Self-assembling tensor networks and holography in disordered spin chains

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Self-assembling tensor networks and holography in disordered spin chains

Andrew M. Goldsborough and Rudolf A. Römer

Phys. Rev. B 89, 214203 – Published 27 June 2014

Abstract

We show that the numerical strong disorder renormalization group algorithm of Hikihara et al. [Phys. Rev. B 60, 12116 (1999)] for the one-dimensional disordered Heisenberg model naturally describes a tree tensor network (TTN) with an irregular structure defined by the strength of the couplings. Employing the holographic interpretation of the TTN in Hilbert space, we compute expectation values, correlation functions, and the entanglement entropy using the geometrical properties of the TTN. We find that the disorder-averaged spin-spin correlation scales with the average path length through the tensor network while the entanglement entropy scales with the minimal surface connecting two regions. Furthermore, the entanglement entropy increases with both disorder and system size, resulting in an area-law violation. Our results demonstrate the usefulness of a self-assembling TTN approach to disordered systems and quantitatively validate the connection between holography and quantum many-body systems.

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Received 17 January 2014

DOI:https://doi.org/10.1103/PhysRevB.89.214203

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

Andrew M. Goldsborough^* and Rudolf A. Römer^†

Department of Physics and Centre for Scientific Computing, The University of Warwick, Coventry CV4 7AL, United Kingdom

^*a.goldsborough@warwick.ac.uk; www.warwick.ac.uk/andrewgoldsborough.
^†r.roemer@warwick.ac.uk; www.warwick.ac.uk/rudoroemer.

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Vol. 89, Iss. 21 — 1 June 2014

