Signature of a quantum dimensional transition in the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on a square lattice and space reduction in the matrix product state

We study the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on an $\infty \times N$ square lattice for even $N$'s up to 14. Previously, the nonlinear sigma model perturbatively predicted that its spin-rotational symmetry breaks asymptotically with $N\to \infty$, i.e., when it becomes two dimensional (2D). However, we identify a critical width $N_c=10$ for which this symmetry breaks spontaneously. It shows the signature of a dimensional transition from one dimensional (1D) including quasi-1D to 2D. The finite-size effect differs from that of the $N\times N$ lattice. The ground-state (GS) energy per site approaches the thermodynamic limit value, in agreement with the previously accepted value, by one order of $1/N$ faster than when using $N\times N$ lattices in the literature. Methodwise, we build and variationally solve a matrix product state (MPS) on a chain, converting the $N$ sites in each rung into an effective site. We show that the area law of entanglement entropy does not apply when $N$ increases in our method and the reduced density matrix of each effective site has a saturating number of dominant diagonal elements with increasing $N$. These two characteristics make the MPS rank needed to obtain a desired energy accuracy quickly saturate when $N$ is large, making our algorithm efficient for large $N$'s. Furthermore, the latter enables space reduction in MPS. Within the framework of MPS, we prove a theorem that the spin-spin correlation at infinite separation is the square of staggered magnetization and demonstrate that the eigenvalue structure of a building MPS unit of $\langle g|g\rangle ,|g\rangle$ being the GS is responsible for order, disorder, and quasi-long-range order.

Figure 1

Spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on an $\infty \times N$ ladder $(N=4$ for example). The circle represents lattice sites, and the lines and curves connecting the nearest-neighboring sites represent the spin-spin interactions. Periodic boundary conditions are assumed. (a) The original lattice. (b) The effective lattice whose single site is indicated by the dashed rectangle enclosing the $N$ lattice sites in each rung of original lattice.

Figure 2

Various designs for the lattice wave function using tensor/matrix product. $\xi$ denotes a tensor. The solid line refers to the bonding index while the dashed line refers to the space index. (a) Tensor network state (TNS). Each tensor has four bonding indices which resemble the same lattice architecture shown in Fig. 1 and one space index which accounts for a lattice site. TNS is employed in iPEPS, etc. (b) Matrix product state (MPS) built on a winding lattice chain. Each tensor has two bonding indices that differ from the lattice-linking architecture and one space index for a site. DMRG wave function after projections are reduced to this form of MPS. (c) MPS built on a lattice chain of a translational symmetry. The two bonding indices resemble the architecture of an effective lattice shown in Fig. 1; the combination of dashed lines, each of which corresponds to a physical site of each $\xi$, is treated as a single space index running from 1 to $2^N$ for spin-$\frac{1}{2}$.

Figure 3

Illustration of MPO for $N=4$. (a) The construction. The local quantum space is represented by rectangles enclosing four physical sites (circle in gray color). Directed out of the page is the conjugate of those spaces. The building units of MPO, symbolized by ${\mathrm{\Gamma }}^1$ and ${\mathrm{\Gamma }}^2$ in (b), are enclosed within the tilted rectangles in-between the spaces. Going from the space to its conjugate, the first four pairs of green-red solid circles are stacked shell by shell in sequence. They account for the on-effective-site interaction, with the green/red circle denoting $f/g$ operator mentioned in the context. The dashed line denotes the contraction of bond index running from 1 to 4. The solid line denotes the inner product between the operators operating sequentially on the same physical lattice site. Following the on-effective-site operators are one shell of $f/g$ operators bonding the $\left(i-1\right)\text{th}/i\text{th}$ effective sites, and another shell of $g/f$ operators bonding the $\left(i-2\right)\text{th}/\left(i-1\right)\text{th}$ effective sites and the $i\text{th}/\left(i+1\right)\text{th}$ effective sites, respectively. They are stacked layer by layer from bottom to top, forming inter-effective-site interactions and hence giving the rise of entanglement in MPO represented schematically by the horizontal dashed lines in (b). The individual space indices $r_{i=1,4}/s_{i=1,4}$ are combined into single indices of vertical solid lines in (b), while the individual bond indices $m_{i=1,4}/n_{i=1,4}$ being combined into horizontal dashed lines in (b).

