Multiscale Space-Time Ansatz for Correlation Functions of Quantum Systems Based on Quantics Tensor Trains

The correlation functions of quantum systems—central objects in quantum field theories—are defined in high-dimensional space-time domains. Their numerical treatment thus suffers from the curse of dimensionality, which hinders the application of sophisticated many-body theories to interesting problems. Here, we propose a multiscale space-time ansatz for correlation functions of quantum systems based on quantics tensor trains (QTTs), “qubits” describing exponentially different length scales. The ansatz then assumes a separation of length scales by decomposing the resulting high-dimensional tensors into tensor trains (also known as matrix product states). We numerically verify the ansatz for various equilibrium and nonequilibrium systems and demonstrate compression ratios of several orders of magnitude for challenging cases. Essential building blocks of diagrammatic equations, such as convolutions or Fourier transforms, are formulated in the compressed form. We numerically demonstrate the stability and efficiency of the proposed methods for the Dyson and Bethe-Salpeter equations. The QTT representation provides a unified framework for implementing efficient computations of quantum field theories.

Popular Summary

The idea of scale separation is essential in simulating physical systems. For example, in a solid, the ions are much slower than the electrons, so from the electron’s point of view, the ions seem almost frozen in time. This in turn allows a physicist to separate out the two timescales, one for the electrons and one for the ions, and approximate the two systems as only weakly coupled. A similar mix of timescales and length scales governs the dynamics of the electronic system itself; however, separating these scales has proved difficult. Without such separation the amount of simulation data to be stored and manipulated is prohibitively large. In this paper, we tackle this challenge with a mathematically rigorous version of scale separation.

We approach the problem using “quantics tensor trains.” Each object is represented as a set of systems (tensors) for exponentially different scales, and tensors for adjacent scales are coupled to each other like carriages on a train. Crucially, this form of scale separation can be performed by a machine with minimal human input and no assumption other than that the scales are indeed separable. We use this mathematical train network to represent the multiple timescales and length scales in the dynamics of electrons. We find that this works well in practice: For several electronic properties of a range of systems, we find reasonable scale separation and thus high ratios of data compression and computational speedup.

This work is an important step in solving the long-standing problem of how to efficiently represent the physics of the interacting electron ensemble and may be applicable in other fields of physics.

Figure 1

Multiscale ansatz for momentum space. Each row, numbered by the bond index $b$, corresponds to a different level of discretization of the 1D momentum $k$ (different length scale). In this way, $k$ can be represented by a set of bits $k_1,\dots ,k_R$ (see text). On the right, the QTT representation of a momentum-dependent function Eq. (2) is shown.

Figure 2

(a) QTT representation in momentum space. The rightmost bits (indices) represent fine structures in momentum space. Low entanglement structures are assumed between different length scales. (b) Schematic illustration of the bond dimensions along the chain representing the momentum dependence. The dashed line indicates the maximum bond dimensions in maximally entangled cases. (c) Fourier transform from momentum space to real space by applying a MPO. The orange diamonds represent the MPO tensors. The structure of the MPO is illustrated in Fig. 27 in Appendix pp3.

Figure 3

Matrix product state representing a 2D space spanned by $x$ and $y$. The expansion can be truncated at the right edge.

Figure 4

Bond dimensions of the MPO for the discrete Fourier transform recursively constructed with truncation cutoff $\epsilon ={10}^{-25}$.

Figure 5

Tensor contraction for the elementwise product (a) and matrix product (b) of two MPSs $A$ and $B$. The filled circles in (a) denote a superdiagonal tensor whose nonzero entries are 1. The dashed squares denote the tensors of the auxiliary MPOs.

Figure 6

Compression of $G(\tau )$ generated by randomly chosen 100 poles with $R=12$ (see the text). (a) Singular values at $b=5$, (b) bond dimensions for $\epsilon ={10}^{-20}$, (c) comparison between the exact and reconstructed data. $G(\tau )$ has three sign changes. In (b), the dashed line indicates the maximum bond dimensions in maximally entangled cases.

