Bose–Einstein Condensation Processes with Nontrivial Geometric Multiplicities Realized via 𝒫𝒯−Symmetric and Exactly Solvable Linear-Bose–Hubbard Building Blocks
Abstract
:1. Introduction
2. Bose-Einstein Condensation
2.1. Symmetric Bose–Hubbard Model
2.2. BEC-Formation Patterns
3. Nontrivial Geometric Multiplicities at Small
3.1. Three Bosons and
3.2. Four Bosons and
4. BEC Models of Any Dimension and Multiplicity
4.1. Canonical Representation
4.2. Transition Matrices
4.3. Geometric Multiplicities : Realization
5. Physics behind the Generalized BH Model
5.1. Change of Phase: Two Alternative Physical Interpretations
5.2. The Role of Symmetry
6. Combinatorics behind the Classification
6.1. Classification Scheme
6.2. An Alternative Notation
7. Discussion
7.1. The Specific Features of Bosons
7.2. The Problem of the Non-Uniqueness of the Model
7.3. A Remark on the Theory of Quantum Phase Transitions
8. Summary
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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N | |||
---|---|---|---|
2 | [2] | 1 | 1 |
3 | [3] | 1 | 1 |
4 | [4], [2,2] | 2 | 2 |
5 | [5], [3,2] | 2 | 2 |
6 | [6], [4,2], [3,3], [2,2,2] | 4 | 3 |
7 | [7], [5,2], [4,3], [3,2,2] | 4 | 3 |
8 | [8], [6,2], [5,3], [4,4], | ||
[4,2,2], [3,3,2], [2,2,2,2] | 7 | 4 |
N | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1 | 2 | 2 | 4 | 4 | 7 | 8 | 12 | 14 | 21 | 24 | 34 | 41 | 55 | 66 | 88 | 105 | 137 |
N | # | K | m | Partition | Set |
---|---|---|---|---|---|
2 | 1 | 1 | 0 | [2] | {1} |
3 | 1 | 1 | 0 | [3] | {1} |
4 | 1 | 1 | 0 | [4] | {1} |
2 | 2 | 1 | [2,2] | {1,3} | |
5 | 1 | 1 | 0 | [5] | {1} |
2 | 2 | 1 | [3,2] | {1,4} | |
3 | 2 | 1 | [3,2] | {2,2} | |
6 | 1 | 1 | 0 | [6] | {1} |
2 | 2 | 1 | [4,2] | {1,5} | |
3 | 3 | 3 | [2,2,2] | {1,3,5} | |
7 | 1 | 1 | 0 | [7] | {1} |
2 | 2 | 1 | [5,2] | {1,6} | |
3 | 2 | 2 | [4,3] | {2,2} | |
4 | 3 | 3 | [3,2,2] | {1,4,6} | |
5 | 3 | 3 | [3,2,2] | {2,2,6} | |
6 | 3 | 3 | [3,2,2] | {3,2,4} |
N | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1 | 2 | 3 | 3 | 6 | 4 | 11 | 6 | 17 | 7 | 32 | 8 | 47 |
N | Partitions (18) of | # | Isospectral–Hamiltonian Indices | # |
---|---|---|---|---|
2 | 2 | 1 | {1} | 1 |
3 | 3 | 1 | {02} | 1 |
4 | 4, 2 + 2 | 2 | {13}, {1}{3} | 2 |
5 | 5, 3 + 2 | 2 | {024}, {02}{4}, {04}{2} | 3 |
6 | 6, 4+2, 3+3, 2+2+2 | 4 | {135}, {13}{5}, {1}{3}{5} | 3 |
7 | 7, 5 + 2, 4 + 3, | 4 | {0246}, {024}{6}, {04}{26}, | 6 |
3+2+2 | {02}{4}{6}, {04}{2}{6}, {06}{2}{4} | |||
8 | 8, 6 + 2, 5 + 3, 4 + 4, 4 + 2 + 2, | 7 | {1357}, {135}{7}, {13}{5}{7}, | 4 |
3 + 3 + 2, 2 + 2 + 2 + 2 | {1}{3}{5}{7} |
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Znojil, M. Bose–Einstein Condensation Processes with Nontrivial Geometric Multiplicities Realized via 𝒫𝒯−Symmetric and Exactly Solvable Linear-Bose–Hubbard Building Blocks. Quantum Rep. 2021, 3, 517-533. https://doi.org/10.3390/quantum3030034
Znojil M. Bose–Einstein Condensation Processes with Nontrivial Geometric Multiplicities Realized via 𝒫𝒯−Symmetric and Exactly Solvable Linear-Bose–Hubbard Building Blocks. Quantum Reports. 2021; 3(3):517-533. https://doi.org/10.3390/quantum3030034
Chicago/Turabian StyleZnojil, Miloslav. 2021. "Bose–Einstein Condensation Processes with Nontrivial Geometric Multiplicities Realized via 𝒫𝒯−Symmetric and Exactly Solvable Linear-Bose–Hubbard Building Blocks" Quantum Reports 3, no. 3: 517-533. https://doi.org/10.3390/quantum3030034