Momentum Selection

Translation invariance of wave functions are different from that in Neural networks, since wave functions have phases.

For state $\vert\psi\rangle$ with momentum $k$, we have the momentum-translation relation

$\langle T_n\sigma\vert\psi\rangle=e^{ikn}\langle \sigma\vert\psi\rangle$

Assume $\langle\sigma\vert\psi\rangle=h(\sum\limits_m e^{-ikm}g(T_m\sigma))$, and wish to obtain the state in desired momentum sector. Usually $g(T_m\sigma)$ correspond to some elemental function applied on the output of a convolution layer. To make g more powerful, we can mix different features but not $m$ (as batch dimension).

The above form also pose some requirements on function $h$.

For $k=0$, momentum-translation relation is automatically satisfied using arbituary $h$.

For $k\neq0​$, let $y(\sigma)=\sum\limits_m e^{-ikm}g(T_m\sigma)​$, we have $y(T_n\sigma)=e^{ikn}y(\sigma)​$ which is already a qualified solution. In order to keep this attribute, $h​$ need to meet the phase keeping requirement $h(e^{ikn}y)=e^{ikn}h(y)​$. The folloing construction are legal

  • linear function that $h(\alpha y_1+\beta y_2)=\alpha h(y_1)+\beta h(y_2)$, let $\beta=0$ and the above result is straight forward,
  • nonlinear function that map $y$ to either $0$ or $\alpha y$ like ReLU,
  • elemental function that act on absolute part like $h(y)=\hat{y}h(\vert y\vert)$ with $\hat{y}\equiv\frac{y}{\vert y\vert}$, $h(e^{ikn}y)=e^{ikn}\hat{y} h(\vert y\vert)=e^{ikn}h(y)$,
  • combination of phase keeping functions are also phase keeping.

Numerical Result

For 1D $J_1-J_2$ model, $L=16, J_2=0.8$, I calculated the RBM ground state in $k=0,\pi$. For $k=\pi$, we have the yellow line, which is the true ground state. And the blue line is the ground state in the $k=0$ block, both of them matches the exact diagonalization results (dashed, straight lines, don’t believe the legend…).

Ground state symmetry analysis

These ground states are obtained using ED.

columns are different sizes for chain varying from $2$ to $20$, and rows are different $J_2$ varying from $0.0$ to $1.0$.

Data element ‘-++’ means system changes sign for translate 1 site operation ($T_1$), keeps sign for spin flip ($F$) and space inversion symmetry $I$.

$J_2$\$\mathcal{N}$ 2 4 6 8 10 12 14 16 18 20
0.0 $—$ $+++$ $—$ $+++$ $—$ $+++$ $—$ $+++$ $—$ $+++$
0.2 $—$ $+++$ $—$ $+++$ $—$ $+++$ $—$ $+++$ $—$ $+++$
0.4 $—$ $+++$ $—$ $+++$ $—$ $+++$ $—$ $+++$ $—$ $+++$
0.6 $—$ $-++$ $+–$ $-++$ $+–$ $+++$ $—$ $+++$ $—$ $-++$
0.8 $—$ $-++$ $+–$ $+++$ $—$ $+++$ $+–$ $-++$ $—$ $+++$
1.0 $—$ $-++$ $+–$ $+++$ $—$ $-++$ $+–$ $+++$ $—$ $+++$

Summary

  • $I,F$ changes sign of wave function when and only when $L/2$ is odd.
  • For $J_2<0.5$, $T_1, F, I$ are all positive for even $L/2$ and negative for odd $L/2$.
  • For $L=16$, $T_1$ have negative eigenvalue! This must be why I fail to get ground state for $N=16$. Is there a phase transition? I suspect even representation power of our Ansaz is enough to describe signn structures of $J_2=0.5$ and $J_2=0.0$, it can fail to describe sign structures for $J_2>0.5$.