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arXiv:1907.06018 · Physical review. B./Physical review. B

Generalization of the exponential basis for tensor network representations of long-range interactions in two and three dimensions

Zhendong Li, Matthew J. O’Rourke, Garnet Kin‐Lic Chan

Published Oct 14, 2019• 12 citations
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Abstract

In one dimension (1D), a general decaying long-range interaction can be fit to a sum of exponential interactions ${e}^{\ensuremath{-}\ensuremath{\lambda}{r}_{ij}}$ with varying exponents $\ensuremath{\lambda}$, each of which can be represented by a simple matrix product operator with bond dimension $D=3$. Using this technique, efficient and accurate simulations of 1D quantum systems with long-range interactions can be performed using matrix product states. However, the extension of this construction to higher dimensions is not obvious. We report how to generalize the exponential basis to two and three dimensions by defining the basis functions as the Green's functions of the discretized Helmholtz equation for different Helmholtz parameters $\ensuremath{\lambda}$, a construction which is valid for lattices of any spatial dimension. Compact tensor network representations can then be found for the discretized Green's functions, by expressing them as correlation functions of auxiliary fermionic fields with nearest-neighbor interactions via Grassmann Gaussian integration. Interestingly, this analytic construction in three dimensions yields a $D=4$ tensor network representation of correlation functions which (asymptotically) decay as the inverse distance (${r}_{ij}^{\ensuremath{-}1}$), thus generating the (screened) Coulomb potential on a cubic lattice. These techniques will be useful in tensor network simulations of realistic materials.