Figure 7

Compression of $G(\mathrm{i}\nu )$ for the same model as in Fig. 6 with $R=12$. (a) Singular values at $b=4$, (b) bond dimensions, (c) comparison between the exact and reconstructed data. In (b), the dashed line indicates the maximum bond dimensions in maximally entangled cases.

Figure 8

Compression of $\tilde{G}(\mathrm{i}\nu )$ for the same model as in Fig. 6 with $R=12$. Note that the tail is subtracted from the data before the compression. (a) Singular values at $b=4$, (b) bond dimensions, (c) comparison between the exact and reconstructed data. In (b), the dashed line indicates the maximum bond dimensions in maximally entangled cases.

Figure 9

Momentum dependence of the Matsubara Green’s function of the 1D tight-binding model (16). (a) Band dispersion, (b) Green’s function at the lowest positive Matsubara frequency, (c) bond dimensions of a MPS for the full BZ, and (d) bond dimensions of patchwise constructed MPSs. We use $\epsilon ={10}^{-10}$ and $2^5$ ($=32$) patches in (d). The vertical lines in (a) denote the boundaries of the patches. $\epsilon (k)$ has 34 roots. The shaded region in (c) corresponds to the patch size. Two out of the 32 patches do not contain a Fermi point. In (d), the dashed line indicates the maximum bond dimensions in maximally entangled cases.

Figure 10

Momentum dependence of the Matsubara Green’s function for the nearest-neighbor tight-binding model on the square lattice. (a) Bond dimensions per patch, (b) maximum value of the bond dimensions, (c) number of patches with large bond dimensions, and (d) bond dimensions along the chain for patches with relatively large bond dimensions ($\ge 25$). In (d), the dashed line indicates the maximum bond dimensions in maximally entangled cases.

Figure 11

Green’s function of the 2D Hubbard model solved within FLEX at the lowest positive Matsubara frequency. (a) Intensity map of the original data and the error in the reconstructed data. The full BZ is divided into 16 patches. (b) Bond dimensions for all 16 patches. The dashed line indicates the maximum bond dimensions in maximally entangled cases. The compression ratios are 10.86 and 45.20 for the patches $A$ and $B$, respectively. Here, the compression ratio is defined as the ratio between the number of elements in a TT and that of the original tensor.

Figure 12

(a) Spin susceptibility of the 2D Hubbard model at zero frequency and an enlarged plot. (b) Reconstructed data and error at $q_y=\pi$ and $q_y=\pi /2$. (c) Bond dimensions of the MPS. The dashed line indicates the maximum bond dimensions in maximally entangled cases. The compression ratio is 269.97.

Figure 13

Compression of the real-frequency spectral functions of an AF insulator, a doped Mott insulator, and an impurity model. See the main text for a more detailed description of the models. (a) Spectral functions of the AF insulator and the doped Mott insulator, (b) spectral function of the impurity model (an enlargement of the low-frequency region is shown in the inset), (c) bond dimensions for $\epsilon ={10}^{-8}$ and $R=18$. The compression ratios are 40.3, 280.7, 318.1, respectively.

Figure 14

Irreducible vertex function ${\mathrm{\Gamma }}_d$ of the Hubbard atom for two values of $U$: (a) $U=3.0$ and (b)–(c) $U=3.56$ and for a given bosonic frequency $2m\pi /\beta$ ($\beta =1$). The middle panels show the reconstructed data from MPSs ($\epsilon ={10}^{-14}$). The right panels show the error in the reconstructed data. Note that the entire three-frequency dependence is fitted by a single MPS.

Figure 15

Dependence of (a) compression ratio and (b) error (accuracy) on the bond dimension for ${\mathrm{\Gamma }}_d$ of the Hubbard atom at $U=3$ ($\beta =1$). We compress the data on a $2^R\times 2^R\times 2^R$ grid with $R=10$ using the cutoffs $\epsilon ={10}^{-6}$, ${10}^{-8}$, ${10}^{-10}$, ${10}^{-12}$, and ${10}^{-14}$. The symbol $|\cdots {|}_{\infty }$ denotes the maximum norm of a tensor, which is the maximum of the absolute values of its elements.

