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MPS-local time-evolution methods

The global Krylov method uses generic features of Krylov subspaces to compute the time evolution of an MPS. However, working ‘globally’ and only using MPS to represent the Krylov vectors comes with the major drawback that those vectors are typically much more entangled than the actual time-evolved state $\ket{\psi(t+\delta)}$. To circumvent this problem, it may be desirable to work in the local basis of reduced sites. From a practical point of view, this is also motivated by the DMRG formulation which always works in local spaces and hence makes it very difficult to formulate a global Krylov method. There are two different ways to obtain such a local basis, one being via an attempt to directly Lie-Trotter decompose the Hamiltonian into local subspaces[1][2][3][4][5], the other by enforcing a constraint that the evolution lies within the MPS manifold[6][7]. Both have been used in the literature, but their precise derivation requires some work. On this page, we will present a very heuristic understanding of the general approach. For a detailed description of the first approach, see the page on the local Krylov method. The second approach is review in detail on the page on the time-dependent variational principle, TDVP.

We start with an MPS $|\psi\rangle$, a Hamiltonian $\hat H$ and a site $j$. The tensors $\psi^{L}_{j-1} \equiv (M_1, \ldots, M_{j-1})$ and $\psi^{R}_{j+1} \equiv (M_{j+1}, \ldots, M_L)$ then constitute a left and right map, respectively, from the joint Hilbert space on sites $1$ through $j-1$ onto the bond space $m_{j-1}$ and from the joint Hilbert space on sites $j+1$ through $L$ onto the bond space $m_j$.

By sandwiching the Hamiltonian between two copies of $\psi^{L}_{j-1}$ and $\psi^{R}_{j+1}$, we can transform it to act on the effective single-site space on site $j$ (of dimension $\sigma_j \times m_{j-1} \times m_j$). Let us call this effective single-site Hamiltonian $\hat H_j^{\mathrm{eff}}$:

medium

Likewise, we can take the MPS $\ket{\psi}$ and act on it with a single copy of $\bar{\psi}^{L}_{j-1}$ and $\bar{\psi}^{R}_{j+1}$ to obtain the effective single-site tensor $\tilde{M}_j$ representing the effective state $\ket{\psi_j^{\mathrm{eff}}}$:

medium

If we move the orthogonality center of the MPS to site $j$ first, the transformation of the state is an identity operation, as the $\bar{A}_1$ of the transformation operator cancels with the $A_1$ in the state etc. This is desirable for many reasons, primarily numerical stability and avoiding a generalized eigenvalue problem in favor of a standard eigenvalue problem.

Instead of mapping onto the space of a single site, we can also map onto the space of two sites. This results in an effective state $|\psi_{(j,j+1)}^{\mathrm{eff}}\rangle$ given by the product of two site tensors $M_j \cdot M_{j+1}$ over their shared bond and likewise an effective Hamiltonian $\hat H_{(j,j+1)}^{\mathrm{eff}}$. On the other hand, if we map onto the space of a single bond between sites $j$ and $j+1$ (without a physical index), we obtain the center matrix. We will use underlined indices ${}_{\underline{j}}$ to refer to bond tensors between sites $j$ and $j+1$ whereas ${}_j$ (without the underline) refers to the site tensor or effective Hamiltonian tensors on site $j$. $C_{\underline{j}}$ to represent the effective state $|\psi_{\underline{j}}^{\mathrm{eff}}\rangle$ and the effective center site Hamiltonian $\hat H_{\underline{j}}^{\mathrm{eff}}$.

In any of these smaller spaces – typically in the basis of left- and right-orthogonal environments as produced naturally during left-to-right and right-to-left sweeps – it is very easy to solve the time-dependent Schrödinger equation exactly from time $t$ to time $t+\delta$ using a local Krylov procedure or any other exponential solver[8]. This results then in a new effective state $|\psi_{j}^{\mathrm{eff}}(t+\delta)\rangle$. Unfortunately, the new state will be represented in a relatively bad basis of the environments $\psi^L_{\vphantom{j}}$ and $\psi^R_{\vphantom{j}}$ which are optimized to represent the original state at time $t$. To alleviate this issue, after evolving the site tensor $M_j$ forwards by $\delta$, one now wishes to obtain a global state $\ket{\psi(t)}$ at the original time but in the new left auxiliary basis. If this is achieved, we may iteratively update the basis transformation tensors to be suitable for the new state $\ket{\psi(t+\delta)}$.

