content_image/422133.jpg content_image/422145.jpg content_image/422137.jpg content_image/422134.jpg
Calculations show that the energy gap △ E between the first and the second Bloch bands decreases with the de- crease of M , and when M is large, the energy gap △ E is independent of M . The energy gap between the first and the second Bloch bands as a function of magnetic field B and frequency ω are plotted in Fig. 1, respectively. At B = 0, raised by Girardeau [2], the energy gap of TG gas is the smallest (not 0). The energy gap increases as B and ω increase. As the Boltzman Distribution Law notes, n$_{k}$ n$_{k}$$_{′}$ ∝ exp( − △ E kT ), with △ E = E$_{k}$ − E$_{k}$ ′ , n$_{k}$ and n$_{k}$$_{′}$ are particle numbers on the corresponding states. So the system have a bigger probability in the ground state with large B and ω than the case B = 0 when the to- tal number N = M . That is to say, one could realize ground-stated TG system more simple with large q and ω than q = 0 when the total number N = M . 

<img src='content_image/422133.jpg'>

## III. PROPERTIES OF THE TG GAS COMPARED WITH THE FERMI GAS 

In this section, we further study properties of the 1D N magnetized hard-core Bosons in the external magnetic field (7), and then compared with the properties of 1D N magnetized Fermions in the “same” state which hard- core Bosons occupy. 

## A. Reduced single-particle density matrix and Single-particle density 

The reduced single-particle density matrix with nor- malization ∫ ρ ( x, x ) d x = N is given by B 

<img src='content_image/422145.jpg'>

For the Fermi gas, the reduced single-particle density matrix can be simplified to 

<img src='content_image/422137.jpg'>

<img src='content_image/422134.jpg'>

Both the reduced single-particle matrices of hard-core bosons and fermions in the ground state in the exter- nal magnetized field (7) are plotted separately in Fig. 2. The multidimensional integral in equation (18) is evalu- ated numerically by Monte Carlo integration using Math- ematica. Because the relationship between x and z is proportional, for the sake of simplicity, we take z as a variable. From the figure we can see the off-diagonal ele- ments of the matrix of both the two kinds of gases decay quickly as N increases; The bright diagonal stripe of TG gas is thinner than Fermi gas under the same condition. This reflects the condensate properties of Bose gas; And 

