Infinite-system Density Matrix Renormalization Group Research Articles

Chiral Spin Liquid Phase of the Triangular Lattice Hubbard Model: A Density Matrix Renormalization Group Study

Motivated by experimental studies that have found signatures of a quantum spin liquid phase in organic crystals whose structure is well described by the two-dimensional triangular lattice, we study the Hubbard model on this lattice at half filling using the infinite-system density matrix renormalization group (iDMRG) method. On infinite cylinders with finite circumference, we identify an intermediate phase between observed metallic behavior at low interaction strength and Mott insulating spin-ordered behavior at strong interactions. Chiral ordering from spontaneous breaking of time-reversal symmetry, a fractionally quantized spin Hall response, and characteristic level statistics in the entanglement spectrum in the intermediate phase provide strong evidence for the existence of a chiral spin liquid in the full two-dimensional limit of the model.

Infinite boundary conditions for response functions and limit cycles within the infinite-system density matrix renormalization group approach demonstrated for bilinear-biquadratic spin-1 chains

Response functions $\langle A_x(t) B_y(0)\rangle$ for one-dimensional strongly correlated quantum many-body systems can be computed with matrix product state (MPS) techniques. Especially, when one is interested in spectral functions or dynamic structure factors of translation-invariant systems, the response for some range $|x-y|<\ell$ is needed. We demonstrate how the number of required time-evolution runs can be reduced substantially: (a) If finite-system simulations are employed, the number of time-evolution runs can be reduced from $\ell$ to $2\sqrt{\ell}$. (b) To go beyond, one can employ infinite MPS (iMPS) such that two evolution runs suffice. To this purpose, iMPS that are heterogeneous only around the causal cone of the perturbation are evolved in time, i.e., the simulation is done with infinite boundary conditions. Computing overlaps of these states, spatially shifted relative to each other, yields the response functions for all distances $|x-y|$. As a specific application, we compute the dynamic structure factor for ground states of bilinear-biquadratic spin-1 chains with very high resolution and explain the underlying low-energy physics. To determine the initial uniform iMPS for such simulations, infinite-system density-matrix renormalization group (iDMRG) can be employed. We discuss that, depending on the system and chosen bond dimension, iDMRG with a cell size $n_c$ may converge to a non-trivial limit cycle of length $m$. This then corresponds to an iMPS with an enlarged unit cell of size $m n_c$.

Erratum: “Ground-State Phase Diagram of Frustrated Antiferromagnetic S = 1 Chain with Uniaxial Single-Ion-Type Anisotropy” [J. Phys. Soc. Jpn. 71, 319 (2002)

In the paper, it was incorrectly mentioned that by the exactdiagonalization calculation for small systems, the ground state of the model (1.4) was found to belong to the subspace of Stotal 1⁄4 0 with even parity.1) Although it is the case for the periodic chains and for a wide parameter region of the open chains, we have noticed that the ground state with odd parity appears in some parameter points of the open chains, as shown in Fig. 1 of this erratum. We note that this is a boundary effect in small systems with frustration, and the infinite-system density-matrix renormalization-group (DMRG) calculation in the original paper, which was performed for the subspace of Stotal 1⁄4 0 with even parity, should capture the ground-state properties in the thermodynamic limit correctly. Indeed, we have newly performed the infinite-system DMRG for the whole subspace of Stotal 1⁄4 0 including both even and odd parity states, and obtained the same results as the previous ones within a numerical accuracy. Therefore, our conclusions remain the same. We also correct misprints, which do not affect our conclusions. (i) In Figs. 2(a), 4(a), and 6(a), the distance r of the chiral correlation C ðrÞ was shifted by one due to a simple error in the data processing. The label for the x-axis should read r þ 1 instead of r. (ii) In Figs. 2(b), 3, 4(b), and 5(a), the sign of the string correlation was reversed erroneously: The minus sign should be removed. (iii) In Fig. 4, the number of kept states m for d 1⁄4 1:10 was not 350 but 300. We have checked that the results with these m’s are almost identical. In Fig. 6(a), in addition to j 1⁄4 0:800, the data for j 1⁄4 0:810 were obtained with m 1⁄4 500. Accordingly, in Fig. 1, a cross, which indicates the point of high-precision calculation, should be added at ðj; dÞ 1⁄4 ð0:810; 0:30Þ.