This is where the local Krylov method and the TDVP diverge and proceed differently. An interesting intermediate between the local Krylov method based on time-step targetting and the TDVP as introduced in Refs. 6, 7 was presented in Refs. 9, 10 where the time-step targetting was combined with a variational principle to solve the local TDSE. However, regardless of how one obtains this old state in the new basis, by sweeping once through the system, all site tensors are incrementally optimized for a representation of the next time step while the global state is kept at the original time. Upon reaching the far end of the system, we keep the local state tensor at the new time and hence obtain a global state at the new time $t+\delta$ written using near-optimal basis transformation tensors.

Instead of solving the forward problem a single site at a time, it is also possible to consider the effective Hamiltonian and state of two neighboring sites. In this two-site scheme, one obtains a forward-evolved tensor $M_{(j,j+1)}$ which needs to be split up using a SVD, with then the single-site tensor on site $j+1$ backwards-evolved to keep the state at the original time. The advantage is the variable bond dimension which allows a successive adaptation of the MPS bond dimension to the entanglement growth in the system. The downside is that an additional truncation error is introduced. In practice, it appears that a hybrid scheme which first uses the two-site method to increase the bond dimension to the usable maximum and then switches to the single-site variant may fare best, at least for some observables[11].

Depending on the particular way in which we obtain the local problems, solving the system of local Schrödinger equations exactly (using a small direct solver each time) is possible. The result is that both the energy and the norm (where applicable) of the state are conserved exactly. This conservation may extend to other global observables[11][12]. In practice, the TDVP achieves this goal.

Finally, one may turn the first-order integrator of sweeping once through the system into a second-order integrator by sweeping once left-to-right and then right-to-left through the system, each time with a reduced time step $\delta/2$.

The content of this page is based on Time-evolution methods for matrix-product states by S. Paeckel, T. Köhler, A. Swoboda, S. R. Manmana, U. Schollwöck and C. Hubig and is licensed under the CC-BY 4.0 license.

References

  1. Time-step targeting methods for real-time dynamics using the density matrix renormalization group, Adrian E. Feiguin, Steven R. White, Phys. Rev. B 72, 020404 (2005)
  2. Time evolution of one-dimensional Quantum Many Body Systems, Salvatore R. Manmana, Muramatsu,Alejandro, Reinhard M. Noack, AIP Conference Proceedings 789, 269-278 (2005)
  3. Coherent matter waves emerging from Mott-insulators, K Rodriguez, S R Manmana, M Rigol, R M Noack, A Muramatsu, New Journal of Physics 8, 169 (2006)
  4. Time evolution of Matrix Product States, J. J. Ripoll, New J. Phys. 8, 305 (2006)
  5. Time-Step Targeting Time-Dependent and Dynamical Density Matrix Renormalization Group Algorithms with ab Initio Hamiltonians, Enrico Ronca, Zhendong Li, Carlos A. Hoyos, Garnet Kin-Lic Chan, J. Chem. Theory Comput. 13, 5560 - 5571 (2017)
  6. Time-Dependent Variational Principle for Quantum Lattices, Jutho Haegeman, J. Ignacio Cirac, Tobias J. Osborne, Iztok Pizorn, Henri Verschelde, Frank Verstraete, Phys. Rev. Lett. 107, 070601 (2011)
  7. Unifying time evolution and optimization with matrix product states, Jutho Haegeman, Christian Lubich, Ivan Oseledets, Bart Vandereycken, Frank Verstraete, Phys. Rev. B 94, 165116 (2016)
  8. Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later, C. Moler, C. Loan, SIAM Review 45, 3-49 (2003)
  9. Entanglement Entropy for the Long-Range Ising Chain in a Transverse Field, Thomas Koffel, M. Lewenstein, Luca Tagliacozzo, Phys. Rev. Lett. 109, 267203 (2012)
  10. Spread of Correlations in Long-Range Interacting Quantum Systems, P. Hauke, L. Tagliacozzo, Phys. Rev. Lett. 111, 207202 (2013)
  11. Performance of the time-dependent variational principle for matrix product states in long time evolution of a pure state, S. Goto, I. Danshita, arxiv:1809.01400
  12. Quantum thermalization dynamics with Matrix-Product States, E. Leviatan, F. Pollmann, J. H. Bardarson, D. A. Huse, E. Altman, arxiv:1702.08894

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