, the dc conductivity for Vd =3t has been most enhanced and slope of the conductivity, σdc/dT , hasnoslope...... 79

5.8 The frequency dependence of the conductivity for Vd = 1 (left top), Vd = 2 (right top), Vd = 3 (left bottom) and Vd = 4 (right bottom) at diﬀerent temperatures. The system size is 16 16, U = 4t, and × the data is averaged over 20 disorder realizations. The low frequency region of σ(ω) for Vd = 3t is obtained by a third order polynomial ﬁt of the current-current correlation function, Λ = ωσ. For low disorder at low T there is the Mott gap for low frequency and with increasing T , the peak in the conductivity moves to the weight at ω = 0. With increasing disorder from Region 1 to Region 2, the spectral weight above the Mott gap is transferred to lower spectral region at low T and σ(ω) shows the Drude like behavior for all T . With increasing disorder from Region 2 to Region 3, this Drude like behavior is less clear. The Drude like behavior shows non-monotonic behavior as a function of T . The conductivity weight in low frequency increases with increasing T up to T =0.8 but with further increasing T it decreases...... 81

xv Tˆ 4 h− i ˆ † ∞ 5.9 N (red square) where T = t i=j,σ ciσcjσ and πe2 0 σ(ω)dω (blue dot) as a function of disorder for T6 = 0 (left top) or a function of a P R temperature for Vd = 1 (right top), Vd = 2 (left bottom), and Vd = 3 (right bottom). The system size is 16 16, U =4t, and the data is av- × eraged over 20 disorder realizations. These values are non-monotonic. Especially, increasing temperature, the values are increasing for lower temperature and decreasing for high temperature. The kinetic energy is getting higher for low disorder because disorder is added to a Mott insulator but at high disorder the kinetic energy decreases with increasing disorder because the mobility of electrons is decreased at strong disordered sites. This is also seen as a function of temperature with ﬁxed disorder. At low temperature, the sum rule is good agreement and the sum rules at ﬁnite temperature are better agreement as increasingdisorder...... 83

5.10 χ(0, 0) and χ(π, π) as a function of a temperature for Vd =0, 1, 2, 3, 4, and 5. The system size is 16 16, U =4t, and the data is averaged over × 20 disorder realizations. The tails of χ(0, 0) are ﬁtted by the function C T +a where C is the measure of local moments and a is the measure of the strength of paramagnetism. These C and a are plotted as a function of the disorder, V. At low temperature, χ(0, 0) and χ(π, π) for low disorder are suppressed and these are most enhanced in Region 2, and with increasing disorder further, χ(0, 0) and χ(π, π) decrease. On the other hand, at high temperature, increasing disorder, χ(0, 0) and χ(π, π)monotonicallydecrease...... 85

A.1 Comparison of DOS between the analytical solution Eq. (A.7) and a numerical result (N =128)...... 90

A.2 Comparison of total energy between the analytical solution (line) and a numerical result (dot) for N = 128 as a function of U...... 93

A.3 Comparison of local moment between the analytical solution (line) and a numerical result (dot) for N = 128 as a function of U...... 94

A.4 Comparison of order parameter between the MFT solution (line) and a numerical result (dot) for N = 128 of staggered magnetization in the z direction as a function of U...... 95

xvi C.1 Possible AF couplings between two singly occupied sites for the nearest neighbor (nn) coupling and the next nearest neighbor couplings, (a,b,c,d,e,f)...... 103

C.2 Comparison between the probability distributions for the coupling for U = 4t at Vd = 2.5 (red), Vd = 3 (green), and Vd = 5 (blue) from the perturbation method. For J, Vd does not eﬀect on the plot as long as V U/2. P (J ) = P (J ), P (J ) = P (J ) and P (J ) = P (J ) from d ≥ a b c e d f thesymmetry...... 106

C.3 Comparison between the probability distributions for the coupling for U = 4t at Vd = 2.5 (red), Vd = 3 (green), and Vd = 5 (blue) from the diagonalization method. For J, Vd does not eﬀect on the plot as long as V U/2. P (J )= P (J ), P (J )= P (J ) and P (J )= P (J ) from d ≥ a b c e d f thesymmetry...... 114

xvii CHAPTER 1

INTRODUCTION

Nature shows phase transitions. These phase transitions are classified as first order or continuous depending on the behavior of the order parameter, which is a physical quantity that is nonzero in one phase and zero in the other phase [1]. For example, magnetization is the order parameter for a ferromagnet. In a first order phase transition, the order parameter jumps discontinuously to zero at the transition temperature, Tc and in a continuous phase transition, the order parameter vanished continuously as the temperature is increased toward the transition temperature, Tc.

One example of the first order phase transition is the ice to water phase transition with the density difference playing the role of the order parameter. On the other hand, an example of a continuous phase transition is a ferromagnet in a zero external magnetic field. Its magnetization is zero for T T , where T is the Curie temperature ≥ c c and nonzero for T < Tc. Other important examples include phase transitions at low temperatures in superfluidities and superconductivities.

In recent years, phase transitions that occur at a zero temperature, called quantum phase transitions (QPTs), have become an important direction of research. QPT occurs at T = 0 by tuning a parameter, such as pressure, the concentration of carriers

1 or doping, or an external magnetic ﬁeld [2]. At the critical point, quantum ﬂuctua-

tions due to the Heisenberg uncertainty principle, not thermal ﬂuctuations, kill the

ordering of one of the phase. Such QPTs have been observed in high temperature

superconductors by changing the doping, in heavy fermion systems by varying the

pressure, or even in the usual BCS quench-condensed superconducting ﬁlms by tuning

the magnetic ﬁeld or ﬁlm thickness.

When at zero temperature, changing some parameters such as the ratio of the strength of the interactions in the Hamiltonian, the phase of the system may be changed due to the quantum ﬂuctuation. On the other hand, the phase may be changed by temperature. Let’s consider the system of Josephson junction arrays which shows both classical and quantum phase transitions. Fig. 1.1 shows the phase diagram of square arrays (solid squares) and triangular arrays (solid triangles), which is obtained by measuring the resistance of the junction [3]. The x-axis shows the coupling strength, the ratio of two interaction strengths and the y-axis shows temperature. The blue line corresponds to a classical phase transition and this is the

Kosterlitz-Thouless-Berezinskii (KTB) phase transition from the superconducting phase to resistive/conductive phase [4]. The red line corresponds to a quantum phase transition from the superconducting phase to insulating phase at T = 0. In this way, the quantum phase transition and the classical phase transition can happen because of the ﬂuctuations even for the same model.

In this paper, thin wires, which undergo transitions into an ordered superconducting state lowering a temperature, and an array of Josephson junctions, where phases on each Josephson junction are ordered or disordered controlling a temperature, are considered for the examples of the classical phase transition. On the other

2 Figure 1.1: Comparison between a quantum phase transition and a classical phase transition using a phase diagram of square arrays (solid squares) and triangular arrays (solid triangles) obtained from experimental data in zero magnetic ﬁeld, f = 0 (from Ref. [3]), where the vertical axis is temperature and horizontal axis, x is a coupling ratio. The solid line and the dotted line are guides to the eye. The resistance per junction by applying a small transport current is measured to characterize these phases. The resistance of the superconducting phase is zero, that of the insulating 3 phase is a ﬁnite value with R > 10 R0 where R0 is the normal state resistance, and 3 that of the resistive/conductive phase is a value with R 10 R0. The quantum phase transition corresponds to the red line on the x-axis≤ at zero temperature and the classical phase transition corresponds to the blue line or green line, which are parallel to the y-axis.

3 hand, for the examples of quantum phase transition, a 2D metal-insulator transition is considered where disorder is the cause of the quantum ﬂuctuation.

The remainder of this paper is organized as follows. In Chapter 2, we review classical and quantum phase transitions using the model of the Josephson junction.

In Chapter 3, the effect of fluctuations on a network of parallel superconducting wires with a perpendicular magnetic field is discussed. In Chapter 4, thermal properties of small Josephson junctions coupled to a single-mode resonant cavity are investigated.

In Chapter 5, the metal-insulator transition on 2D disordered Hubbard model is considered.

4 CHAPTER 2

BRIEF REVIEW OF CLASSICAL AND QUANTUM PHASE TRANSITIONS

We give an illustrative explanation for the quantum phase transition, comparing the classical XY model, an example of a classical thermal phase transition, with the

Josephson Junction Arrays (JJA) model, an example of the quantum phase transition

[2].

2.1 Model of the JJA

An exact Hamiltonian, which can describe a real system, is likely to be too complex to handle. At a low temperature, a few degrees of freedom of the exact Hamiltonian need to be considered because only low excited states contributes the physical properties of the system. We consider an example of eﬀective Hamiltonians, the model of the JJA. The important degrees of freedom are the phase of Cooper pairs and the number, which is a conjugate variable of the phase.

A Josephson junction is a junction between two superconductors separated by a thin insulating layer or an empty space as illustrated in Fig. 2.1 [5]. When this layer is thin enough for Cooper pairs to jump from one superconductor to another, a quantum tunneling can occur and the Cooper pairs go and back between two islands.

5 V

INSULATOR

L R

SUPERCONDUCTOR

Figure 2.1: A model of single Josephson junction connected by the voltage

Because the fundamental length scale of the Cooper pairs, the coherent length ξ, is much larger than an atomic scale, this quantum tunneling behavior is much stronger than a usual quantum tunneling between two metals separated by an insulator.

Let’s consider two superconductors where Cooper pairs on the left have the wave function, ψL and those of the right have the wave function, ψR in Fig. 2.1 [6]. The

Schr¨odinger equations for each superconductor are written by [7]

∂ψ ∂ψ i~ L = V ψ + T ψ , i~ R = V ψ + T ψ , (2.1) ∂t L L R ∂t R R L where T is the Cooper pair hopping strength from one side to another. Supposing that there is a battery, V connecting across the junction and 0 voltage at the center

of the junction, V = 2e V and V = 2e V are taken where e is the unit of electron L 2 R − 2

6 charge. Thus, we can get two coupling equations.

∂ψ ∂ψ i~ L = eV ψ + T ψ , i~ R = eV ψ + T ψ . (2.2) ∂t L R ∂t − R L

The following notations are used for the complex wave functions, ψR and ψL.

iθL iθR ψL = √ρLe , ψL = √ρRe , (2.3)

where ρL and ρR are the density and θL and θR are phases on the left or right superconductor, respectively. Substituting Eqs. (2.3) to Eqs. (2.2), the real and imaginary part of Eqs. (2.2) become

∂ρ 2 ∂ρ 2 L =+ T √ρ ρ sin ∆θ, R = T √ρ ρ sin ∆θ, (2.4) ∂t ~ L R ∂t −~ L R ∂θ T ρ eV ∂θ T ρ eV L = R cos ∆θ , R = L cos ∆θ + , (2.5) ∂t − ~ ρ − ~ ∂t − ~ ρ ~ r L r R where ∆θ is a phase diﬀerence, θ θ . The ﬁrst two equations show that I = R − L S ∂ρL = ∂ρR where I is the current of the Cooper pairs from L to R. Supposing ∂t − ∂t S n ρ ρ , we can get the following two equations. ∼ L ∼ R d∆θ 2eV I = I sin ∆θ, = , (2.6) S C dt ~

2 2 where IC = T √ρLρR T n. It is these two equations that Josephson predicted in ~ ∼ ~ 1962 [8].

The energy stored for this supercurrent, the Josephson energy, can be obtained

by ∆θ ~ W = I V dt = I sin ∆θ d(∆θ)= E cos ∆θ (2.7) S C 2e − J Z Z0 ~IC where EJ = 2e . The lowest energy state is ∆θ = 0, where both phase are the same. EJ is called the Josephson junction coupling and depends on the shape of the junction.

7 The charging energy need to be considered when Cooper pairs move from one side to another [5]. Let us suppose that n Cooper pairs on the left side move to the right side at 0 voltage. The charge of Cooper pairs on the left is Q = 2en and that of − the right is Q =2en. Thus, the charging energy can be written as

Q2 1 = Un2, (2.8) 2C 2

where C is the junction capacitance and U =4e2/C.

Thus, the model of the single Josephson junction can be written as

1 H = Unˆ2 E cos(∆ˆθ) (2.9) JJ 2 − J

where n and ∆ become operators because these are canonically conjugate.

The single Josephson junction can show the quantum phase transition due to the

competition of the Josephson energy and the charging energy at zero temperature

[9]. At U/E 1, the charging energy need to be minimized by ﬁxing the number of J ≫ the Cooper pairs on the junction and the phase is uncertain. On the other hand, at

U/E 1, the phases are fixed and the supercurrent can flow due to the fluctuation J ≪ of the Cooper pairs. For k T E , thermal fluctuations kill the phase oder and B ≫ J suppress the supercurrent.

A natural extension from the model of single Josephson junction to an array of

N Josephson junctions shown as Fig. 2.2 may be written by the following model

Hamiltonian:

1 H = U (ˆn n¯)(ˆn n¯) E cos(θˆ θˆ ), JJA 2 ij i − j − − J i − j ij ij X Xh i

wheren ˆj and θˆj are operators representing the number of Cooper pairs and phase

on the jth junction,n ¯ is related to the gate voltage and ij means nearest neighbor h i 8 Figure 2.2: A model of 2D square Josephson junction arrays.

pairs in the lattice. At 0 voltage,n ¯ is zero. The phase and number are canonically

conjugate and satisfy the commutation relation [ˆnj, θˆk] = i~δjk, which is satisﬁed if

∂ we use the representationn ˆj = i . − ∂θj The JJA undergoes a superconducting phase as a whole if Cooper pairs can move

from a junction to another through the array, which is called a global coherence.

However, the JJA is not a superconducting phase if phases are disordered.

2.2 Classical Thermal Phase Transition

The competition between the entropy, S and the energy, E gives a classical thermal phase transition. At a ﬁnite temperature, the free energy, F = E TS needs to be − minimized. At a low temperature, the contribution of the second term of F is small,

so the lowest energy state is favored. On the other hand, at a high temperature,

9 Temperature

Figure 2.3: Phase states as a function of a temperature on classical XY model.

the contribution of the second term is important, so the system try to become a disordered state, which has larger entropy. Therefore, as a temperature is decreased, the system becomes from a disordered phase to an ordered phase and somewhere there is a phase transition.

In order to see the example of classical thermal phase transitions, we consider

classical 3D XY model, which is written as

H = E cos(θ θ ). (2.10) XY − JXY i − j i,j Xh i This is the model of superﬂuidity 4He, which is well studied, and the same as the JJA

model at Uij = 0. In this model, it is well known that there is a order-disorder phase

transition as a temperature is increased as shown in Fig. 2.3. At a low temperature,

an ordered phase is preferred. The phases on ith site and jth site are unlikely to

10 take diﬀerent values even when i and j are much apart. On the other hand, at a

high temperature, a disordered phase is preferred because of a large entropy and the

phases on ith site and jth site are not correlated when they are apart enough. From

a lower to a higher temperature, thermal ﬂuctuations kill the ordering of the phases.

This competition between ordering of the phases and the thermal ﬂuctuations triggers

the classical thermal phase transition.

2.3 Quantum Phase Transition

The competition of two parameters, U and EJ in the JJA model, gives a quantum phase transition at a zero temperature. We consider simpliﬁed JJA model (Uii =

U, Uij = 0 otherwise) at the two limit cases, EJ = 0 and U = 0.

U HS = nˆ2 E cos(θˆ θˆ ). (2.11) JJA 2 i − J i − j i ij X Xh i At U = 0, the Hamiltonian becomes classical XY model,

HS (U =0)= E cos(θˆ θˆ ). (2.12) JJA − J i − j ij Xh i So, the lowest ground state is a phase ordered state, which is written by

θ θ , ..., θ = , , ..., . (2.13) | 1 2 N i |↑ ↑ ↑i

On the other hand, at EJ = 0, the model can be written as

U HS (E =0)= nˆ2 (2.14) JJA J 2 i i X The ground state of this Hamiltonian becomes, in the representation of number states,

n , n , ..., n = 0, 0, ..., 0 (2.15) | 1 2 N i | i

11 Now, we have the uncertainty principle betweenn ˆ and θˆ, so if the number of Cooper

pairs in each junction is fixed, the fluctuation of phase at the site is infinity. Therefore,

this ground state is a state with disordered phases.

θ θ , ..., θ = , , ..., . (2.16) | 1 2 N i |↑ → ւi

U At ﬁnite values of U and EJ , it is assumed that when K = 1, the system EJ ≪ q achieves a superconducting phase because of ordered phases as shown in Fig. 2.4.

On the other hand, when K 1, the system achieves a normal phase because of ≫ disordered phases. The ﬂuctuation of Cooper pairs and phases are competing. Thus, when the order of Cooper pairs (the ﬁrst term of Eq. (2.11)) wins, the system becomes an insulator phase and when a phase order (the second term of Eq. (2.11)) wins, the system becomes a superconductor phase. Therefore, somewhere in between, there is the quantum critical point where a superconducting phase changes to an insulator phase.

Fig. 2.5 shows the experimental date of the resistance per junction of six different aluminum arrays with different coupling strengths, x = EC =8K2 = 8U as a function EJ EJ of a temperature [3]. Three arrays (x = 0.56, 1.1, 1.25) become a superconducting phase near a zero temperature and two arrays (x = 1.8, 4.6) become an insulating phase. The resistance of x = 1.7 has a finite value at zero temperature. The experimental value of this resistance is about 16.4kΩ as shown in the dashed horizontal line. This describes the quantum phase transition.

Fig. 2.6 shows the measured phase diagram of square arrays (solid squares) and triangular arrays (solid triangles) [3]. The x-axis shows the coupling strength and the y-axis shows temperature. There is a classical phase transition and this is the

Kosterlitz-Thouless-Berenzinskii (KTB) phase transition from the superconducting

12 K

Figure 2.4: Phase states as a function of K = U on Josephson junction arrays EJ model. q

Figure 2.5: Experimental data of the resistance per junction as a function of temperature for six diﬀerent coupling strengths. The dashed horizontal line is 16.4kΩ, which corresponds to the universal conductivity that is independent of the microscopic details of the system. (From Ref. [3]).

13 Figure 2.6: A phase diagram of square arrays (solid squares) and triangular arrays (solid triangles) obtained from experimental data in zero magnetic ﬁeld, f = 0, where the vertical axis is temperature and horizontal axis, x is a coupling ratio. The solid line and the dotted line are guides to the eye. The resistance per junction by applying a small transport current is measured to characterize these phases. The resistance of the superconducting phase is zero, that of the insulating phase is a 3 ﬁnite value with R> 10 R0 where R0 is the normal state resistance, and that of the resistive/conductive phase is a value with R 103R . (from Ref. [3]). ≤ 0

14 phase to resistive/conductive phase and there is a quantum phase transition from the superconducting phase to insulating phase at T = 0. Near x x =1.7, there is the ∼ c quantum phase transition.

15 CHAPTER 3

THEORY OF FLUCTUATIONS IN A NETWORK OF PARALLEL SUPERCONDUCTING WIRES

We show how the partition function of a network of parallel superconducting wires weakly coupled together by the proximity eﬀect, subjected a vector potential along the wires can be mapped onto N-distinguishable two dimensional quantum-mechanics problem with a perpendicular imaginary magnetic ﬁeld. Then, we show, using a mean

ﬁeld approximation, that, for a given coupling, there is a critical temperature for onset of inter-wire phase coherence. The transition temperature Tc is plotted on both cases for non-magnetic and a magnetic ﬁeld perpendicular to the wires.

3.1 Introduction

There has been considerable recent interest in thin wires that undergo transitions into an ordered state, such as superconducting or ferromagnetic. For example, a recent experiment [10] has suggested that single-walled carbon nanotubes (which have diameters of only about 4 A)˚ are superconducting up to temperatures as high as

20 K. Because these tubes are so thin, they behave very much like one-dimensional superconductors. It was therefore proposed [10] that they could be described by a

16 complex order parameter ψ(z) which varies only in one dimension, say the z direction, i.e. along the tube. ψ(z) might represent the complex energy gap, or, in a diﬀerent normalization, it could represent the condensate wave function in a BCS superconductor.

Moreover, there have been many experiments for investigating superconductivity on nanowires. Ropes of carbon nanotubes between superconducting electrodes can show superconductivity due to the proximity eﬀect of the electrodes [11, 12, 13].

