We consider a frustrated anti-ferromagnetic triangular lattice Hamiltonian and show that the properties of the manifold of its degenerated ground state are represented by a novel type of tensor networks. These tensor networks are not based on ancillary maximally entangled pairs, but rather on triangular W-like simplices. Anti-ferromagnetic triangular frustration is then related to ancillary W-states in contrast to ferromagnetic order which emerges from the contraction of GHZ-like triangular simplices. We further discuss the outwards entangling power of various simplices. This analysis suggests the emergence of distinct macroscopic types of order from the classification of entanglement residing on the simplices that define a tensor network.
We consider a frustrated anti-ferromagnetic triangular lattice Hamiltonian and show that the properties of the manifold of its degenerated ground state are represented by a novel type of tensor networks. These tensor networks are not based on ancillary maximally entangled pairs, but rather on triangular W-like simplices. Anti-ferromagnetic triangular frustration is then related to ancillary W-states in contrast to ferromagnetic order which emerges from the contraction of GHZ-like triangular simplices. We further discuss the outwards entangling power of various simplices. This analysis suggests the emergence of distinct macroscopic types of order from the classification of entanglement residing on the simplices that define a tensor network.
We have a strong interest in the interplay between geometrical frustrations and quantum fluctuations in the strongly correlated matter because it causes the emergence of new exotic quantum phases. For example, in the past decade, we have found a number of new Mott insulator materials on a triangular layer: Cs2CuCl4 [1] and organic charge transfer salts [2], such as κ-(BEDT-TTF)2 X and β-Z[Pd(dmit)2]2. There are geometrical frustrations and quantum fluctuations in these materials. Probably, the interplay has an important role for the disordered behaviors at very low temperatures in some materials. In order to study the low temperature property of strongly correlated matter in low-dimensional systems, we need a theoretical tool beyond the mean-field theory. The numerical approach is a candidate. Unfortunately, there are serious drawbacks in the conventional numerical methods. For example, quantum Monte Carlo (QMC) methods are powerful tools because they are unbiased. However, the weight of QMC samples can be negative in quantum frustrated magnets, leading to a cancellation in sign, and the accuracy of simulations fatally decreases (this is the so-called negative sign problem.) Exact diagonalization gives us the detail of a ground state, but it can only be applied to small systems. On the other hand, the variational method is flexible, and it is powerful if we choose the appropriate variational wave function. In the past decade, based on the study of entanglement, we discussed the new types of wave functions for quantum many-body systems. We define the probability amplitudes of these wave functions by the tensor contractions of small tensors. Since the tensor contractions can be drawn as a network, it is called a tensor network state. In this paper, we will report the performance of tensor network state for the variational study of quantum frustrated magnets. There are two major tensor network states: projected entangled pair state(PEPS) [3] and multiscale entanglement renormalization ansatz(MERA) [4]. In the following, we will focus on the MERA, and we will report the performance for S = 1/2 spatially anisotropic triangular antiferromagnets [5]. We organize this paper as follows: In Sec. 2, we will briefly introduce the tensor network. In addition, we will explain a MERA tensor network for triangular lattice models in detail. In Sec. 3, we will show the performance of MERA for the S = 1/2 antiferromagnetic Heisenberg model on a
Frustration is a ubiquitous phenomenon in many-body physics that influences the nature of the system in a profound way with exotic emergent behavior. Despite its long research history, the analytical or numerical investigations on frustrated spin models remain a formidable challenge due to their extensive ground-state degeneracy. In this paper, we propose a unified tensor network theory to numerically solve the frustrated classical spin models on various two-dimensional (2D) lattice geometry with high efficiency. We show that the appropriate encoding of emergent degrees of freedom in each local tensor is of crucial importance in the construction of the infinite tensor network representation of the partition function. The frustrations are thus relieved through the effective interactions between emergent local degrees of freedom. Then the partition function is written as a product of a one-dimensional (1D) transfer operator, whose eigenequation can be solved by the standard algorithm of matrix product states rigorously, and various phase transitions can be accurately determined from the singularities of the entanglement entropy of the 1D quantum correspondence. We demonstrated the power of our general theory by numerically solving 2D fully frustrated XY spin models on the kagome, square, and triangular lattices, giving rise to a variety of thermal phase transitions from infinite-order Brezinskii-Kosterlitz-Thouless transitions, second-order transitions, to first-order phase transitions. Our approach holds the potential application to other types of frustrated classical systems like Heisenberg spin antiferromagnets.