Erratum: “Ground State Phase Diagram of Frustrated S = 1 XXZ Chains: Chiral Ordered Phases” [J. Phys. Soc. Jpn. 69, 259 (2000)

In the paper, it was incorrectly mentioned that the exactdiagonalization calculation for small systems found that the ground state of the model (1.1) belonged to the subspace of S total 1⁄4 0 with even parity.1) (We call this subspace PE below.) While it is the case for the periodic chains2) and for a wide parameter region of the open chains, we have noticed that the ground state with odd parity appears in some narrow parameter regions of the open chains, as shown in Fig. 1 of this erratum. We note that this is a boundary effect in small systems with frustration, and the infinite-system density-matrix renormalization-group (DMRG) calculation for the subspace PE performed in the original paper should capture the groundstate properties in the thermodynamic limit correctly. Indeed, we have newly performed the infinite-system DMRG for the whole subspace of S total 1⁄4 0 including both even and odd parity states (we call this subspace PEO), and obtained essentially the same results as the previous ones for the subspace PE. For example, as for the estimates of jc1 and jc2 shown in Table II, those for 1⁄4 0, 0.2, and 0.4 remain the same, while comparing the results for PE and PEO, we modify the errors of the estimates for 1⁄4 0:6 (0.8) from ðjc1; jc2Þ 1⁄4 ð0:642 0:002; 0:660 0:003Þ [ð0:715 0:003; 0:720 0:002Þ] to ð0:642þ0:004 0:002; 0:660þ0:009 0:003Þ [ð0:715þ0:005 0:003; 0:720þ0:009 0:002Þ]. The other results in the paper, including the phase diagram and the estimates of critical exponents, remain the same within a numerical accuracy. Therefore, our conclusions remain valid. We also correct misprints and typographical errors, which do not affect our conclusions. (i) In Figs. 2(b) and 5, the distance r of the chiral correlation C ðrÞ was shifted by one due to a simple error in the data processing. The label for the x-axis should read r + 1 instead of r. (ii) Equation (3.2) should read

Erratum: “Spin and Chiral Orderings of Frustrated Quantum Spin Chains” [J. Phys. Soc. Jpn. 68, 3185 (1999)

In p. 3186, below Eq. (3), it was incorrectly stated that by the exact-diagonalization (ED) calculation for the open chains with even N [8 N 20 (S = 1/2) and 8 N 16 (S = 1)], the ground state was found to belong to the subspace of S total 1⁄4 0 with even parity.1) In fact, for the S = 1/2 XY and Heisenberg open chains, the ground state belongs to the subspace of S total 1⁄4 0 with even (odd) parity when N = 4n (4n + 2), where n is an integer. For the S = 1 case, the ground state for a wide parameter region belongs to the subspace of S total 1⁄4 0 with even parity, while the ground state with odd parity appears in some parameter regions, whose position is confined within a range 0:3 . j . 0:6 and changes depending on the system size N and on whether XY or Heisenberg. This correction has no effect on the other parts of the paper. The data of the Binder parameter shown in Fig. 1 were obtained by the ED calculation for the whole subspace of S total 1⁄4 0 including both the even and odd parity states, and therefore, they are not affected by the correction. As for the density-matrix renormalization-group (DMRG) calculation for the S = 1 XY chains, we note that the appearance of the odd-parity ground state found by the ED is a boundary effect in small systems with frustration, and the infinite-system DMRG for the subspace of S total 1⁄4 0 with even parity should capture the ground-state properties in thermodynamic limit correctly. Indeed, we have newly performed the DMRG calculation for the whole subspace of S total 1⁄4 0 and obtained the same results as the previous ones within a numerical accuracy. As to the data shown in Fig. 2, for example, the differences between the previous and newly-obtained results averaged on the distance r are at most 1:2 10 3 (chiral correlation), 1:5 10 4 (string correlation), and 4:8 10 4 (spin correlation), except for the chiral and string correlations at the transition point j 1⁄4 jc1 1⁄4 0:473 for which the averaged differences are 1:2 10 2 or less (chiral) and 1:1 10 3 (string). These differences have no effect on our argument on the behaviors of the correlation functions, and therefore, our conclusions in the paper remain the same. We also correct a misprint in Fig. 2(a): The distance r of the chiral correlation C ðrÞ was shifted by one due to a simple error in the data processing. Therefore, the label for the x-axis should read r + 1 instead of r. Finally, we note that, although the ED data of the Binder parameter in Fig. 1 suggested the absence of the chiral order in the S = 1/2 XY chains, it has been revealed by the DMRG that the chiral order indeed appears in the S = 1/2 chains with easy-plane anisotropy. The DMRG results and the reason why the ED analysis missed the chiral order in the S = 1/2 chains have already been reported in Ref. 2.