Furthermore, superconductivity on carbon nanowires connected to normal contacts, has been observed [14, 15]. On the other hand, superconductivity of nanowires of Zn or Sn has been investigated [16, 17].

Fluctuations are, of course, especially important in one-dimensional systems. It was shown many years ago by Scalapino et al [18] that classical ﬂuctuations in one dimension could be treated exactly, within the context of a Ginzburg-Landau (GL) free energy functional. Their treatment involved mapping the GL functional onto a single-particle quantum mechanics problem, using an exact connection between the classical partition function and a path integral treatment of the quantum mechanics problem. These authors showed that classical ﬂuctuations could give rise to a non- zero order parameter even above the GL transition temperature. This mapping was extended to treat Josephson-coupled thin wires [19, 20].

However, in the mapping, the effect of a magnetic field was ignored. In the case of a non-zero perpendicular magnetic field, we show that the partition function for the wires maps onto a certain zero-temperature quantum mechanics problem in two dimensions with an effective imaginary perpendicular magnetic field, which brings to a non-Hermitian quantum mechanics problem.

17 The non-Hermitian problem in physics has not been new recently. Nonequilibrium processes can be described by non-Hermitian Liouville operators [21, 22, 23]. The non-Hermitian quantum mechanics are well studied in order to study the pinning of magnetic ﬂux lines in high temperature superconductors [24, 25, 26].

The remainder of this paper is organized as follows. In Section 3.2, we describe our formalism and mapping. In Section 3.3, we give our numerical results including phase diagrams. This is followed by a concluding discussion and an outline of possible future research.

3.2 Formalism

3.2.1 Mapping to a quantum mechanics problem for interacting superconducting wires when B~ is perpendicular to the wires

Let us consider a network of N parallel superconducting wires in a non-zero vector

potential. We assume, for convenience, that these wires all have the same GL param-

eters, though the formalism can easily be generalized to the case when the parameters

are diﬀerent. Then the partition function can be written as a functional integral over

the N complex order parameters ψ1(z1), ...ψN (zN ):

Z = ψ (z )... ψ (z ) exp βF [ψ (z ), ...ψ (z )] . (3.1) D 1 1 D N N {− 1 1 N N } Z We assume that the free energy functional is the sum of two parts: a single-wire

term Fs and a term describing inter-wire interactions, which we denote Fint. The

single-wire term will just be the sum of general GL equation for each wire:

Fs = FGL[ψi(zi)]. (3.2) i=1 X

18 Here,

zmax 1 ~ e A~ HB F [ψ (z )] = [ ∗ ψ(z) 2 + α ψ(z) 2 + γ ψ(z) 4 + Σ]dz, GL i i 2m | i ∇ − c | | | | | 8π Z0 ∗ ! (3.3) where α, γ, and m∗ are material-dependent (and possibly temperature-dependent)

coeﬃcients. Commonly, it is assumed that γ is positive and that α = α′(T T ), − c

where T is the temperature, Tc is the critical temperature, and α′ is greater than

zero. Also, Σ is the cross-sectional area of the sample, but for one-dimensional wire

we may ignore this term. For the interaction term, we assume a form similar to that

used by Lawrence and Doniach for interacting superconducting layers [27], namely

zmax F = K ψ (z) ψ (z) 2. (3.4) int ij| i − j | ij 0 Xh i Z

where zmax is the length of the wires. Basically, we are assuming that there is a

Josephson coupling of strength Kij between diﬀerent wires, but at the same point

along the length, z. We choose a gauge such that the vector potential is parallel

to the superconducting wires, has only z component and independent of z. When

a wire is a loop, a vector potential is related to the total ﬂux Φ through the loop,

Az = Φ/zmax. In this case, using ψi(z)= ψiR(z)+ iψiI (z), Fs and Fint take the forms

zmax~2 ~ 2 2 e∗ Az 1 e∗ 2 2 4 F = [ ψ′ (ψ ψ′ ψ′ ψ )+ α + A ψ +γ ψ ]dz, s 2m | i| − m c iR iI − iR iI 2m c z | i| | i| i Z0 ∗ ∗ ( ∗ ) X (3.5) and

zmax F = K ( ψ (z) 2 + ψ (z) 2 2(ψ ψ + ψ ψ )), (3.6) int ij | i | | j | − iR jR iI jI ij 0 Xh i Z where ψ′(z)= dψ(z)/dz. Finally, the partition function takes the form

Z = ψ ψ exp( βF [ψ , ψ ]), (3.7) D iRD iI − iR iI i Z X 19 where we use ψi = ψiR + iψiI .

We now show that eqs. (3.5), (3.6) and (3.7) for Z are actually equivalent to a

quantum mechanical problem of a N distinguishable particles in N distinct quantum

wells in two dimensions in the presence of a perpendicular magnetic ﬁeld. In order

to simplify our argument, we consider the case of single particle with mass, m and a charge e∗ subjected to a 2D potential, V (x, y). The density matrix of a two- dimensional system, using ψI and ψF are boundary condition at initial and ﬁnal time, can be written [28]

ψF F S/~ I 1 ψ e− ψ = x(τ) y(τ) exp S[x(τ),y(τ)] , (3.8) h | | i I D D {−~ } Zψ where

βeff ~ m 2 2 e∗ S = [ (x′ + y′ )+ V (x, y) i A~ ~v]dτ. (3.9) 2 − c · Z0

For the given B~ = Beff zˆ with the gage

B A~ = eff (xyˆ yxˆ), (3.10) eff 2 −

this S becomes

βeff ~ m 2 2 e∗Beff S = dτ[ (x′ + y′ )+ V (x, y) i (xy′ yx′)]. (3.11) 2 − 2c − Z0 This is a similar equation to the partition function of the superconducting wires.

In order to simplify this mapping, we use the suitable dimensionless form. τ =

~ β z, ψ˜ = ξ3/2ψ , and ψ˜ = ξ3/2ψ . Then we can make the identiﬁcations of ξ0 ix 0 iR iy 0 iI Table 3.1.

We find that the magnetic field has two effects: (i) it determines an effective per-

pendicular magnetic ﬁeld in which the equivalent quantum-mechanical particle moves;

20 Q.M. S.C. τ z ~ρ = x (u),y (u) ψ~ = ψ˜ (z), ψ˜ (z) i { i i } i { ix iy } E F ξ0 zmax βeff βzmax/ξ0 ∗ α 1 e Az 2 ˜ 2 γ ˜ 4 Vi(xi,yi) ξ2 + 2m∗ξ2 ( c ) ψi + ξ5 ψi 0 0 | | 0 | | ~4β2 m ∗ 4 m ξ0 2 2~ Azβ Beff i m∗ξ3 − 0 Kij Jij 2 ξ0

Table 3.1: Correspondence on the mapping between Q.M. and S.C. (in each case, the left-hand variable corresponds to the parameters on the quantum mechanics problem and the right hand variable corresponds to the parameters on the superconductor wires)

and (ii) it changes the quadratic part of the eﬀective potential. The Hamiltonian for the analogous quantum problem is N 1 e B 2 1 e B 2 H = [ p + ∗ eff y + p ∗ eff x + V (~ρ )]+ 2J ~ρ ~ρ 2, 2m ix 2c 2m iy − 2c i i ij| i − j | i=1 X X (3.12) where pix and piy are momentum operators of x and y components of ith particle, respectively.

3.2.2 Probability distribution of the order parameter

We consider the probability distribution of the order parameter, which corresponds to the probability distribution of particles in quantum mechanics. In order to simplify our discussion, we consider single wire case. The probability distribution function of the order parameter can be deﬁned as

H H 1 F (Lτ τ) τ I P (~ρ(τ)) = ψ e− ~ − ~ρ(τ) ~ρ(τ) e− ~ ψ , (3.13) Z h | | ih | | i 21 H F Lτ I I where Z = ψ e− ~ ψ and ψ represents the boundary condition at τ = 0 and h | | i | i ψF represents the boundary condition at τ = L . Using the eigenstates of the h | τ Hamiltonian, H n = E n , the probability can be written as | i n| i

Em En 1 F I (Lτ τ) τ P (~ρ(τ)) = ψ m m ~ρ(τ) ~ρ(τ) n n ψ e− ~ − e− ~ (3.14) Z h | ih | ih | ih | i m,n X with

F I En τ Z = ψ n n ψ e− ~ . (3.15) h | ih | i n X Explicitly, the expectation value of operator, ρ at the distance τ from the bottom of

the wires is given by

corresponds to ψI and summed over all possible this conﬁguration, the density matrix

can be written as.

Em En 1 I I (Lτ τ) τ P (~ρ(τ)) = ψ m m ~ρ(τ) ~ρ(τ) n n ψ e− ~ − e− ~ Z h | ih | ih | ih | i m,n X XI En 1 Lτ = n ~ρ(τ) ~ρ(τ) n e− ~ , Z h | ih | i n X En En I I Lτ Lτ where Z = ψ n n ψ e− ~ = e− ~ . So, if the wire is actually in the n I h | ih | i n

form of a loop,P P which means the boundaryP conditions ψ(0) = ψ(zmax), our problem

corresponds to this statistical mechanics. Of course, in the limit of a very long wire,

the periodic boundary condition imposed by the loop should become unimportant.

In the case of the periodic boundary condition for single wire, we can see qual-

itative behavior of order parameter. The average gap in the GL problem (denoted

22 ∆(˜ t)) corresponds to the mean distance ρ in the quantum-mechanical problem, i.e. h i

ρ ∆(˜ t), (3.17) h i↔

˜ 3/2 where ∆(t) = ∆(t)/ξ0 . At much lower temperature than the critical temperature

0 Tc , the mean distance from the origin of the particle approaches the value predicted

for the quantum problem in the limit of inﬁnite mass, i.e. the value of ρ for which the

quartic potential is a minimum although the magnitude of these gaps at T = 0 are

different. The function √1 t, is the classical solution, i.e., in the case when thermal − fluctuations in the GL case are negligible. These fluctuations do indeed become very

small when T 0, because in this regime, the eﬀective potential rises steeply above → its minimum, and the ρ becomes very close to the value that minimizes the GL free h i energy. When ρ has this value, the corresponding value for ∆(˜ t) is h i

3 2 2 α T 0ξ ∆(˜ t)= ψ˜ + ψ˜ = 0 c 0 √1 t = ∆(0)˜ g(t), (3.18) R I 2γ − q s where ∆(0)˜ is the gap at T = 0. These considerations may suggest that we can approximate g(t)= ∆(˜ t)/∆(0)˜ = √1 t. − 3.2.3 Phase only model and mean-ﬁeld approximation

This system will undergo a phase transition into a phase-ordered state below a critical temperature Tc which is distinct from (and lower than) the single wire mean-

0 ﬁeld transition temperature Tc . To do this, we consider a simpliﬁed, “phase-only”

version of this Schr¨odinger equation (3.12). We assume that the magnitudes ρi of the

variables xi are ﬁxed at the values which minimize the single-wire GL free energy, i.e.

ρ ρ (3.18). All terms in the Hamiltonian involving ∂/∂ρ can be ignored in this i ≡ 0 i

23 phase-only model. The eﬀective Hamiltonian (3.12) then becomes ~2 2 ~ ∂ e∗Beff ∂ 2 H = 2 2 +2 Jijρ0(1 cos(φi φj)), (3.19) − 2mρ ∂φ − 2mc i ∂φi − − i 0 i i ij X X Xh i where the sum runs over distinct nearest neighbor pairs. This is the well-known

quantum XY model, which exhibits a quantum phase transition at a critical value.

The mean ﬁeld approximation can be applied to this Hamiltonian, assuming that

Jij = J for only nearest neighbors, by replacing the second term according to the

prescription

cos(φ φ ) = 2 cos φ cos φ cos φ 2, (3.20) i − j ih i−h i where we are supposing sin φ = 0 because of the symmetry. Thus, h i 2 Jρ2(1 cos(φ φ )) = 2 Jρ2 1 2 cos φ cos φ + cos φ 2 0 − i − j 0 − ih i h i ij ij Xh i Xh i = 4z Jρ2 cos φ cos φ +2 Jρ2(1 + cos φ 2) − n 0h i i 0 h i i ij X Xh i = 4z Jρ2 cos φ cos φ +2z NJρ2(1 + cos φ 2), − n 0h i i n 0 h i i X where zn is the number of nearest neighbors in the lattice. Thus, the eﬀective Hamil-

tonian corresponding to eq. (3.19) becomes a following Schr¨odinger equation:

2 2 ~ ∂ e∗B ~ ∂ − eff 4z ρ2J cos φ cos φ +2z Jρ2(1+ cos φ 2) ψ (φ )=E ψ (φ ). 2mρ2 ∂φ2− 2mc i ∂φ − n 0 h i i n 0 h i n i n n i 0 i i (3.21)

We consider the self consistent equation for cos φ on the periodic boundary condi-

tion. The mean field theory is defined by the self-consistency requirement on cos φ : h i β En e− eff ψ (φ ) cos φ ψ (φ ) cos φ = n h n i | i| n i i. (3.22) h i e βeff En P n − For example, when the wires are sufficientlyP long where only the ground state contribution may be important, the self-consistent condition becomes

cos φ = ψ (φ ) cos φ ψ (φ ) . (3.23) h i h 0 i | i| 0 i i 24 These equations may be solved for cos φ and T , where the critical temperature can h i c be determined by cos φ 0. h i→ 3.3 Results and Discussion

We have considered long-range phase coherence among wires in the bundle in order to see whether the phases on the wires are coherent and the bundle as a whole is superconducting or not. The self-consistent equation gives rise to a phase diagram exhibiting superconductivity, which can be deﬁned as the greatest temperature and ﬁeld such that cos θ takes on a non-zero value [29]. Here, we assume that the

Josephson coupling is independent of a temperature. We consider the temperature dependence, √1 t for ρ. In order to simplify our calculations, we consider the case − of the periodic boundary condition.

3.3.1 No magnetic ﬁeld

We consider the following self-consistent equation, substituting Beff = 0 for the diﬀerential eq. (3.21),

~2 ∂2 4z ρ2J cos φ cos φ +2z Jρ2(1 + cos φ 2) ψ (φ )= E ψ (φ ). −2mρ2 ∂φ2 − n 0 h i i n 0 h i n i n n i 0 i (3.24)

This equation can be reduced to the standard Mathieu equation [30], using v = φ/2,

y(v)= ψn(φi/2), d2y (v) n +(a 2q cos2v)y (v)=0, (3.25) dv2 n − n where the characteristic value of the Mathieu equation and q are written as

2mρ2 E B(1 + cos φ 2) a = 4(E 2z Jρ2(1 + cos φ 2)) 0 = n − h i , n n − n 0 h i ~2 A 2mρ2 B q = 8z ρ2J cos φ 0 = cos φ , − n 0 h i ~2 − A h i 25 ~2 2 where we deﬁne A = 2 and B =2znJρ0. The eigenvalues are explicitly written as 8mρ0

E = Aa + B(1 + cos φ 2). (3.26) n n h i

The allowed eigenfunctions are determined by the condition that the wave functions

be single-valued, i.e., that ψn(φ+2π)= ψn(φ), or equivalently, that yn(v+π)= yn(v).

The allowed three lowest solutions, up to the order of q2, are [30]

1 q cos4v 1 q2 y (v, q) = 1 cos2v + q2 , a = , 0 √π − 2 32 − 16 0 − 2 2 cos4v 1 cos6v 19cos2v 5q2 y (v, q) = cos2v q +q2 , a =4+ , 2 √π − 12 − 4 384 − 288 2 12 2 2 sin 4v 2 sin 6v sin 2v q y 2(v, q) = sin 2v q + q , a 2 =4 , − √π − 12 384 − 288 − − 12 2π where these are normalized like 0 ψn(φ)dφ = 1. Thus, the matrix elements for cos θ on the corresponding bases, n =0R , 2, and 2, are − q 1 0 − 2 √2 cos φ = 1 5q 0 . (3.27) √2 12 h i  q  0 0 12  −  From the mapping, we can get the self-consistent condition for the critical tem-

perature of the phase ordering in terms of the parameters of the GL equation for

suﬃcient or inﬁnite long wires eq. (3.23), which corresponds to the only consideration

of the ground state (n = 0) in the quantum mechanics problem, and it takes the

following form. A q cos φ = q = . (3.28) h i −B −2 The temperature dependence of order parameter obtained by eq. (3.28) for inﬁnite long wires is shown in Fig. 3.1. This ﬁgure clearly shows that there is a second order phase transition at t = 0.5 because the order parameter continuously becomes zero at the critical point. As expected, the critical temperature of the whole wires is lower

26 1 L=Infinity

0.8 > φ

0.6

0.4

Order parameter

0 0 0.2 0.4 0.6 t=T/Tc

Figure 3.1: Temperature dependence of the phase-ordering for inﬁnite long wires.

than the critical temperature of a single wire. The transition temperature of phase

ordering can be calculated by ﬁnding the temperature where cos φ becomes zero. h i ∗ 4 ˜ 2 m ξ0 2znK∆ (t) Thus, because with A ~2 2 ˜ 2 and B ξ2 , → 8 β ∆ (t) → 0 2 2 4 2 B 16zn~ β K∆˜ (t) (1 t) 6 =2α −2 , (3.29) A → m∗ξ0 t

2 4 8zn~ ∆˜ (0)K where α = ∗ 6 0 2 , the condition becomes m ξ0 (kB Tc )

cos φ =0 t = √αg2(t ). (3.30) h i → c c

Therefore, using g(t)= √1 t, this critical temperature becomes − √α t = . (3.31) c 1+ √α

On the other hand, for ﬁnite length wires, contributions from excited states in the quantum mechanics problem need to be considered because the eﬀective temperature

27 is not zero. Using up to the order n 2 for the solution of Mathieu’s equation, | | ≤ using eq. (3.22), the following self-consistent condition can be obtained,

q β E0 5q β E2 q β E−2 A e− eff + e− eff e− eff q = − 2 12 − 12 , (3.32) β E0 β E2 β E−2 −B e− eff + e− eff + e− eff where β E = β (Aa + B(1 + cos φ )), but the second term can be canceled. eff n eff n h i Therefore, with the mapping, we can get

2 2 4x t t 1 e− 1−t = α − 3 , (3.33) 4x t 1 t 1−t − 1+2e− where we use the following mapping

2 0 2 t zmax t m∗ξ0 kBTc ξ0 zmax t βeff A x = x0 = . (3.34) ~2 2 → 1 t ξ0 1 t 8 ∆˜ (0) ξ0 1 t − − − 4 Using the numerical values according to Tang et al [10], x 1.4 10− . A plot of 0 ≈ ×

Tc versus α for several lengths (100ξ0, 1000ξ0, 2000ξ0, and 5000ξ0) and inﬁnite length

are given in Fig. 3.2. This ﬁgure shows that as the length of the wires has increased,

the phase critical temperature has increased.

3.3.2 Perpendicular magnetic ﬁeld

The critical temperature for the presence of a magnetic ﬁeld on the wires can be obtained by solving the non-Hermitian eq. (3.21).