Geometric frustration in quantum systems can stabilize unconventional phases of matter that avoid traditional magnetic ordering at low temperatures. Here, we observe this phenomenon while mapping out the finite temperature phase diagram of the spin-1/2 Heisenberg antiferromagnet on the ruby lattice with next-nearest-neighbor interactions. Using an infinite tensor network state (iTNS) optimized and measured with belief propagation (BP) and corrections to BP, we observe the low temperature formation of stable magnetic plateaus at various magnetic field strengths. We find these plateaus host a novel `simplex liquid state' -- a disordered phase involving strongly paired spin simplices that retains non-zero residual entropy due to an exponentially large subspace of crystalline configurations. We accurately quantify the energy gap associated with these states and show that, as the temperature of the system is lowered, it does not go through a phase transition to reach them: the heat capacity remains finite and continuous at all observed temperatures. Our work demonstrates how BP-based tensor network techniques provide a powerful route to understanding frustrated quantum magnets at finite temperature.
In spin systems, geometrical frustration describes the impossibility of minimizing simultaneously all the interactions in a Hamiltonian, often giving rise to macroscopic ground-state degeneracies and emergent low-temperature physics. In this thesis, combining tensor network (TN) methods to Monte Carlo (MC) methods and ground-state energy lower bound approaches, we study two-dimensional frustrated classical Ising models. In particular, we focus on the determination of the residual entropy in the presence of farther-neighbor interactions in kagome lattice Ising antiferromagnets (KIAFM). In general, using MC to determine the residual entropy is a significant challenge requiring ad-hoc updates, a precise evaluation of the energy at all temperatures to allow for thermodynamic integration, and a good control of the finite-size scaling behavior. As an alternative, we turn to TNs; however, we argue that, in the presence of frustration and macroscopic ground-state degeneracy, standard algorithms fail to converge at low temperatures on the usual TN formulation of partition functions. Inspired by methods for constructing ground-state energy lower bounds, we propose a systematic way to find the ground-state local rule using linear programming. Characterizing the rules as tiles that can be tessellated to form ground states of the model gives rise to a natural contractible TN formulation of the partition function. This method provides a direct access to the ground-state properties of frustrated models and, in particular, allows an extremely precise determination of their residual entropy. We then study two models inspired by artificial spin systems on the kagome lattice with out-of-plane (OOP) anisotropy. The first model is motivated by experiments on an array of chirally coupled nanomagnets. We argue that the farther-neighbor to nearest-neighbor couplings ratios in this system are much smaller than in the dipolar case, J2/J1 being of the order of 2%. A comparison of the experimental correlations with the results of extensive TN and MC simulations shows that (1) the experimental second- and third-neighbor correlations are inverted as compared to those of a pure nearest-neighbor model at equilibrium (even with a magnetic field), and (2) second-neighbor couplings as small as 1% of the nearest-neighbor couplings will affect the spin-spin correlations even at fairly high temperatures. Motivated by dipolar coupled artificial spin systems, we turn to the progressive lifting of the ground-state degeneracy of the KIAFM. We provide a detailed study of the ground-state phases of this model with up to third neighbor interactions, for arbitrary J2, J3 such that J1 >> J2, J3, obtaining exact results for the ground-state energies. When all couplings are antiferromagnetic, we exhibit three macroscopically degenerate ground-state phases and establish their residual entropy using our TN approach. Furthermore, in the phase corresponding to the dipolar KIAFM truncated to third neighbors, we use the ground-state tiles to establish the existence of a mapping to the ground-state manifold of the triangular Ising antiferromagnet.
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.