We present several improvements of the infinite matrix product state (iMPS) algorithm for finding ground states of one-dimensional quantum systems with long-range interactions. As a main ingredient, we introduce the superposed multioptimization method, which allows an efficient optimization of exponentially many MPS of different lengths at different sites all in one step. Here, the algorithm becomes protected against position-dependent effects as caused by spontaneously broken translational invariance. So far, these have been a major obstacle to convergence for the iMPS algorithm if no prior knowledge of the system's translational symmetry was accessible. Further, we investigate some more general methods to speed up calculations and improve convergence, which might be partially interesting in a much broader context, too. As a more special problem, we also look into translational invariant states close to an invariance-breaking phase transition and show how to avoid convergence into wrong local minima for such systems. Finally, we apply these methods to polar bosons with long-range interactions. We calculate several detailed Devil's staircases with the corresponding phase diagrams and investigate some supersolid properties.
We present several improvements of the infinite matrix product state (iMPS) algorithm for finding ground states of one-dimensional quantum systems with long-range interactions. As a main ingredient, we introduce the superposed multioptimization method, which allows an efficient optimization of exponentially many MPS of different lengths at different sites all in one step. Here, the algorithm becomes protected against position-dependent effects as caused by spontaneously broken translational invariance. So far, these have been a major obstacle to convergence for the iMPS algorithm if no prior knowledge of the system's translational symmetry was accessible. Further, we investigate some more general methods to speed up calculations and improve convergence, which might be partially interesting in a much broader context, too. As a more special problem, we also look into translational invariant states close to an invariance-breaking phase transition and show how to avoid convergence into wrong local minima for such systems. Finally, we apply these methods to polar bosons with long-range interactions. We calculate several detailed Devil's staircases with the corresponding phase diagrams and investigate some supersolid properties.
An algorithm is presented which computes a translationally invariant matrix product state approximation of the ground state of an infinite one-dimensional (1D) system. It does this by embedding sites into an approximation of the infinite ``environment'' of the chain, allowing the sites to relax and then merging them with the environment in order to refine the approximation. By making use of matrix product operators, our approach is able to directly model any long-range interaction that can be systematically approximated by a series of decaying exponentials. We apply these techniques to compute the ground state of the Haldane-Shastry model [Phys. Rev. Lett. 60, 635 (1988) and Phys. Rev. Lett. 60, 639 (1988)] and present the results.
Obtaining quantitative ground-state behavior for geometrically-frustrated quantum magnets with long-range interactions is challenging for numerical methods. Here, we demonstrate that the ground states of these systems on two-dimensional lattices can be efficiently obtained using state-of-the-art translation-invariant variants of matrix product states and density-matrix renormalization-group algorithms. We use these methods to calculate the fully-quantitative ground-state phase diagram of the long-range interacting triangular Ising model with a transverse field on six-leg infinite-length cylinders and scrutinize the properties of the detected phases. We compare these results with those of the corresponding nearest neighbor model. Our results suggest that, for such long-range Hamiltonians, the long-range quantum fluctuations always lead to long-range correlations, where correlators exhibit power-law decays instead of the conventional exponential drops observed for short-range correlated gapped phases. Our results are relevant for comparisons with recent ion-trap quantum simulator experiments that demonstrate highly-controllable long-range spin couplings for several hundred ions.
Response functions $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}}_{x}(t){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}}_{y}(0)\ensuremath{\rangle}$ 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 $|x\ensuremath{-}y|<\ensuremath{\ell}$ 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 $\ensuremath{\ell}$ to $2\sqrt{\ensuremath{\ell}}$. (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 $|x\ensuremath{-}y|$. 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 nc may converge to a nontrivial limit cycle of length m. This then corresponds to an iMPS with an enlarged unit cell of size mnc.
