Two-Dimensional Tensor Product Variational Formulation

Nishino, Tomotoshi; Hieida, Yasuhiro; Okunishi, Kouichi; Maeshima, Nobuya; Akutsu, Yasuhiro; Gendiar, Andrej

doi:10.1143/ptp.105.409

Cited by 163 publications

(180 citation statements)

References 9 publications

(16 reference statements)

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“…Our calculation on 16 × 16 lattice shows that the ground states below δ ∼ 0.12 and above δ ∼ 0.2 show the stripe orders with (l c , l s ) = (8,16) and (l c , l s ) = (4, 8), respectively, while roughly in the region 0.12 < δ < 0.2 the energy is convex implying the phase separation. However, one can speculate that other periodicities or structures of the stripes with the period between (8,16) and (4,8) that are not compatible with this size may have slightly lower energy filling the convexity like t-J model 51 and precludes the phase separation in this region. From the energy curve in Fig.…”

Section: Summary and Discussionmentioning

confidence: 98%

“…14. The peak of spin structure factor is at 3π 4 , π , and the peak of the charge structure factor is at π 2 , 0 , which indicate that the ground state has a stripe order with (l c , l s ) = (4,8). The reduction of the period with the increasing hole concentration is intuitively understood from the deceasing mean hole distance with doping and also consistent with the experimental indications of the diffuse peak in neutron scattering in the cuprates 47,48 .…”

Section: Spin and Charge Correlationsmentioning

confidence: 92%

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Variational Monte Carlo method for fermionic models combined with tensor networks and applications to the hole-doped two-dimensional Hubbard model

et al. 2017

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The conventional tensor-network states employ real-space product states as reference wave functions. Here, we propose a many-variable variational Monte Carlo (mVMC) method combined with tensor networks by taking advantages of both to study fermionic models. The variational wave function is composed of a pair product wave function operated by real space correlation factors and tensor networks. Moreover, we can apply quantum number projections, such as spin, momentum and lattice symmetry projections, to recover the symmetry of the wave function to further improve the accuracy. We benchmark our method for one-and two-dimensional Hubbard models, which show significant improvement over the results obtained individually either by mVMC or by tensor network. We have applied the present method to hole doped Hubbard model on the square lattice, which indicates the stripe charge/spin order coexisting with a weak d-wave superconducting order in the ground state for the doping concentration less than 0.3, where the stripe oscillation period gets longer with increasing hole concentration. The charge homogeneous and highly superconducting state also exists as a metastable excited state for the doping concentration less than 0.25.

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Section: Summary and Discussionmentioning

confidence: 98%

Section: Spin and Charge Correlationsmentioning

confidence: 92%

Variational Monte Carlo method for fermionic models combined with tensor networks and applications to the hole-doped two-dimensional Hubbard model

et al. 2017

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“…This approach, however, suffers from a slow decay in eigenvalues of the reduced density matrix, where the target scheme in the RG transformation was not appropriate from the modern view point. Complementary direct variational approaches based on the 2D tensor product state 18,19 , or the 2D projected entangled pair state [20][21][22][23][24] has been applied to the higher-dimensional problems. Recently, Xie et al proposed an improved tensor RG method 25 combined with the higher order singular value decomposition (HOSVD) 26 , which has been abbreviated as HOTRG 27 .…”

Section: Introductionmentioning

confidence: 99%

Doubling of entanglement spectrum in tensor renormalization group

2014

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We investigate the entanglement spectrum in HOTRG -tensor renormalization group (RG) method combined with the higher order singular value decomposition-for two-dimensional (2D) classical vertex models. In the off-critical region, it is explained that the entanglement spectrum associated with the RG transformation is described by 'doubling' of the spectrum of a corner transfer matrix. We then demonstrate that the doubling actually occurs for the square-lattice Ising model by HOTRG calculations up to D = 64, where D is the cut-off dimension of tensors. At the critical point, we also find that a non-trivial D scaling behavior appears in the entanglement entropy. We mention about the HOTRG for the 1D quantum system as well.

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“…In recent years, a new class of numerical methods which combine the renormalization group techniques [9][10][11][12][13][14][15][16] with the tensor-network representation of quantum many-body states or partition functions of statistical models [17][18][19][20][21][22] has been developed. These methods have played an important role in the study of strongly correlated problems, especially in two or higher dimensions [16,23,24].…”

Section: Introductionmentioning

confidence: 99%

Optimized contraction scheme for tensor-network states

et al. 2017

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In the tensor-network framework, the expectation values of two-dimensional quantum states are evaluated by contracting a double-layer tensor network constructed from initial and final tensornetwork states. The computational cost of carrying out this contraction is generally very high, which limits the largest bond dimension of tensor-network states that can be accurately studied to a relatively small value. We propose an optimized contraction scheme to solve this problem by mapping the double-layer tensor network onto an intersected single-layer tensor network. This reduces greatly the bond dimensions of local tensors to be contracted, and improves dramatically the efficiency and accuracy of the evaluation of expectation values of tensor-network states. It almost doubles the largest bond dimension of tensor-network states whose physical properties can be efficiently and reliably calculated, and it extends significantly the application scope of tensornetwork methods.

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Two-Dimensional Tensor Product Variational Formulation

Cited by 163 publications

References 9 publications

Variational Monte Carlo method for fermionic models combined with tensor networks and applications to the hole-doped two-dimensional Hubbard model

Variational Monte Carlo method for fermionic models combined with tensor networks and applications to the hole-doped two-dimensional Hubbard model

Doubling of entanglement spectrum in tensor renormalization group

Optimized contraction scheme for tensor-network states

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