2 2 ~ ∂ e∗B ~ ∂ − eff 4z ρ2J cos φ cos φ +2z Jρ2(1+ cos φ 2) ψ (φ )=E ψ (φ ). 2mρ2 ∂φ2− 2mc i ∂φ − n 0 h i i n 0 h i n i n n i 0 i i (3.35)

pv Using ψn(φ)= e F (v) with and v = φ/2 and

2e ρ2B p = i ∗ 0 eff , (3.36) c~

again this equation reduces to the standard Mathieu equation:

d2F (v) (2q cos2v)F (v)= a F (v), (3.37) dv2 − − ν 28 0.6 Infinity L=5000ξ ξ0 L=2000 0 L=1000ξ 0 0

c ξ L=100 0 /T c 0.4 Normal

0.2

Critical temperature T SC

0 0 0.2 0.4 0.6 0.8 1 α

0 Figure 3.2: Phase diagram of tc = Tc/Tc as a function of α for several values of length of the wires, 100ξ , 1000ξ , 2000ξ , 5000ξ , and where ξ = 42A.˚ 0 0 0 0 ∞ 0

where

2mρ2 E B(1 + cos φ 2) a p2 = 4(E 2z Jρ2(1 + cos φ 2)) 0 = n − h i , ν − n − n 0 h i ~2 A 2mρ2 B q = 8z ρ2J cos φ 0 = cos φ . − n 0 h i ~2 − A h i

The allowed eigenvalues are determined by the boundary condition that ψn(φ +

2π)= ψ (φ), or equivalently F (v + π) = exp( pπ)F (v). Thus we are interested only n − in the Floquet solutions of the Mathieu equation with Floquet exponent ν =2n + ip, where n =0, 1, 2, ..... These solutions are explicitly written as [30] ± ± e2iv e 2iv F (v)= c eiνv 1 q − , (3.38) ν 0 − 4(ν + 1) − 4(ν 1) − where c is a normalization constant. The eigenvalues are, using q = B cos θ, 0 − A q2 a = ν2 + . (3.39) ν 2(ν2 1) − 29 The allowed three lowest solutions, up to the order of q2, are [30]

1 cos2v + p sin 2v q2 ψ (v)= 1 q , a = , ip π − 2(1 + p2) ip −2(1 + p2) r 1 q e4iv 1 q2 ψ (v)= e2iv , a =4(1 + ip)+ , 2+ip π − 4 3+ ip − 1+ ip 2+ip 2( p2 +4ip + 3) r − 4iv 2 1 2iv q e− 1 q ψ 2+ip(v)= e− + , a 2+ip =4(1 ip)+ . − π 4 ip 3 − ip 1 − − 2(3 p2 4ip) r − − − − L R Left wave functions can be obtained from right wave function with ψ (v,p)= ψ (v, p)∗. n n − The self-consistent condition for long wires becomes the following form,

A q cos φ = q = , (3.40) h i −B −2(1 + p2) because the matrix elements for cos θ corresponding to n = 0, 2, and 2 are, using − q = B cos θ , − A h i q 1 1 1 − 1+p2 1 q 1 cos φ =  3+4ip p2 2(1+p2)  . (3.41) h i 2 1 − q 1 2(1+p2) 3 4ip p2  − −  Again, we can determine the transition temperature of the phase ordering, where

cos φ becomes zero. h i α cos φ =0 t = g2(t ). (3.42) h i → c 1+ p2(t ) c r c The approximation g(t)= √1 t is again used for this case and then we can get − α f 2 tc = − , (3.43) 1+ α f 2 p − where we deﬁne f as p

2 2 2 2 fg (t) Azξ0 g (t) 8π~ ∆˜ (0) Azξ0 1 t p = f0 = 0 2 2 − , (3.44) → t Φ0 t kBTc m∗ξ0 ξ0 Φ0 t where Φ0 = hc/e∗. When f = 0, this solution corresponds to the previous case. A plot of Tc versus α for f = 0, 0.2, 0.4, 0.6, and 0.8 is given in Fig. 3.3. This ﬁgure

30 0.6 f=0 f=0.2 f=0.4

0 f=0.6 Normal c f=0.8 /T c 0.4

0.2 SC Critical temperature T

0 0 0.2 0.4 0.6 0.8 1 α

0 Figure 3.3: Phase diagram of tc = Tc/Tc as a function of α for several values of magnetic ﬁeld strength, f = 0.2, f = 0.4, f = 0.6, and f = 0.8 for inﬁnity length wires

shows that the critical temperatures have the minimum values for the interaction

between the wires. These values can be calculated by

8π2~2 A ξ 2 α f 2 z K z 0 . (3.45) ≥ → n ≥ m ξ2 Φ ∗ 0 0

The critical fc, which is related to the maximum ﬂux in the wires can be obtained, α(1 t)2 t2 f = − − . (3.46) c 1 t p − Near the critical temperature of phase ordering, using t = √α δt, this can be 1+√α − written

f √2(1 + √α)α1/4√δt. (3.47) c ≈

Fig. 3.4 shows that this critical magnetic ﬁeld fc for α =0.2, 0.4, 0.6, 0.8, and 1 as a

function of a temperature.

31 α=1 α 1 =0.8 α=0.6 α=0.4 α=0.2 0.8

0.6 Normal c f

0.4

0.2 SC

0 0 0.2 0.4 0.6 0 Temperature T/Tc

Figure 3.4: Temperature dependence of the critical ﬁeld strength, fc for α = 0.2, 0.4, 0.6, 0.8, and 1 for inﬁnity long wires.

3.4 Summary

We have presented a mapping between a one-dimensional GL problem in the

presence of a vector potential along wires and a two-dimensional quantum mechanics

problem with a perpendicular magnetic ﬁeld. Moreover, in the case of weak links

between wires, we have obtained, using the mean-ﬁeld approximation, the phase

diagrams for the presence of a magnetic ﬁeld and absence of it.

Next, we discuss the parameters used in this paper. Using the numerical values of the various parameters appropriate to those of a single-walled carbon nanotube, which according to Tang et al [10], where superconducting with a relatively high

0 0 0 ˚ transition temperature Tc = 15K, kBTc = 1.3meV, α0Tc = 6meV, γ = 1.3meVA,

32 ~ ˚ m∗ = 0.36me, and ξ0 = ∗ 0 = 42A, we can obtain the following values for √2m α0Tc

znK 4 Azξ0 α and f, α = −6 and f = 1.7 10 . The Josephson coupling energy 8.6 10 meV Φ0 × × ~2 K is approximated by 2 where s is the distance between nearest wires. If we 2mcs 2 0 ξ0 α0Tc use m = m∗, K can be written as . Thus, supposing s 5A,˚ K is order c s2 ≈ of 100 1000[meV]. Therefore, our values used in the ﬁgures are well suited for ∼ describing real systems.

We discuss about the use of the GL free energy functional. In principle, this

free energy functional is applicable only near the critical temperature, T T 0 T 0. − c ≪ c 0 Besides near the critical temperature Tc , the qualitative description of this functional may not be reasonable, although we can employ higher order expansions of the order parameter in the G.L equation.

We want to comment the eﬀect on the interaction term by a magnetic ﬁeld. When

there is a magnetic field, the phase difference needs to be replaced by φ φ i − i+1 − 2π A~ d~l where the integration is between different wires. However, because the Φ0 · directionR of vector potential is taken in the direction of the wires, z, there is no contribution from the integral on the phase difference.

In this paper, we only consider the periodic boundary condition for simpliﬁcation.

When wires are suﬃcient long, the eﬀect of the boundary conditions may not change

the physical properties of the system. However, these boundary conditions may aﬀect

the properties of the system because of ﬁnite length of wires. Moreover, our theory

neglects the eﬀects of disorder, which plays an important role on balk superconduc-

tors. With these degrees of freedom, the properties of the system may be changed.

Thus, it might be an interest to consider these cases for our future research.

33 CHAPTER 4

STATISTICAL MECHANICS OF SMALL JOSEPHSON JUNCTIONS COUPLED TO A SINGLE-MODE RESONANT CAVITY

We calculate the thermodynamic properties of a collection of N small Josephson junctions coupled to a single-mode resonant electromagnetic cavity, at finite temperature T , using several approaches. In the first approach, we include all the quantum- mechanical levels of the junction, but treat the junction-cavity interaction using a mean-field approximation developed previously for T = 0. In the other approaches, the junctions are treated including only the two lowest energy levels per junction, but with two different Hamiltonians. The first of these maps onto the Dicke model of quantum optics. The second is a modified Dicke model which contains an additional

XY-like coupling between the junctions. The modiﬁed Dicke model can be treated using a mean-ﬁeld theory, which in the limit of zero XY coupling gives the solution of the Dicke model in the thermodynamic limit using Glauber coherent states to represent the cavity. In all cases, for an N-independent junction-cavity coupling, there is

a critical junction number N above which there is a continuous transition from inco-

herence to coherence with decreasing T . If the coupling scales with N so as to give a well-behaved thermodynamic limit, there is a critical minimum coupling strength for

34 the onset of coherence. In all three models, the cavity photon occupation numbers have a non-Bose distribution when the system is coherent.

4.1 Introduction

When a two-dimensional array of Josephson junctions is driven by an applied current, it can radiate coherently. Experiments showing this behavior have emphasized current-driven arrays of overdamped Josephson junctions [31, 32]. The radiated coherent power from such arrays has been predicted to be proportional to the square of the number N of Josephson junctions in the array [33]. More recently, coherent emission from underdamped one-dimensional Josephson arrays coupled to a single-mode electromagnetic cavity has been experimentally studied [34, 35, 36, 37]. In this work, it was shown that no coherent radiation is emitted below a threshold number Nc of junctions, but above this threshold the array can radiate coherently, with emitted power again proportional to N 2. Such behavior had already been predicted much earlier, on the basis of an analogy between a one-dimensional voltage biased series array and a collection of two-level atoms coupled to an electromagnetic cavity [38]; the analogy suggests that this radiation is the Josephson analog of the population inversion that leads to coherent emission in a laser.

A simple model Hamiltonian to describe this coherent radiation, taking into ac- count the quantum-mechanical nature of both the junctions and the cavity, was suggested recently [39]. In this paper, the ground state of the model Hamiltonian is obtained within a mean-ﬁeld theory (MFT). In agreement with experiment, the MFT predicts that there is a critical threshold number Nc of junctions for the onset of coherence at ﬁxed coupling strength.

35 The mechanism for coherent radiation from a Josephson junction array resembles that of superradiance in a system of N two-level atoms coupled to a electromagnetic

field [40]. The latter system can be treated the Dicke model [41], which describes the system of identical two-level atoms in a single-mode radiation field. Emission and absorption within the Dicke model have been extensively studied [42, 43]. The predicted response of this two-level atom/radiation system agrees qualitatively with that of an array of Josephson junctions [44, 45]. It has also been shown [45], that a modified Dicke Hamiltonian, which contains an additional term resembling a dipole- dipole interaction between the junctions, is a better approximation to the cavity- junction system than is the original Dicke model.

In contrast to the Josephson/cavity system, the Dicke model can be solved in the

thermodynamic limit, i.e. N , volume V , and N/V const [46, 47, 48]. →∞ →∞ → The solution yields a continuous transition from a normal to a superradiant state at a critical coupling strength, for ﬁxed temperature T = 0.

In this paper, we extend the model [39] to ﬁnite T . We ﬁnd, again within MFT,

that there is a critical threshold number Nc(T ) for coherence at a suﬃciently low T .

If T is increased at ﬁxed N > Nc(0), there is a continuous transition from coherence

to incoherence at an N-dependent temperature Tc(N). To test the MFT, we compare

its predictions with those of the Dicke model [47, 48] and of the modiﬁed Dicke

model [45]. Our Hamiltonian does not map exactly onto these models, because the

individual Josephson junctions have more than two quantum levels, whereas the Dicke

and modiﬁed Dicke models assume two-level systems interacting with a single-mode

cavity. Nonetheless, when the parameters of our model systems are such that the

lowest two levels of the junction are well-separated from the higher levels, the MFT

36 agrees well with the two-level model predictions. We also show that, when applied to

the Dicke model, the MFT is equivalent to a coherent state expansion, an approach

known to give the solution of the Dicke model at large N.

The remainder of this paper is organized as follows. In Section 4.2, we describe

the MFT for N Josephson junctions interacting with a cavity at ﬁnite T . In Section

4.3, we review the coherent state treatment of the Dicke model, present a MFT for

both the Dicke model and the modiﬁed Dicke model, and show that, when applied to

the Dicke model, the MFT is equivalent to the coherent state approach. In Section

4.4, we give numerical results for all three models. In Section 4.5, we show how the

Josephson-cavity model of Section 4.2 can be mapped onto the Dicke model when the

Josephson coupling is small compared to the charging energy; we also discuss how

the parameters of all three models must scale in the thermodynamic limit. Section

4.6 presents a concluding discussion.

4.2 Many Josephson Junctions Interacting with a Single- mode Cavity

4.2.1 Model Hamiltonian

An array of N underdamped voltage-biased Josephson junctions in a lossless elec-

tromagnetic cavity having a single electromagnetic mode of frequency ω may be described by the following idealized model Hamiltonian:

H = Hphoton + HJj. (4.1) j=1 X Here 1 H = ~ω(a†a + ) (4.2) photon 2

37 is the photon Hamiltonian, a† and a being the photon creation and annihilation op-

erators for photons having angular frequency ω, which satisfy the usual commutation

th relations. [a, a†]=1, [a, a] = [a†, a†] = 0. The Hamiltonian of the j Josephson junction can be expressed as

1 H = U(n n¯ )2 J cos γ . (4.3) Jj 2 j − j − j

Here U = 4e2/C is the capacitive energy of the junction, e is the electronic charge,

C is the junction capacitance. nj is an operator representing the diﬀerence in the number of Cooper pairs on the two superconducting islands forming the junction,n ¯j

th is related to the gate voltage across the j junction, J = ~Ic/(2e) is the Josephson

coupling energy of the junction, Ic is the junction critical current, and ﬁnally γj is

the gauge-invariant phase diﬀerence across the junction.

Explicitly, γj may be written 2π γ = φ A dl, (4.4) j j − Φ · 0 Zj

where φj is the phase diﬀerence across the junction in a particular gauge, and A is the vector potential due to the cavity mode (given explicitly below) in the same gauge, Φ0 = hc/(2e) is the ﬂux quantum, and the line integration is carried out across the junction. The operators nj and φj are canonically conjugate and satisfy

the commutation relation [nk,φl]= iδkl, which is satisﬁed if we use the representation

∂ nk = i . ∂φk In the Coulomb gauge, A = 0, A can be expressed as ∇· hc2 A = (a + a†)E(x), (4.5) rωV where V is the cavity volume, and E(x) is proportional to the local electric ﬁeld

of the cavity mode, normalized so that E(x) 2d3x = 1. Introducing a coupling V | | R38 parameter 2π hc2 g = E(x) dl, (4.6) j Φ ωV · 0 r Zj

we may rewrite γj as

γ = φ g (a + a†). (4.7) j j − j

Eq. (4.6) suggests that typically g 1/√V for a given mode (provided that the j ∝ cavity shape does not change as the volume increases.

4.2.2 Mean-ﬁeld approximation

We now develop a suitable mean-ﬁeld approximation for the Hamiltonian (4.1), for both zero and ﬁnite T . We consider only the case of identical Josephson junctions, so

that all gj = g andn ¯j =n ¯; the extension to non-identical junctions is straightforward

[49]. We also assume that the coupling parameters g are weak. In this case, we can

expand the cosine in eq. (4.3), retaining only the term of ﬁrst order in g(a + a†), so

that cos γ cos φ + g(a + a†) sin φ . Within this approximation, the only part of j ∼ j j the Hamiltonian that depends on both cavity and junction variables is

H gJ(a + a†) sin φ . (4.8) int ∼ − j j X The eigenvalues and eigenfunctions of H can now be found if we make the following

mean-ﬁeld approximation for Hint:

m H H gJ(a + a†) sin φ int ∼ int ≡ − h ji j X gJ a + a† sin φ − h i j j X + gJ a + a† sin φ . (4.9) h i h ji j X

39 Here ... denotes a canonical average at temperature T with respect to the mean-ﬁeld h i Hamiltonian Hm. Hm is now given by

m m m m H = Hphoton + HJj + Hc , (4.10) j X where

m 1 H = ~ω(a†a + ) gJ(a + a†) sin φ , (4.11) photon 2 − h ji j X m U 2 H = (n n¯) J cos φ gJ a + a† sin φ , (4.12) Jj 2 j − − j − h i j and

m H = gJ a + a† sin φ . (4.13) c h i h ji j X Evidently, the ﬁrst term depends only on the photon variables, the second is a sum

of single-junction terms, and the third is simply a c-number.

We now introduce the variable λ = sin φ = λ, since λ is independent of j. In j h ji j terms of λ, we may rewrite the photon term (4.11) as

m 1 H = ~ω(a†a + ) gJ(a + a†)Nλ. (4.14) photon 2 −

This is simply the Hamiltonian of a displaced harmonic oscillator. Its eigenvalues

2 2 2 2 E and normalized eigenfunctions ψ (x) are just E = ~ω(n + 1 ) J g N λ and n n n 2 − ~ω ip x /~ 1 ω 1/4 ω (x x )2 ω ψ (x) = x e− h i n = e− 2~ −h i H (x x ) . Here H is a n h | | i √2nn! π~ n ~ −h i n ~ω gJNλ 2 Hermite polynomial, p = i (a† a) is a momentump operator, x = 2 − h i ω ~ω q 2JgNλ q is the mean displacement, and we have used the relation a + a† = . The h i ~ω 2 ~ m m eβξ /( ω) canonical partition function corresponding to Hphoton is just Zphoton = 2 sinh β~ω/2 , where ξ = gJNλ and β =1/(kBT ).

m Next, we consider the Josephson junction Hamiltonian HJj [eq. (4.12)]. With 2 g2JNλ in¯(φ α) the deﬁnitions K(λ) = J 1+4 and ψ(φ ) = e− j − u(φ α), where ~ω j − r 40 2Jg2Nλ tan α = ~ω , the Schr¨odinger equation for the junctions, HJjψ(φj) = Eψ(φj), reduces to the standard Mathieu equation [30]:

2 d y(vj) 2 +(a 2q cos2vj)y(vj)=0, (4.15) dvj − where v =(φ α)/2, y(v )= u((φ α)/2), q = 4K(λ)/U, the characteristic value j j − j j − − of the Mathieu equation is a =8E/U, and we have used the representation n = i ∂ . j ∂φj

The allowed eigenvalues are determined by the condition that ψ(φj +2π) = ψ(φj),

or equivalently, that y(vj + π) = exp(2inπ¯ )y(vj). The allowed solutions yν(vj) are

therefore the Floquet (Bloch) functions of vj, with Floquet exponent ν = 2¯n +2k,

where k = 0, 1, 2, .... The corresponding eigenvalues of H are labeled by the ± ± Jj quantum number ν = 2¯n +2k and the parameter q, and may be denoted E(ν =

2¯n +2k; q). For 0 n¯ 0.5, the lowest eigenvalue corresponds to k = 0, followed in ≤ ≤ order by k = 1, 1, 2, 2, ... Including only these Floquet solutions, we can formally − − express the junction partition function as

m βE(2¯n+2k;q) m Z = e− Z , (4.16) Jj ≡ J k=0, 1, 2,... X± ± where the last identity holds for identical junctions.

We now determine the properties of the junction-cavity system within MFT. At

T = 0, the system is in its ground state, and the approximate ground state properties

can be obtained analytically as shown in Ref. [39]. For 0.5 n¯ 0.5, the ground − ≤ ≤ state energy is ~ω (gJNλ)2 U E (λ)= + + N a(ν = 2¯n; q), (4.17) g 2 ~ω 8 where a(ν; q) is the eigenvalue of eq. (4.15) corresponding to characteristic exponent

ν and parameter q. If we use the approximate analytical expression [39] q2 a(ν = 2¯n; q) 4 1 4¯n2 (1 4¯n2)2 + +4¯n2, (4.18) ∼ − − r − 4 ! 41 we obtain

~ω (gJNλ)2 U¯ 4J 2 g2JNλ 2 Un¯2 E (λ) + + N 1 1+ 1+4 + N , g ~  v ¯ 2 ~  ∼ 2 ω 2 − u U " ω # 2  u  t (4.19)   where U¯ = U(1 4¯n2). λ is determined by the condition dE0(λ) = 0, which leads to − dλ ~ω 2 U¯ 2 λ2(T = 0) 1 1+ . (4.20) ∼ − 2g2JN 4J 2 Because the right hand side of this equation must be non-negative, the critical junction number Nc(0) for a non-zero λ at T = 0 is

~ω N (T = 0) U¯ 2 +4J 2. (4.21) c ∼ 4g2J 2 p When N N (0), λ = 0 corresponds to a minimum of the energy, but when N ≤ c ≥ N (0), λ = 0 is a local maximum; the energy minimum occurs at λ = 0. c 6

More generally, the exact solution for Nc(T = 0) can be calculated from eq. (4.17),

dEg(λ) supplemented by the condition dλ = 0. The result is

2 2 ~ 2 (a′g JN) ( ω) λ(T =0)= 2 − , (4.22) p 2g JN

da(ν=2¯n;q) where a′(λ)= dq q= 4K(λ) . The corresponding critical number is | − U ~ω N (T =0)= , (4.23) c a (0) g2J | ′ |

where a′(0) = a′(λ) . |λ=0 For T = 0, because Hm is the sum of several commuting terms, the total parti- 6 tion function Zm = Zm (Zm)N Zm, where Z = exp( βHm). The corresponding photon J c c − c Helmholtz free energy F m is

F m = kT ln Zm = F m + NF m + F m, (4.24) − photon J c 42 2 where F m = kT ln (2 sinh (β~ω/2)) (gJNλ) , F m = k T ln Zm, and F m = photon − ~ω J − B J c 2 2 (gJNλ) . When the coherence order parameter λ = 0, the Helmholtz free energy, for ~ω 6 ﬁxed g, is quadratic in the number of the junctions N. This quadratic dependence is a hallmark of the coherent state.

