1 Introduction to Many-Body Physics in Utracold Atomic Gases . . 1
1.1 Motivation: Many-Body Physics . . 1
1.2 Motivation: General Atomic Physics . . 4
1.3 History and Introduction to Many-Body Physics in Cold Atomic Systems . . 5
1.4 Challenges and Outline of Thesis . . 7
References . . 9

2 Theoretical and Experimental Techniques Used to Explore Many-Body Physics in Cold Atoms, 
  Especially Optical Lattices . . 11
2.1 Experimental Techniques . . 11
2.2 Theoretical Techniques . . 18
References . . 31

3 Radio-Frequency Spectroscopy: Broad Introduction . . 33
3.1 Motivation and Background . . 33
3.2 Two Differing Pictures of RF Spectroscopy . . 34
References . . 35

4 RF Spectra: A Sum Rule Approach to Trapped Bosons in an Optical Lattice . . 37
4.1 Chapter Overview . . 37
4.2 Introduction . . 37
4.3 Spectrum of Harmonically Trapped Gas . . 39
4.4 Refinements . . 46
4.5 Summary . . 49
References . . 50

5 RF Spectra: Multiple Peaked Spectrum in a Homogeneous System . . 51
5.1 Chapter Overview . . 51
5.2 Introduction . . 52
5.3 Bose-Hubbard Model . . 54
5.4 Random Phase Approximation . . 55
5.5 Conclusions and Discussion . . 61
References . . 62

6 Radio-Frequency Spectra at Finite Temperature, Fluctuation-Response Relations, 
  and Proposed Applications . . 63
6.1 Chapter Overview . . 63
6.2 Introduction . . 63
6.3 RF Spectra Introduction . . 64
6.4 Finite Temperature Superfluid . . 65
6.5 Applications . . 67
6.6 Determining Density Profiles from RF Spectra . . 75
6.7 Conclusions . . 83
References . . 84

7 RF Spectra: Summary, Conclusions, and the Future . . 85
References . . 86

8 Rotation, Inducing Gauge Fields, and Exotic States of Matter in Cold Atoms . . 87
8.1 Physics of Rotating Particles/Particles in Gauge Fields . . 88
8.2 Rotation . . 92
8.3 Other Methods of Inducing Gauge Fields . . 93
8.4 On-Site Correlations . . 94
References . . 94

9 Stirring up Fractional Quantum Hall Puddles . . 97
9.1 Chapter Overview . . 97
9.2 Introduction . . 97
9.3 Summary . . 103
References . . 104

10 Incorporating Arbitrarily Strong On-Site Correlations into Lattice Models . . 105
10.1 Chapter Overview . . 106
10.2 Body . . 106
References . . 113

11 Quantitative Calculation of Parameters for a Model Sufficiently 
   General to Capture all On-Site Correlations . . 115
11.1 Background . . 115
11.2 Introduction, Notation, and Set Up . . 115
11.3 Quantitative Estimates of the Hamiltonian Parameters with Quantum Monte Carlo . . 116
11.4 Solutions for Various Values of t(mn) and Em . . 118
11.5 Note on Temperature Dependence of Response Functions . . 129
Reference . . 129

12 Summary, Conclusions, and the Future of Induced Gauge 
   Fields and Lattices with On-Site Correlations . . 131
References . . 131

13 Techniques to Measure Quantum Criticality in Cold Atoms . . 131
13.1 Introduction . . 133
13.2 Note and Summary . . 139
References . . 140

14 Quantum Criticality: More Detailed Information . . 143
14.1 Bose-Hubbard Model . . 143
14.2 Finite Temperature Gutzwiller Theory . . 144
14.3 Non-Universal Contributions . . 144
14.4 Time of Flight Expansion . . 145
14.5 Finite Density O(2) Model . . 146
14.6 Calculating Density Profiles of One-Dimensional Hardcore Bosons . . 147
14.7 Other Cold Atoms Systems Displaying Quantum Criticality . . 148
14.8 Precise Definition of Universality . . 148
14.9 Quantum Monte Carlo Parameters and Signal-to-Noise . . 150
14.10 Finite-Size Scaling . . 151
References . . 151

15 Systems Other than Cold Atoms . . 153

16 Film Mediated Interactions Alter Correlations and Spectral 
   Shifts of Hydrogen Adsorbed on Helium Films . . 155
16.1 Chapter Overview . . 155
16.2 Results . . 156
References . . 162

17 Candidate Theories to Explain the Anomalous Spectroscopic 
   Signttures of Atomic H in Molecular H2 Crystals . . 165
17.1 Introduction and Motivation . . 165
17.2 Experiments . . 166
17.3 Scenarios . . 168
17.4 Other Observations . . 175
17.5 Summary, Conclusions, and Consequences . . 176
References . . 177

18 Helium and Hydrogen (super?)Solids . . 179
18.1 Background . . 179
18.2 Chapter Overview . . 179
18.3 Introduction . . 180
18.4 Torsional Oscillator NCRI . . 181
18.5 Two Supersolid Features . . 181
18.6 Blocked Annulus Torsional Oscillators . . 182
18.7 Dissipation Peaks . . 182
18.8 H2 Experiments . . 183
18.9 Frequency Dependence . . 186
18.10 Thermodynamics: Specific Heat and Pressure . . 187
18.11 3He Doping . . 188
18.12 Anomalous Critical Velocity . . 191
18.13 DC Flow . . 191
18.14 Shear . . 191
18.15 Implications for Mechanism . . 192
18.16 Future Directions . . 192
18.17 Conclusions . . 193
References . . 193

Appendix A: Relating Scattering Amplitudes and T-Matrix . . 195
Appendix B: Ward Identities for the RF Spectrum for the Bose-Hubbard Model—Vertex Corrections, 
            Symmetries, and Conservation Laws . . 197

Biographical Sketch . . 229
Curriculum Vitae . . 231