Bridging Perturbative Expansions with Tensor Networks

We demonstrate that perturbative expansions for quantum many-body systems can be rephrased in terms of tensor networks, thereby providing a natural framework for interpolating perturbative expansions across a quantum phase transition. This approach leads to classes of tensor-network states parametrized by few parameters with a clear physical meaning, while still providing excellent variational energies. We also demonstrate how to construct perturbative expansions of the entanglement Hamiltonian, whose eigenvalues form the entanglement spectrum, and how the tensor-network approach gives rise to order parameters for topological phase transitions.

Figure 1

Magnetization along the $Z$ direction in the TFIM as a function of the magnetic field for different variational Ansätze: order zero (blue) with one parameter ($\alpha$), order 1 (red) with three parameters ($\alpha ,{\beta }_0,{\beta }_1$), order 2 (yellow) with five parameters ($\alpha ,{\beta }_0,{\beta }_1,{\gamma }_0,{\gamma }_1$) (see Table 2 for details). Inset: behavior of the virtual order parameter $\alpha$ near the critical point.

Figure 2

Virtual order parameter $v$ (blue), the magnetization along the $X$ direction (red), and the expectation value of the operator $B_p$ (yellow) in the TCX model as a function of the magnetic field obtained with the $D=4$–PEPS Ansatz (15 parameters). The vertical line (dashed green) indicates the critical point ${\lambda }_0/{\lambda }_1\simeq 0.328$. Inset: behavior of the virtual order parameter $v$ near the critical point.