The study of low-dimensional, strongly-interacting quantum systems has been of interest in the past few decades due to new and exotic physics that show great potential in the advancement of fundamental physics as well as technological applications. In order to investigate such systems, a vast number of techniques have been developed throughout the years, all of which have their respective strengths and weaknesses. In the world of one-dimensional (1D) quantum systems, the density-matrix renormalization group (DMRG) algorithm and matrix product states (MPS) have proven to be powerful and reliable tools that provide invaluable information while providing a wide range of data that support both theoretical and experimental studies. Coupled with the development and refinement of tensor networks - an ansatz for quantum many-body wavefunctions, in particular MPS, the DMRG algorithm has seen a spike of improvements that has pushed the boundaries of its application into more sophisticated and complex systems.This thesis broadly covers the study of several 1D many-body quantum systems represented by infinite MPS (iMPS) - a class of ansatz that represent translationally-invariant, 1D many-body quantum states. More specifically, it is divided into two main parts.The first part investigates the 1D topological Kondo insulator (TKI). This is an effective model designed to investigate how strong interactions between conduction electrons and localized moments result in a symmetry-protected topological (SPT) phase, specifically the Haldane phase. By studying the effect of the electron repulsion as well as the competition between the local and non-local couplings between the electrons and the local moments, it is found that a topological phase transition from the SPT phase to a topologically trivial phase occurs with the assistance of electron interaction. The properties of both topologically trivial and nontrivial groundstates were revealed via means of quantum information theoretic quantities such as the entanglement entropy, entanglement spectrum and non-local order parameters. These quantities unveil the nature, structure and the effect that the different interactions have on the groundstate properties. Besides the phases, the properties of the critical point was studied and classified. By obtaining the critical exponents and central charge, the universality class of this transition was found to be that of 1D free fermions/bosons. Furthermore, the presence of electron interactions was found not to alter this universality class.The second part of this thesis is a project that stems from the necessity to overcome the inherent limitation of an iMPS in representing a system at a cusp of a quantum phase transition (QPT), specifically, in determining the critical point and exponents. In the thermodynamic limit, the laws of statistical mechanics dictates that the system’s correlation length diverges at the critical point. However, when representing a critical state with an iMPS, the finite bond dimension implies that the correlation length of the iMPS remains finite, albeit large. This presents a fundamental challenge in extracting critical data since the critical exponents are defined for systems in the thermodynamic limit. To overcome this shortcoming, finite-entanglement scaling (FES) - the analog of finite-size scaling, has previously been used to locate the critical point and extract the critical exponents for systems represented by finite-bond dimensional iMPS. While FES serves its purpose sufficiently well, the computational cost is high for just about any moderately complex, real-world system. To overcome such difficulties, the second part of this thesis combines new ideas of using higher-order cumulants of the order parameter with previously-known iMPS techniques to determine the critical point and extract the exponents, thus relating statistical mechanics of QPTs and tensor networks of critical states. Some of the main methods used and developed here are FES, the application of the Binder cumulant in iMPS and the new cumulant exponent relation. Putting these tools together provides a scheme that can be used to efficiently locate the critical point and extract the critical exponents at a lower computational cost than previously-known methods. This is shown through comparisons of several exemplary models such as the 1D and quasi-1D transverse field Ising model, the 1D TKI, the S = 1 Heisenberg chain with single-ion anisotropy and the 1D Bose-Hubbard model.
We present a matrix product state (MPS) algorithm to approximate ground states of translationally invariant systems with periodic boundary conditions. For a fixed value of the bond dimension D of the MPS, we discuss how to minimize the computational cost to obtain a seemingly optimal MPS approximation to the ground state. In a chain with N sites and correlation length $\ensuremath{\xi}$, the computational cost formally scales as $g(D,\ensuremath{\xi}/N){D}^{3}$, where $g(D,\ensuremath{\xi}/N)$ is a nontrivial function. For $\ensuremath{\xi}\ensuremath{\ll}N$, this scaling reduces to D3, independent of the system size N, making our method N times faster than previous proposals. We apply the algorithm to obtain MPS approximations for the ground states of the critical quantum Ising and Heisenberg spin-1/2 models as well as for the noncritical Heisenberg spin-1 model. In the critical case, for any chain length N, we find a model-dependent bond dimension D(N) above which the polynomial decay of correlations is faithfully reproduced throughout the entire system.
We present some exact results for the optimal matrix product state (MPS) approximation to the ground state of the infinite isotropic Heisenberg spin-1/2 chain. Our approach is based on the systematic use of Schmidt decompositions to reduce the problem of approximating for the ground state of a spin chain to an analytical minimization. This allows one to show that results of standard simulations, e.g. density matrix renormalization group and infinite time evolving block decimation, do correspond to the result obtained by this minimization strategy and, thus, both methods deliver optimal MPS with the same energy but, otherwise, different properties. We also find that translational and rotational symmetries cannot be maintained simultaneously by the MPS ansatz of minimum energy and present explicit constructions for each case. Furthermore, we analyze symmetry restoration and quantify it to uncover new scaling relations. The method we propose can be extended to any translational invariant Hamiltonian.