Finite-temperature line shapes of hard-core bosons in quantum magnets: A diagrammatic approach tested in one dimension

The dynamics in quantum magnets can often be described by effective models with bosonic excitations obeying a hard-core constraint. Such models can be systematically derived by renormalization schemes such as continuous unitary transformations or by variational approaches. Even in the absence of further interactions the hard-core constraint makes the dynamics of the hard-core bosons nontrivial. Here, we develop a systematic diagrammatic approach to the spectral properties of hard-core bosons at finite temperature. Starting from an expansion in the density of thermally excited bosons in a system with an energy gap, our approach leads to a summation of ladder diagrams. Conceptually, the approach is not restricted to one dimension, but the one-dimensional case offers the opportunity to gauge the method by comparison to exact results obtained via a mapping to Jordan-Wigner fermions. In particular, we present results for the thermal broadening of single-particle spectral functions at finite temperature. The line shape is found to be asymmetric at elevated temperatures and the bandwidth of the dispersion narrows with increasing temperature. Additionally, the total number of thermally excited bosons is calculated and compared to various approximations and analytic results. Thereby, a flexible approach is introduced that can also be applied to more sophisticated and higher dimensional models.

Figure 1

Ladder diagrams for the one-particle self-energy.

Figure 2

Scattering amplitude $\Gamma$.

Figure 3

Bethe-Salpeter equation for the scattering amplitude.

Figure 4

Calculation of the self-energy.

Figure 5

Comparison between the spectral function obtained from the fermionic approach and from the diagrammatic expansion. Parameters are $p=0$ and $p=\pi$ with $W=0.5\Delta$ and $T=0.434\Delta$, i.e., $exp(-\beta \Delta )=0.100$.

Figure 6

Comparison between the spectral function obtained from the fermionic approach and from the diagrammatic expansion. Parameters are $p=0$ and $p=\pi$ with $W=4\Delta$ and $T=0.434\Delta$, i.e., $exp(-\beta \Delta )=0.100$.

Figure 7

Peak shift for $W=0.5\Delta$ as a function of the inverse temperature. Crosses represent results obtained in the fermionic approach, while boxes and circles represent results obtained by the diagrammatic expansion. The function ${log}_{10}[Aexp(-2\beta \Delta )]$ has been fitted to the data using the parameter $A$.

Figure 8

Peak shift for $W=4\Delta$ as a function of the inverse temperature. Crosses represent results obtained in the fermionic approach, while boxes and circles represent results obtained by the diagrammatic expansion. The function ${log}_{10}[Aexp(-2\beta \Delta )]$ has been fitted to the data using the parameter $A$.

Figure 9

Full width at half maximum of the peak for $W=0.5\Delta$ as a function of the inverse temperature. Crosses represent results obtained in the fermionic approach, while boxes and circles represent results obtained by the diagrammatic expansion. The function ${log}_{10}[Aexp(-\beta \Delta )]$ has been fitted to the data using the parameter $A$.

Figure 10

Full width at half maximum of the peak for $W=4\Delta$ as a function of the inverse temperature. Crosses represent results obtained in the fermionic approach, while boxes and circles represent results obtained by the diagrammatic expansion. The function ${log}_{10}[Aexp(-\beta \Delta )]$ has been fitted to the data using the parameter $A$.

Figure 11

Weight of the spectral function for $W=0.5\Delta$ as a function of the inverse temperature. Crosses represent results obtained in the fermionic approach, while boxes and circles represent results obtained by the diagrammatic expansion. The function ${log}_{10}[Aexp(-\beta \Delta )]$ has been fitted to the data using the parameter $A$.

Figure 12

Weight of the spectral function for $W=4\Delta$ as a function of the inverse temperature. Crosses represent results obtained in the fermionic approach, while boxes and circles represent results obtained by the diagrammatic expansion. The function ${log}_{10}[Aexp(-\beta \Delta )]$ has been fitted to the data using the parameter $A$.

Figure 13

Temperature dependence of the occupation function for $W=0.5\Delta$. Comparison between a simple bosonic approximation, the non-self-consistent solution, the self-consistent solution, the approximate statistics from Ref. [45], and the exact fermionic expression.

Figure 14

Temperature dependence of the occupation function for $W=4\Delta$. Comparison between a simple bosonic approximation, the non-self-consistent solution, the self-consistent solution, the approximate statistics from Ref. [45], and the exact fermionic expression.

Figure 15

Real and imaginary parts of the self-energy $\Sigma (p,\omega )$ at various temperatures for the gap mode $p=0$ for $W=0.5\Delta$. The vertical line indicates the energy of the $p=0$ mode at zero temperature.

Figure 16

Real and imaginary parts of the self-energy $\Sigma (p,\omega )$ at various temperatures for the maximum mode $p=\pi$ for $W=0.5\Delta$. The vertical line indicates the energy of the $p=\pi$ mode at zero temperature.

Figure 17

Spectral function $A(p,\omega )$ at various temperatures for the gap mode $p=0$ for $W=0.5\Delta$.

Figure 18

Comparison between Lorentzian line shape and asymmetric spectral function at $p=0$ for $W=0.5\Delta$.

Figure 19

Spectral function $A(p,\omega )$ at various temperatures for the maximum mode $p=\pi$ for $W=0.5\Delta$.

Figure 20

Spectral function for various momenta at fixed temperature $T=0.8\Delta$ for $W=0.5\Delta$. Note the offset of the $y$ axis.

Figure 21

Dispersion determined from the position of the maximum of the spectral function for $W=0.5\Delta$ for various temperatures.

Figure 22

Dispersion determined from the position of the maximum of the spectral function for $W=4\Delta$ for various temperatures. The inset shows a more detailed plot for the gap mode and for the maximum mode.