Quantum criticality on a chiral ladder: An SU(2) infinite density matrix renormalization group study

In this paper we study the ground-state properties of a ladder Hamiltonian with chiral $\text{SU}\left(2\right)$-invariant spin interactions, a possible first step toward the construction of truly two-dimensional nontrivial systems with chiral properties starting from quasi-one-dimensional ones. Our analysis uses a recent implementation by us of $\text{SU}\left(2\right)$ symmetry in tensor network algorithms, specifically for infinite density matrix renormalization group. After a preliminary analysis with Kadanoff coarse graining and exact diagonalization for a small-size system, we discuss its bosonization and recap the continuum limit of the model to show that it corresponds to a conformal field theory, in agreement with our numerical findings. In particular, the scaling of the entanglement entropy as well as finite-entanglement scaling data show that the ground-state properties match those of the universality class of a $c=1$ conformal field theory (CFT) in $(1+1)$ dimensions. We also study the algebraic decay of spin-spin and dimer-dimer correlation functions, as well as the algebraic convergence of the ground-state energy with the bond dimension, and the entanglement spectrum of half an infinite chain. Our results for the entanglement spectrum are remarkably similar to those of the spin-$\frac{1}{2}$ Heisenberg chain, which we take as a strong indication that both systems are described by the same CFT at low energies, i.e., an $\text{SU}{\left(2\right)}_1$ Wess-Zumino-Witten theory. Moreover, we explain in detail how to construct matrix product operators for $\text{SU}\left(2\right)$-invariant three-spin interactions, something that had not been addressed with sufficient depth in the literature.

Figure 1

Two-leg ladder made of triangles, for the model in Eq. (1).

Figure 2

Different orientations of the chiral triple product result in different models. In the first two cases, the orientation is chosen to be the same whereas it is opposite in the last two cases.

Figure 3

Expectation values of the link currents for the first excited state of $H_1$ with 16 spins and $\langle {\mathbf{S}}^2\rangle =2,S^z=-1$ for open boundary conditions (top) and periodic boundary conditions (bottom). The color refers to the strength of the currents normalized to the maximal current ${\mathcal{J}}^{\max }=0.107$ in Figs. 3 and 4. Both cases have counterpropagating edge currents, with periodic boundary conditions showing translation invariance and no currents on the rung and slash links. The first excited state with $\langle {\mathbf{S}}^2\rangle =2$ and $S^z=0$ does not show ${\mathcal{J}}^z$ current expectation values, for $S^z=+1$ the current patterns are inverted.

Figure 4

Expectation values of the link currents for the ground state of $H_3$ with 16 spins and $\langle {\mathbf{S}}^2\rangle =12,S^z=-3$ for open boundary conditions (top) and periodic boundary conditions (bottom). The color refers to the strength of the currents normalized to the maximal current ${\mathcal{J}}^{\max }=0.107$ in Figs. 3 and 4. Both cases have copropagating edge currents, with periodic boundary conditions showing translation invariance and no oscillations in the strength of the currents. The ground state with $\langle {\mathbf{S}}^2\rangle =12$ and $S^z=0$ does not show ${\mathcal{J}}^z$ current expectation values, for $S^z=+3$ the current patterns are inverted.

Figure 5

Dispersion for different coupling strengths. The color indicates the polarization of the bands. In the case we study in the paper, both bands are fully polarized and the kinetic dispersion is $\omega \left(k\right)=\pm \frac{1}{2}sin\left(k\right)$. If we tune $\frac{J_2}{J_1}>-1$, we allow for a mixing between $A$ and $B$, and only one of the avoided crossings is preserved.

Figure 6

At the Fermi energy $E_F$ the linearized spectrum consists of left- and right-moving species for each flavor $A,B$.

Figure 7

Convergence of the ground-state energy $E_0$ with the MPS bond dimension $\chi$. Here, we use the discarded weight (see also Fig. 8) as an estimate of the relative error for the ground-state energy in each simulation. The inset shows the convergence of the error ${\mathrm{\Delta }}_{E_0}=E_0-E_0(\chi \to \infty )$ between the energy and its extrapolated value.

Figure 8

Convergence of the ground-state energy $E_0$ with the iDMRG discarded weight. The inset shows the convergence of the error ${\mathrm{\Delta }}_{E_0}=E_0-E_0(\chi \to \infty )$ between the energy and its extrapolated value.

Figure 9

Scaling of the entanglement entropy $S\left(L\right)$ for a block of size $L$ and different bond dimensions $\chi$. Computing the entanglement entropy of a block is restricted to only moderate bond dimensions due to a higher computational cost of $O\left({\chi }^5\right)$ compared to the semi-infinite chain which scales like $O\left({\chi }^3\right)$.