The actual value of λ is obtained from the Helmholtz free energy, using the condition dF m(λ) =0. (4.25) dλ

We have obtained λ by solving eqs. (4.24) and (4.25) self-consistently. These equations

may allow for several possible values of λ, of which we choose that value which gives

the lowest F m.

In all the above discussion, we have assumed implicitly that g is independent of

N. The expected behavior when g depends on N is discussed below.

4.3 Dicke Model and Generalized Dicke Model

4.3.1 Model Hamiltonians

In the previous section, we described a simple mean-ﬁeld approximation for the

statistical mechanics of the junction-cavity system. This approximation includes all

the junction levels, but treats the junction-cavity interaction only approximately. We

now describe an alternative approach, which retains only the two lowest energy levels

of each junction. In this case, the Hamiltonian reduces to the well-known Dicke

model of quantum optics. The Dicke Hamiltonian [41] is a simple model describing

the interaction of N two-level systems with a single harmonic oscillator mode. It can

43 be written (omitting the cavity zero-point energy)

Dicke ~ 1 j j j H = ωa†a + ǫjσz + ξja†σ + ξj∗aσ+ . (4.26) 2 − j X th th Here ǫj > 0 is the energy level splitting of the j two-level system at the j junction,

and ξj is a parameter characterizing the strength of the coupling between the harmonic

th oscillator and the j two-level system. The quantities a† and a are raising and lowering

j operators, as above. The quantities σα are Pauli spin-1/2 spin operators and satisfy

j k i [σα, σβ]=2iδjkσγ , where α, β, and γ are cyclic commutations of (x,y,z). In order for the thermodynamic limit to exist, we must assume that ξ 1/√N for large N. j ∝ Besides the Dicke model, we also consider a modiﬁed Dicke model[45], which is an extension of eﬀective two-qubit model[50, 51, 52] to the case of N coupled two-level systems,

MDicke ~ 1 j j j j k j k H = ωa†a + ǫjσz + ξja†σ + ξj∗aσ+ + Ωjk(σ+σ + σ σ+), (4.27) 2 − − − j jk X Xh i where the last sum runs over all distinct pairs jk (i. e., not including j = k). The last term in eq. (4.27) is an eﬀective direct junction-junction interaction. As discussed in

Ref. [45], the Hamiltonian (4.27) generally gives levels in closer agreement with the

Hamiltonian (4.1) than does the pure Dicke Hamiltonian (4.26), when there is more than one photon excited in the cavity. A simple derivation of this term is given in

Ref. [45].

In order for the thermodynamic limit to exist in the modiﬁed Dicke model, we require not only that ξ 1/√N, and but also that Ω 1/(N 1), as further j ∝ jk ∝ − explained below.

44 4.3.2 Statistical mechanics of Dicke model using Glauber coherent state expansion

The thermodynamics of the Hamiltonian (4.26) can be calculated in the limit

N , using a product basis consisting of the Glauber coherent states α for the → ∞ | i i photons and eigenstates of σz for the two-level systems [47, 48]. In this section, we brieﬂy review this solution.

The states α are eigenstates of the lowering operator, i. e., | i

a α = α α . (4.28) | i | i where the eigenvalue α is generally complex, since a is a non-Hermitian operator. The eigenfunctions satisfy the completeness relation (1/π) ∞ ∞ dRe(α)dIm(α) α α = −∞ −∞ | ih | 1, the integral running over the entire complex α plane.R R

In terms of this basis, the partition function of the Dicke Hamiltonian takes the form

βHDicke ZDicke = Tre− (4.29)

∞ ∞ dRe(α)dIm(α) βHDicke = ... σ ...σ α e− α σ ...σ π h 1 N |h | | i| 1 N i σ1= 1 σ = 1 X± NX± Z−∞ Z−∞

∞ ∞ dRe(α)dIm(α) β~ω α 2 βh = e− | | σ e− j σ (4.30) π  h j| | ji j σ = 1 Z−∞ Z−∞ Y Xj ±   where

1 j j j hj = ǫσz + ξjα∗σ + ξj∗ασ+. (4.31) 2 −

In order to evaluate the sums, following Refs. [47] and [48], we expand the operator

Dicke exp( βH ) in a Taylor series, assume that a/√N and a†/√N exist in the limit − N , and ﬁnally, within individual terms of the Taylor expansion, interchange the →∞

45 order of the double limits as follows:

R ( βHDicke)r R ( βHDicke)r lim lim − = lim lim − . (4.32) N R r! R N r! →∞ →∞ r=0 →∞ r=0 →∞ X X Each sum in eq. (4.30) can easily be evaluated, since it is just the trace of the

operator exp( βh ), and the resulting partition function can be expressed as − j

2 dRe(α)dIm(α) β~ω α 2 βǫj 4 ξj α Z = e− | | 2 cosh 1+ | || | Dicke π  2 ǫ  j s j Z Y  2  ∞ β~ωr2 βǫj 4 ξj r = 2 rdre− 2 cosh 1+ | | , (4.33)  2 ǫ  0 j s j Z Y   dRe(α)dIm(α) ∞ ∞ ∞ where we have introduced r = α and written π =2 0 rdr. | | −∞ −∞ To complete the evaluation of the free energy,R weR make the changesR of variables

′ ξ 2 ξ = j and y = r . Then Z takes the form | j| √N N Dicke

∞ ZDicke = N dy exp [Nφ(y)] , (4.34) Z0 where 2 1 βǫ 16ξ′ y φ(y)= β~ωy + ln (2 cosh j 1+( j )) . (4.35) − N 2 ǫ2 j s j ! X The last integral can be evaluated accurately using Laplace’s method [53]. This method makes use of the fact that, if N 1, the integral should be dominated by ≫ values of y near the maximum of φ(y), which is determined by the condition

φ′(y)=0. (4.36)

4ξ′2β 16ξ′2y ~ 1 j βǫj j In our case, φ′(y) = β ω + ′ tanh( 1+( 2 )); so eq. (4.36) N j 16ξ 2 2 ǫ − j j ǫj 1+ y ǫ2 r P s j becomes 2 ~ω 1 ξ′ βǫ = j tanh j η , (4.37) 4 N ǫ η 2 j j j j X 46 ′2 16ξj y βǫj where ηj = 1+ ǫ2 , and lies in the range 1 < ηj < . For large β, tanh 2 ηj j ∞ → r ′2 ~ ξ 1. Thus, when ω > 1 j , φ(y) is maximum at y = 0. On the other hand, when 4 N j ǫj ξ′2 ~ω < 1 j , the allowedP solutions to eq. (4.36) depend on the value of β. When β 4 N j ǫj

is smallerP than a critical value βc given by

2 ~ω 1 ξ′ β ǫ = j tanh c j , (4.38) 4 N ǫ 2 j j X then again φ(y) is maximum for y = 0. However, if β > βc, there is a non-zero solution y0 for y determined by the equation

2 2 ~ω 1 ξ′ βǫ 16ξ′ y0 = j tanh j 1+ j . (4.39) 4 N ′2 2 ǫ2 j 16ξj y0 s j ! X ǫj 1+ ǫ2 r j Therefore, we can discuss the statistical mechanics of this model in three diﬀerent

′2 ′2 ′2 ξj ~ω ξj ~ω ξj ~ω regimes: (i) ; (ii) and T Tc; and (iii) and j Nǫj ≤ 4 j Nǫj ≥ 4 ≥ j Nǫj ≥ 4 T T , whereP T is determinedP by eq. (4.38). In the regimes (i)P and (ii), the free ≤ c c energy is given simply by

F 1 βǫj lim = kBTφ(y)y=0 = kBT ln (2 cosh ) (4.40) N N − − N 2 →∞ j X and the moments of the photon occupation number by

a a k † =(yk) = δ , (4.41) h N i y=0 k0 where the last result is obtained once again by using Laplace’s method. On the other

hand, in regime (iii), F lim = kBTφ(y)y=y0 (4.42) N →∞ N − and a a k † =(yk) yk (4.43) h N i y=y0 ≡ 0 47 where y0 is determined by eq. (4.39).

Assuming that all junctions are identical, ǫ ǫ, ξ ξ,λ λ, the conditions j → j → j → for the critical junction number and the critical temperature become

~ωǫ N Dicke = , (4.44) c 4 ξ 2 | | 1 ǫ k T Dicke = = . (4.45) B c ~ βc 1 ωǫ 2 tanh− 4N ξ 2 | | k k Furthermore, the moments (a†a) = Ny are obtained from h i 0

2 2 ~ωǫ 16Nξ y0 βǫ 16Nξ y0 2 1+ 2 = tanh 1+ 2 . (4.46) 4Nξ r ǫ 2 r ǫ 4.3.3 Statistical mechanics of modiﬁed Dicke model using a mean-ﬁeld approximation

Next, we consider the modiﬁed Dicke model, eq. (4.27). The coherence transition

can be obtained if we make the following mean-ﬁeld approximation:

j j j j a†σ a† σ + σ a† σ a† , (4.47) − ∼ h i − h −i −h −ih i aσj a σj + σj a σj a , (4.48) + ∼ h i + h +i −h +ih i j k j k k j j k σ+σ σ+ σ + σ σ+ σ+ σ , (4.49) − ∼ h i − h −i −h ih −i j k j k k j j k σ σ+ σ σ+ + σ+ σ σ σ+ . (4.50) − ∼ h −i h i − −h −ih i

With the additional assumption that the ξj’s are real, the Hamiltonian (4.27) sepa-

rates into a sum of three terms as follows:

MDicke MDicke MDicke MDicke H = Hphoton + HJj + Hc , (4.51) j X

48 where

MDicke 1 j j H = ~ω(a†a + )+ ξ ( σ (a + a†)+ i σ (a a†)), (4.52) photon 2 j h xi h yi − j X MDicke ǫj j j j j k k j HJj = σz + ξj a† σ + a σ+ + Ωjk( σ+ σ + σ (4.53)σ+), 2 h i − h i h i − h −i j j j j k=j X X X X X6 MDicke j j j k Hc = ξj a† σ + a σ+ Ωjk σ+ σ , (4.54) − h ih −i h ih i − h ih −i j jk X Xh i j j j j j j and we have used the relations σ+ = σx + i σy and σ = σx i σy . h i h i h i h −i h i − h i The free energy associated with each term in the above Hamiltonian can be eval-

MDicke uated separately. Hphoton is the Hamiltonian of a harmonic oscillator displaced in both momentum and position space. Introducing the operators

~ ~ω u = (a + a†), p = i (a† a), (4.55) r2ω r 2 − MDicke we find that Hphoton can be rewritten as 1 mω2 HMDicke = (p p )2 + (u u )2 photon 2m − 0 2 − 0 ξ ξ ( σj σk + σj σk ) jk j k h xih xi h yih y i ~ . (4.56) −P ω where p = 2 σj ξ and u = 2 σj ξ . The corresponding eigenval- 0 ~ω jh yi j 0 − ~ω3 jh xi j q q ues are P P j k j k 1 ξjξk( σ σ + σ σ ) MDicke ~ jk h xih xi h yih y i Ephoton (n)= ω(n + ) ~ . (4.57) 2 − P ω j ξ σ− j j h i Also, from the fact that p = p0 and u = u0, we obtain a = ~ω , a† = h i h i h i − P h i ξ σj j j h +i ~ω . − P MDicke HJj is the Hamiltonian of a collection of non-interacting spin-1/2 particles in an applied effective magnetic field (which is not parallel to the z axis). The two

MDicke eigenvalues of HJj are readily found to be

ǫ2 MDicke j Ωjk k Ωjk k EJj = +4 ξj a† + σ+ ξj a + σ . (4.58) ±v 4 h i 2 h i h i 2 h −i u k=j ! k=j ! u X6 X6 t 49 MDicke MDicke Finally, Hc is just a c-number whose expectation value is just Ec =

MDicke Hc .

For N identical junctions, ǫ ǫ, ξ ξ, and Ω Ω. Then, after some algebra, j → j → jk → one ﬁnds that the ground state energy can be written in terms of a single expectation

2 2 1/2 value σ = [ σx + σy ] . The result is h ⊥i h i h i

2 2 2 ~ 2 2 2 MDicke N ξ σ (N 1) ωΩ ǫ 2 (N 1) Nξ E0 ( σ )= h ⊥i 1 − N +4 σ − Ω . h ⊥i ~ω − 2Nξ2 − 4 h ⊥i 2 − ~ω s (4.59)

MDicke σ is again determined by the requirement that E0 ( σ ) be a minimum with h ⊥i h ⊥i respect to σ , which leads to h ⊥i 1 ~ωǫ 2 σ 2 =1 . (4.60) h ⊥i − 4 2Nξ2 (N 1)~ωΩ − − The condition σ 2 > 0 leads to the critical number of junctions h ⊥i ~ωǫ 1 2Ω N MDicke = − ǫ , (4.61) c 4ξ2 1 Ω~ω − 2ξ2 above which σ 2 is non-negative. When Ω = 0, this critical number exactly corre- h ⊥i sponds to the critical number obtained in Section 4.3. Thus, for the Dicke model, this

MFT yields the same critical junction number as obtained from the coherent state analysis.

At ﬁnite T , the properties of the modiﬁed Dicke model are obtained from the

Helmholtz free energy F MDicke. An analysis similar to that at T = 0 again allows

F MDicke to be written as the sum of three terms, which for N identical junctions may be written

F MDicke = k T ln ZMDicke Nk T ln(ZMDicke)+ EMDicke, (4.62) − B photon − B J constant

50 2 MDicke 1 MDicke ǫ2 2 (N 1) Nξ2 where Z = ~ ; Z = 2 cosh β +4 σ − Ω , photon 2 sinh β ω J 4 2 ~ω 2 h ⊥i − ! r MDicke N 2ξ2 σ⊥ 2 (N 1)Ω~ω and E = h i 1 − , which also includes constant contributions constant ~ω − 2Nξ2 from photon and junction terms.

As at T = 0, the optimal value of σ at ﬁnite T is obtained by minimizing h ⊥i F MDicke with respect to σ , which leads to the following relation for σ : h ⊥i h ⊥i ~ωǫη βǫ = tanh η , (4.63) (N 1)~ωΩ 4Nξ2 1 − 2 − 2Nξ2 2 16N 2ξ4 σ⊥ 2 (N 1)~ωΩ MDicke where η = 1+ h i 1 − . The critical temperature T for ǫ2(~ω)2 − 2Nξ2 c r this modiﬁed Dicke model is again determined by the requirement that σ 2 > 0, h ⊥i and is given by

MDicke ǫ kBTc = . (4.64) 1 ~ωǫ 1 2 tanh− 2 ~ − 4Nξ 1 ω(N 1)Ω − 2Nξ2 “ ” MDicke kBTc = 0 for 2Nξ2 ~ω Ω > Ω = 1 . (4.65) max (N 1)~ω − 4Nξ2 − When Ω = 0, eq. (4.64) for T reduces to eq. (4.45), provided we assume a† a = c h ih i 2 2 ~ω a†a and use the relation σ = a† a . Thus, both MFT and the coherent h i h ⊥i Nξ h ih i state expansion lead to the same thermodynamic properties for the Dicke model.

4.4 Numerical Results

We have carried out several illustrative numerical calculations using the models and approximations of Sections 4.2.2 and 4.3. For the MFT of Section 4.2.2, these results are obtained by minimizing the mean-ﬁeld Helmholtz free energy, eq. (4.24), with respect to λ at each T for ﬁxed g andn ¯. In all our calculations, we have taken

~ω =0.15U, J =0.2U, g = 0.1, and N = 110; other parameters are described below.

51 0.8 t=0, n=0.5- t=0.12, n=0.5- t=0, n=0- t=0.12, n=0- 0.6 λ

0.4

Order Parameter 0.2

0 60 80 100 120 140 160 180 Number of Junctions N

Figure 4.1: Coherence order parameter λ(N, t), plotted as a function of the number of junctions N, forn ¯ =0 and 0.5 at values of the scaled temperature t =0 and 0.12, as indicated in the legend, using J =0.2U, ~ω =0.15U, and g =0.1.

It is convenient to introduce a dimensionless temperature t kBT . Except for g, these ≡ U are in the same ratios as in recent eperiments of Ref.[54] using the correspondence,

(~ω,U/8,J) in our notation to (~ω =6.0GHz, E 5.0GHz, and E 8.0GHz) r C ∼ J,max ∼ in the experiment. However, we have used a much larger value of g, in order to see the transition to a coherent state at a reasonable value of N.

Figs. 4.1, 4.2, 4.3, 4.4, and 4.5 show mean-ﬁeld results for the Hamiltonian of

Section 4.2, including all Josephson levels. First, we consider the coherence order parameter λ(N, T ) assuming g independent of N. Fig. 4.1 shows λ(N, T ) forn ¯ = 0 and 0.5 at t = 0 and 0.12. In all cases, there is obviously a threshold number of junctions N (t) below which λ vanishes, and above which λ = 0. For a suﬃciently c 6 large N, λ 1, signaling complete phase locking. Fig. 4.1 shows that both λ(t) and → 52 0.6

t=0 t=0.12

λ t=0.14 0.4 t=0.16

0.2 Order Parameter

0 -0.4 -0.2 0 0.2 0.4 n-

Figure 4.2: Coherence order parameter λ(¯n, t), at several values of the scaled temperature t = 0, 0.12, 0.14 and 0.16, as indicated in the legend, using J = 0.2U, ~ω =0.15U, g =0.1, and N = 110.

Nc(t) decrease with increasing t, and thatn ¯ =0.5 leads to a larger λ at ﬁxed t than doesn ¯ = 0. Both features are intuitively reasonable, since atn ¯ =0.5, the two lowest states of the junction have only a small gap, making it easier to couple the junction to the cavity.

Fig. 4.2 shows λ(¯n, t) as a function ofn ¯, which is related to the voltage across the

Josephson array, at t = 0, 0.12, 0.14 and 0.16. Since λ(¯n, t) is a periodic function of n¯ with period unity, we plot only the range 0.5 n¯ 0.5. All the plots of Fig. 4.2 − ≤ ≤ show that, for any choice of the other parameters, λ is maximum atn ¯ = 0.5. The plots also show that there exist values of N such that the array is coherent for some non-zero values ofn ¯ even if it is incoherent atn ¯ = 0. Finally, Fig. 4.2 shows, as

53 λ 0.4

0.2 -

Order Parameter n=0.5 n=0.4- n=0.3- n=0.2- n=0.1- n=0- 0 0 0.05 0.1 0.15 Temperature t

Figure 4.3: Temperature dependence of coherence order parameter λ(t) forn ¯ = 0, 0.1, 0.2, 0.3, 0.4 and 0.5, using J =0.2U, ~ω =0.15U, g =0.1, and N = 110.

expected and as is also shown in Fig. 4.1, that the eﬀect of increasing t at ﬁxed N and g is to suppress λ.

The temperature dependence of λ(¯n, t) is plotted versus t in Fig. 4.3 for diﬀerent values ofn ¯. In all cases, there is a critical temperature tc above which λ = 0. Note also that, as t t from below, λ 0 continuously. This behavior is a hallmark → c → of a continuous phase transition. However, since there are only a ﬁnite number of junctions, the transition is not a true thermodynamic phase transition.

