In this study, we explore the geometric construction of the Klein bottle and the real projective plane (RP^{2}) within the framework of tensor networks, focusing on the implementation of crosscap and rainbow boundaries. Previous investigations have applied boundary matrix product state techniques to study these boundaries. We introduce an approach that incorporates such boundaries into the tensor renormalization group methodology, facilitated by an efficient representation of a spatial reflection operator. This advancement enables us to compute the crosscap and rainbow free energy terms and the one-point function on RP^{2} with enhanced efficiency and for larger system sizes. Additionally, our method is capable of calculating the partition function under isotropic conditions of space and imaginary time. The versatility of this approach is further underscored by its applicability to constructing other (non)orientable surfaces of higher genus.
In this study, we explore the geometric construction of the Klein bottle and the real projective plane (RP^{2}) within the framework of tensor networks, focusing on the implementation of crosscap and rainbow boundaries. Previous investigations have applied boundary matrix product state techniques to study these boundaries. We introduce an approach that incorporates such boundaries into the tensor renormalization group methodology, facilitated by an efficient representation of a spatial reflection operator. This advancement enables us to compute the crosscap and rainbow free energy terms and the one-point function on RP^{2} with enhanced efficiency and for larger system sizes. Additionally, our method is capable of calculating the partition function under isotropic conditions of space and imaginary time. The versatility of this approach is further underscored by its applicability to constructing other (non)orientable surfaces of higher genus.
Universal thermal data in conformal field theory (CFT) offer a valuable means for characterizing and classifying criticality. With improved tensor network techniques, we investigate the universal thermodynamics on a nonorientable minimal surface, the crosscapped disk (or real projective plane, RP2). Through a cut-and-sew process, RP2 is topologically equivalent to a cylinder with rainbow and crosscap boundaries. We uncover that the crosscap contributes a fractional topological term 21lnk related to a nonorientable genus, with k a universal constant in two-dimensional CFT, while the rainbow boundary gives rise to a geometric term $\frac{c}{4}ln\ensuremath{\beta}$, with $\ensuremath{\beta}$ the manifold size and c the central charge. We have also obtained analytically the logarithmic rainbow term by CFT calculations, and discuss its connection to the renowned Cardy-Peschel conical singularity.
Thermal quantum critical systems, with partition functions expressed as conformal tensor networks, are revealed to exhibit universal entropy corrections on nonorientable manifolds. Through high-precision tensor network simulations of several quantum chains, we identify the universal entropy SK=lnk on the Klein bottle, where k relates to quantum dimensions of the primary fields in conformal field theory (CFT). Different from the celebrated Affleck-Ludwig boundary entropy lng(g reflects noninteger ground-state degeneracy), SK has no boundary dependence or surface energy terms accompanying it, and can be very conveniently extracted from thermal data. On the M\"obius-strip manifold, we uncover an entropy SM=21(lng+lnk) in CFT, where 21lng is associated with the only open edge of the M\"obius strip and 21lnk is associated with the nonorientable topology. As a useful application, we employ the universal entropy to accurately pinpoint the quantum phase transitions, even for those without local order parameters.
Tensor networks are a class of methods for studying many-body systems. They give a geometrical description of the internal structure of many-body states, operators, and partition functions, that can be used to implement efficient algorithms to simulate them numerically. In this thesis, after providing a very brief overview of the field, we present new tensor network methods for three different use cases. \n \nFirst, we present a new real-space renormalization group algorithm for tensor networks. Compared to existing methods, its advantages are its low computational cost, simplicity of implementation, and applicability to any network. We benchmark it on the 2D classical Ising model and find accuracy comparable with the best existing tensor network methods. Due to its simplicity and generalizability, we consider our algorithm to be an excellent candidate for implementation of real-space renormalization in higher dimensions, and discuss some of the details and the remaining challenges in 3D. \n \nSecond, we show how certain topological conformal defects of critical lattice theories can be represented as tensor networks, using the 2D classical Ising model as an example. Furthermore, we show how such tensor network descriptions, combined with a renormalization group algorithm, can be used to obtain accurate estimates of the universal properties of these defects. We also show how coarse-graining of defects can be applied to any conformal defect (i.e. not just topological ones), and yields a set of associated scaling dimensions. \n \nFinally, we leave behind the focus on renormalization group methods, and present a method for spatially resolving the overlap between two tensor network states, to obtain localized information about the similarities and differences between them. For a given region, the similarity of two states in this region can be quantified by the Uhlmann fidelity of their reduced density matrices, and we show how such fidelities can be efficiently computed in many cases when the two states are represented as tensor networks. We demonstrate the usefulness of evaluating such subsystem fidelities with three example applications: studying local quenches, comparing critical and non-critical states, and quantifying convergence in tensor network simulations.
