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Continuous Matrix Product Ansatz for One-dimensional Fermi Systems

Sangwoo S. Chung

Published Jan 1, 2016• 0 citations
OpenAlex

Abstract

The central topic of this thesis is on the fermionic continuous matrix product states (cMPS) variational ansatz for many-body quantum systems in a one-dimensional (1D) continuum.The matrix product states (MPS) is a variational ansatz which has been tremendously successful for nearly two decades in efficiently approximating groundstate properties of low-dimensional quantum lattice systems.The MPS is intrinsically designed for discrete lattice systems, however, and the cMPS has been introduced in 2010 as a continuum analogue of the MPS, enabling the possibility to directly tackle systems of quantum gases and liquids without resorting to discretization of space.

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The central topic of this thesis is on the fermionic continuous matrix product states (cMPS) variational ansatz for many-body quantum systems in a one-dimensional (1D) continuum.The matrix product states (MPS) is a variational ansatz which has been tremendously successful for nearly two decades in efficiently approximating groundstate properties of low-dimensional quantum lattice systems.The MPS is intrinsically designed for discrete lattice systems, however, and the cMPS has been introduced in 2010 as a continuum analogue of the MPS, enabling the possibility to directly tackle systems of quantum gases and liquids without resorting to discretization of space.
The fertile new field of quantum information theory is inspiring new ways to study correlated quantum systems by providing fresh insights into the structure of their Hilbert spaces. One of the latest developments in this direction was the extension of the ubiquitous matrix-product-state constructions, epitomized by the density-matrix renormalization-group algorithm, to continuous space-time; so as to be able to describe low-dimensional field theories within a variational approach. Following the earlier success achieved for bosonic theories, we present the first implementation of a continuous matrix product state (cMPS) for spinfull non-relativistic fermions in 1D. We propose a construction of variational matrices with an efficient parametrization that respects the translational symmetry of the problem (without being overly constraining) and readily meets the regularity conditions that arise from removing the ultraviolet divergences in the kinetic energy. We tested the validity of our approach on an interacting spin-1/2 system with spin imbalance. We observe that the ansatz correctly predicts the ground-state magnetic properties for the attractive spin-1/2 Fermi gas, including a phase-oscillating pair correlation function in the partially polarized regime (the 1D correlate of the FFLO state). We shall also discuss how to generalize the cMPS ansatz to other situations.
By combining the continuous matrix product state (cMPS) representation for quantum fields in the continuum with standard optimization techniques for matrix product states (MPS) on the lattice, we obtain an approximation $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$, directly in the continuum, of the ground state of nonrelativistic quantum field theories. This construction works both for translation-invariant systems and in the more challenging context of inhomogeneous systems, as we demonstrate for an interacting bosonic field in a periodic potential. Given the continuum Hamiltonian HHH, we consider a sequence of discretized Hamiltonians ${{H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})}}_{\ensuremath{\alpha}=1,2,\ensuremath{\cdots},p}$ on increasingly finer lattices with lattice spacing ${\ensuremath{\epsilon}}_{1}>{\ensuremath{\epsilon}}_{2}>\ensuremath{\cdots}>{\ensuremath{\epsilon}}_{p}$. We first use energy minimization to optimize an MPS approximation $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{1})\ensuremath{\rangle}$ for the ground state of $H({\ensuremath{\epsilon}}_{1})$. Given the MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})\ensuremath{\rangle}$ optimized for the ground state of $H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})$, we use it to initialize the energy minimization for Hamiltonian $H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}+1})$, resulting in the optimized MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{\ensuremath{\alpha}+1})\ensuremath{\rangle}$. By iteration we produce an optimized MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{p})\ensuremath{\rangle}$ for the ground state of $H({\ensuremath{\epsilon}}_{p})$, from which we finally extract the cMPS approximation $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$ for the ground state of HHH. Two key ingredients of our proposal are as follows: (i) a procedure to discretize HHH into a lattice model where each site contains a two-dimensional vector space (spanned by vacuum $|0\ensuremath{\rangle}$ and one boson $|1\ensuremath{\rangle}$ states), and (ii) a procedure to map MPS representations from a coarser lattice to a finer lattice.
The generalization of matrix product states (MPS) to continuous systems, as proposed in the breakthrough Letter of Verstraete and Cirac [Phys. Rev. Lett. 104, 190405 (2010).PRLTAO0031-900710.1103/PhysRevLett.104.190405], provides a powerful variational ansatz for the ground state of strongly interacting quantum field theories in one spatial dimension. A continuous MPS (cMPS) approximation to the ground state can be obtained by simulating a Euclidean time evolution. In this Letter we propose a cMPS optimization algorithm based instead on energy minimization by gradient methods and demonstrate its performance by applying it to the Lieb-Liniger model (an integrable model of an interacting bosonic field) directly in the thermodynamic limit. We observe a very significant computational speed-up, of more than 2 orders of magnitude, with respect to simulating a Euclidean time evolution. As a result, a much larger cMPS bond dimension D can be reached (e.g., D=256 with moderate computational resources), thus helping unlock the full potential of the cMPS representation for ground state studies.
We present a way of constructing continuous matrix product states (cMPS) for coupled fields. The cMPS is a variational ansatz for the ground state of quantum field theories in one dimension. Our proposed scheme is based on a physical interpretation in which the cMPS class can be produced by means of a dissipative dynamic of a system interacting with a bath. We study the case of coupled bosonic fields. We test the method with previous DMRG results in coupled Lieb-Liniger models. Besides, we discuss a novel application for characterizing the Luttinger liquid theory emerging in the low-energy regime of these theories. Finally, we propose a circuit QED architecture as a quantum simulator for coupled fields.
We discuss various properties of the variational class of continuous matrix product states, a class of Ansatz states for one-dimensional quantum fields that was recently introduced as the direct continuum limit of the highly successful class of matrix product states. We discuss both attributes of the physical states, e.g., by showing in detail how to compute expectation values, as well as properties intrinsic to the representation itself, such as the gauge freedom. We consider general translation noninvariant systems made of several particle species and derive certain regularity properties that need to be satisfied by the variational parameters. We also devote a section to the translation invariant setting in the thermodynamic limit and show how continuous matrix product states possess an intrinsic ultraviolet cutoff. Finally, we introduce a new set of states, which are tangent to the original set of continuous matrix product states. For the case of matrix product states, this construction has recently proven relevant in the development of new algorithms for studying time evolution and elementary excitations of quantum spin chains. We thus lay the foundation for similar developments for one-dimensional quantum fields.
Just as matrix product states represent ground states of one-dimensional quantum spin systems faithfully, continuous matrix product states (cMPS) provide faithful representations of the vacuum of interacting field theories in one spatial dimension. Unlike the quantum spin case, however, for which the density matrix renormalization group and related matrix product state algorithms provide robust algorithms for optimizing the variational states, the optimization of cMPS for systems with inhomogeneous external potentials has been problematic. We resolve this problem by constructing a piecewise linear parameterization of the underlying matrix-valued functions, which enables the calculation of the exact reduced density matrices everywhere in the system by high-order Taylor expansions. This turns the variational cMPS problem into a variational algorithm from which both the energy and its backwards derivative can be calculated exactly and at a cost that scales as the cube of the bond dimension. We illustrate this by finding ground states of interacting bosons in external potentials and by calculating boundary or Casimir energy corrections of continuous many-body systems with open boundary conditions.