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Density fluctuations in the 1D Bose gas
Panfil, M.K.
Publication date
2013
Link to publication
Citation for published version (APA):
Panfil, M. K. (2013). Density fluctuations in the 1D Bose gas.
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Chapter 7
Conclusions and Future Ideas
The focus of this thesis was aimed at understanding collective, emergent behavior of the 1D Bose gas. The tool was the density-density correlation function. The main objective was to recover the full dynamic correlation function,S(k, ω).
We have started with the low-energy limit of the theory in Chapter4. We have seen how the mixture of the universal, emergent description in the form of Luttinger liquid and the exact microscopic description provide us with a detailed prediction of the correlations. However the universal character of the results means that this depiction is limited: for real space correlation only to the large distance, long time asymptotes, for momentum space to a vicinity of the thresholds.
In the next Chapter (see Chapter 5) we have investigated effects of finite temperatures on the correlations. We have seen how the critical, T = 0, behavior is smoothed in a non-uniform way and still carries signatures of the criticality, like traces of the threshold singularities. We hope that in the future the presented result will allow for a comparison with the experimental realizations of the 1D Bose gas.
Finally in the previous chapter (see Chapter 6) we embarked on a cruise towards an exotic phase of the 1D Bose gas, the super Tonks-Girardeau gas. We have shown that despite its metastability the sTG gas is still a critical fluid: it fits into the Luttinger universality class. This shows that emergent properties are not per se dependent on an equilibrium situation.
The most direct expansion of the results presented in Chapters 5 and 6 would be by supplementing the density-density correlation function with other local correlations. The natural candidates would be Green’s functionhΨ(x, t)Ψ†(0, 0)i and the one-body function hΨ†(x, t)Ψ(0, 0)i. This would allow to access different characteristics of the 1D Bose gas
such as probability of particles tunneling into/out the gas, the momentum distribution 133
Chapter 7: Conclusions and Future Ideas 134
or the phase coherence. This would certainly enrich our understanding of the many-body quantum effects.
Specifically in the context of the sTG gas an interesting problem would be dynamics of the bound states in the sea of single particles.
The results presented in Chapter 4 invite for a discussion of feasibility of an analytic computation of the correlation functions. An interesting approach would be to consider low energy limit directly at the microscopic, Lieb-Liniger Hamiltonian, level. This would allow to determine the correlation function in the regime that is beyond the (non-linear) Luttinger liquid, particularly, far from the threshold singularities.