Figure 10

Scaling of the entanglement entropy $S\left(\xi \right)$ for half an infinite chain with the MPS correlation length $\xi$ for up $\chi =1184$.

Figure 11

Scaling of the MPS correlation length $\xi$ with the bond dimension $\chi$.

Figure 12

Spin-spin correlation function between spins in the same chain. The two-point correlation function is expected to follow $C\left(r\right)\sim {(logr)}^{1/2}/r$, which is in good agreement with the numerical data [25, 26]. The exponent for the short-distance term is fixed.

Figure 13

Spin-spin correlation function between spins in different chains. The two-point correlation function is expected to follow $C\left(r\right)\sim {(logr)}^{1/2}/r$, which is in good agreement with the numerical data [25, 26]. The exponent for the short-distance term is fixed.

Figure 14

Dimer-dimer correlation function between vertical dimers with logarithmic corrections. The correlations for smaller bond dimensions show an exponential tail due to the final amount of entanglement in the MPS. For larger bond dimension, the correlation is expected to follow the fitted function to even larger separation distances.

Figure 15

Entanglement spectrum for the triangle ladder model (TM, squares) and the Heisenberg spin chain (HM, dots), with multiplets organized according to their spin sector $S$. Every point is a $(2S+1)$-plet.

Figure 16

All terms of the spin current operator contributing at each site of the ladder. We define incoming currents as being positive and outgoing currents as being negative. The current in each link of the ladder is a three-site observable which includes a scalar product (indicated by arrows) between two sites, and the multiplication with $\mathit{S}$ on the third site (depicted as a blue colored circle).

Figure 17

Internal $\text{SU}\left(2\right)$ structure of a generic MPO tensor with four indices.

Figure 18

Matrices for the MPO tensor with the left bond index fixed to spin-0 and the right bond index fixed to spin-1, leaving the freedom of choosing $m=-1,0,+1$. Physical indices have spin $S=\frac{1}{2}$.

Figure 19

Matrices for the MPO tensor with the left bond index fixed to spin-1 and the right bond index fixed to spin-0, leaving the freedom of choosing $m=-1,0,+1$. Physical indices have spin $S=\frac{1}{2}$.

Figure 20

The contraction of the MPO tensors in Figs. 18 and 19 produces the desired two-site Heisenberg interaction with a $-\frac{4}{3}$ prefactor. The sum is over the values of $m$ for the spin-1 channel of the bond index.

Figure 21

MPO tensor with SU(2) symmetry for the Heisenberg quantum spin chain. The degeneracy tensors go together with the rank-4 structural tensor for the spin sectors shown. All other spin sectors have vanishing degeneracy tensors. The internal spin for every block is shown in bold font, the physical spin is always $S=\frac{1}{2}$. The labels of the MPO indices are in the order $D,L,R,U$.

Figure 22

Possible coefficients of the central MPO tensor for the scalar triple product. Here, the internal leg of the MPO can have the two spin representations $\frac{1}{2}$ and $\frac{3}{2}$.

Figure 23

General construction of a three-site MPO with the linear combination $\overleftrightarrow{M}=\alpha \overleftrightarrow{A}+\beta \overleftrightarrow{B}$. Summation over the common indices $m$ and $m^{\prime }$ is assumed.

Figure 24

MPO tensor with $\text{SU}\left(2\right)$ symmetry for the quantum spin chain Heisenberg and chiral three-spin interactions. The degeneracy tensors go together with the rank-4 structural tensor for the spin sectors shown. All other spin sectors have vanishing degeneracy tensors. The operator $\overleftrightarrow{M}$ is the combination of the operators $\overleftrightarrow{A}$ and $\overleftrightarrow{B}$ with proper weights. Again, the labels of the MPO indices are in the order $D,L,R,U$.

Figure 25

Finite state machine for the MPO of the spin-$\frac{1}{2}$ nearest-neighbor Heisenberg model with chiral three-spin interactions. The different states correspond to the different spin sectors for the bond dimensions of the MPO.

Figure 26

Snake pattern for the MPO and also coarse graining of two ladder sites into one site of the MPS.

Figure 27

Scaling of the entanglement spectrum for $S=0$ with the MPS symmetric bond dimension $\chi$, in normalized units.

Figure 28

Scaling of the entanglement spectrum for $S=1$ with the MPS symmetric bond dimension $\chi$, in normalized units.

Figure 29

Scaling of the entanglement spectrum for $S=2$ with the MPS symmetric bond dimension $\chi$, in normalized units.

Figure 30

Scaling of the entanglement spectrum for $S=3$ with the MPS symmetric bond dimension $\chi$, in normalized units.