We have also calculated the average Cooper pair diﬀerence n across the jth h ji junction, within the mean-ﬁeld approximation. Since all the junctions are assumed identical, n is independent of j and may be denoted n . n is related to the h ji h i h i voltage drop V across a junction by CV/2e = n , and can be calculated from the h i

54 0.4 t=0 t=0.12 t=0.16 t=0.2 0.2 t=0.24

-0.2

-0.4

-0.4 -0.2 0 0.2 0.4 n-

Figure 4.4: Average Cooper pair number diﬀerence n at t = kBT/U = 0, 0.12, 0.16, 0.2, and 0.24. In all cases, J =0.2U, ~ω =0.15hU,i g =0.1, and N = 110 where tc(¯n =0)=0.124 and tc(¯n =0.5)=0.181.

relation βE(ν=2¯n+2k,q) k=0, 1 2... e− n k ± ± h i n = βE(ν=2¯n+2k,q) . (4.66) h i k=0, 1, 2,... e− P ± ± where n , the expectation valueP of the operator n in state k, is h ik 2π π ∂ dyν=2¯n+2k(v) n = ψ∗ (φ) i ψ (φ)dφ =n ¯+i y (v) dv. h ik ν=2¯n+2k ∂φ ν=2¯n+2k ν=2¯n+2k dv Z0 Z0 (4.67)

∂ i ∂ The last expression is obtained using the relations n = i ∂φ = 2 ∂v and ψν (φ) =

in¯(φ α) 2inv¯ e− − u (φ α)= e− y (v). In Fig. 4.4, we show this calculated n (¯n, t) versus ν − ν h i n¯ for several values of t. Note that, t t , n n¯. ≫ c h i ∼ Next, we discuss the temperature dependence of the photon probability distribu-

tion P (n, t) calculated in this mean-ﬁeld approximation. P (n, t) is deﬁned simply as

55 the probability that the cavity contains exactly n photons at temperature t. Since there is no coherence for t > tc, P (n) is given simply by the usual Bose distribution with zero chemical potential:

e β~ω(n+1/2) e β~ω(n+1/2) P (n)= − = − (4.68) β~ω(l+1/2) ~ l∞=0 e− 2 sinh β ω/2 [eq. (4.14)] P

P (n)= n ρ n . (4.69) h | | i

m Using the solutions of Hphoton,

1 ∞ βEl ip x /~ ip x /~ ρ = m e− e− h i l l e h i , (4.70) Zphoton | ih | Xl=0 m m where Zphoton is the partition function corresponding to Hphoton. Thus, we obtain,

∞ ∞ m βE ip x /~ ip x /~ ip x /~ 2 βE P (n)Z = e− l n e− h i l l e h i n = n e− h i l e− l , photon h | | ih | | i |h | | i| l=0 l=0 X X (4.71) where

ip x /~ ∞ ip x /~ ∞ n e− h i l = n x x e− h i l = ψ∗(x + x )ψ (x)dx. (4.72) h | | i h | | ih | | | i n h i l Z−∞ Z−∞ For the consideration of only l = 0 term, the probability function P (n) corresponds

where tc = 0.131. From low t up to near tc, P (n, t) is substantial over a wide range

of n, but for t t , the population of the photon state with n = 0 rapidly increases. ≥ c Next, we compare the mean-ﬁeld results of Section 4.2 (which includes all junction

levels) to the results of Section 4.3 for the Dicke model and the modiﬁed Dicke model.

56 t=0 t=0.12 0.6 t=0.13 t=0.14

0.4

0.2 Photon Distribution P(n)

0 0 5 10 15 Photon Number

Figure 4.5: Photon number distribution P (n) for the Josephson junction model at various temperatures: t = 0, 0.12, 0.13, and 0.14. In all cases, J = 0.2U, ~ω = 0.15U, g = 0.1, N = 110, andn ¯ = 0. For these parameters, tc = 0.131. P (n) represents the probability that there are exactly n photons in the cavity mode.

We consider speciﬁcallyn ¯ = 0.5; at this value ofn ¯, the two-level approximation may be best, because the two lowest junction levels are separated by the largest gap from the higher levels. In order to compare the three models, we plot tc(N) in

Fig. 4.6(a), and the average photon number a†a (t) in Fig. 4.6(b). For comparison h i purposes, we choose the Dicke parameter ξ = gJ , and the Dicke parameter ǫ = − √2 E(2 2¯n, q) E( 2¯n, q)=(U/8)[a(2 2¯n, q) a( 2¯n, q)] [55]. Also, we treat Ω − − − − − − simply as a parameter determined by best ﬁtting to the results of the MFT. We

denote the critical number of junctions and the critical temperature of the MFT by

m m Dicke Dicke Nc (t), tc ; of the Dicke model, by Nc (t), tc ; and of the modiﬁed Dicke model,

MDicke MDicke m by Nc (t), tc . The tc ’s are obtained numerically; the other tc’s are obtained

Dicke MDicke from eq. (4.45) for tc and from eq. (4.64) for tc . In the mean-ﬁeld case, when

57 the coherence order parameter vanishes we just have the Bose result for the average

photon occupation number:

m 1 a†a m(t > tc )= ~ω . (4.74) h i e kB T 1 − On the other hand, in the coherent state, we have

∞ m a†a (t < t )= nP (n). (4.75) h im c n=0 X

For the other two models, a†a and a†a are calculated from the condi- h iDicke h iMDicke 2 ~ω 2 tions (4.46) and (4.63) with σ =( ) a†a , respectively. h ⊥i Nξ h i

In Fig. 4.6, we plot t (N) and the average photon number a†a (t), for the three c h i models. To compare the MFT with the modiﬁed Dicke model, we have considered

four choices for Ω, corresponding to Ω~ω/ ξ 2 =0.2, 0.4, 0.6 and 0.8. All three models | | show the same qualitative behavior, i. e., a transition from coherence to incoherence

with decreasing N or increasing t. The solutions of the Dicke and modiﬁed Dicke

models are qualitatively in good agreement with that of the MFT; however, the

solution of the modiﬁed Dicke model with Ω = 0.8 ξ 2/(~ω) agrees better with MFT | | than do any of the other three.

The behavior of a†a (t) diﬀers somewhat among the three models. For the MFT h i m model, a†a (t) 0 as t t from below, and remains very small, but non-zero, h i → → c m for t > t . On the other hand, in both the Dicke and modiﬁed Dicke models, a†a c h i

reaches exactly zero at t = tc and remains zero for t > tc. The most conspicuous

qualitative diﬀerence between the two models occurs at large t, where the Dicke and

modiﬁed Dicke models give a†a (t) = 0, while a†a (t) in the MFT increases with h i h i increasing t according to the Bose distribution. This discrepancy probably occurs

58 Dicke MDicke 0.2 MDicke 0.4 MDicke 0.6 MDicke 0.8 c 0.2 MFT (a)

0.1 Critical Temperature t

0 40 60 80 100 Number of Junctions N

Dicke 30 MDicke 0.2 MDicke 0.4 MDicke 0.6 MDicke 0.8 MFT

20 (b)

10 Average Photon Number

0 0 0.05 0.1 0.15 Temprature t

Figure 4.6: Comparison between the predictions of the mean-field approximation, the Dicke model, and the modified Dicke model for the critical temperature [part (a)] and the average photon number, a†a (t) for N = 70 [part (b)], atn ¯ = 0.5. For the Josephson junction model, we use theh parametersi J =0.2U, ~ω =0.15U, and g =0.1; m m these lead to Nc (0) = 64.0 and tc = 0.0681. The corresponding critical numbers Dicke Dicke for the Dicke model are Nc (0) = 37.1 and tc =0.168. For the modified Dicke model, we show plots with Ω~ω/ ξ 2 =0.2, 0.4, 0.6 and 0.8. | |

59 because the ﬁrst two models include only two levels per junction, while the MFT of

Section 4.2 treats a many-level system.

4.5 Thermodynamic Limit

We now make more precise the connection between the Josephson-cavity model

and the Dicke and modiﬁed Dicke models in the thermodynamic limit. We ﬁrst

consider the Dicke model at T = 0. For N identical two-level systems, the condition

for the onset of coherence at T = 0 is given by eq. (4.44). With the assumption

ξ = ξ/˜ √N, this condition becomes

4ξ˜2 = ~ω. (4.76) ǫ

To map the Josephson-cavity model onto the Dicke model, we assume that the

Josephson coupling parameter g 1/√N. As mentioned earlier, this assumption ∝ seems reasonable in the thermodynamic limit, since according to eq. (4.6), g 1/√V ∝ for ﬁxed cavity shape. Writing g =g/ ˜ √N, we may express the Josephson coherence

condition (4.23) as

2 a′(0) g˜ J = ~ω. (4.77) | |

We now show that this condition reduces to eq. (4.76) in the limit J U. The ≪ eigenvalue derivative a′(0) can be obtained approximately in this limit by diﬀeren- | | tiating the right-hand side of eq. (4.18) with respect to q. Substituting back into eq.

(4.77), we obtain 4(˜gJ)2 = ~ω. (4.78) U 2(1 4¯n2)2 +4J 2 − We can also compute the splittingp between the ground and ﬁrst excited states. If 0 < n<¯ 1/2, and J U, it is easily shown that the splitting ∆E between the ground and ≪

60 ﬁrst excited states of the Josephson junction is equal ∆E 1 [U 2(1 2¯n)2 +4J 2]1/2[56]. ∼ 2 − Using the approximation, 1 2¯n 1 4¯n2, we can rewrite the coherence condition − ∼ − (4.78) as 2˜g2J 2 = ~ω. (4.79) ∆E

This condition is identical to eq. (4.76), with the identiﬁcation ǫ ∆E, ξ˜ gJ˜ . ↔ ↔ − √2 The parameter Ω has primarily a quantitative eﬀect on the coherence transition in

this model. As discussed earlier, in order for the modiﬁed Dicke model to be well-

behaved in the thermodynamic limit, Ω must vary as 1/N. Therefore, we write

Ω=2Ω˜/(N 1); we use N 1 rather than N since each two-level system interacts with − − N 1 others. Substituting this relation into eq. (4.60), and again using ξ = ξ/˜ √N, − we ﬁnd that eq. (4.76) is replaced by

4(ξ˜2 ~ωΩ)˜ − = ~ω. (4.80) ǫ

If the left-hand side is larger than ~ω, the system is coherent at T = 0 in the thermodynamic limit; otherwise, it is not. Thus, a positive Ω˜ actually inhibits coherence at T = 0 (not unexpectedly, since a positive Ω˜ represents a repulsive interaction).

MDicke Similarly, one can recalculate kBTc [eq. (4.64) using the above N-dependence

of ξ and Ω, with the result

MDicke ǫ kBTc = . (4.81) 1 ~ωǫ 1 2 tanh− ~ ˜ 4ξ˜2 1 ωΩ − ξ˜2 Thus, for given values of ω, ǫ, and ξ˜, the coherence transition temperature is reduced by a ﬁnite Ω,˜ showing that this form of direct interaction between junctions inhibits coherence in the modiﬁed Dicke model.

61 4.6 Discussion

In this paper, we have calculated the equilibrium properties of an array of identical

Josephson junctions coupled to a single-mode electromagnetic cavity at temperature

T , by generalizing a T = 0 MFT [39]. Within the MFT, this system shows a continuous transition between coherence and incoherence at a critical temperature Tc, provided that the number of junctions N > Nc. We have also compared our mean-

ﬁeld results to the solutions of the Dicke and modiﬁed Dicke models. When the parameters of the Dicke model are adjusted to match those of the Josephson-cavity system, the two approaches agree qualitatively.

Next, we briefly discuss the expected accuracy of our mean-field approach, used in Sections 4.2 and 4.3. The MFT appears reasonable, because for both models, all the junctions (or all the two-level systems) interact with the same harmonic mode and hence, in effect, with all the other junctions or two-level systems. Since each junction or two-level system effectively has many “neighbors”, there are only small

fluctuations in the environment of each about its mean, provided that N is sufficiently large. Thus, the mean-field approach should work well at large N. In support of this picture, we have shown that, when the mean-field approach is applied to the Dicke model, it produces the exact result (obtained from a coherent state expansion).

Besides the mean-ﬁeld approximation in Section 4.2, we have expanded the Joseph- son coupling in powers of the interaction parameter g(a+a†). The value of this quan-

2 2 tity can be estimated as follows. g(a+a†) g a+a† =2Jg Nλ/(~ω)=2Jg˜ λ/(~ω) ∼ h i if we assume g =g/ ˜ √N. Since λ 1 is small near T , this approximation is accurate ≪ c in this regime, but may break down deep in the coherent regime. Therefore, a more

62 accurate approach than that used in Section 4.2, may be desirable in order to treat

the entire regime 0

We brieﬂy comment on the nature of the coherence transition emerging from

our mean-ﬁeld approach. This approach produces a continuous transition, i.e., the

coherence order parameter λ varies continuously with t. By contrast, another recent

calculation [57] ﬁnds a ﬁrst-order transition, in which there is a discontinuous jump

in the order parameter at the superradiant transition. Their Hamiltonian also has

the form of a generalized Dicke model, but slightly different from ours: N λ j j ǫ j j j+1 Hlee = a†a + (a + a†)(σ+ + σ )+ σz Jσyσy , (4.82) √ − 2 − j=1 2 N X j j j th where σ+, σi , and σz are the usual Pauli spin operators for the j two-level system. Our mean-field treatment of our own generalized Dicke model does not give a first-

order transition. We speculate that the diﬀerence is due to a real distinction between

the two models: Hlee has only nearest-neighbor interactions between spins, in addi-

tion to the usual Dicke-type model, whereas our modiﬁed Dicke Hamiltonian has an

additional term which is long-range. Perhaps the long-range nature of this additional

term helps to maintain the continuous nature of the coherence transition, as well as

the accuracy of MFT.

Finally, we discuss how our model could be generalized in order to make it more

a more realistic basis for treating Josephson arrays in a cavity. For real systems,

there are other factors aﬀecting Josephson junctions besides those included here. For

example, there are eﬀects due to dissipation, either due to the ﬁnite Q of the cavity,

or a ﬁnite dissipation within individual junctions. Both eﬀects can be treated by

considering the Josephson junctions as coupled to appropriate baths of harmonic

oscillators [58, 59]. When these dissipative degrees of freedom are properly included,

63 the nature of the coherence transition may be changed. We plan to include some of these eﬀects, as well as the eﬀects of disorder, in a future publication.

64 CHAPTER 5

CONDUCTIVITY AND SPIN SUSCEPTIBILITY FOR THE DISORDERED 2D HUBBARD MODEL

The eﬀect of disorder on a class of transition metal oxides described by a single orbital Hubbard model at half ﬁlling is investigated. The phases are characterized by the nature of the electronic and spin excitations. The frequency and temperature- dependent conductivity and spin susceptibility as functions of disorder are calculated.

The interplay of disorder and electron-electron interaction produces unusual behavior in this system. For example, the dc conductivity, which is vanishingly small at low disorder in the Mott phase and at high disorder in the localized phase, gets surprisingly enhanced at intermediate disorder in a ”metallic” phase. Moreover, the spin susceptibility in this ”metallic” phase is not the expected Pauli-behavior but

Curie-1/T due to the presence of local moments.

5.1 Introduction

Based on the scaling theory, it is expected that non-interacting electrons are localized for any weak disorder in two dimensions in zero magnetic field [60]. This theoretical result was questioned when transport experiments in 2D silicon metal- oxide-semiconductor field-effect transistors (MOSFET’s) showed a metallic phase

65 dρ with dT > 0, where ρ is the resistivity, at higher density of carriers than the crit-

dρ ical density of carriers, nc and an insulating phase with dT < 0 at lower density of carriers [61]. Further experiments supported the observation of a MIT in 2D systems

[62, 63, 64, 65, 66, 67, 68, 69].

The interplay of disorder and electron-electron interaction can have an important eﬀect on the MIT in 2D systems. The recent experiments have shown that electron- electron interaction can lead to a possible metallic phase in the 2D electron gas.

Motivated by these experiments, we investigate the 2D disordered Hubbard model at half-ﬁlling [70]. In previous work, a possible metallic phase with extended wave functions sandwiched between a Mott insulator for low disorder and a localized insulator for large disorder at T = 0 was found. In the metallic phase, the spectral gap closed

but antiferromagnetism was found to persist. The existence of a metallic phase was

conﬁrmed by ﬁnite size scaling behavior of the inverse participation ratio (IPR). At

intermediate disorder, the IPR for U = 0 is extrapolated to a ﬁnite value at L , →∞ thus the localization length being a ﬁnite value and the system being a localized state

and for the interacting system the IPR as a function of 1/L is extrapolated to zero,

meaning that the localization length diverges for the inﬁnite system and the system

is a metallic phase. On the other hand, the non-interacting system with the same

disorder realization had an IPR that extrapolated to a ﬁnite value implying a ﬁnite

localization length in the macroscopic system. Thus interactions clearly played an

important role in delocalizing the wave functions or at least in making these much

more extended than the wave function in the absence of interaction.

The study of the Hubbard model has been stimulated after the discovery of cuprate

superconductors for the example, YBa2Cu3O7 x, where x is the doping concentration −

66 Figure 5.1: Systematic phase diagram of cuprate.

[71]. x = 0 corresponds to a half-ﬁlling with 1 electron per siteandx=1 n when the − concentration of electron, n = Nelec . A systematic phase diagram of the copper oxide Nsite material is shown in Fig. 5.1. Upon doping, the N´eel temperature, TN decreases. On

the other hand, Tc shows a dome shaped dependence on x. This doping not only

gives superconductivity but also introduces disorder. At x = 0 the system is a Mott

insulator with anti-ferromagnetic order. A realistic modeling will therefore need to

consider both eﬀects of electron correlation and disorder.

Our focus is to understand the competition between random disorder and electron- electron interaction on the Mott phase with long-range AF order in 2D. In order to characterize the phases, we calculate the temperature and frequency dependent

67 conductivity and spin susceptibility. The calculation are done for the two-dimensional

disordered Hubbard model on lattices of size up to 32 32 at T = 0 and as function × of temperature and disorder using a numerical Hartree-Fock self-consistent method.

Our results are reported at half ﬁlling for interaction strength, U =4t.

The remainder of this paper is organized as follows. In Section 5.2, we give the model and the Hartree-Fock method. In Section 5.3, the self-consistent numerical method is presented. In Section 5.4, the important results from previous work on the disordered Hubbard model are discussed. In Section 5.5, we give our results of the conductivity and spin susceptibility. This is followed by a concluding discussion and an outline of possible future research.

5.2 Model

We consider the disordered strongly correlated electron system described by the

2D one-band Hubbard model with site disorder:

H = t ciσ† cjσ + cjσ† ciσ + Uni ni + (Vi µ)ciσ† ciσ (5.1) − ↑ ↓ − ij ,σ i i,σ hXi X X where t is the near-neighbor hopping amplitude, U is the on-site repulsion between electrons, ciσ† (ciσ) is the creation (destruction) operator for an electron with spin σ on site ~ri, njσ = cjσ† cjσ is the number operator of electrons with spin σ at site ~ri, and

µ is the chemical potential. Vi is local disorder potential at site ~ri and is chosen from the uniformly distributed interval, [ V,V ], where V is the measure of the strength − of disorder.

The Hubbard model with U 10t describes the essential physics of the CuO ∼ 2 plane of the copper oxide perovskites upon doping. At half ﬁlling, n = 1, with no h i

68 disorder, the system at T = 0 has long-range antiferromagnetic (AF) order and is a

Mott-type insulator.

