Multiparticle spectral properties in the transverse field Ising model by continuous unitary transformations

The one-dimensional transverse field Ising model is solved by continuous unitary transformations in the high-field limit. A high accuracy is reached due to the closure of the relevant algebra of operators, which we call string operators. The closure is related to the possibility to map the model by Jordan-Wigner transformation to noninteracting fermions, but it is proven without referring to this mapping. The effective model derived by the continuous unitary transformations is used to compute the contributions of one, two, and three elementary excitations to the diagonal dynamic structure factors. The three-particle contributions have, so far, not been addressed analytically, except close to the quantum critical point.

Figure 1

Ground-state energy per site as function of $J$. Comparison of the exact result to CUT results in various orders of $x=J/\left(2\Gamma \right)$. The inset shows the absolute difference between the exact result and the CUT calculation on a logarithmic scale. The critical point is located at ${\log }_{10}\left(\frac{J}{2\Gamma }\right)=0$.

Figure 2

Transverse magnetization as a function of $J$. Comparison of the exact result with the CUT calculation. The inset shows a close view on the critical point $J=2\Gamma$.

Figure 3

One-particle and three-particle spectral weights as functions of the parameter $J$. Comparison of the exact expression for the one-particle weight (8a) with the CUT results.

Figure 4

One-particle and three-particle spectral weights as functions of the parameter $J$. Comparison of the exact expression for the one-particle weight (8b) with the CUT results.

Figure 5

One-particle equal time structure factor $S_1^{xx}\left(Q\right)$ for the parameters $J=\Gamma$, $1.5\Gamma$, and $1.9\Gamma$. Comparison of the exact expression (8a) for the one-particle weight with the CUT results.

Figure 6

One-particle equal-time structure factor $S_1^{yy}\left(Q\right)$ for the parameters $J=\Gamma$, $1.5\Gamma$, and $1.9\Gamma$. Comparison of the exact expression (8b) with the CUT results.

Figure 7

Energy dispersion for $J=2.0\Gamma$ (top) and $J=1.9\Gamma$ (bottom). Comparison of the CUT calculations to the exact results.

Figure 8

Qualitative sketch of the band-edge singularities in the DSF.

Figure 9

The DSF $S^{zz}(\omega ,Q)$ for the parameters $J=\Gamma$ (top), $1.5\Gamma$ (center), and $1.9\Gamma$ (bottom). The maximum range for the Lanczos algorithm is $d_{\max }=1000$ sites, the continued fraction was evaluated to a depth of 50 and then terminated by the square root terminator. The color indicates the spectral density, see legend to the right. The upper and lower edges of the two-particle continuum are indicated by white lines.

Figure 10

DSF $S^{zz}(\omega ,Q)$ for the parameters $J=1.5\Gamma$ (left) and $1.9\Gamma$ (right) for three total momenta $Q$. The maximum range for the Lanczos algorithm is $d_{\max }=4000$ sites, the continued fraction was evaluated to a depth of 100 and then terminated by the square root terminator.

Figure 11

Absolute difference between the continued fraction coefficients and their final values for the case $J=1.5\Gamma$ and total momenta $Q=0$ and $\pi /2$.

Figure 12

The DSF $S_3^{xx}(\omega ,Q)$ for the parameters $J=\Gamma$ (top), $1.5\Gamma$ (center), and $1.9\Gamma$ (bottom). The maximum range for the Lanczos algorithm is $d_{\max }=100$ sites, the continued fraction was evaluated to a depth of 50 and then terminated by the square root terminator. The color indicates the spectral density, see a legend to the right. The dispersion is indicated by the white solid line. The upper and lower edges of the three-particle continuum are indicated by white dashed lines. All results are computed in order 38.

Figure 13

DSF $S_3^{xx}(\omega ,Q)$ for the parameter $J=1.9\Gamma$ for three chosen total momenta $Q$. The maximum range for the Lanczos algorithm is $d_{\max }=300$ sites, the continued fraction was evaluated to a depth of 100 and then terminated by the square root terminator.

Figure 14

Continued fraction coefficients for the case $J=1.5\Gamma$ and total momentum $Q=0$. The upper panel shows the coefficients $a_n$ and the lower panel shows the coefficients $b_n$. The limit values are indicated by horizontal lines. The red/green lines indicated linear fits in $1/n^2$; note the scale of the $x$ axis.

Figure 15

DSF $S_3^{yy}(\omega ,Q)$ for the parameters $J=\Gamma$ (top), $1.5\Gamma$ (center), and $1.9\Gamma$ (bottom). The maximum range for the Lanczos algorithm is $d_{\max }=100$ sites, the continued fraction was evaluated to a depth of 50 and then terminated by the square root terminator. The color indicates the spectral density, see a legend to the right. The dispersion is indicated by the white solid line. The upper and lower edges of the three-particle continuum are indicated by white dashed lines. All results are computed in order 38.

Figure 16

DSF $S_3^{yy}(\omega ,Q)$ for the parameter $J=1.9\Gamma$ for three chosen total momenta $Q$. The maximum range for the Lanczos algorithm is $d_{\max }=300$ sites, the continued fraction was evaluated to a depth of 100 and then terminated by the square root terminator.