Efficient matrix product state methods for extracting spectral information on rings and cylinders

Based on the matrix product state (MPS) formalism, we introduce an ansatz for capturing excited states in finite systems with open boundary conditions, providing a very efficient method for computing, e.g., the spectral gap of quantum spin chains. This method can be straightforwardly implemented on top of an existing density-matrix renormalization group or MPS ground-state code. Although this approach is built on open-boundary MPSs, we also apply it to systems with periodic boundary conditions. Despite the explicit breaking of translation symmetry by the MPS representation, we show that momentum emerges as a good quantum number and can be exploited for labeling excitations on top of MPS ground states. We apply our method to the critical Ising chain on a ring and the classical Potts model on a cylinder. Finally, we apply the same idea to compute excitation spectra for 2D quantum systems on infinite cylinders. Again, despite the explicit breaking of translation symmetry in the periodic direction, we recover momentum as a good quantum number for labeling excitations. We apply this method to the 2D transverse-field Ising model and the half-filled Hubbard model; for the latter, we obtain accurate results for, e.g., the hole dispersion for cylinder circumferences up to eight sites.

Figure 1

The energy density of the four lowest-lying spin-1 magnon excitations on top of the ground state of the spin-1 Heisenberg chain with 100 sites. The ground state has bond dimension $D=64$.

Figure 2

The lowest-lying energy levels for the critical quantum Ising chain on a circle with circumferences $N=20$. The blue dots are results from the MPS excitation ansatz with $D=50$ and the red crosses are results from exact diagonalization. The energies were shifted and rescaled such that the ground state is at $e_0=0$ and the gap ${\mathrm{\Delta }}_{\epsilon }=1$. We see that the MPS with open boundary conditions restores translation invariance, and that degeneracies agree with exact results.

Figure 3

The rescaled spectrum of the transfer matrix of the three-state Potts model with system size $N=28$, as obtained with the excitation ansatz with bond dimension $D=150$. We take the negative logarithm of the transfer-matrix eigenvalues, $f=-ln\lambda$, such that $f$ corresponds to a free energy. We have rescaled the values such that fixed-point free energy is set at $f=0$ and the first excited state at $f=2/15$.

Figure 4

The momentum per rung of the MPS ground state [Eq. (16)] for the square-lattice Ising model on an $N=12$ cylinder as a function of bond dimension for three values of the field, $\lambda =2.5$ (blue), $\lambda =2.9$ (red) and $\lambda =3.04438$ (orange).

Figure 5

The dispersion relation of the square-lattice Ising model on a cylinder of circumference $N=12$ for three values of the field, $\lambda =2.5$ (blue), $\lambda =2.9$ (red) and $\lambda =3.04438$ (orange). The top panel show the BZ cut at transversal momentum $p_y=0$, nicely capturing the spectrum becoming gapless at the critical point. The bottom panel is for $p_y=\pi$, where the dispersion changes very little as the field is tuned. Here, all simulations were done with an MPS bond dimension $D=256$.

Figure 6

The gap at $\lambda =3.04438$ for cylinders $N=8$ (blue) and $N=12$ (red). We can converge the value of the gap with increasing bond dimension in both cases, although the larger cylinder needs higher bond dimension. The nonzero gap is due to the finite circumference of the cylinder, and clearly goes down as $N$ increases.

Figure 7

The dispersion relation of the Hubbard model with $t=1$ and $U=12$ on a cylinder of circumference $N=4$, in three cuts of the Brillouin zone $p_y=0$ (blue), $p_y=\pi /2$ (red), and $p_y=\pi$ (orange). The top panel shows the dispersion of the magnon excitation (charge-neutral spin-1 excitation) in three momentum cuts, and the bottom panel shows the hole dispersion (charged spin-1/2 excitation). Here, the total bond dimension is around $D=3500$, where the largest $\mathrm{SU}\left(2\right)\otimes \mathrm{U}\left(1\right)$ symmetry block is $D_{\max }=154$.

Figure 8

The minimum of the dispersion relation at the $S$ point [momentum $(\pi /2,\pi /2$)] for the $N=4$ (blue) and the $N=8$ (red) cylinder as a function of truncation threshold of the MPS.