A set of nearly good real numbers to specify the ground states associated with a Hamiltonian containing non-commutable terms and the effect of the odd-channel of a pair of different bosons emerging in multi-species systems

TL;DR

This work addresses ground-state spin textures in binary mixtures of spin-1 bosons with an odd-channel coupling that renders the spin interactions noncommuting. By reducing the spin-dependent Hamiltonian to a 4-parameter form in terms of $a$, $b$, $c$, and $d$ and performing exact diagonalization in a complete basis built with fractional parentage coefficients, the authors reveal that the odd channel mixes components and breaks the separate conservation of $S_A$ and $S_B$ while preserving total spin $S$. They identify coherent and cyclic mixing patterns and show that the ground state can be characterized by nearly good real numbers $ar{S}_A$ and $ar{S}_B$ that vary in a stepwise fashion with the parameters; the sign and magnitude of $d'$ control the attraction or repulsion from the odd channel and shift phase boundaries between p-p and f-f-like states. These findings advance understanding of noncommuting terms in many-body spin systems and demonstrate how nearly good quantum numbers can specify the ground state in regimes governed by odd-channel physics. The results have potential implications for engineering and interpreting spin textures in multi-species quantum gases where noncommuting interactions play a significant role.

Abstract

A distinguishing feature of multi-species boson systems is the appearance of the odd channel, in which the spins of two different bosons are coupled to an odd integer. Through exact numerical solutions of the Schrodinger equation for a medium-body cold system containing two kinds of spin-1 atoms, the effect of the odd channel on the ground state (GS) has been studied. It was found that the odd-channel causes two types of fluctuation (a mixing of various components). (i) coherent mixing, where all the components have the same sign. In this way, the probability of an odd-pair emerging in the spin-state would be smaller; thus, this way would be adopted by the GS when the odd channel is repulsive. (ii) cyclic mixing, where half selected components have the + sign while the other half have the - sign. In this way, the probability of an odd-pair is larger; thus, this way would be adopted by the GS when the odd channel is attractive. It was further found that the terms in the Hamiltonian are no longer all commutable. Accordingly, the spin of a single species S_X (X=A, B) is no longer conserved. However, its average \overline{S_X} is well defined. It turns out that S_X and \overline{S_X} vary with the strengths in a similar way. The former jumps step-by-step from a good quantum number (an even integer) to the next good quantum number, the latter jumps also in a step-by-step way, but from a "real number" to another well-separated "real number". Exactly speaking, each of these real numbers is not exactly a number but an interval with a very narrow width at the real axis. Thus, the GS can be specified by these real numbers. It was found that, when the strengths of the two intraspecies interactions are not remarkably different, and/or the particle numbers are larger, the widths of the intervals are narrower, the above picture holds more nicely, and the GS can be well-specified by these real numbers.

Figures (3)

• Figure 1: (color online) The phase diagram of the g.s. against $a'$ and $b'$, $N_A=12$, $N_B=8$, and $c<0$ are assumed. (a) is for $d'=0$, where four domains for p-p, f-f, f-q, and q-f are marked. Where f-q denotes that the $A$-species is in the f-phase while the $B$-species is in the q-phase, and so on. In (b), $d'$ is given at five values (i.e., $-2$, $-1$, $0$, $1$, and $2$), and only the associated boundary between the p-p phase and other phases is plotted to demonstrate how the domains vary with $d'$.
• Figure 2: (color online) (a) When $N_A=12$, $N_B=8$, $c<0$ and $b'=-1$, the variation of $S_A$ and $S_B=N_B$ (both in solid line for the case $d'=0$) and $\overline{S_A}$ and $\overline{S_B}$ (both in dashed line for the case $d'=-2$) against $a'$. (b) Similar to (a) but with $b'$ being fixed at $+1$.
• Figure 3: (color online) The same as Fig.\ref{['fig2']} but with $d'=0$ and $2$.