Planar parity-preserving tensor networks (ppTNs) admit a fermionic formulation in which the Gaussian, efficiently contractible limit is deformed by local interactions. We investigate how many-body methods can be used to analyze such contractions in a minimal two-parameter ppTN. Its contraction is simultaneously an interacting fermionic partition function, a loop gas, a deformed toric-code norm, and a quartic Ising model, allowing the same phase diagram to be approached with complementary tools. At its center lies a `topological island', characterized beyond the Gaussian limit by a boundary-twist $\mathbb Z_2$ indicator that reduces to Chern-number parity and admits a loop-winding interpretation under duality. Fermionic perturbation theory predicts the interacting phase boundaries, tensor-network numerics establish their critical behavior, and the loop and spin descriptions reveal a self-dual line with a $c=1$ four-state-Potts multicritical fixed point.
Planar parity-preserving tensor networks (ppTNs) admit a fermionic formulation in which the Gaussian, efficiently contractible limit is deformed by local interactions. We investigate how many-body methods can be used to analyze such contractions in a minimal two-parameter ppTN. Its contraction is simultaneously an interacting fermionic partition function, a loop gas, a deformed toric-code norm, and a quartic Ising model, allowing the same phase diagram to be approached with complementary tools. At its center lies a `topological island', characterized beyond the Gaussian limit by a boundary-twist Z2 indicator that reduces to Chern-number parity and admits a loop-winding interpretation under duality. Fermionic perturbation theory predicts the interacting phase boundaries, tensor-network numerics establish their critical behavior, and the loop and spin descriptions reveal a self-dual line with a c=1 four-state-Potts multicritical fixed point.
This work proposes a minimal model extending the duality between classical statistical spin systems and fermionic systems beyond the case of free fermions. A Jordan-Wigner transformation applied to a two-dimensional tensor network maps the partition sum of a classical statistical mechanics model to a Grassmann variable integral, structurally similar to the path integral for interacting fermions in two dimensions. The resulting model is simple, featuring only two parameters: one governing spin-spin interaction (dual to effective hopping strength in the fermionic picture), the other measuring the deviation from the free fermion limit. Nevertheless, it exhibits a rich phase diagram, partially stabilized by elements of topology, and featuring three phases meeting at a multicritical point. Besides the interpretation as a spin and fermionic system, the model is closely related to loop gas and vertex models and can be interpreted as a parity-preserving (non-unitary) circuit. Its minimal construction makes it an ideal reference system for studying non-linearities in tensor networks and deriving results by means of duality.
The ground state of the toric code, that of the two-dimensional class D superconductor, and the partition sum of the two-dimensional Ising model are dual to each other. This duality is remarkable inasmuch as it connects systems commonly associated to different areas of physics -- that of long range entangled topological order, (topological) band insulators, and classical statistical mechanics, respectively. Connecting fermionic and bosonic systems, the duality construction is intrinsically non-local, a complication that has been addressed in a plethora of different approaches, including dimensional reduction to one dimension, conformal field theory methods, and operator algebra. In this work, we propose a unified approach to this duality, whose main protagonist is a tensor network (TN) assuming the role of an intermediate translator. Introducing a fourth node into the net of dualities offers several advantages: the formulation is integrative in that all links of the duality are treated on an equal footing, (unlike in field theoretical approaches) it is formulated with lattice precision, a feature that becomes key in the mapping of correlation functions, and their possible numerical implementation. Finally, the passage from bosons to fermions is formulated entirely within the two-dimensional TN framework where it assumes an intuitive and technically convenient form. We illustrate the predictive potential of the formalism by exploring the fate of phase transitions, point and line defects, topological boundary modes, and other structures under the mapping between system classes. Having condensed matter readerships in mind, we introduce the construction pedagogically in a manner assuming only minimal familiarity with the concept of TNs.
The ground state of the toric code, that of the two-dimensional class D superconductor, and the partition sum of the two-dimensional Ising model are dual to each other. This duality is remarkable in as much as it connects systems commonly associated to different areas of physics—that of long-range entangled topological order, (topological) band insulators, and classical statistical mechanics, respectively. Connecting fermionic and bosonic systems, the duality construction is intrinsically nonlocal, a complication that has been addressed in a plethora of different approaches, including dimensional reduction to one dimension, conformal field theory methods, and operator algebra. In this paper, we propose a unified approach to this duality, whose main protagonist is a tensor network (TN) assuming the role of an intermediate translator. Introducing a fourth node into the net of dualities offers several advantages: the formulation is integrative in that all links of the duality are treated on an equal footing, (unlike in field theoretical approaches) it is formulated with lattice precision, a feature that becomes key in the mapping of correlation functions, and their possible numerical implementation. Finally, the passage from bosons to fermions is formulated entirely within the two-dimensional TN framework where it assumes an intuitive and technically convenient form. We illustrate the predictive potential of the formalism by exploring the fate of phase transitions, point and line defects, topological boundary modes, and other structures under the mapping between system classes. Having condensed-matter readerships in mind, we introduce the construction pedagogically in a manner assuming only minimal familiarity with the concept of TNs. Published by the American Physical Society 2024
The combination of quantum effects and interactions in quantum many-body systems can result in exotic phases with fundamentally entangled ground state wavefunctions -- topological phases. Topological phases come in two types, both of which will be studied in this thesis. In topologically ordered phases, the pattern of entanglement in the ground state wavefunction encodes the statistics of exotic emergent excitations, a universal indicator of a phase that is robust to all types of perturbations. In symmetry protected topological phases, the entanglement instead encodes a universal response of the system to symmetry defects, an indicator that is robust only to perturbations respecting the protecting symmetry.Finding and creating these phases in physical systems is a motivating challenge that tests all aspects - analytical, numerical, and experimental - of our understanding of the quantum many-body problem. Nearly three decades ago, the creation of simple ansatz wavefunctions - such as the Laughlin fractional quantum hall state, the AKLT state, and the resonating valence bond state - spurred analytical understanding of both the role of entanglement in topological physics and physical mechanisms by which it can arise. However, quantitative understanding of the relevant phase diagrams is still challenging. For this purpose, tensor networks provide a toolbox for systematically improving wavefunction ansatz while still capturing the relevant entanglement properties.In this thesis, we use the tools of entanglement and tensor networks to analyze ansatz states for several proposed new phases. In the first part, we study a featureless phase of bosons on the honeycomb lattice and argue that this phase can be topologically protected under any one of several distinct subsets of the crystalline lattice symmetries. We discuss methods of detecting such phases with entanglement and without.In the second part, we consider the problem of constructing fixed-point wavefunctions for intrinsically fermionic topological phases, i.e. topological phases contructed out of fermions with a nontrivial response to fermion parity defects. A zero correlation length wavefunction and a commuting projector Hamiltonian that realizes this wavefunction as its ground state are constructed. Using an appropriate generalization of the minimally entangled states method for extraction of topological order from the ground states on a torus to the intrinsically fermionic case, we fully characterize the corresponding topological order as Ising x (p - i p). We argue that this phase can be captured using fermionic tensor networks, expanding the applicability of tensor network methods.