Figure 4

(a) Energy observation in the presence of MPS and MPO. The quantities enclosed in each ellipse are combined into new tensors $A$ and $B$, each of which has a larger rank of $P^{2}Q$ with $P$ and $Q$ being the MPS and MPO ranks, respectively. (b) Normalization of MPS wave function. The quantities enclosed in each ellipse are combined into new tensors $C$ and $D$, each of which has a rank of $P^2$.

Figure 5

Schematic of applicable and inapplicable scenarios of the area law. Applicable: (a) isolations embedded in a large 2D lattice whose sites (circles) interact along both dimensions (solid lines). The boundaries in dashed lines are enclosed; (b) the boundary of isolation embedded in a decoupled lattice shows no difference with or without the horizontal dotted lines to enclose them. Inapplicable: from (c) to (d), the width $N$ of the ladder is increasing. They are described by two distinct Hamiltonians. For $N=\infty$ in both (a) and (d), the boundaries are topologically different. The former is single connected while the latter is disconnected.

Figure 6

(a) Owing to the antiferromagnetic nature, the tensors of MPO and MPS have a bipartite checkerboard symmetry in the original spin basis. (b) After applying the checkerboard transformation on any sublattice, for instance the blank lattice as shown, those tensors need not be distinguished anymore in the new spin basis.

Figure 7

(a) Relative error versus MPS rank $P$ for decoupled ladders of $N=1,2,3,4,5,6,7$, and 8, ordered from left to right. (b) Ratio of $P_N$ and $P_{N-1}$ with respect to $\frac{1}{N}$. At both $P_N$ and $P_{N-1}$ the same accuracy is obtained. The comparison is made for accuracies $99.9\%,99.7\%$, and $99\%$ from top to bottom.

Figure 8

Entanglement entropy of a rung versus $P^{-1/3}$ for (a) decoupled ladders and (b) coupled ladders with OBC in the rung. (c) Entanglement entropy versus $N$. Open and solid circles represent coupled and decoupled ladders, respectively.

Figure 9

(a) Sum of squared residuals versus GS energy per site and (b) log-log view of relative error versus MPS rank $P$ for an isotropically coupled ladder of $N=5$ with OBC along the rungs. The minimum of curve in (a) gives the extrapolation at $P\to \infty$ for GS energy per site needed to calculate the relative error in (b).

Figure 10

Comparison of GS energy per site for spin ladders of (a) $N=4$ and (b) $N=6$ both in the energy scale of 0.003, by iqEPT, DMRG, and the MC loop algorithm. The short dotted line indicates the DMRG extrapolation. Circled iqEPT results converge to its dashed extrapolation.

Figure 11

(a) The convergence of GS energy per site with respect to the MPS rank $P$, for $N=4,6,8,10$, and 12 from bottom to top. The inset is for $N=2$ in a distinct energy scale. (b) The convergence of energy with respect to $1/P$. The inset is for $N=4$ in a distinct energy scale. They give extrapolations used in (c). (c) The GS energy per site approaches the thermodynamic value as a fourth-order function of $\frac{1}{N}\to 0$, one order faster than the approach from $N\times N$ lattices. Only data for $N=6,8,10$, and 12 are shown due to the very fast decay of $N^{-4}$.

Figure 12

(a) Ratio of the MPS rank $P_N$ to $P_{N-2}$, with respect to $\frac{1}{N}$. At both $P_N$ and $P_{N-2}$ the same accuracy is obtained. The tendency of $\frac{P_N}{P_{N-2}}\to 1$ for $\frac{1}{N}\to 0$ implies the saturating MPS rank with increasing $N$. The inset shows a larger scale starting from $\frac{P_2}{P_1}$. (b) The horizontal dashed line intercepts the curves of “relative error versus $1/P$,” giving $P_N$'s used in (a), given certain relative error, say $1.9\times {10}^{-3}$. The comparison between $N=12$ and 14 is made for the relative error of $5.3\times {10}^{-3}$, where $P=195$ for $N=12$ and $P=350$ for $N=14$, respectively.

Figure 13

Entanglement entropy of an effective site for coupled ladders with PBC in the rung. (a) Entanglement entropy versus $P^{-1/3}$. The steady tendency to $P^{-1/3}=0$ gives extrapolation for $N=8,10$, and 12 shown as rectangles, circles, and triangles, respectively. Those of $N=2,4$, and 6 from bottom to top in the inset exhibit sudden convergence. (b) Entanglement entropy versus $N$.

