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.
Inspired by the recent progress in the simulation of tight-binding Hamiltonians on large lattices using tensor networks, we introduce a systematic matrix product operator (MPO) construction for single-particle Hamiltonians on recursively structured lattices. Taking advantage of this recursive structure, we encode the lattice geometry in a finite-state automaton and, adapting ideas from Abelian-symmetric tensor networks, obtain an exact and analytical MPO representation of the Hamiltonian, where the number of tensors grows logarithmically with the system size and the bond dimension is set by the number of automaton states. Several examples demonstrate the generality of the framework: regular lattices, fractals, Cayley trees, hyperbolic lattices, and one- and two-dimensional Fibonacci quasicrystals. Combined with the kernel polynomial method, these MPO representations enable large-scale calculations of spectral properties without explicitly constructing the full Hamiltonian. This provides a unified route to the exploration of electronic properties across a broad class of lattices at exponentially large system sizes.
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.
This work is concerned with tree tensor network operators (TTNOs) for representing quantum Hamiltonians. We first establish a mathematical framework connecting tree topologies with state diagrams. Based on these, we devise an algorithm for constructing a TTNO given a Hamiltonian. The algorithm exploits the tensor product structure of the Hamiltonian to add paths to a state diagram, while combining local operators if possible. We test the capabilities of our algorithm on random Hamiltonians for a given tree structure. Additionally, we construct explicit TTNOs for nearest neighbour interactions on a tree topology. Furthermore, we derive a bound on the bond dimension of tensor operators representing arbitrary interactions on trees. Finally, we consider an open quantum system in the form of a Heisenberg spin chain coupled to bosonic bath sites as a concrete example. We find that tree structures allow for lower bond dimensions of the Hamiltonian tensor network representation compared to a matrix product operator structure. This reduction is large enough to reduce the number of total tensor elements required as soon as the number of baths per spin reaches 3.
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=1/2 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)-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
Z2 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=3/2 on the
z=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 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.
We show how to construct relevant families of matrix product operators (MPOs) in one and higher dimensions. These form the building blocks for the numerical simulation methods based on matrix product states and projected entangled pair states. In particular, we construct translationally invariant MPOs suitable for time evolution, and show how such descriptions are possible for Hamiltonians with long-range interactions. We show how these tools can be exploited for constructing new algorithms for simulating quantum spin systems.