Figure 16

Bond dimensions of MPSs representing the vertex function ${\mathrm{\Gamma }}_d$ with cutoff $\epsilon ={10}^{-14}$. The dashed line indicates the maximum bond dimensions in maximally entangled cases.

Figure 17

Local generalized susceptibility of ${\mathrm{CeB}}_6$ computed by $\mathrm{DFT}+\mathrm{DMFT}$. (a) MPS, (b),(c) intensity map of the reconstructed $X_{\mathrm{loc}}$ and error for $\epsilon ={10}^{-8}$, (d) bond dimensions along the MPS. In (d), the dashed line indicates the maximum bond dimensions in maximally entangled cases. The estimated compression ratio is around 608.

Figure 18

Multipolar susceptibility of ${\mathrm{CeB}}_6$ computed by $\mathrm{DFT}+\mathrm{DMFT}$. (a) MPS, (b) bond dimensions along the MPS for $\epsilon ={10}^{-8}$. In (b), the dashed line indicates the maximum bond dimensions in maximally entangled cases. The estimated compression ratio is around 134.

Figure 19

Real-time and mixed Green’s functions computed for the equilibrium paramagnetic system. Exact data for the imaginary parts of (a) the lesser, (b) the retarded, and (c) the left-mixing component are shown in the left panels. Cuts through these functions at $t^{\prime }/t_{\max }=1/2$ or $\tau /\beta =1$ (horizontal lines in the left panels) are shown in the middle panel. The right panels show the logarithm of the error for cutoff $\epsilon ={10}^{-10}$ together with the bond dimension $D$ automatically set by $\epsilon$. The symbol $|\cdots {|}_{\infty }$ denotes the maximum norm of a tensor, which is the maximum of the absolute values of its elements.

Figure 20

Nonequilibrium Green’s functions computed for the photoexcited antiferromagnetic system. See the caption of Fig. 19 for the description of the panels.

Figure 21

Scaling of the relative accuracy [panels (a) and (c)] and compression ratio [panels (b) and (d)] for the different components of the Green’s function. The left panels are for the equilibrium case and the right panels for the nonequilibrium case. The symbol $|\cdots {|}_{\infty }$ denotes the maximum norm of a tensor.

Figure 22

Fast Fourier transform of the one-particle Green’s function, where $G(\mathrm{i}\nu )$ is transformed from $G(\tau )$ [$\mathrm{i}\nu =i(2n+1)\pi /\beta$]. We only plot every ${10}^5$th data point.

Figure 23

Fourier transform of the Green’s function to the $\tau$ domain for $R=8$, 12, 16. The other parameters are the same as in Fig. 22.

Figure 24

Solving the Dyson equation for the nearest-neighbor tight-binding model on the 1D lattice for $\beta =100$. (a) $\epsilon (k)$ computed by Fourier transformation in the QTT form. (b) [(c)] Green’s functions computed for $\nu =(1/\beta )\pi$ [$\nu =(11/\beta )\pi$].

Figure 25

(a) Error in the reconstructed full vertex by solving the BSE, (b) timings of the one-shot solution of the BSE for $U=3$ and $R=7$, 8, 9. The BSE is solved effectively on a $2^R\times 2^R\times 2^R$ grid. The horizontal dashed line in (a) denotes the error level set by the finite-size effects of the grid (see the text). The symbol $|\cdots {|}_{\infty }$ denotes the maximum norm of a tensor.

Figure 26

(a) Decomposing a tensor into a tensor train or matrix product state by SVD, yielding a MPS of the dependence on $i$. The filled square denotes a diagonal matrix consisting of singular values. The singular values are absorbed into the right singular matrix. (b) MPO. (c) Multiplication of a MPO and a MPS. We do not show the dummy virtual bonds at the edges for simplicity.

Figure 27

MPO for the discrete Fourier transform (c2) of a MPS.