The mean-ﬁeld approximation is applied for the electron-electron interaction with the following prescription:

ni ni ni ni + ni ni ni ni ci† ci ci† ci ci† ci ci† ci + ci† ci ci† ci . ↑ ↓ ≃h ↓i ↑ h ↑i ↓ −h ↓ih ↑i − ↑ ↓h ↓ ↑i − ↓ ↑h ↑ ↓i h ↑ ↓ih ↓ ↑i (5.2)

Then the eﬀective Hamiltonian, ignoring constant terms, iU ni ni + ci† ci ci† ci , − {h ↑ih ↓i h ↑ ↓ih ↓ ↑i} becomes P

+ Heff = t ciσ† cjσ + (Vi µ)ciσ† ciσ + U niσ¯ ciσ† ciσ + (hi−ci† ci + hi ci† ci ), − − h i ↑ ↓ ↓ ↑ ij ,σ i,σ i,σ i hXi X X X (5.3)

+ ¯ ¯ where hi = U ci† ci , hi− = U ci† ci , = , and = . In our calculation, we − h ↑ ↓i − h ↓ ↑i ↑ ↓ ↓ ↑ + + assume hi and hi− are real so hi = hi = hi−. We, therefore, have 3N variational parameters, n and h that must be determined self-consistently. h iσi i Explicitly, the eﬀective Hamiltonian for N lattice sites in the bases, 1 , 1 , 2 , 2 | ↑i | ↓i | ↑i | ↓i ... N , N with periodic boundary condition is: | ↑i | ↓i

W1 h1 t 0 . . . t 0 ↑ − − h1 W1 0 t ... 0 t  ↓ − −  t 0 W2 h2 . . . 0 0 − ↑  0 t h2 W2 . . . 0 0   − ↓   ......  , (5.4)    ......     ......     t 0 0 0 ...WN hN   − ↑   0 t 0 0 ... hN WN   − ↓    where W = V + U n µ. iσ i h iσ¯ i −

69 5.3 Method

The inhomogeneous Hartree-Fock approximation for the disordered Hubbard model

generates a matrix with local densities, n , and local magnetic ﬁelds, h , which are h iσi i variational parameters and must be solved self-consistently for each µ [70, 72]. For

half ﬁlling, µ = U/2 with no disorder but in the presence of disorder, µ is adjusted

to satisfy the half ﬁlling condition. The initial input parameters for n and h are h iσi i determined by assuming speciﬁc phases. For low disorder we start the self-consistent

procedure with an AF initial condition for n . On the other hand, for high dis- h iσi

order a paramagnetic state is used for the stating conﬁguration. For both cases, hi

are chosen from an AF initial condition from 0.1t or 0.1t. The initial condition −

for hi should not affect the results. The Broyden method is used for achieving self- consistency efficiently [73]. Input and output fields are compared after each iteration

4 and if the difference of the fields at all sites is less than 10− , the self-consistent loop is exited. Using these final self-consistent fields, n and h , and eigenvalues, inter- h iσi i ested physical quantities such as conductivity are calculated. The above method is tested for the 1D Hubbard model in Appendix. We also test that the self-consistent values of the parameters are not dependent on the initial starting conditions for the parameters.

5.4 Review for Numerical Simulation of the 2D Disordered Hubbard Model

In this section, I review the previous work on the disordered Hubbard model

[70, 72]. The most signiﬁcant result is that electron-electron interaction, U, leads to

a possible metallic phase for the 2D disordered Hubbard model at half ﬁlling. For

70 the speciﬁc case studied of U = 4t, a metallic phase with extended wave functions around V =3t sandwiched between a Mott insulator for low disorder and an Ander-

son insulator for large disorder is formed. The metallic phase is characterized by a

vanishing spectral gap but with long-range AF. Most importantly, the wave functions

as quantiﬁed by the inverse participation ration (IPR) was found to be extended when

subjected to a ﬁnite size scaling analysis.

The density of states (DOS) in Fig. 5.2 (a) shows a ﬁnite Mott gap in the spectrum

for low disorder, V =1t. This spectral gap closes around a critical disorder strength,

V 2t. Even when the spectral gap closes, there is still some structure at low c1 ≃ energies in the DOS. Increasing the disorder generates more states at low energy and

ﬁlls up the DOS. The AF order parameter, m† , is obtained from the long distance h i behavior of the spin-spin correlation function. As seen in Fig. 5.2 (b), m† is ﬁnite h i up to critical disorder, V 3.4t. The critical disorder where AF order is lost, V , c2 ≃ c2

is larger than Vc1 where the spectral gap closes. Thus, the phase with Vc1

has an AF order without a spectral gap. In summary, the 2D Hubbard system is

classiﬁed to 3 regions. Region 1 (0 V < 2t) has a gap and AF long-range order, ≤ Region 2 (2t V 3.4t) has no spectral gap and long-range AF order, and Region ≤ ≤ 3 (3.4t

insulator and Region 3 is a localized Anderson type insulator. The authors propose

Region 2 is an inhomogeneous AF metal based on further analysis of wave functions

[70].

The plot of the density, ψ (~r) 2, for a particular eigen index, n, can indicate σ | nσ | whether electrons are extendedP or localized. In Fig. 5.3, the electron density corre-

sponding to the Fermi energy is plotted. In Region 1 (e.g. Vd =2t) and Region 3 (e.g.

71 Figure 5.2: The above figure (a) is density of states as a function of E = ǫ µ where − µ is the chemical potential and the below figure (b) is spectral gap from the density of states (blue line) and the AF order parameter (red line) as a function of disorder, V. U =4t at half-filling and the system size is 32 32. The data is averaged over 10 × disorder realizations. (From Ref. [70]).

72 Figure 5.3: Plots of the density, ψ (~r) 2 at the Fermi energy. The density is σ | nσ | high at the dark gray area. U =4t at half-ﬁlling and the system size is 32 32. The data is averaged over 20 disorder realizations.P (From Ref. [70]). ×

Vd = 5t), the plots of the density corresponding to ǫ = ǫF are fairly localized. On

the other hand, in Region 2 (e.g. Vd =3t), the wave function is extended throughout the entire system. In the Mott phase, the electrons are localized due to strong interactions. Disorder, in a rather contradictory manner, competes with the interactions and generates puddles of local density ﬂuctuations. These puddles increase in size and with increasing disorder. At a critical disorder strength, the tuning between the puddles generates a connected pathway across the system. The system thus becomes a metallic.

To further quantify the extent of the wave function, the inverse participation ratio

(IPR) deﬁned by

4 2 IPR ψ (~r ) ξ− (n) (5.5) ≡ | n i | ∝ loc X~ri is calculated. The IPR is a useful quantity since it is also related to the localization length, ξloc. In Fig. 5.4, the IPR corresponding to the Fermi energy is plotted for

73 2 Figure 5.4: Plots of the density, σ ψn(~r) at the Fermi energy. The density is high at the dark gray area. U =4t at half| ﬁlling| and the system size is 32 32. The data is averaged over 20 disorder realizations.P (From Ref. [70]). ×

systems of various sizes. This is done mainly to test whether the apparent extended nature of states at finite L persists to the L limit. For comparison, we also plot →∞ the IPR for U = 0at V =3t. The IPR for U = 0 extrapolates finite values at L . d →∞ A finite intercept implies that the localization length is finite in the thermodynamic limit. For the same disorder strengths, the IPRs for U = 4t extrapolate to zero in the L limit, meaning that the localization length diverges and the system is → ∞ metallic with extended wave function in the thermodynamic limit.

5.5 Results

Our main eﬀort in this part of the thesis is to calculate the frequency and temperature dependent conductivity and spin susceptibility, which are experimentally

74 measurable and provide further characterization of these phases. The formulas used

in this part are summarized in Appendix.

Summary of results are the following. In Region 2, the conductivity near ω = 0

is most enhanced and shows the Drude like behavior and the peak frequency, where

the conductivity has the maximum value, is lowest. The Drude model can explain

non-monotonic behavior of the dc conductivity as a function of T . The result of

ac conductivity shows non-Fermi liquid type behavior in Region 2. The sum rule is

justiﬁed except for high T at low disorder. The temperature dependent behavior of spin susceptibility, χ, shows the nature of moments. At high T , the moments are free as indicated by their Curie behavior; at low T , χ is strongly suppressed because electrons get paired by forming singlet on low disorder and by on-site pairing on high disorder sites.

5.5.1 Frequency dependent conductivity at T =0

The real part of frequency dependent conductivity, σ(ω), is related to the imaginary part of the current-current correlation function [74],

ImΛ(ω) Re [σ(ω)] = , (5.6) ω where Λ(ω) is the Fourier transform of Λ(τ)= j(τ)j(0) and j is the paramagnetic h i current density operator. Fig. 5.5 shows σ(ω) for various disorder strengths and

Fig. 5.6 shows the gap of the conductivity as a function of disorder and the value of

ω where the conductivity has a peak as a function of disorder. For low disorder,

Region 1, there is a gap in the conductivity which is twice the spectral gap seen in

the DOS and a strong peak in the absorption at ω U. At intermediate disorder, ≃ the gap in σ(ω) closes at ω = 0 in Fig. 5.6 (a) but the peak in Re[σ(ω)] persists.

75 V=1 0.6 V=2 V=3 V=4 V=5 )

2 0.4 t 2 /(e

σ 0.2

0 0 2 4 6 ω/t

Figure 5.5: The frequency dependence of the conductivity, σ(ω), for Vd = 1, 2, 3, 4, and 5. The system size is 32 32, U =4t, and the data is averaged over 20 disorder × realizations. The low frequency region of σ(ω) for Vd = 3t is obtained by a third order polynomial ﬁt of the current-current correlation function, Λ = ωσ. For Vd =1t there is clearly a ﬁnite Mott gap at the low frequency. Thus, σ(ω) is suppressed at low frequency and there is a peak in the frequency when ω is around the Mott gap. With increasing disorder the spectral weight above the Mott gap is transfered to lower spectral region and the peak of σ(ω) moves to the low frequency. At V 2.2t, the d ≃ gap in the conductivity closes because energy scale of disorder is comparable with the gap. Near Vd =3t, the conductivity near ω = 0 is most enhanced and the frequency where the conductivity has the maximum value is lowest. For Vd = 4t and Vd = 5t, the dc conductivity, σ(0), is zero due to the Anderson insulating regime. In Region 2, the peak location of σ(ω) moves to the low frequency as the Mott gap closes but with increasing disorder the peak location again moves to the high frequency in Region 3

76 Figure 5.6: (a) The gap in Re[σ(ω)] as a function of disorder, Vd. (b) ωpeak as a function of disorder. The system size is 32 32, U = 4t, and the data is averaged × over 20 disorder realizations. ωpeak is obtained by ﬁtting Re[σ(ω)] in Fig. 5.5 near the peak using Lorentzian. The gap in the conductivity clearly closes at V 2.2t. ω d ≃ peak in Region 2 is lower than that in Region 1 and 3.

As seen in Fig. 5.6 (b), the peak moves to lower values of ω as disorder increases in Region 2. In Region 3, the peak starts moving out to higher frequencies but the conductivity at high T is still steeper than Re[σ(ω)] 1/ω2. With increasing ∼ disorder, the spectral weight above the Mott gap is transfered to lower spectral region, enhancing the absorption at a low frequency. At Vd =2.2t, the gap of the conductivity closes because energy scale of disorder is comparable with the gap. Near Vd = 3t,

Re[σ(ω = 0)] is most enhanced and ωpeak, frequency where the conductivity has the

maximum value, is lowest. The optical conductivity at Vd = 3t shows a Drude-like behavior. As disorder increases, this Drude-like behavior disappears. The origin of this Drude-like behavior at Vd =3t may be the result of the screening of the strongly disordered paramagnetic sites with 2 electrons on a site. The disordered potentials

77 are screened and an electron feels the following density dependent eﬀective potentials.

V˜i = Vi µ + U ni ↑ − h ↓i

V˜i = Vi µ + U ni ↓ − h ↑i

where the eﬀect to an electron comes from the density with opposite spin. Ignoring

spin-polarization, this eﬀective potential becomes

1 U V˜ = (V˜i + V˜i )= Vi µ + ( ni + ni ) . (5.7) 2 ↑ ↓ − 2 h ↑i h ↓i

For Vd =4t and Vd =5t, although the Mott gap is completely ﬁlled, the conductivity is

suppressed due to the Anderson insulating regime. This insulating behavior continues

and is enhanced more as disorder increases. It is diﬃcult to ﬁt the conductivity at

high frequency by the form (1/ωα) and σ(ω) for all case has a much faster decay with

α = 2, which is a conventional Fermi liquid result.

5.5.2 DC conductivity

DC conductivity is obtained by

ImΛ(ω) σdc = limω 0 . (5.8) → ω

Fig. 5.7 shows the temperature dependence of σdc for various V . In the Mott insulator

(e.g. Vd =1t or Vd =2t), σdc is exponentially small at low T , it peaks at a temperature

of order U, and at higher T , σdc decreases with T . In the intermediate disorder regime

the system with Vd = 3t is more conducting than Vd = 1t at low T but at high T it is the opposite way. Increasing the site disorder further drives the system from a metallic to an Anderson insulating phase and the low T , σdc drops again. The dc conductivity at low T and at low disorder e.g. 0

78 6 V=1 5 V=2 V=3 ) V=4 2 4 t 2 3 /(e

dc 2 σ 1

0 0 0.5 1 1.5 2 T/t

Figure 5.7: Temperature dependence of the dc conductivity, σdc(ω), for Vd =1, 2, 3, and 4. The system size is 16 16, U =4t, and the data is averaged over 20 disorder × realizations. The dc conductivity at low T for Vd = 3t is obtained by a third order polynomial ﬁt of the current-current correlation function,Λ= ωσ. At low T , the dc conductivity in Region 1 is zero because of the Mott gap, the system with Vd = 3t is more conducting than Vd = 1t due the close of the Mott gap, and with increasing disorder, σdc is suppressed because of an Anderson localization of electrons. On the other hand, at high T , σdc in Region 1 is higher than the others because energy scale of temperature is comparable with the Mott gap and as disorder increases, σdc decreases due to the increase of transport scattering of electrons and disorder. It is demonstrated that disorder can drive a metallic phase in the system. At low temperature, the dc conductivity in Region 1 and 2 decreases as lowering temperature, signifying an insulating phase. However, the dc conductivity for Vd = 3t has been most enhanced and slope of the conductivity, σdc/dT , has no slope.

79 Mott gap, σdc at moderate disorder e.g. 2

The dc conductivity shows non-monotonic behavior as a function of T which can be

understood using the Drude model:

ne2τ σDrude = , (5.9) dc m where τ is the relaxation time and n is the density of electrons. At high temperature

σdc is dominated by decreasing τ. For increasing disorder, τ decreases because of enhanced scattering between electrons and the disorder potential. At low temperature σdc is dominated by the temperature dependence of n. The number of electrons

∆ /T need to be excited beyond the Mott gap so the density is proportional to e− Mott .

Therefore, the conductivity at low disorder increases rapidly with increasing temperature. For both low and high disorder, σ vanishes as T 0, signifying an insulating dc → phase, either a Mott insulator or an Anderson insulator. However, as T 0 the dc → conductivity for Vd = 3t is unusual; it is ﬁnite with negligible T -dependence at low

T .

5.5.3 Frequency dependence of the conductivity for ﬁnite temperature

Fig. 5.8 shows the frequency dependence of the conductivity, Re[σ(ω)], for various

disorder strengths. For low disorder at low T there is a Mott gap but with increasing

T this gap closes and σ(ω = 0) has large weight. In Region 2, for all T , the Drude like

conductivity is seen and in Region 3, this behavior is less obvious. AC conductivity

from the Drude theory is σ σ(ω)= 0 , (5.10) 1 iωτ − 80 1.5 1.8 T=0 T=0 T=0.4 1.6 T=0.4

) T=0.8 ) 1.4 T=0.8

2 T=1.2 2 T=1.2 t 1 t 1.2 2 2 1

)/(e )/(e 0.8

ω 0.5 ω 0.6 ( (

σ σ 0.4 0.2 0 0 0 1 2 3 4 5 6 0 2 4 6 ω ω

0.8 T=0 T=0 T=0.4 T=0.4 0.4 ) 0.6 T=0.8 ) T=0.8 2 T=1.2 2 T=1.2 t t 2 2 0.4 )/(e )/(e 0.2 ω ω ( ( σ 0.2 σ

0 0 0 2 4 6 0 2 4 6 ω ω

Figure 5.8: The frequency dependence of the conductivity for Vd = 1 (left top), Vd = 2 (right top), Vd = 3 (left bottom) and Vd = 4 (right bottom) at diﬀerent temperatures. The system size is 16 16, U = 4t, and the data is averaged over 20 × disorder realizations. The low frequency region of σ(ω) for Vd = 3t is obtained by a third order polynomial ﬁt of the current-current correlation function, Λ = ωσ. For low disorder at low T there is the Mott gap for low frequency and with increasing T , the peak in the conductivity moves to the weight at ω = 0. With increasing disorder from Region 1 to Region 2, the spectral weight above the Mott gap is transferred to lower spectral region at low T and σ(ω) shows the Drude like behavior for all T . With increasing disorder from Region 2 to Region 3, this Drude like behavior is less clear. The Drude like behavior shows non-monotonic behavior as a function of T . The conductivity weight in low frequency increases with increasing T up to T =0.8 but with further increasing T it decreases.

81 ne2τ where σ0 = m . The real part is

σ σDrude(ω)= 0 . (5.11) 1+ ω2τ 2

At high frequency, σDrude(ω) = ne2/(mω2τ). It is diﬃcult to ﬁt the conductivity at high frequency by the form (1/ωα) because σ(ω) for all cases has a much faster decay with α = 2. The sudden increase of the conductivity at low ω with increasing T for

Vd = 1 and Vd = 2 may be explained by the exponential increase of the DOS at the

Mott insulator as a function of temperature.

5.5.4 Sum rule for the conductivity

In order to check for our numerical results, we check the consistency of the f-sum

rule [75]. The frequency integral of Re[σ(ω)] is related to the total kinetic energy:

Tˆ 4 ∞ h− i = Re[σ(ω)]dω, (5.12) N πe2 Z0 ˆ where T = t i=j,σ ciσ† cjσ. We calculate and compare the left-hand side and the right- 6 hand side ofP Eq. (5.12), independently in Fig. 5.9. These values are non-monotonic.

Especially, increasing temperature, the values are increasing for lower temperature

and decreasing for high temperature. The kinetic energy is getting higher for low

disorder because disorder is added to a Mott insulator but at high disorder the ki-

netic energy decreases with increasing disorder because the mobility of electrons is

decreased at strong disordered sites. This is also seen as a function of temperature

with ﬁxed disorder. At low temperature, the sum rule is good agreement and the

sum rules at ﬁnite temperature are better agreement as increasing disorder. The

disagreement at high temperature for Vd = 1 comes from the diﬃculty of calculating exactly the Drude weight, a singular behavior at ω = 0.

82 1.4 1.4

1.3 1.3

1.2 1.2 <-T>/N <-T>/N π ∫∞ σ ω ω π ∫∞ σ ω ω (4/ ) 0 ( )d (4/ ) 0 ( )d 1.1 1.1 0 1 2 3 4 5 0 0.5 1 V T 1.4 1.4

1.3 1.3

1.2 1.2 <-T>/N <-T>/N π ∫∞ σ ω ω π ∫∞ σ ω ω (4/ ) 0 ( )d (4/ ) 0 ( )d 1.1 1.1 0 0.5 1 0 0.5 1 T T

Tˆ 4 h− i ˆ † ∞ Figure 5.9: N (red square) where T = t i=j,σ ciσcjσ and πe2 0 σ(ω)dω (blue dot) as a function of disorder for T = 0 (left top)6 or a function of a temperature P R for Vd = 1 (right top), Vd = 2 (left bottom), and Vd = 3 (right bottom). The system size is 16 16, U =4t, and the data is averaged over 20 disorder realizations. These values are× non-monotonic. Especially, increasing temperature, the values are increasing for lower temperature and decreasing for high temperature. The kinetic energy is getting higher for low disorder because disorder is added to a Mott insulator but at high disorder the kinetic energy decreases with increasing disorder because the mobility of electrons is decreased at strong disordered sites. This is also seen as a function of temperature with ﬁxed disorder. At low temperature, the sum rule is good agreement and the sum rules at ﬁnite temperature are better agreement as increasing disorder.