DOI
10.1103/physrevb.100.155121

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In one dimension (1D), a general decaying long-range interaction can be fit to a sum of exponential interactions e\ensuremath−\ensuremathλrij{e}^{\ensuremath{-}\ensuremath{\lambda}{r}_{ij}}e\ensuremath−\ensuremathλrij​ with varying exponents \ensuremathλ\ensuremath{\lambda}\ensuremathλ, each of which can be represented by a simple matrix product operator with bond dimension D=3D=3D=3. Using this technique, efficient and accurate simulations of 1D quantum systems with long-range interactions can be performed using matrix product states. However, the extension of this construction to higher dimensions is not obvious. We report how to generalize the exponential basis to two and three dimensions by defining the basis functions as the Green's functions of the discretized Helmholtz equation for different Helmholtz parameters \ensuremathλ\ensuremath{\lambda}\ensuremathλ, a construction which is valid for lattices of any spatial dimension. Compact tensor network representations can then be found for the discretized Green's functions, by expressing them as correlation functions of auxiliary fermionic fields with nearest-neighbor interactions via Grassmann Gaussian integration. Interestingly, this analytic construction in three dimensions yields a D=4D=4D=4 tensor network representation of correlation functions which (asymptotically) decay as the inverse distance (rij\ensuremath−1{r}_{ij}^{\ensuremath{-}1}rij\ensuremath−1​), thus generating the (screened) Coulomb potential on a cubic lattice. These techniques will be useful in tensor network simulations of realistic materials.
We describe a practical and efficient approach to represent physically realistic long-range interactions in two-dimensional tensor network algorithms via projected entangled-pair operators (PEPOs). We express the long-range interaction as a linear combination of correlation functions of an auxiliary system with only nearest-neighbor interactions. To obtain a smooth and radially isotropic interaction across all length scales, we map the physical lattice to an auxiliary lattice of expanded size. Our construction yields a long-range PEPO as a sum of ancillary PEPOs, each of small, constant bond dimension. This representation enables efficient numerical simulations with long-range interactions using projected entangled pair states.
We consider the representation of operators in terms of tensor networks and their application to the ground-state approximation and time evolution of systems with long-range interactions. We provide an explicit construction to represent an arbitrary many-body Hamilton operator in terms of a one-dimensional tensor network (i.e., as a matrix product operator). For pairwise interactions, we show that such a representation is always efficient and requires a tensor dimension growing only linearly with the number of particles. For systems obeying certain symmetries or restrictions we find optimal representations with minimal tensor dimension. We discuss the analytic and numerical approximation of operators in terms of low-dimensional tensor operators. We demonstrate applications for time evolution and the ground-state approximation, in particular for long-range interaction with inhomogeneous couplings. The operator representations are also generalized to other geometries such as trees and two-dimensional lattices, where we show how to obtain and use efficient tensor network representations respecting a given geometry.
Abstract We provide an exact construction of particular Hamitonians on a one-dimensional lattice as matrix product operators, a type of tensor network. Namely, we consider Hamiltonians describing interactions between degrees of freedom at lattice sites whose strength grows with the lattice site separation as a polynomial multiplied by an exponential. We show that the bond dimension is ( k + 3) for a polynomial of order k , independent of the system size and the number of particles. Our construction is manifestly translationally invariant, and so may be used in finite- or infinite-size variational matrix product state algorithms. Our results provide new insight into the correlation structure of many-body quantum operators, and may also be practical in simulations of many-body systems whose interactions are exponentially screened at large distances, but may have complex short-distance structure.
Simulating many-body open quantum systems is an extremely challenging problem, with methods often restricted to either models with nearest-neighbor interactions or semi-classical approximations. In particular, modeling two-dimensional systems with realistic long-range interactions, in addition to dissipation, is of vital importance to the development of modern quantum computing and simulation platforms. In this paper, we present a construction of the time-evolution operator, as a projected entangled pair operator (denoted tePEPO), that can be used to evolve a tensor network ansatz through time. Interactions beyond nearest-neighbor, including interactions between sites not collinear in the lattice, can be represented efficiently as a tePEPO. Furthermore, we obtain approximations to realistic radial long-range interactions decaying with a power-law, that give accurate results with small tePEPO bond dimension. Finally, we consider a physical example of a Rydberg atom Hamiltonian with long-range dipolar interactions, and show evidence of a dipole-dipole blockading effect in presence of dissipation. This work demonstrates the applicability of tensor networks to two-dimensional systems widely studied in experiments, but previously inaccessible to non-semi-classical methods.
This thesis treats the classical simulation of strongly-interacting many-body quantum-mechanical systems in more than one dimension using matrix product states and the more general tensor product states. Contrary to classical systems, quantum many-body systems possess an exponentially larger number of degrees of freedom, thereby significantly complicating their numerical treatment on a classical computer. For this thesis two different representations of quantum many-body states were employed. The first, the so-called matrix product states (MPS) form the basis for the extremely successful density matrix renormalization group (DMRG) algorithm. While originally conceived for one-dimensional systems, MPS are in principle capable of describing arbitrary quantum many-body states. Using concepts from quantum information theory it is possible to show that MPS provide a representation of one-dimensional quantum systems that scales polynomially in the number of particles, therefore allowing an efficient simulation of one-dimensional systems on a classical computer. One of the key results of this thesis is that MPS representations are indeed efficient enough to describe even large systems in two dimensions, thereby enabling the simulation of such systems using DMRG. As a demonstration of the power of the DMRG algorithm, it is applied to the Heisenberg antiferromagnet with spin S=1/2S = 1/2S=1/2 on the kagome lattice. This model's ground state has long been under debate, with proposals ranging from static spin configurations to so-called quantum spin liquids, states where quantum fluctuations destroy conventional order and give rise to exotic quantum orders. Using a fully SU(2)SU(2)SU(2)-symmetric implementation allowed us to handle the exponential growth of entanglement and to perform a large-scale study of this system, finding the ground state for cylinders of up to 700 sites. Despite employing a one-dimensional algorithm for a two-dimensional system, we were able to compute the spin gap (i.e. the energy gap to the first spinful excitation) and study the ground state properties, such as the decay of correlation functions, the static spin structure factors, and the structure and distribution of the nearest-neighbor spin-spin correlations. Additionally, by applying a new tool from quantum information theory, the topological entanglement entropy, we could also with high confidence demonstrate the ground state of this model to be the elusive gapped Z2Z_2Z2​ quantum spin liquid with topological order. To complement this study, we also considered the extension of MPS to higher dimensions, known as tensor product states (TPS). We implemented an optimization algorithm exploiting symmetries for this class of states and applied it to the bilinear-biquadratic-bicubic Heisenberg model with spin S=3/2S=3/2S=3/2 on the z=3z=3z=3 Bethe lattice. By carefully analyzing the simulation data we were able to determine the presence of both conventional and symmetry-protected topological order in this model, thereby demonstrating the analytically predicted existence of the Haldane phase in higher dimensions within an extended region of the phase diagram. Key properties of this symmetry-protected topological order include a doubling of the levels in the entanglement spectrum and the presence of edge spins, both of which were confirmed in our simulations. This finding simultaneously validated the applicability of the novel TPS algorithms to the search for exotic order.
We present a tree-tensor-network-based method to study strongly correlated systems with nonlocal interactions in higher dimensions. Although the momentum-space and quantum-chemistry versions of the density-matrix renormalization group (DMRG) method have long been applied to such systems, the spatial topology of DMRG-based methods allows efficient optimizations to be carried out with respect to one spatial dimension only. Extending the matrix-product-state picture, we formulate a more general approach by allowing the local sites to be coupled to more than two neighboring auxiliary subspaces. Following [Y. Shi, L. Duan, and G. Vidal, Phys. Rev. A 74, 022320 (2006)], we treat a treelike network ansatz with arbitrary coordination number zzz, where the z=2z=2z=2 case corresponds to the one-dimensional (1D) scheme. For this ansatz, the long-range correlation deviates from the mean-field value polynomially with distance, in contrast to the matrix-product ansatz, which deviates exponentially. The computational cost of the tree-tensor-network method is significantly smaller than that of previous DMRG-based attempts, which renormalize several blocks into a single block. In addition, we investigate the effect of unitary transformations on the local basis states and present a method for optimizing such transformations. For the 1D interacting spinless fermion model, the optimized transformation interpolates smoothly between real space and momentum space. Calculations carried out on small quantum chemical systems support our approach.