Topological defects in quantum field theory with matrix product states

Topological defects (kinks) in a relativistic ${\varphi }^4$ scalar field theory in $D=(1+1)$ are studied using the matrix product state tensor network. The one kink state is approximated as a matrix product state and the kink mass is calculated. The approach used is quite general and can be applied to a variety of theories and tensor networks. Additionally, the contribution of kink-antikink excitations to the ground state is examined and a general method to estimate the scalar mass from equal time ground state observables is provided. The scalar and kink mass are compared at strong coupling and behave as expected from universality arguments. This suggests that the matrix product state can adequately capture the physics of defect-antidefect excitations and thus provides a promising technique to study challenging nonequilibrium physics such as the Kibble-Zurek mechanism of defect formation.

Figure 1

The field expectation value for various effective couplings using a uMPS approximation to the ground state. In the left-hand plot, weak-couplings are displayed with ${\lambda }_0=0.1$ and $(d,\chi )=(40,32)$ (blue triangles). These can be compared to low $\chi =4$ approximations (hollow red circles) which look almost identical on this scale. This contrasts the low $d=12$ approximations (hollow black squares) for which the expectation value appears truncated agreeing with the higher $d$ approximations only when the field expectation value is relatively low. The insets show the convergence for the selected inverse coupling $g_0^{-1}=-1.5$ (the best estimate has been subtracted so that the plots tend to zero) and the difference in scales between the two indicate that $d$ is the relevant parameter to achieve a good approximation in the weak coupling regime. The semiclassical result is also shown in the main figure and lower-left inset (solid red line) with $v=\sqrt{-6{\mu }^2/{\lambda }_0}$ where ${\mu }^2={\mu }_0^2-m_C^2$ and $m_C^2=-0.019$ is determined by fitting to the data. The stronger coupling data (right-hand plot) with $(d,\chi )=(18,32)$ and ${\lambda }_0=2$ (blue diamonds) shows the usual symmetry breaking pattern. Here, the relevant parameter to achieve a good approximation is $\chi$ as shown by the scale difference between the insets and also by the fact that the low $d$ approximation agrees well with the data on the main plot while the low $\chi$ data fails to agree near the critical point.

Figure 2

The field expectation value of the finite size lattice MPS approximation to the one kink state for weak coupling $g_0\approx -0.33$ (left-hand plot) and stronger coupling $g_0\approx -2.58$ (right-hand plot) with lattice sizes $L=32$, 64 and $28\le d\le 32$, $14\le d\le 18$ respectively. These can be compared with the corresponding field expectation values of a uMPS approximation to the ground state $\pm v$ (solid black lines) with $(d,\chi )=(40,32)$ and (18,32) for the weak coupling and stronger coupling respectively. For the low $\chi =6$ runs (solid red line) a classical kinklike profile is visible for both couplings which interpolates between $\pm v$ such that the correct $⟨\varphi {⟩}_K=0$ is not found. Increasing $\chi$ delocalizes the kink and translational invariance can be approximated in the stronger coupling case such that $⟨\varphi {⟩}_K\approx 0$.

Figure 3

The ${\varphi }^2$ expectation value corresponding to the one kink approximations in Fig. 2. As with $⟨\varphi {⟩}_K$, translational invariance can only be approximated in the stronger coupling case. Nevertheless, since the operator is ${\mathbb{Z}}_2$ invariant, the spatial average is well behaved and its convergence can be studied as shown in the insets.

Figure 4

The spatial average (red dots) of the $\varphi$ and ${\varphi }^2$ expectation values for the finite size lattice MPS approximation to the one kink state at weak coupling $g_0\approx -0.33$ (left-hand plots) and stronger coupling $g_0\approx -2.58$ (right-hand plots) with the same parameters as in Fig. 2. The spatial standard deviation of the expectation values, shown as error bars, changes strongly with $\chi$ and becomes small only for strong couplings. However, the spatial averages changes only weakly and the value of $⟪{\varphi }^2{⟫}_K$ can be reliably approximated in both regions. Note that, due to its antisymmetry, the value of $⟪\varphi {⟫}_K$ is only correct and equal to zero (indicated by the black line) in the stronger coupling case.

Figure 5

The kink mass $M_K$ (left-hand plot) and scalar mass $m_S$ (right-hand plot) for various weak couplings (data in blue triangles). The kink mass is calculated from the energy expectation value of the finite size lattice MPS approximation of the one kink state with $\chi =14$ and the approximation of the ground state energy density obtained from a uMPS approximation with $\chi =32$. The scalar mass is extracted from the uMPS approximation to the ground state connected equal time two point function $G_2(r)$ via a Bessel function ansatz (45). Both are compared with the semiclassical continuum results (solid red lines) $M_K=4\sqrt{2}{\mu }^3/{\lambda }_0$ and $m_S=\sqrt{2}\mu$ where ${\mu }^2={\mu }_0^2-m_C^2$ and $m_C^2$ is determined by fitting to the data to give $-0.025$, $-0.037$ respectively. The convergence of the approximations with $d$ and $\chi$ is shown in the insets.

Figure 6

Estimate of the scalar mass $m_S$ extracted from the uMPS approximation to the ground state connected equal time two point function $G_2(r)$ via a Bessel function ansatz (45) ($\chi =32$ red dots, $\chi =64$ red diamonds). This can be compared with the estimate extracted from the excitation ansatz $m_S^A$ (dashed red line) and twice the kink mass $2M_K$ (blue triangles) calculated from the $L=64$ finite size lattice MPS approximation to the one kink state. The left-hand plot shows a larger range of bare coupling $2.5\lesssim g_0\lesssim 5$ while the right-hand plot focuses on the strong coupling region with $2.95\lesssim g_0\lesssim 3.39$. A qualitative change can be seen when entering the strong coupling region at $m_S\approx 2M_K$ but the $\chi =32$ Bessel ansatz (45) does not provide a good quantitative agreement with the expected behavior $m_S\approx 2M_K$ in the strong coupling region. The higher $\chi =64$ Bessel ansatz does improve the estimate but the $\chi =64$ Bessel-squared ansatz (49) (black squares, only in right-hand plot) improves the estimate further in agreement with the $\chi =32$ excitation ansatz estimate (dashed red line). As mentioned in Sec. 5, the scalar mass is estimated from averaging over a “uniform” region of $m_S(r)$ chosen here by a single tolerance for all values of $g_0^{-1}$. While this has the advantage of being “blind” using a single tolerance can lead to somewhat anomalous points (e.g. see the point at $g_0^{-1}\approx -0.331$) and instead one can choose the tolerance adaptively for each $g_0^{-1}$ which eliminates such points, improving the estimates of $m_S$.