Excitation spectrum and density matrix renormalization group iterations

We show that, in certain circumstances, exact excitation energies appear as locally site-independent (or flat) modes if one records the excitation spectrum of the effective Hamiltonian while sweeping through the lattice in the variational matrix-product-state formulation of the density matrix renormalization group, a remarkable property since the effective Hamiltonian is only constructed to target the ground state. Conversely, modes that are very flat over several consecutive iterations are systematically found to correspond to faithful excitations. We suggest to use this property to extract accurate information about excited states using the standard ground-state algorithm. The results are spectacular for critical systems, for which the low-energy conformal tower of states can be obtained very accurately at essentially no additional cost, as demonstrated by confirming the predictions of boundary conformal field theory for two simple minimal models: the transverse-field Ising model and the critical three-state Potts model. This approach is also very efficient to detect the quasidegenerate low-energy excitations in topological phases and to identify localized excitations in systems with impurities. Finally, using the variance of the Hamiltonian as a criterion, we assess the accuracy of the resulting matrix-product-state representations of the excited states.

Figure 1

Graphical notations for matrix-product-state representation. (a) MPS representation of a state $|\psi \rangle$ in terms of left- (green) and right-normalized (blue) tensors and of the diagonal matrix $S$ whose entries are given by the Schmidt values. The tensors are contracted over connected bonds. The vertical unconnected bonds correspond to the physical (spin) index. (b), (c) Graphical representation of left and right normalizations.

Figure 2

Decay of the Schmidt values for gapped and critical systems. Here we show singular values computed in the middle of the spin-1/2 transverse-field Ising chain with $N=100$ spins at the critical point $h=0.5J$ and inside the gapped phase $h=0.7J$.

Figure 3

Graphical representation of the total and effective Hamiltonians: The row of on-site MPOs (yellow boxes) contracted through the auxiliary bonds is a $d^N\times d^N$ matrix that represents the total Hamiltonian. Contracting through all bonds except the physical indices at sites $n$ and $n+1$ gives rise to a local effective Hamiltonian that acts in a Hilbert space of dimension ${\left(dD\right)}^2\ll d^N$. The collection of $A_n$ and $B_n$ tensors approximate the basis via rotation and truncation.

Figure 4

Results of the diagonalization of the effective Hamiltonian with all $D=256$ kept states for the transverse-field Ising chain of Eq. (2) with $J=-1$ and $h=0.7$, in the gapped phase with a nondegenerate ground state. (a) Energy of the lowest 50 states as a function of iteration during left-to-right half sweep. The exact results are provided for reference and shown with grey lines. (b) The difference between the energy obtained by diagonalizing the effective Hamiltonian and the exact values of the energy as a function of iteration. The difference vanishes for three points in the middle of the chain for all energy levels.

Figure 5

Energy of the 30 low-energy states in the critical Ising model in a transverse field with $N=100$ sites as a function of iterations. The flattening of the energies in the middle of the chain is an indicator of convergence. The periodic increase of the energy that occurs close to the chain boundary is the result of the reduced Hilbert space by MPS construction. Exact results are provided for reference and shown with grey lines. The plot starts with the first full sweep. The result from infinite-size DMRG and warmup are not shown.

Figure 6

Excitation energies obtained during one DMRG sweep in the (a)–(e) critical and (f), (g) gapped transverse-field Ising model for an open chain with $N=100$ sites. Exact results are provided for reference and shown with grey lines. (a) DMRG results at the center of the chain agree with exact energies. (b)–(e) Enlarged parts of (a) around different excitation levels. The ordinal numbers of the states are indicated in each plot, with 1 corresponding to the ground state. The number at the left side of plots (b)–(e) indicates the energy window around the selected energy level. (f), (g) Far from criticality, DMRG results do not reproduce the exact energy levels: some states are missing, and strong oscillations appear.