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Figure 1
Schematic diagrams of the various SDRG variants. Horizontal lines indicate the one-dimensional spin system. (a) Traditional MDH SDRG [19], spins ${\vec{s}}_{i}, {\vec{s}}_{i + 1}$ (arrows) with the greatest coupling strength, $J_{i} > J_{k} \forall k \neq i$ , are removed and replaced by an effective coupling $\tilde{J}$ . (b) SDRG of Westerberg et al. [24]; spin pairs are renormalized for the largest energy gap $Δ_{i}$ and replaced by an effective spin $\tilde{S}$ . (c) SDRG variant of Hikihara et al. [25]; the chain is decomposed into blocks of spins described by block Hamiltonians $H^{B}$ (shaded rectangles), with left and right spins, respectively, $s^{L}$ and $s^{R}$ (dark dots) on the boundaries of the blocks forming the coupling Hamiltonians, $H^{C}$ .
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Figure 2
(a) Tensor network diagram of the matrix product operator. The (vertical) $σ$ and $σ^{'}$ legs denote physical indices and couple to the tensor network wave function and conjugate. The $b$ 's are virtual indices (in horizontal direction) and couple the local tensors (blue-shaded squares) of the MPO to each other. (b) The pair of sites with the largest gap $Δ_{i_{m}}$ is found, the MPO tensors for these sites are contracted, and the physical indices fused to form a matrix. (c) Contracting the matrices of eigenvectors $V_{χ}$ (red-shaded rectangle) and $V_{χ}^{†}$ creates a new MPO for a coarse-grained system.
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Figure 3
Schematic representation of the contraction step (8) for the Heisenberg Hamiltonian (1). Circles (and ellipses) denote (combined) operator entries in the Heisenberg MPO $W^{[i, i + 1]}$ (see Appendix pp2-s2 for details). (a) Renormalizing the on-site components has the effect of creating a new on-site component, which is a diagonal matrix of the lowest eigenvalues $Λ_{χ}$ . (b) Contracting $V_{χ}$ and $V_{χ}^{†}$ has the effect of renormalizing the coupling spins in the same way as the Hikihara method, storing them as the coupling components of the new MPO tensor.
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Figure 4
(a) Schematic representation of the isometric property $w w^{†} = 1$ given by Eq. (11). (b) One step in the MPO SDRG algorithm in terms of isometric tensors $w$ . Triangles (red-shaded) denote the isometries; squares are as in Fig. 2.
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Figure 5
The SDRG algorithm as a TTN for a chain of $L = 20$ sites. The squares are the MPOs (i.e., the spin operators), triangles are isometric tensors, and solid lines denote summations over physical (vertical) and virtual (horizontal) indices as before. The circle indicates the top tensor, i.e., the ground-state eigenvector of the coarse-grained system. Lattice and holographic dimensions are indicated by the dashed arrows.
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Figure 6
Diagram showing the TTN form of (a) the correlation function $〈 {\vec{s}}_{3} \cdot {\vec{s}}_{15} 〉$ and (b) the reduced density matrix $ρ_{A}$ (for the block $A$ indicated by the dashed rectangle ten sites long) of the 20 site system from Fig. 5. Lines and symbols as in Fig. 5. The causal cone in both panels is indicated by a light-blue-shaded region. The bold line in panel (a) shows the path length through the TTN connecting the two sites, whereas the bold line in (b) shows the minimal surface in the TTN between regions $A$ and $B$ (the rest of the chain). The diagram in the right-hand side of (b) has been reduced in the horizontal direction to highlight the reduction in complexity due to the isometries.
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Figure 7
Ground-state energy per site $E_{g} / L$ as a function of disorder $Δ J$ for system size $L = 100$ for tSDRG (solid lines) and variational MPS (dashed). The error bars correspond to the standard error on the mean obtained from averaging over 200 different disorder configurations and various values of $χ$ . Lines are guides to the eye. Inset: System size dependence of $E_{g} / L$ for $Δ J = 2^{-}$ . Sizes $L = 10$ –80 have been averaged over 500 disorder configurations, 90, 100, and 120 over 1000, 150 and 200 over 2000 configurations, respectively.
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Figure 8
Correlation function for $L = 150$ and $Δ J = 2^{-}$ averaged over 2000 samples for the direct calculation of $〈 〈 {\vec{s}}_{x_{1}} \cdot {\vec{s}}_{x_{2}} 〉 〉$ (black circles) and also via the holographic approach (13) using $D_{TTN}$ (dashed red line with error of mean indicated by the gray shading) such that $〈 〈 {\vec{s}}_{x_{1}} \cdot {\vec{s}}_{x_{2}} 〉 〉 \approx (5.81 \pm 0.93) exp [- (0.62 \pm 0.02) D_{TTN}]$ . The expected thermodynamic scaling $| x_{2} - x_{1} |^{- 2}$ is also shown (solid blue line) while the dashed orange line denotes a power-law fit up to $| x_{2} - x_{1} | = 50$ with slope $1.64$ . The (brown) crosses show $〈 〈 {\vec{s}}_{x_{1}} \cdot {\vec{s}}_{x_{2}} 〉 〉 / 4$ (for clarity) with all values for even distances $| x_{2} - x_{1} |$ multiplied by $1.25$ . Inset: The holographic path length $D_{TTN}$ connecting sites $x_{1}$ and $x_{2}$ averaged over the 2000 TTNs (black) and a fit in the logarithmic regime (red).
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Figure 9
The scaling parameter $a$ from Eq. (13) as a function of system size $L$ for different values of $χ$ at $Δ J = 2^{-}$ . The solid lines are guides to the eye only. The asymptotic value of $a = 2$ is indicated by the horizontal dashed line.
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Figure 10
The typical spin correlation function averaged over 2000 samples for $L = 500$ (green circles) and $L = 150$ (triangles) and $χ$ values as given in the legend. Error bars are within symbol size throughout. The dashed lines are fits to the linear regimes for $L = 150$ , $χ = 50$ (blue) and $L = 500$ , $χ = 20$ (green). The vertical dotted line indicates half the system size for $L = 150$ .
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Figure 11
Entanglement entropy $S_{A | B}$ for all possible bipartitions (cf. Fig. 6) for $L = 30$ as a function of $Δ J$ averaged over 100 disorder configurations using vMPS and tSDRG. Solid lines indicate the arithmetic mean over disorder configurations while dashed lines denote the mean of the maximal $S_{A | B}$ values at the chosen $Δ J$ . Lines connecting symbols are guides to the eye only. Error bars denote standard error of the mean when larger than symbol size. The two vertical dotted lines highlight $Δ J = 0.5$ and $1.2$ as discussed in the text.
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Figure 12
Entanglement entropy $S_{A | B}$ as a function of $L$ averaged over 100 samples and all possible bipartitions (as in Fig. 11) for $χ = 4$ and $Δ J = 2^{-}$ . The dashed blue line is the fit $(0.358 \pm 0.005) {log}_{2} L + (0.41 \pm 0.03)$ ; the solid black line is $(0.096 \pm 0.008) {log}_{2} L + (0.67 \pm 0.04)$ . Error bars denote the standard error of the mean for the $S_{A | B}$ values when larger than symbol size while gray shaded regions show the standard error of the indicated fits.
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Figure 13
The entanglement entropy $S_{A, B}$ (black) averaged over 500 samples as a function of the size of a block $L_{B}$ placed in the middle of a chain with $L = 50$ for $χ = 10$ and $Δ J = 2^{-}$ . The fitting (red, solid line) gives $S_{A | B} = (0.22 \pm 0.02) {log}_{2} L_{B} + (1.12 \pm 0.05)$ for $L_{B} \leq 25$ , above which finite size effects dominate. The gray-shaded region indicates the accuracy of the fit. The (green) dashed line shows the entanglement scaling (15) from Ref. [42] with the vertical position fitted to the point $L_{B} = 2$ . The straight black lines are a guide to the eye only. At the bottom, we show the failure rate in percent (crosses) for different $L_{B}$ .
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Figure 14
Entanglement entropy $S$ (black circles) and entanglement entropy per bond $S / n_{A}$ (red diamonds) for bipartitions $A | B$ (top, open symbols) and blocks $A, B$ (bottom, filled symbols) with $χ = 10$ and $Δ J = 2^{-}$ . The entanglement per bond saturates to $0.47 \pm 0.02$ for bipartitions and $0.48 \pm 0.02$ for blocks (gray-shaded regions).
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Figure 15
(a) Tensor network diagram of the MPS ket. The circles represent the $M$ tensors in Eq. (B2), the lines are the indices, and connected lines represent tensor contractions. The bra state is the same apart from the vertical lines point upwards and the contents of the tensors are the complex conjugate. (b) Tensor network diagram of the matrix product operator. The $σ$ and $σ^{'}$ legs are physical indices and couple to the tensor network wave function and conjugate. The $b$ are virtual indices and couple the local tensors of the MPO (squares) to each other.
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Figure 16
The left, center, and right are MP diagrammatic forms of Eqs. (B6), (B7), and (B8), respectively. The circles represent the virtual indices $(b_{i - 1}, b_{i})$ of the MPO tensor and arrows show the corresponding operator. The dashed arrows highlight the possibility of an additional magnetic field operator $h S^{z}$ not present in (B6), (B7), and (B8).
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Figure 17
MP diagram of the contraction of a four-site MPO of the Heisenberg Hamiltonian (B5). Symbols and lines as in Fig. 16. The shaded ellipses linking tensor entries 2, $3,$ and 4 corresponds to the simplifications employed in Fig. 3.
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