83 5.5.5 Uniform static spin susceptibility

Frequency-dependent spin susceptibility is calculated as

β χzz(~q, ω)= eiωτ χzz(~q, τ)dτ, (5.13) Z0 where

zz 1 i~q (~i ~j) z z χ (~q, τ)= S (~q, τ)S ( ~q, 0) = e · − S (τ)S (0) (5.14) h z z − i N h i j i i,j X and Sz is z component of spin operator. Fig. 5.10 shows the uniform spin susceptibility, χ(0, 0), and staggered spin susceptibility, χ(π, π), for various disorder strengths as a function of T . At low temperature, for low disorder, χ(0, 0) shows spin gap and

χ(π, π) is ﬁnite due to AF order in the system. These are most enhanced for Vd =3t

but with increasing disorder further, these are suppressed. At high temperature,

with increasing disorder, χ(0, 0) and χ(π, π) decrease. Also, the ﬁtted parameters,

C C and a, where T +a is used for the ﬁtting the tail of χ(0, 0), are considered to see local moments and the strength of paramagnetism. On the other hand, the Curie susceptibility of the spin at k T gµ H for localized electrons is B ≫ B n(gµ )2S(S + 1) χCurie = B , (5.15) 3kBT where n is the density, g is 2, and S is a local moment. At high T increasing disorder, the number of on-site coupling of up and down spin electrons increases because electrons can occupy the low disordered sites and not occupy the strongly disordered sites. These coupled sites and unoccupied sites have no contribution on local moments, so χ is more suppressed with increasing disorder at high T . The plot of C justify this description. These are increased with increasing disorder as expected. On the other hand, at low T , we consider the Pauli paramagnetic susceptibility for free

84 0.5

0.3 0.4 ) π 0.2 , 0.3 π (0,0) ( s

V=1 s 0.2

χ V=0 0.1 V=2 χ V=1 V=3 V=2 V=4 0.1 V=3 V=5 V=4 0 0 V=5 0 0.4 0.8 1.2 1.6 2 0 0.4 0.8 1.2 1.6 2 Temperature/t Temperature/t 0.9 3 2.5 0.8 a C 2

0.7 1.5 1 0 1 2 3 4 5 0 1 2 3 4 5 V V

Figure 5.10: χ(0, 0) and χ(π, π) as a function of a temperature for Vd =0, 1, 2, 3, 4, and 5. The system size is 16 16, U =4t, and the data is averaged over 20 disorder × C realizations. The tails of χ(0, 0) are ﬁtted by the function T +a where C is the measure of local moments and a is the measure of the strength of paramagnetism. These C and a are plotted as a function of the disorder, V. At low temperature, χ(0, 0) and χ(π, π) for low disorder are suppressed and these are most enhanced in Region 2, and with increasing disorder further, χ(0, 0) and χ(π, π) decrease. On the other hand, at high temperature, increasing disorder, χ(0, 0) and χ(π, π) monotonically decrease.

85 electrons:

Pauli 2 χ = µBD(ǫF ), (5.16) where µB is the Bohr magneton and ǫF is the DOS at the Fermi level. For low disorder, there is a spin gap because DOS(ǫF ) = 0. Increasing disorder, χ is enhanced due to the increase of available states. Increasing disorder further (e.g. Vd = 4 or

Vd = 5), this enhancement disappears because of the suppressed local moments at strong disordered sites.

5.6 Conclusion

In summary, we have calculated Re[σ(ω, T )], the frequency and temperature dependence of conductivity, χ(~q = 0,ω,T ), uniform spin susceptibility, and χs(~q =

(π, π),ω,T ), staggered spin susceptibility for the two-dimensional Hubbard model with disorder at half ﬁlling using an inhomogeneous self-consistent Hartree-Fock numerical method. There is a gap in the spectrum of the conductivity at low disorder.

Increasing disorder, spectral weight is shifted to a low frequency region. The intensity of the dc conductivity grows quickly with disorder. From the conductivity plots, at moderate disorder, σ is most enhanced although σ /dT 0 for low temperature. dc dc ∼ From the temperature dependence of the conductivity, at low T , the system is a insu-

dσ(0) dσ(0) lating phase with dT > 0 and at high T , the system is a metal phase with dT < 0.

The Mott insulator with low disorder, such as Vd = 1t or Vd = 2t, have a large conductivity at a high temperature. The enhancement of dc conductivity is found with increasing disorder strength but the dc conductivity is decreased for strong enough disorder strength because of the Anderson localization. The results of uniform static

86 spin susceptibility are expected. For the small systems, the coupling constants are considered in order to understand spin conﬁgurations more in Appendix.

We have investigated the effect of an inhomogeneous magnetic field, hi = U ci† ci − h ↑ ↓i considering the case with hi = 0 at all sites. We could have not seen any significant difference between two cases, h = 0 for all sites and h = 0. Thus, the inhomogeneous i i 6 field can only select the z direction on the 2D system and physical quantities that do not depend on z direction are not depending on hi.

It may be interesting to contrast the behavior of the conductivity for the positive-

U and negative-U disordered Hubbard model. Moreover, away from half ﬁlling is interested.

87 APPENDIX A

COMPARISON BETWEEN ANALYTICAL SOLUTIONS AND NUMERICAL RESULTS FROM THE INHOMOGENEOUS HARTREE-FOCK METHOD IN SEVERAL LIMITING CASES OF 1D DISORDERED HUBBARD MODEL

We consider analytical solutions for several limiting cases of the 1D disordered

Hubbard model and compare these analytical solutions at half ﬁlling with the numerical method for the 1D Hubbard model.

A.1 Tight-binding model, (U =0 and Vi =0)

We consider the free electron model in 1D,

H = t c† c = ǫ c† c , (A.1) tb − iσ jσ k kσ kσ i,j ,σ k,σ hXi X where

ǫ = 2t cos ka. (A.2) k − The density of states can be calculated by

1 g(E)= lim ImGii(z = E + iǫ), (A.3) ε 0 −π →

where Gii(z) is the Green’s function deﬁned by

d~k G (z)= i G(z) i = . (A.4) ii h | | i z ǫ ZBZ − k 88 Using the energy formula, Eq. (A.2),

π π a a 1 a a 1 Gii(z)= dk = dk . (A.5) 2π π z +2t cos ka π 0 z +2t cos ka Z− a Z At z > 2t , | | | |

2 2 k 2 z (2t) tan 2 π π Gii(z)= arctan − 0 = . (A.6) z2 (2t)2 z +2t !| z2 (2t)2 − p − On the other hand,p at z < 2t , p | | | |

2 2 k 1 (2t) z tan 2 + z +2t π Gii(z) = ln − π (2t)2 z2 (2t)2 z2 tan k (z +2t)|0 − p − 2 − 1 i = p (ln1p ln ( 1)) = − . π (2t)2 z2 − − (2t)2 z2 − − Therfore, the density of statesp per spin becomes p

1 g(E)= , ( 2t 2t , g(E) = 0. Thisp analytical solution of g(E) (black) and a numerical | | result (red) for N = 128 are plotted for t = 1 on Fig. A.1.

A.2 The disordered Hubbard model with t =0

We consider the model,

Ht=0 = Uni ni + (Vi µ)ciσ† ciσ. (A.8) ↑ ↓ − i i,σ X X This may be a limit of very strong disorder and no metallic phase exists because of

the loss of the hopping term.

The partition function can be written as

N β(Vi µ) β(2(Vi µ)+U) Z(β) = Πi=1 1+2e− − + e− − . (A.9) 89 DOS 1

0.8

0.6

0.4

0.2

Ît -3 -2 -1 1 2 3

Figure A.1: Comparison of DOS between the analytical solution Eq. (A.7) and a numerical result (N = 128).

Using the partition function, we can get the grand potential, Ω, the average density,

N , the energy, E, the magnetism, M, and the susceptibility, χ. h i N β(V µ) β(2(V µ)+U) Ω= k T ln Z = k T ln(1+2e− i− + e− i− ). − B − B i=1 N X ∂Ω 2(e β(Vi µ) + e β(2(Vi µ)+U)) N = = − − − − . h i − ∂µ 1+2e β(Vi µ) + e β(2(Vi µ)+U) β i − − − − X N β(V µ) β(2(V µ)+U) 2(V µ)e− i− + (2(V µ)+ U)e− i− E µN = i − i − . − 1+2e β(Vi µ) + e β(2(Vi µ)+U) i=1 − − − − X N β(V µ µ B) β(V µ+µ B) e− i− − B e− i− B M = µB − 1+ e β(Vi µ+µB B) + e β(Vi µ µB B) + e β(2(Vi µ)+U) i − − − − − − − XN β(Vi µ) 2e− − sinh (βµBB) = µB . 1+2e β(Vi µ) cosh (βµ B)+ e β(2(Vi µ)+U) i − − B − − X N β(Vi µ) β(2(Vi µ)+U) β(Vi µ) ∂M 2 e− − cosh (βµBB)(1 + e− − )+2e− − χ = = 2βµB { 2 }. ∂B β(Vi µ) β(2(Vi µ)+U) i (1+2e− − cosh (βµBB)+ e− − ) X 90 At small value of B, the susceptibility becomes

N 2µ2 e β(Vi µ) χ = B − − . (A.10) k T 1+ e β(2(Vi µ)+U) +2e β(Vi µ) i B − − − − X On the other hand, at high temperature limit,

N µ2 U χ B 1+ . (A.11) ∼ 2k T 4k T i B B X A.3 Non-disordered case at a half ﬁlling of the Hubbard model with t =0

When we consider non-disordered case (Vi = 0), Eq. (A.10) is simpliﬁed as the follows. βµ β(2µ U) 2(e + e − ) N = N βµ β(2µ U , (A.12) 1+2e + e − )

U which gives µ = 2 at any temperatures. The energy is E µN U 1 − = βU/2 . (A.13) N − 2 1+ e−

At the limit Uβ , we can get →∞ E µN U − = . (A.14) N − 2

Thus, the energy approaches 0 at Uβ 0. →

The susceptibility and speciﬁc heat, CV become

χ 1 1 2 = βU/2 , (A.15) NµB kBT e− +1

and 2 2 ∂E 2 ∂E kBβ U 1 CV = = kBβ = N . (A.16) ∂T − ∂β 42 βU 2 cosh 4 At the limit βU , →∞ 2 2 CV kBβ U βU e− 2 , (A.17) N ∼ 4 91 and χ 1 2 . (A.18) NµB ∼ kBT

A.4 Hubbard model at half ﬁlling (No disorder)

We consider the Hubbard model for non-disordered case at half ﬁlling,

H = t ciσ† cjσ + Uni ni µ ciσ† ciσ. (A.19) − ↑ ↓ − i=j,σ i i,σ X6 X X This 1D Hubbard model can be solved exactly using the Bethe Ansatz method [76]

and the application gives several exact results for physical quantities.

A.4.1 Total energy

The analytical solution for the total energy (eq.(20) of [76]) is

E ∞ J0(ω)J1(ω) = 4 dω ωu , (A.20) N − ω(1+ e 2 ) Z0 where Jn is the Bessel function of the order n and u = U/t. The analytical solution of total energy and a numerical result for N = 128 are plotted for t = 1 as a function of U on Fig. A.2.

A.4.2 Local Moment

The local moment is deﬁned as from eq.(2.12) - eq.(2.16) of [77]

1 L = S~ S~ , (A.21) 0 N h i · ii i X 1 + + 2 2 2 where S~ S~ = S S + (S S− + S−S )= S + S + S . · z z 2 x y z

3 ∞ J0(ω)J1(ω) L0 = 1 dω . (A.22) Uω 2 4 − 0 cosh Z 4 ! 92 Total Energy U 2 4 6 8 10 t -0.2 -0.4 -0.6 -0.8 -1 -1.2

Figure A.2: Comparison of total energy between the analytical solution (line) and a numerical result (dot) for N = 128 as a function of U.

For several limiting cases,

3 3 L (u =0)= =0.375, L (u )= =0.75. (A.23) 0 8 0 →∞ 4

At strong U limit, 3 8ln2 L 1 . (A.24) 0 ∼ 4 − u2 The analytical solution of local moment and a numerical result for N = 128 are

plotted for t = 1 as a function of U on Fig. A.3.

A.4.3 Gap from MFT

The gap equation for the mean ﬁeld theory can be written as [78]

1 ′ 1 1 = , (A.25) 2 2 N ǫk + ∆ U X p 93 LO 0.8

0.7

0.6

0.5

0.4

U 2 4 6 8 10 t

Figure A.3: Comparison of local moment between the analytical solution (line) and a numerical result (dot) for N = 128 as a function of U.

z π π where ∆ = US = U S and ′ on the sum means sum only

94 mÖ 0.5

0.4

0.3

0.2

0.1

U 2 4 6 8 10 t

Figure A.4: Comparison of order parameter between the MFT solution (line) and a numerical result (dot) for N = 128 of staggered magnetization in the z direction as a function of U.

95 APPENDIX B

FORMALISM OF THE EXPECTATION VALUES ON THE NUMERICAL SIMULATION

This appendix includes the detail formalisms of the expectation values used in the

numerical simulation that are not given in the paper [72].

B.1 Spin degree of a freedom: Magnetic Properties

B.1.1 Local moments

Local moments is 1 L = S~2 . (B.1) 0 N h i i i X

2 x 2 y 2 z 2 z 2 1 + + S~ = S~ S~ = (S ) +(S ) +(S ) = (S ) + (S S− + S−S ) , (B.2) h i i h i · ii h i i i i h i 2 i i i i i where

+ 2 n m m n Si Si− = ci† ci ci† ci = ci† ci + ψi ∗ψi ψi ∗ψi f(ǫn) ni ni h i h ↑ ↓ ↓ ↑i h ↑ ↓i ↑ ↓ ↓ ↑ −h ↑ih ↓i n,m X 2 n m m n = ci† ci + f(ǫn)(1 f(ǫm))ψi ∗ψi ψi ∗ψi h ↑ ↓i − ↑ ↓ ↓ ↑ n,m + X = Si Si− ni ni + ni , h ih i−h ↑ih ↓i h ↑i

96 + 2 n m m n Si−Si = ci† ci ci† ci = ci† ci + ψi ∗ψi ψi ∗ψi f(ǫn) ni ni h i h ↓ ↑ ↑ ↓i |h ↑ ↓i| ↓ ↑ ↑ ↓ −h ↑ih ↓i n,m X 2 n m m n = ci† ci + f(ǫn)(1 f(ǫm))ψi ∗ψi ψi ∗ψi |h ↑ ↓i| − ↓ ↑ ↑ ↓ n,m + X = Si Si− ni ni + ni , h ih i−h ↑ih ↓i h ↓i

and

1 z 2 ′ (Si ) = σσ′ niσniσ h i 4 ′ h i Xσσ 1 n m m n = σσ′ n n ′ + ψ ∗ψ ψ ∗′ ψ ′ f(ǫ ) c† c ′ c† ′ c 4 h iσih iσ i iσ iσ iσ iσ n −h iσ iσ ih iσ iσi ′ nm ! Xσσ X 1 + 1 = S S− n n + ( n + n ) . 2 h ih i−h ↑ih ↓i 4 h ↑i h ↓i Thus,

~2 ~ ~ 3 2 Si = Si Si = ci† ci ni ni h i h · i 2 h ↑ ↓i −h ↑ih ↓i 1 m m + ( ni +2 ni ) ψi ∗ψi 4 h ↑i h ↓i ↑ ↑ ( m ) X 1 m m m m + ni +2 ni ) ψi ∗ψi 2 ci† ci ψi ∗ψi 4 h ↓i h ↑i ↓ ↓ − h ↑ ↓i ↑ ↓ ( m m ) X X 3 3 3 + = ( ni + ni ) ni ni + Si Si− . 4 h ↑i h ↓i − 2h ↑ih ↓i 2h ih i B.1.2 Spin-spin correlation function

The spin-spin correlation functions are

1 zz ~ z z ′ C (l = ~ri ~rj) = Si Sj = σσ′ niσnjσ − h i 4 ′ h i Xσσ 1 ′ ′ ′ ′ = σσ′ niσ njσ + δijδσσ ciσ† cjσ ciσ† cjσ cjσ† ′ ciσ , 4 ′ h ih i h i−h ih i Xσσ where at i = j 6

z z 1 Si Sj = ( ni ni )( nj ni ) h i 4 { h ↑ − ↓i h ↑ − ↓i } 1 2 2 2 2 ci† cj ci† cj ci† cj + ci† cj , − 4 |h ↑ ↑i| − |h ↑ ↓i| − |h ↓ ↑i| |h ↓ ↓i| 97 and at i = j

z z 1 Si Si = ( ni ni )( ni ni )+ ci† ci + ci† ci h i 4 h ↑ − ↓i h ↑ − ↓i h ↑ ↑i h ↓ ↓i 1 n 2 2 2 2o ci† ci ci† ci ci† ci + ci† ci − 4 |h ↑ ↑i| − |h ↑ ↓i| − |h ↓ ↑i| |h ↓ ↓i| 1 2 2 = 2 ni ni + ci† ci + ci† ci + ni + ni . 4 − h ↑ih ↓i |h ↑ ↓i| |h ↓ ↑i| h ↑i h ↓i B.1.3 Magnetic structure factor

The Fourier transform of spin-spin correlation function gives the magnetic struc-

ture factor.

zz zz i~q ~l 1 i~q (~i ~j) z z S (~q) = C (~l)e · = e · − S S N h i j i ij X~l X 1 i~q (~i ~j) = e · − (ni ni )(nj nj ) 4N h ↑ − ↓ ↑ − ↓ i i,j X 1 i~q (~i ~j) = e · − σσ′ n n ′ . 4N h iσ jσ i i,j ′ X Xσσ The magnetic structure factor with ~q corresponding to the dominant magnetic ar- rangements shows peaks. For example, AF system has a peak on Szz at ~p =(π, π).

B.1.4 Spin susceptibility for ω =0

Spin susceptibility is calculated as

β χzz(~q, 0) = χ(~q, τ)dτ. (B.3) Z0 zz z z 1 i~q (~i ~j) z z χ (~q, τ)= S (~q, τ)S ( ~q, 0) = e · − S (τ)S (0) , (B.4) h − i N h i j i i,j X where we used

z 1 z i~q ~i S (τ)= S (τ)e− · , (B.5) √ i N i X and 1 z z ′ Si (τ)Sj (0) = σσ′ ciσ† (τ)ciσ(τ)cjσ† ′ cjσ . (B.6) h i 4 ′ h i Xσ,σ 98 Thus,

β zz 1 i~q (~i ~j) z z χ (~q, 0) = e · − S (τ)S dτ N h i j i ij 0 X Z β = χ(~q, τ)dτ Z0 β i~q (~i ~j) = e · − σσ′ niσ njσ′ 4N ′ h ih i ij,σσX

β i~q (~i ~j) n m m n + e · − σσ′ ψiσ∗ψiσψjσ∗′ ψjσ′ f(ǫn)(1 f(ǫm)) 4N ′  −  ij,σσX nm(Xǫn=ǫm)   β i~q (~i ~j) n m m n f(ǫn) f(ǫm) + e · − σσ′ ψiσ∗ψiσψjσ∗′ ψjσ′ − 4N ′  β(ǫm ǫn)  ij,σσ nm(ǫn=ǫm) X X6 −   β i~q ~i i~q ~j = σσ′ e · n e− · n ′ 4N h iσi h jσ i ′ i ! ′ ! Xσσ X Xjσ β i~q ~i n m i~q ~j m n + σσ′ e · ψ ∗ψ e− · ψ ∗′ ψ ′ f 4N iσ iσ jσ jσ nm ′ nm i ! j ! Xσσ X X X β i~q (~i ~j) = e · − σσ′ n n ′ 4N h iσih jσ i ij ′ X Xσσ β n m m n + σσ′ cos(~q ~i)ψ ∗ψ cos(~q ~j)ψ ∗′ ψ ′ f 4N · iσ iσ · jσ jσ nm nm ′ i ! j ! X Xσσ X X β n m m n + σσ′ sin (~q ~i)ψ ∗ψ sin (~q ~j)ψ ∗′ ψ ′ f , 4N · iσ iσ · jσ jσ nm nm ′ i ! j ! X Xσσ X X where

f = f(ǫ )(1 f(ǫ )) , (B.7) nm(ǫn=ǫm) n − m and

f(ǫn) f(ǫm) fnm(ǫn=ǫm) = − . (B.8) 6 β(ǫ ǫ ) m − n

99 The following relation between the uniform magnetic structure factor, Szz(~q = 0) and the static spin susceptibility, χ(ω = 0) are satisﬁed [79].