Figure 7

Finite-size scaling of the energies of the critical Ising model in open chains with different boundary conditions: (a), (e), (i) free at both edges; (b), (f), (j) fixed with both edge spins pointing up; (c), (g), (k) fixed with one edge spin pointing up and the other pointing down; and (d), (h), (l) fixed at one edge and free at the other one. (a)–(d) Finite-size scaling of the ground-state energy after removing the ground-state energy in the thermodynamic limit ${\varepsilon }_0$ and the boundary terms ${\varepsilon }_1$. (e)–(h) Conformal towers of the excitation spectra. Blue dots are the DMRG data for $n\equiv (E_n-E_0)/\left(\pi vN\right)$ with $v=v_{\text{Ising}}=1/2$. The CFT predictions are shown with grey lines for reference. The multiplicities of the levels are indicated on the right of each tower. (i)–(l) Finite-size scaling of the excitation energies. Blue dots are DMRG data, and red, green, and magenta lines are conformal towers for the identity $I$, energy $\epsilon$, and spin $\sigma$ fields, respectively.

Figure 8

Finite-size scaling of the energies of the critical three-state Potts model in open chains with different boundary conditions: (a), (d), (g) free; (b), (e), (h) fixed with the same state at both edges; and (c), (f), (i) fixed with different states on the left and right edges. (a)–(c) Finite-size scaling of the ground-state energy after removing the ground-state energy in the thermodynamic limit ${\varepsilon }_0$ and the boundary terms ${\varepsilon }_1$. (d)–(f) Conformal towers of the excitation spectra. Blue dots are the DMRG data for $n\equiv (E_n-E_0)/\left(\pi vN\right)$ with $v=v_{\mathrm{A}-\mathrm{A}}=0.857$. The CFT predictions are shown with grey lines for reference. The multiplicities of the levels are indicated on the right of each tower. (g)–(i) Finite-size scaling of the excitation energies. Blue dots are DMRG data, and red and blue lines are conformal towers for $I$ and $\psi$ fields, respectively.

Figure 9

Finite-size scaling of the energies of the critical three-state Potts model in open chains with partially fixed boundary conditions: (a), (d), (g) only two states are allowed at the edges and the excluded state is the same for left and right edges; (b), (e), (h) only two states are allowed at the edges and the excluded state is different for left and right edges; and (c), (f), (i) only two states are allowed at one edge while the other edge remains free. (a)–(c) Finite-size scaling of the universal term in the ground-state energy. (d)–(f) Conformal towers of the excitation spectra. Blue dots are the DMRG data for $n\equiv (E_n-E_0)/\left(\pi vN\right)$ with $v=v_{\text{A-A}}=0.857$. The CFT predictions are shown with grey lines for reference. The multiplicities of the levels are indicated on the right of each tower. (g)–(i) Finite-size scaling of the excitation energies. Blue dots are DMRG data, and lines of different colors correspond to different conformal towers.

Figure 10

Finite-size scaling of the energies of the critical three-state Potts model in open chains with mixed boundary conditions: (a), (d), (g) only two states are allowed at one edge and only one of those is allowed at the other edge; (b), (e), (h) only two states are allowed at one edge and the third one is the only state allowed at the other edge; and (c), (f), (i) only one state is allowed at one edge while the other edge remains free. (a)–(c) Finite-size scaling of the universal term in the ground-state energy. (d)–(f) Conformal towers of the excitation spectra. Blue dots are the DMRG data for $n\equiv (E_n-E_0)/\left(\pi vN\right)$ with $v=v_{\text{A-A}}=0.857$. The CFT predictions are shown with grey lines for reference. The multiplicities of the levels are indicated on the right of each tower. (g)–(i) Finite-size scaling of the excitation energies. Blue dots are DMRG data, and lines of different colors correspond to different conformal towers.

Figure 11

Energy of the ground state and of 20 low-lying excited states in the critical three-state Potts model as a function of iterations. The results are obtained for open chains with $N=100$ spins with mixed boundary A-free conditions: at the left edge (site 1) the state is fixed while at the right edge (site $N$) the boundary spin remains free. The flattening of the energies in the middle of the chain is taken as an indication of convergence. A periodic increase of the energy occurs close to the chain boundary and is the result of the reduced Hilbert space by MPS construction. Remarkably, the boundary effect on the convergence disappears faster close to the free edge than in the vicinity of the fixed one.

