We investigate the J1-J2 Heisenberg model on the triangular lattice with an additional scalar chirality term and show that a chiral spin liquid is stabilized in a sizeable region of the phase diagram. This topological phase is situated in between a coplanar 120 • Néel ordered and a non-coplanar tetrahedrally ordered phase. Furthermore we discuss the nature of the spin-disordered intermediate phase in the J1-J2 model. We compare the groundstates from Exact Diagonalization with a Dirac spin liquid wavefunction and propose a scenario where this wavefunction describes the quantum critical point between the 120 • magnetically ordered phase and a putative Z2 spin liquid. arXiv:1604.07829v1 [cond-mat.str-el]

We investigate the stability and the nature of the chiral spin liquids which were recently uncovered in extended Heisenberg models on the kagome lattice. Using a Gutzwiller projected wave function approach -i.e. a parton construction -we obtain large overlaps with ground states of these extended Heisenberg models. We further suggest that the appearance of the chiral spin liquid in the time-reversal invariant case is linked to a classical transition line between two magnetically ordered phases.Introduction -The quest for quantum spin liquids [1] is currently a very active endeavour in condensed matter physics. This elusive state of quantum matter comes in various forms and is theoretically intensely studied, however was difficult to pin down in computational studies of realistic quantum spin Hamiltonians and hard to characterise unambigously in experiments on quantum magnets.The S = 1/2 Heisenberg antiferromagnet on the kagome lattice has emerged as one of the paradigmatic systems where quantum spin liquid phases are expected. A plethora of theoretical proposals have been put forward, ranging from valence bond crystals [2-6], algebraic spin liquids [7][8][9][10], Z 2 spin liquids [11][12][13][14][15][16][17], to chiral spin liquids [2,18,19]. Despite tremendous theoretical and computational progress [20][21][22][23][24][25][26][27][28][29][30][31][32], the true nature of the ground state and the low-lying excited states of the nearest neighbour Heisenberg model on the kagome lattice is still not settled completely.Chiral spin liquids (CSL) are a particular family of spin liquids in which time-reversal symmetry (TRS) and parity symmetry are (spontaneously or explicitly) broken [33,34]. The scalar chirality S i · ( S j × S k ) is non-zero and uniform and manifests the breaking of time-reversal and parity symmetries, analogous to the presence of an orbital magnetic field. In a favorable situation the breaking of these symmetries could conceivably lead to a spin analogue of the Fractional Quantum Hall Effect, although other types of ground states are possible as well [35,36]. Historically Kalmeyer and Laughlin envisioned such a scenario by considering lattice versions of the bosonic ν = 1/2 Laughlin wave function as candidate ground state wave functions for the triangular lattice Heisenberg model [37,38].In two recent papers [39,40], two forms of chiral spin liquids have been discovered, which are stabilised away from the nearest neighbour Heisenberg model upon adding further neighbour Heisenberg interactions or scalar chirality terms to the Hamiltonian. Both studies numerically demonstrate the required ground state degeneracy and characterize the underlying topological order by computing the modular matrices.This breakthrough lays the foundation for further investigations of chiral spin liquids. Several pressing, important questions arise: i) are the two chiral spin liquids phases distinct or are they related ? ii) is there a simple physical (latticebased) picture or a variational wave function that describes the ch...

We introduce NetKet, a comprehensive open source framework for the study of many-body quantum systems using machine learning techniques. The framework is built around a general and flexible implementation of neural-network quantum states, which are used as a variational ansatz for quantum wavefunctions. NetKet provides algorithms for several key tasks in quantum many-body physics and quantum technology, namely quantum state tomography, supervised learning from wavefunction data, and ground state searches for a wide range of customizable lattice models. Our aim is to provide a common platform for open research and to stimulate the collaborative development of computational methods at the interface of machine learning and many-body physics. I. MOTIVATION AND SIGNIFICANCERecent years have seen a tremendous activity around the development of physics-oriented numerical techniques based on machine learning (ML) tools [1]. In the context of many-body quantum physics, one of the main goals of these approaches is to tackle complex quantum problems using compact representations of many-body states based on artificial neural networks. These representations, dubbed neural-network quantum states (NQS) [2], can be used for several applications. In the supervised learning setting, they can be used, e.g., to learn existing quantum states for which a non-NQS representation is available [3]. In the unsupervised setting, they can be used to reconstruct complex quantum states from experimental measurements, a task known as quantum state tomography [4]. Finally, in the context of purely variational applications, NQS can be used to find approximate ground-and excited-state solutions of the Schrödinger equation [2, 5-9], as well as to describe unitary [2, 10, 11] and dissipative [12-15] many-body dynamics. Despite the increasing methodological and theoretical interest in NQS and their applications, a set of comprehensive, easyto-use tools for research applications is still lacking. This is particularly pressing as the complexity of NQS-related approaches and algorithms is expected to grow rapidly given these first successes, steepening the learning curve.The goal of NetKet is to provide a set of primitives and flexible tools to ease the development of cuttingedge ML applications for quantum many-body physics. NetKet also wants to help bridge the gap between the latest and technically demanding developments in the field and those scholars and students who approach the subject for the first time. Pedagogical tutorials are provided to this aim. Serving as a common platform for future research, the NetKet project is meant to stimulate the open and easy-to-certify development of new methods and to provide a common set of tools to reproduce published results.A central philosophy of the NetKet framework is to provide tools that are as simple as possible to use for the end user. Given the huge popularity of the Python programming language and of the many accompanying tools gravitating around the Python ecosystem, we have built NetKet as a full...

The thermodynamic properties of the Shastry-Sutherland model have posed one of the longestlasting conundrums in frustrated quantum magnetism. Over a wide range on both sides of the quantum phase transition (QPT) from the dimer-product to the plaquette-based ground state, neither analytical nor any available numerical methods have come close to reproducing the physics of the excited states and thermal response. We solve this problem in the dimer-product phase by introducing two qualitative advances in computational physics. One is the use of thermal pure quantum (TPQ) states to augment dramatically the size of clusters amenable to exact diagonalization. The second is the use of tensor-network methods, in the form of infinite projected entangled pair states (iPEPS), for the calculation of finite-temperature quantities. We demonstrate convergence as a function of system size in TPQ calculations and of bond dimension in our iPEPS results, with complete mutual agreement even extremely close to the QPT. Our methods reveal a remarkably sharp and low-lying feature in the magnetic specific heat around the QPT, whose origin appears to lie in a proliferation of excitations composed of two-triplon bound states. The surprisingly low energy scale and apparently extended spatial nature of these states explain the failure of less refined numerical approaches to capture their physics. Both of our methods will have broad and immediate application in addressing the thermodynamic response of a wide range of highly frustrated magnetic models and materials.