2 χ(~q =0)= βµ2 S S 2 = βS(~q = 0), (B.9) B h i i−h ii  i ! i X X   where S = 0 is used to prove. h i ii P B.1.5 Spin susceptibility for ω =0 6 Spin susceptibility is calculated as

β χzz(~q, 0) = eiωM τ χ(~q, τ)dτ, (B.10) Z0 with

zz z z 1 i~q (~i ~j) z z χ (~q, τ)= S (~q, τ)S ( ~q, 0) = e · − S (τ)S (0) , (B.11) h − i N h i j i i,j X where we used

z 1 z i~q ~i S (τ)= S (τ)e− · , (B.12) √ i N i X and 1 z z ′ Si (τ)Sj (0) = σσ′ ciσ† (τ)ciσ(τ)cjσ† ′ cjσ . (B.13) h i 4 ′ h i Xσ,σ Therfore,

β β zz 1 i~q (~i ~j) iω τ z z iω τ χ (~q, ω)= e · − e M S (τ)S dτ = e M χ(~q, τ)dτ N h i j i ij 0 0 X Z Z β β i~q (~i ~j) n m m n iωM τ (ǫn ǫm)τ = e · − σσ′ ψiσ∗ψiσψjσ∗′ ψjσ′ f(ǫn)(1 f(ǫm)) e e − dτ 4N ′ − 0 ij,σσ nm(ǫn=ǫm) X X6 Z

β i~q (~i ~j) n m m n f(ǫm) f(ǫn) = e · − σσ′ ψiσ∗ψiσψjσ∗′ ψjσ′ − . 4N ′  iωM + ǫn ǫm  ij,σσ nm(ǫn=ǫm) X X6 −  

100 Thus,

zz βπ i~q (~i ~j) n m m n Imχ (~q, ω)= e · − σσ′ ψiσ∗ψiσψjσ∗′ ψjσ′ (f(ǫn) f(ǫm)) δ(ω+ǫn ǫm) 4N ′ − − ij,σσ nm(ǫn=ǫm) X X6 βπ ~ n m ~ m n = σσ′ cos(~q i)ψiσ∗ψiσ cos(~q j)ψjσ∗′ ψjσ′ fnm 4N ′ · · nm(ǫn=ǫm),σσ i ! j ! X6 X X βπ ~ n m ~ m n + σσ′ sin (~q i)ψiσ∗ψiσ sin (~q j)ψjσ∗′ ψjσ′ fnm, 4N ′ · · nm(ǫn=ǫm),σσ i ! j ! X6 X X where f =(f(ǫ ) f(ǫ )) δ(ω + ǫ ǫ ). nm n − m n − m

101 APPENDIX C

PROBABILITY DISTRIBUTION OF THE EXCHANGE COUPLING CONSTANT, J

The spin Hamiltonian can be written as

Hspin = JS~ S~ , J = E E (C.1) − 1 · 2 s − t

where S~i is the spin operator at the site i and Es and Et are eigenvalues in the singlet

state and the triplet state, respectively. If J is positive, parallel spins, ferromagnetic,

state is favored. On the other hand, if J is negative, anti-parallel, anti-ferromagnetic

state is favored. For many-body system the total spin Hamiltonian is simply the sum

of all pairs

Hspin = J S~ S~ (C.2) − ij i · j ij Xh i The coupling constant of 1D disordered Hubbard model is considered via a perturbation expansion [80] and exact diagonalization for small spin systems of 2D disordered Hubbard model. Exact diagonalization is employed to calculate the coupling constants at small systems on Fig. C.1. In this appendix, we use the same notations of [72]. For example, the site 1 has the disordered potential, V1.

102 12 1 2 1 2 (nn) 43 4 3 (c) (d) 1 2 3 (a) 12 1 2

1 2 3 4 3 4 3 (b) (e) (f)

Figure C.1: Possible AF couplings between two singly occupied sites for the nearest neighbor (nn) coupling and the next nearest neighbor couplings, (a,b,c,d,e,f).

103 C.1 The probability distribution of coupling constants based on the perturbation theory

The calculation of coupling constant using the perturbation theory for 2D dis-

ordered Hubbard model is given in [72]. The results for coupling constants for each

conﬁguration are the following (The notations used are the same as [72]). The nearest

neighbor coupling is 4t2 U J = 2 . (C.3) (V1 V2) 1 − − U 2 The next nearest neighbor couplings are the followings.

t4 t4 J = + a (V V )2(U + V V ) (V V )2(U V + V ) 1 − 2 1 − 3 3 − 2 − 1 3 t4 1 1 2 + + . U + V + V 2V V V V V 1 3 − 2 1 − 2 3 − 2

t4 t4 J = + b (V V )2(U V + V ) (V V )2(U + V V ) 2 − 1 − 1 3 2 − 3 1 − 3 t4 1 1 2 + + . U +2V V V V V V V 2 − 1 − 3 2 − 1 2 − 3

t4 t4 J = J +J (2 4)+ + c a b → (V V )(U +V V )(V V ) (V V )( U V +V )(V V ) 1− 2 1 − 3 4− 3 1− 2 − − 2 4 4− 3 t4 t4 + + . (V V )( U +V V )(V V ) (V V )(U V +V )(V V ) 4 − 1 − 4 − 2 3 − 2 4 − 1 − 1 3 3 − 2

J = J (2 4). e c ↔

t4 t4 J =J +J (2 4)+ + . d b b → (V V )(U V +V )(V V ) (V V )(U +V V )(V V ) 2− 1 − 1 3 4− 1 4− 3 1 − 3 2− 3 t4 t4 J =J +J (2 4)+ + . f a a → (V V )(U +V V )(V V ) (V V )(U V +V )(V V ) 1− 2 1 − 3 1− 4 3− 4 − 1 3 3− 2 For the distribution of the potential V , a box type is used. Sites can equally have a potential between V and V and other potential realizations are not possible. − d d 104 Sites with the potential V V U are doubly occupied, those with U V U d ≤ ≤ − 2 − 2 ≤ ≤ 2 are singly occupied, and sites with V V U are empty sites. The probability d ≥ ≥ 2 distributions for the nn coupling, J and the next nearest neighbor couplings, Ja, Jc,

Jd are plotted on Fig. C.2. As Vd is increased, the possibility of large coupling is

decreased. For all cases, there is no possibility of negative couplings.

C.2 The probability distribution of coupling constants based on exact diagonalization

The coupling constant is deﬁned as the energy diﬀerence between the lowest triplet

and the lowest singlet.

J =(E E ) (C.4) triplet − singlet

We consider the matrix elements of the Hamiltonian for the small systems. These

matrices are diagonalized and the eigenvalues for the triplet and single states are

obtained. The probability distribution for J are obtained for the distribution of the

boxed type potential V used in the above.

C.2.1 Case of two nearest neighbor sites

There are 3 singlet states:

1, 1 , 2, 0 , 0, 2 . (C.5) | is | i | i

where

1 1, 1 s = ( 1 2 ), (C.6) | i √2 | i−| i 1 1, 1 t = ( 1 + 2 ), (C.7) | i √2 | i | i (C.8)

105 PHJL PHJa L 5 7 4 6 5 3 4 2 3 2 1 1 J J -0.5 0.5 1 1.5 2 2.5 3 -0.5-0.25 0.25 0.5 0.75 1 1.25 1.5 a

PHJc L PHJd L 2 2

1.5 1.5

1 1

0.5 0.5

J J 1 2 3 4 5 c 1 2 3 4 5 d

Figure C.2: Comparison between the probability distributions for the coupling for U = 4t at Vd = 2.5 (red), Vd = 3 (green), and Vd = 5 (blue) from the perturbation method. For J, V does not eﬀect on the plot as long as V U/2. P (J ) = P (J ), d d ≥ a b P (Jc)= P (Je) and P (Jd)= P (Jf ) from the symmetry.

106 where s ant t means singlet and triplet, respectively. The matrix becomes

V + V √2t √2t 1 2 − − √2t U +2V 0 . (C.9)  − 1  √2t 0 U +2V − 2   There are 3 triplet states:

, , , , 1, 1 . |↑ ↑i |↓ ↓i | it

The matrix for the triplet states have the same diagonal terms, V1+V2 and oﬀ-diagonal terms are the same as Eq. (C.9)

C.2.2 Case: (b)

There are 6 singlet states:

2, 0, 0 , 0, 2, 0 , 0, 0, 2 , 1, 1, 0 , 1, 0, 1 , 0, 1, 1 . (C.10) | i | i | i | is | is | is

The matrix for these singlet states is

U +2V 0 0 √2t 0 0 1 − 0 U +2V2 0 √2t 0 √2t  − −  0 0 U +2V 0 0 √2t 3 − . (C.11)  √2t √2t 0 V + V t 0   1 2   − 0− 0 0 t V −+ V t   − 1 3 −   0 √2t √2t 0 t V + V   2 3   − − −  9 triplet states consist of 3 S = 1 states, 3 S = 1 states, and 3 S = 0 states: z z − z

, , 0 , , 0, , 0, , , |↑ ↑ i |↑ ↑i | ↑ ↑i , , 0 , , 0, , 0, , , |↓ ↓ i |↓ ↓i | ↓ ↓i 1, 1, 0 , 1, 0, 1 , 0, 1, 1 . | it | it | it

107 The matrix becomes

V1+V2 t 0000000 t V−+V t 000000 − 1 3 −  0 t V +V 000000  − 2 3  0 0 0 V1+V2 t 0 0 0 0   −   0 0 0 t V1+V3 t 0 0 0 . (C.12)  − −   0 0 0 0 t V +V 0 0 0   − 2 3   000000 V +V t 0   1 2 −   000000 t V +V t   − 1 3 −   0000000 t V +V   − 2 3    C.2.3 Case: (c)

There are 6 singlet states:

2, 2, 0 , 2, 0, 2 , 0, 2, 2 , 1, 1, 2 , 1, 2, 1 , 2, 1, 1 . | i | i | i | is | is | is

The matrix is 2(U+V +V )0 0 00 √2t 1 2 − 0 2(U+V1+V3) 0 √2t 0 √2t  − −  0 0 2(U+V +V ) √2t 0 0 2 3 − , (C.13)  0 √2t √2t U +V t 0   − − 0 3   0 0 0 t U +V t   0 2   √ √   2t 2t 0 0 t U0 +V1   − −  where U0 = U + V1 + V2 + V3.

Triplet states consist of 3 of S = 1 states, 3 of S = 1 states, and 3 of S = 0 z z − z states:

, , 2 , , 2, , 2, , , , , 2 , , 2, , 2, , , 1, 1, 2 , 1, 2, 1 , 2, 1, 1 . |↑ ↑ i |↑ ↑i | ↑ ↑i |↓ ↓ i |↓ ↓i | ↓ ↓i | it | it | it (C.14)

Each spin subspace gives the same following matrix.

V0 + V3 t 0 t V + V t . (C.15)  0 2  0 t U0 + V1   108 C.2.4 Case: (d)

There are 10 singlet states:

2, 0, 0, 0 , 0, 2, 0, 0 , 0, 0, 2, 0 , 0, 0, 0, 2 , | i | i | i | i 1, 1, 0, 0 , 1, 0, 1, 0 , 1, 0, 0, 1 , 0, 1, 1, 0 , 0, 1, 0, 1 , 0, 0, 1, 1 . | is | is | is | is | is | is The corresponding matrix is

U+2V 0 0 0 √2t 0 √2t 0 0 0 1 − − 0 U+2V 0 0 √2t 0 0 √2t 0 0  2 − −  0 0 U+2V3 0 0 0 0 √2t 0 √2t  U V √ t − −√ t   0 0 0 +2 4 0 0 2 0 0 2   t t V V t − t −   √2 √2 0 0 1+ 2 0 0 0 .  − − t V −V t t − t   0 0 0 0 1+ 3 0   t t − t V −V − t −   √2 0 0 √2 0 1+ 4 0 0   − − − −   0 √2t √2t 0 0 t 0 V2+V3 t 0   − − − −   0 0 0 0 t 0 t t V2+V4 t   − − − −   0 0 √2t √2t 0 t 0 0 t V +V   − − − − 3 4   (C.16)

Triplet states consist of 6 of S = 1 states, 6 of S = 1 states, and 6 of S = 0 z z − z states:

, , 0, 0 , , 0, , 0 , , 0, 0, , 0, , , 0 , 0, , 0, , 0, 0, , |↑ ↑ i |↑ ↑ i |↑ ↑i | ↑ ↑ i | ↑ ↑i | ↑ ↑i , , 0, 0 , , 0, , 0 , , 0, 0, , 0, , , 0 , 0, , 0, , 0, 0, , |↓ ↓ i |↓ ↓ i |↓ ↓i | ↓ ↓ i | ↓ ↓i | ↓ ↓i 1, 1, 0, 0 , 1, 0, 1, 0 , 1, 0, 0, 1 , 0, 1, 1, 0 , 0, 1, 0, 1 , 0, 0, 1, 1 . | it | it | it | it | it | it

The matrix is

V1 + V2 t 0 0 t 0 t V −+ V t t 0 t − 1 3 − −  0 t V + V 0 t 0  − 1 4 − . (C.17)  0 t 0 V2 + V3 t 0   − −   t 0 t t V + V t   − − 2 4 −   0 t 0 0 t V + V   − 3 4   

109 C.2.5 Case: (e)

There are 21 singlet states:

2, 2, 0, 0 , 2, 0, 2, 0 , 2, 0, 0, 2 , 0, 2, 2, 0 , 0, 2, 0, 2 , 0, 0, 2, 2 | i | i | i | i | i | i 2, 1, 1, 0 , 2, 1, 0, 1 , 2, 0, 1, 1 , 1, 2, 1, 0 , 1, 2, 0, 1 , 0, 2, 1, 1 | is | is | is | is | is | is 1, 1, 2, 0 , 1, 0, 2, 1 , 0, 1, 2, 1 , 1, 1, 0, 2 , 1, 0, 1, 2 , 0, 1, 1, 2 | is | is | is | is | is | is 1 1 1 ( , , , + , , , ), ( , , , + , , , ), ( , , , + , , , ). √2 |↑ ↓ ↑ ↓i |↓ ↑ ↓ ↑i √2 |↑ ↑ ↓ ↓i |↓ ↓ ↑ ↑i √2 |↑ ↓ ↓ ↑i |↓ ↑ ↑ ↓i

The corresponding matrix element is

0 0 0 0 0 √2t 0 0 0 − 0 0000 √2t 0 √2t 0  − − 00 00000 √2t 0  000 000000−   0000 00000   00000 0000   √2t √2t 0 0 0 0 t 0 t   − − −  000000 t t 0  √ √ − −  0 2t 2t 0 0 0 0 t 0  − − −  000000 t 0 0   √2t 0 0 0 √2t 0 0 t 0 t ...  − − −  0 0 0 √2t √2t 0 000 t  − − −  0 √2t 0 √2t 0 0 000 t  − −  0 √2t 0 0 0 √2t 0 0 0 0  − −  0 00 00 00000   0 0 √2t 0 √2t 0 0 t 0 0  − −  00000000 t 0   0 0 0 0 √2t √2t 0 0 0 0  − −  000000 t 0 t 0   000000 t 0− 0 0  −  00000000 t 0  

110 √2t 0 0 00000000 − 0 0 √2t √2t 0 0 0 0 000 − −  0 0 0 00 √2t 0 0 000 −  0 √2t √2t 0 0 0 0 0 000  − −  √2t √2t 0 0 0 √2t 0 √2t 0 0 0  − − − −  0 0 0 √2t 0 0 0 √2t 0 0 0  − −  0 0 0 00000 t t 0  −  t 0 0 00 t 0 0 000   0 0 0 000 t 0 t 0 t  −  t t t 0 0 0 0 0 000  − −  ... 0 0 00000 t t 0  , (C.18) −  0 00 t 0 0 0 t 0 t  −  00 0 t 0 0 0 t 0 t  − −  0 0 0 t 0 t 0 t t 0  − −  0 t t t 0 0 t 0 0 0  − −  0 0 0 00 t 0 t 0 t  − −  0 0 0 t 0 t t 0 0 0  − −  0 0 0 0 t 0 t t t 0  − −  t t t t 0 t 0 t 0 0  − − −  t 0 0 t 0 0 0 t 0 0  − − −  0 t t 0 0 t 0 0 00   where the diagonal terms are omitted for the better looking. 

Triplet states consist of 16 of S = 1 states, 16 of S = 1 states, and 15 of S =0 z z − z states. First, we consider 15 Sz = 0 states:

2, 1, 1, 0 , 2, 1, 0, 1 , 2, 0, 1, 1 , 1, 2, 1, 0 , 1, 2, 0, 1 , 0, 2, 1, 1 | it | it | it | it | it | it 1, 1, 2, 0 , 1, 0, 2, 1 , 0, 1, 2, 1 , 1, 1, 0, 2 , 1, 0, 1, 2 , 0, 1, 1, 2 | it | it | it | it | it | it 1 1 1 ( , , , , , , , ), ( , , , , , , , ), ( , , , , , , , ). √2 |↑ ↓ ↑ ↓ i−|↓ ↑ ↓ ↑i √2 |↑ ↑ ↓ ↓ i−|↓ ↓ ↑ ↑i √2 |↑ ↓ ↓ ↑ i−|↓ ↑ ↑ ↓i

111 The matrix is t 0 t 00000000 t t 0 − − − t t 0 t 0 0 0 0 t 00000  −0 t − 0000000− t 0 t 0 t  − − − −  t 0 0 t t t 00000000   −   0 t 0 t 0000000 t t 0   − −   0 0 0 t 0 00 t 0 0 0 t 0 t   − −   0 0 0 t 00 0 t 0 0 0 t 0 t   −   0000000 t 0 t 0 t t 0  . (C.19)  − −   00000 t t t 0 0 t 0 0 0   −   0 t 0000000 t 0 t 0 t   − − −   0 0 t 0 0 0 0 t 0 t t 0 0 0   − − −   00000000 t 0 t t t 0   − − −   t 0 t 0 t t t t 0 t 0 t 0 0   − − − − − −   t 0 0 0 t 0 0 t 0 0 0 t 0 0   − − − −   0 0 t 0 0 t t 0 0 t 0 0 0 0   − −    Next, we consider 16 Sz = 1 states:

2, , , 0 , 2, , 0, , 2, 0, , , , 2, , 0 , , 2, 0, , 0, 2, , | ↑ ↑ i | ↑ ↑i | ↑ ↑i |↑ ↑ i |↑ ↑i | ↑ ↑i , , 2, 0 , , 0, 2, , 0, , 2, , , , 0, 2 , , 0, , 2 , 0, , , 2 |↑ ↑ i |↑ ↑i | ↑ ↑i |↑ ↑ i |↑ ↑ i | ↑ ↑ i , , , , , , , , , , , , , , , . |↓ ↑ ↑ ↑i |↑ ↓ ↑ ↑i |↑ ↑ ↓ ↑i |↑ ↑ ↑ ↓i

The matrix is 0 0 t 00000000 t 0 0 t 0 00 t 0 0 00 t 0 000 0− 0 −  00 0000000 t 0 t t 0 0  − −  t 00 00 t 000 0000 00     0 t 00 00000000 t t 0   −   00000 00 t 0 0 0 t t 0 0   −   0 0 0 t 00 0000000 t t   −   0000000 00 t 0 0 t t 0   −  . (C.20)  0 0 000 t 00 00 t 00 0 0     0 t 0000000 0000 t t   − −   0 0 t 00 0 0 t 00 00000   −   00000000 t 0 0 t 0 0 t   −   t 0 t 0 0 t 0 000 0 t 0 0 0     0 0 t 0 t t 0 t 00000 00   − −   0 0 00 t 0 t t 0 t 0000 0   − −   t 0 000 0 t 0 0 t 0 t 0 0 0   − − − −    112 J = E E distributions for the nn coupling, J and the next couplings, J , J , t − s a c

Jd from the above diagnalizations for the boxed disorder potential are plotted on

Fig. C.3.

113 PHJL PHJa L 5 7

4 6 5 3 4 2 3 2 1 1 J J -0.5 0.5 1 1.5 2 2.5 3 -0.5-0.25 0.25 0.5 0.75 1 1.25 1.5 a

PHJc L PHJd L 5 2

4 1.5 3 1 2 0.5 1

J J -0.6-0.4-0.2 0.2 0.4 0.6 c 1 2 3 4 5 d

Figure C.3: Comparison between the probability distributions for the coupling for U =4t at Vd =2.5 (red), Vd = 3 (green), and Vd = 5 (blue) from the diagonalization method. For J, V does not eﬀect on the plot as long as V U/2. P (J ) = P (J ), d d ≥ a b P (Jc)= P (Je) and P (Jd)= P (Jf ) from the symmetry.

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