Understanding many-body quantum systems is one of the most challenging problems in contemporary condensed-matter physics. Tensor network representation of quantum states and operators are taking central stage in this pursuit and beyond. They prove to be a powerful numerical and conceptual tool, and indeed a new language altogether. This thesis investigates various aspects of these representations by focusing on two specific problems: the first half of the thesis is devoted to examining how 'stable' a tensor network representation is for two-dimensional quantum states with topological order, and the second half explores the representability of various unitary loop operators with tensor networks. In the numerical usage of the tensor networks, the tensor is varied as to find the representation of the ground states of the given Hamiltonian. In chapter two and three of this thesis we show that such a numerical program for topological phases can be 'ill-posed'. We show that tensor network can be an unstable representation for a topological phase: even an infinitesimal variation in the representation results in the loss of topological order, completely or partially. We diagnose this problem by identifying the exact causes of this instability, and find that it is only tensor variations in certain directions that result in instability, because they result in the condensation of bosonic quasi-particles of the phase. Such unstable variations are characterized by two properties: (1) they can replace a tensor in the tensor network without making the network collapse, and (2) their presence in the network represents the presence of a non-trivial topological charge. We prove that the general tensor representation of all string-net models suffer with such instabilities. We propose an exact mathematical operator to project out all such unstable variations and show its efficacy for a few models by direct calculations. Such an operator can be useful in numerical programs involving such tensor representations. We also point out that such variations play a crucial role in simulating topological phase transitions and their presence can be vital in an accurate simulation. In chapter four and five of this thesis we focus on the representability of unitary loop operators by tensor networks. Such operators not only provide an important tool in the study of dynamical process in one-dimensional systems, but also in understanding and classification of symmetry protected topological phases in two dimensions. To characterize all such operators, we find a necessary and sufficient condition for any loop tensor network operator of a given length to represent a unitary operator. In particular, it is shown that all unitary operators that map local operators to local operators (locality-preserving) can always be represented by a tensor network. Locality-preserving unitary loop operators are classified by a rational index called the GNVW index defined in Ref. [1] which measures how much information 'flows' along the loop. We define Rank-Ratio index for tensor network operators and show that it is completely equivalent to the GNVW index. Therefore, GNVW index of a unitary operator can be easily extracted from its tensor network representation. We find that, other than representing locality-preserving unitary maps, tensor networks can also represent unitary operators that map local operators to global (non-local) operators. These tensor network operators are found to have a long-ranged order similar to tensors that represent topological tensor network states in two dimensions.
The focus of our investigations in this thesis is quantum spin systems in two dimensions. We examine phase transitions between topologically distinct phases of quantum matter using the framework of tensor network states. We study the phenomena of anyon condensation and confinement in the context of tensor network states where these two notions offer a robust probe to characterize the universal features of quantum phase transitions. We do a comprehensive study of the numerical methods enabled by the tensor network formalism for the study of quantum phase transitions. We map out the phase diagram of certain exotic phases of quantum matter (namely the D(mathbbZ4) quantum double, the toric code, and the double semion model) and identify the order and the universality classes of the phase transitions between these distinct topological phases. We analyze the phase boundaries of the toric code and the double semion model which exhibit first and second order phase transitions. We investigate the class of mathbbZ2-invariant tensor network states while taking in to account the spin rotation and lattice symmetries. The resulting tensor network states allow us to map the phase diagram of mathbbZ2 spin liquids. The behavior of the system at the phase boundaries of mathbbZ2 spin liquid is governed by the condensation of spinons and visions. We also present our preliminary findings regarding the mathbbZ4-invariant tensors with SU(2) symmetry. This approach enables us to study the spin liquid states of the toric code and the double semion model in a unified framework. We give a local tensor description for the approximate ground states of the Heisenberg antiferromagnet on the kagome lattice by using the variational manifolds of just three and five parameters in the gapped mathbbZ2 spin liquid phase. The approximation of the ground states we construct have an energy density that is remarkably close to the results from the state-of-the-art density matrix renormalization group and the exact diagonalization method. By analyzing the deconfinement of anyonic excitations, we also present our findings regarding the vicinity of the variational ground state in gapped mathbbZ2 spin liquid phase.