This dissertation contributes to the ongoing effort of understanding the origins and applications of real-space renormalization group methods in tensor network representations of classical and quantum many-body systems. First, we construct a matrix product operator ansatz to coarse-grain real-space transfer matrices of matrix product state descriptions of one-dimensional quantum spin chains. By treating the physical spin as an impurity, we unravel the virtual entanglement degrees of freedom of matrix product states into a layered structure to reveal an inherent renormalization group scale. Secondly, we rephrase tensor network renormalization for two-dimensional classical lattice models in a manifestly nonnegative way. The resulting real-space renormalization group flow preserves positivity and hence yields an interpretation in terms of Hamiltonian flows, reconciling modern real-space tensor network renormalization methods with traditional block-spin approaches. Thirdly, we study non-local symmetries in tensor networks by expressing two-dimensional classical partition functions in terms of strange correlators of judiciously chosen product states and string-net wave functions. We exhibit and exploit the emerging non-local symmetries of the partition function at criticality and highlight parallels between topological sectors and conformal primary fields in the shared framework of matrix product operator symmetries. Additionally, we provide a complementary perspective on real-space renormalization by recognizing known tensor network renormalization methods as the approximate truncation of an exactly coarse-grained strange correlator.
We propose a real-space renormalization group algorithm for accurately coarse-graining two-dimensional tensor networks. The central innovation of our method lies in utilizing variational boundary tensors as a globally optimized environment for the entire system. Based on this optimized environment, we construct renormalization projectors that significantly enhance accuracy. By leveraging the canonical form of tensors, our algorithm maintains the same computational complexity as the original tensor renormalization group (TRG) method, yet achieves higher accuracy than existing approaches that do not incorporate entanglement filtering. Our work offers a practical pathway for extending TRG methods to higher dimensions while keeping computational costs manageable.
The simulation and numerical study of large, strongly correlated quantum systems containing Fermions or using real-time evolution in finite dimensions is still an essentially unsolved problem, primarily due to the exponential growth of the Hilbert state space with system size and the occurrence of the so-called sign problem in Monte Carlo studies. In this area, the use of tensor-network methods, for one-dimensional systems chief among them the density matrix renormalisation group (DMRG) and matrix-product states (MPS), has grown in importance in recent years. This thesis first recapitulates the use of non-abelian symmetries such as SU(2)-Spin in arbitrary tensor networks with an extensive review of the published literature including detailed algorithms and implementation hints. Implementing such symmetries can lead to a considerably more efficient representation of states in the tensor network. This part is intended to be suitable as an implementation-oriented introduction to tensor networks in general and the implementation of non-abelian symmetries in particular. Second, it introduces a series of technical improvements for the MPS methods. These improvements include a faster convergence scheme for MPS-DMRG, a systematic approach to the construction of matrix-product operators and an improved Krylov time evolution method as well as the combination of several well-known techniques into a single tensor network toolkit, SYTEN. The effectiveness of these improvements is demonstrated in numerical examples. Third, the toolkit is applied to the study of two models of current research interest: A one-dimensional spin chain in a staggered external magnetic field is studied and confinement of the elementary spinon excitations, as predicted by analytical arguments, found numerically using real-time evolution and evaluation of the dynamical structure factor. Additionally, the Hubbard model in two dimensions is studied extensively at various system sizes, geometries, interaction strengths U and filling factors n using up to 30'000 SU(2)-Spin-symmetric states equivalent to approx. 100'000 states in other MPS-DMRG implementations. Hints of a possible phase coexistence in the region 0.85 < n < 0.95 are found at intermediate interaction strengths U = 4 and U = 6 as well as a consistently striped ground state in the region n ≈ 0.875.