Figure 14

$\text{ln}{C}_r$ versus $r$. The linear tilted lines in logarithmic scale suggest the exponential decay of correlations with respect to separations. They approach the fixed one from bottom when $P=4$ to top when $P=28$ (every augment of 4 for $P)$ for $N=2$ in (a); from bottom when $P=40$ to top when $P=200$ (every 40) for $N=4$ in (b). However, the beginning lines are flat at top in (c) for $N=6$, starting from top when $P=190$ to the last flat one in the middle zone when $P=250$ (every 20). It suddenly jumps down to the tilted line at the bottom when $P=270$ and approaches the fixed tilted line when $P=560$ (every 20).

Figure 15

$C_r$ and ${\tau }_r\equiv \text{ln}\left(\text{ln}{C}_r-\text{ln}{C}_{r+1}\right)$ for $N=8$. (a) ${\tau }_r$ versus $P^{-3/4}$ at separations $r=10,20,30,40$, and 50 from top to bottom. (b) ${\tau }_r$ versus $r$ at various $P$'s. They asymptotically approach the dashed curve obtained by the extrapolation in (a). Trace along the asymptotic curve in (c), starting from $\text{ln}{C}_1=\text{ln}\frac{{\overline{\epsilon }}_0}{6}$, yields the asymptotic curve of $\text{ln}{C}_r$ versus $r$.

Figure 16

$C_r$ and ${\tau }_r\equiv \text{ln}\left(\text{ln}{C}_r-\text{ln}{C}_{r+1}\right)$ for $N=10$. (a) ${\tau }_r$ versus $P^{-3/4}$ at $r=10,20,30,40$, and 50 from top to bottom. (b) $\left({\tau }_r-{\tau }_{r+10}\right)/\left({\tau }_{10}-{\tau }_{20}\right)$ versus $P^{-3/4}$ at $r=20,30$, and 40. (c) ${\tau }_r$ versus $r$ at various $P$'s. They asymptotically approach the dashed curve obtained by the extrapolation in (a). (d) Trace along the asymptotic curve in (c) yields the asymptotic curve of $\text{ln}{C}_r$ versus $r$.

Figure 17

Effect of space reduction in MPS for $N=10$. (a) GS energy per site versus MPS rank $P$. Open circles denote the solution before reduction. Result at $P_1=100$ is used to reduce the space rank to 128, 256, 384, and 512, yielding new solutions shown as dotted-dashed, dotted, dashed, and solid curves. (b) GS energy versus $1/P$. Tangents of convergence yield the extrapolated energies for various space ranks, 128, 256, 384, 512, and 1024 (unreduced) from top to bottom. (c) Extrapolation of the energy in unreduced spaces using those obtained in reduced spaces.

Figure 18

Use of space reduction in MPS for $N=14$. Tangents of the convergence of GS energy per site versus $1/P$ yield extrapolated energies for various space ranks, 512, 1024, 1536, and 2048 from top to bottom. The inset extrapolates GS energy per site in the unreduced space using those extrapolations obtained in reduced spaces.

Figure 19

Relative error versus (a) $M/2^N$ (reduction ratio) and (b) $2^N/M$ (inverse ratio). In (a), the dotted line gives a reference of zero error, while the dashed line intercepts each curve to give the reduction ratio at a certain accuracy. (b) Energies obtained in reduced spaces for lattices of larger $N$ approach more linearly to those in unreduced space. Figures 17 and 18 show such examples for $N=10$ and 14, respectively.

Figure 20

Ratio of basis vector numbers kept for $N$ and $N-2$, respectively, versus $1/N$. The linear fit overlaps with the guiding dashed line to 1 when $N\to \infty$.

Figure 21

QLRO correlations of a spin chain. (a) The largest few eigenvalues, which are shown in inset to be nearly degenerate with ${\varrho }_1$, have less significant $F_j$'s. They make small contributions to $C_r$ that slowly decay with $r$. The eigenvalues which have significant $F_j$'s are definitely smaller than ${\varrho }_1$. They make contributions that are large at smaller $r$'s but decay rapidly with $r$. (b) The solid line in the log-log view of $C_r$ versus $r$ collects contributions to $C_r$ from all eigenvalues. Rectangles collect contributions only from those mentioned in (a). (c) ${\tau }_r\equiv \text{ln}\left(\text{ln}{C}_r-\text{ln}{C}_{r+1}\right)$ versus $r$. The inset takes the odd branch as an example to show ${\tau }_r$ is linear with $\text{ln}\left(r\right)$, in sharp contrast to the linear dependence of ${\tau }_r$ on $r$ for both disorder and order.

Figure 22

Eigenvalue structure of $B$ for (a) the disordered spin ladder of $N=4$ and (b) the ordered ladder of $N=10$.