Figure 12

Ground-state and excitation energies calculated in the sectors $S_{\text{tot}}^z=0$ (blue) and $S_{\text{tot}}^z=1$ (red) in an open Haldane chain with (a) $N=100$ and (b) $N=101$ sites. (c), (d) Enlarged parts of (a) and (b) close to the ground-state energy.

Figure 13

Transformation of an open system with edge states (left) into a system where two quasidecoupled spins are brought next to each other (right). (a), (c) valence-bond solid (VBS) sketches. (b) MPS tensor network for (a). The tensor network for (c) can be obtained either (d) by shift of the MPS with respect to the MPO or (e) by breaking a selected bond in the MPO for a periodic Hamiltonian. From the physical point of view, (d) and (e) are equivalent. However, the implementation sketched in (e) has lower complexity.

Figure 14

(a) Ground-state and excitation energies sectors of $S_{\text{tot}}^z=0$ (blue) and $S_{\text{tot}}^z=1$ (red) in a periodic Haldane chain with $N=100$ sites and one broken bond between the 25th and 26th sites. (b) Enlarged part of (a) around the ground-state energy. The energy of the first excited state is flat over seven spins in the vicinity of the broken bond.

Figure 15

Local excitation energy in the dimerized Heisenberg chain with bond impurities. (a) Energy spectrum as a function of the number of iterations for the alternating Heisenberg chain with $J_{\mathrm{odd}}=1$ and $J_{\mathrm{even}}=0.1$ and weak first ($J_1=0.3$) and last ($J_N=0.8$) bonds for $N=28$ in the sector of $S_{\text{tot}}^z=0$. Results obtained with exact diagonalizations are shown with grey lines for reference. The number of states increases during the warmup and the first five sweeps up to $D^{max}=1000$ and is kept constant over the last three sweeps. (b) Energy spectrum in the sector $S_{\text{tot}}^z=0$ for different numbers of kept states $D^{max}=100$ (green), 500 (magenta), and 1000 (blue) with the same growing procedure as in (a).

Figure 16

Energy spectrum as a function of iteration for the alternating Heisenberg chain with $J_{\mathrm{odd}}=1$ and $J_{\mathrm{even}}=0.1$ and weak first ($J_1=0.3$) and last ($J_N=0.8$) bonds for $N=100$ in the sector of $S_{\text{tot}}^z=0$. The energy of the two localized excited states only appears in the narrow window close to the edges.

Figure 17

Graphical representation of the tensor network that corresponds to the variance: $\langle {\mathrm{\Psi }}_j|{\widehat{H}}^2|{\mathrm{\Psi }}_j\rangle -\langle {\mathrm{\Psi }}_j|\widehat{H}{|{\mathrm{\Psi }}_j\rangle }^2$. Green and blue boxes correspond to the left and right normalized on-site tensors of the ground state. Yellow boxes denote the on-site MPO. The operator ${\widehat{H}}^2$ is obtained by doubling the line of the MPO. The magenta box corresponds to the various eigenstates of the effective Hamiltonian. The term in brackets is equal to a real number (energy). Thus taking the square of this network is a trivial algebraic operation.

Figure 18

Variance of the Hamiltonian in the first 30 low-energy MPS states for the critical transverse-field Ising model as a function of iterations during one sweep (right to left and left to right). The energy spectrum for this model is shown in Fig. 5. Inset: Variance in the middle of the chain as a function of the sequential number of the excited states.

Figure 19

Accuracy of the MPS for the ground state (blue), low-lying in-gap state (red), and the bulk excitation (yellow) of a Haldane chain with $N=50$ sites. (a) Energy of the three lowest states as a function of iterations during one sweep, with 800 states kept. (b) Enlarged part of (a). (c) Variance of the Hamiltonian for these three states as a function of iterations. The MPS representation of the ground state and of the in-gap state are of the same order of accuracy. The variance for the bulk excitation is much higher.

Figure 20

Same as Fig. 5 obtained with at most 200 Lanczos iterations. The number of states increases linearly from $D=50$ (in the warmup) up to $D\approx 200$. When the number of Lanczos iterations is sufficiently large, the noise in the higher excited states is suppressed as shown in Fig. 5 obtained with at most 500 Lanczos iterations.