In many physical scenarios, close relations between the bulk properties of quantum systems and theories associated to their boundaries have been observed. In this work, we provide an exact duality mapping between the bulk of a quantum spin system and its boundary using Projected Entangled Pair States (PEPS). This duality associates to every region a Hamiltonian on its boundary, in such a way that the entanglement spectrum of the bulk corresponds to the excitation spectrum of the boundary Hamiltonian. We study various specific models, like a deformed AKLT 1 , an Ising-type 2 , and Kitaev's toric code 3 , both in finite ladders and infinite square lattices. In the latter case, some of those models display quantum phase transitions. We find that a gapped bulk phase with local order corresponds to a boundary Hamiltonian with local interactions, whereas critical behavior in the bulk is reflected on a diverging interaction length of the boundary Hamiltonian. Furthermore, topologically ordered states yield non-local Hamiltonians. As our duality also associates a boundary operator to any operator in the bulk, it in fact provides a full holographic framework for the study of quantum many-body systems via their boundary.

No abstract

We study the energy and the static spin structure factor of the ground state of the spin-1/2 quantum Heisenberg antiferromagnetic model on the kagome lattice. By the iterative application of a few Lanczos steps on accurate projected fermionic wave functions and the Green's function Monte Carlo technique, we find that a gapless (algebraic) U (1) Dirac spin liquid is competitive with previously proposed gapped (topological) Z2 spin liquids. By performing a finite-size extrapolation of the ground-state energy, we obtain an energy per site E/J = −0.4365(2), which is equal, within three error bars, to the estimates given by the density-matrix renormalization group (DMRG). Our estimate is obtained for a translationally invariant system, and, therefore, does not suffer from boundary effects, like in DMRG. Moreover, on finite toric clusters at the pure variational level, our energies are lower compared to those from DMRG calculations.PACS numbers: 75.10. Jm, 75.10.Kt, 75.40.Mg, 75.50.Ee Introduction. The spin-1/2 quantum Heisenberg antiferromagnet (QHAF) on the kagome lattice provides a conducive environment to stabilize a quantum paramagnetic phase of matter down to zero temperature, [1][2][3] a fact that has been convincingly established theoretically from several studies, including exact diagonalization, 4-8 series expansion, 9,10 quantum Monte Carlo, 11and analytical techniques. 12 The question of the precise nature of the spin-liquid state of the kagome spin-1/2 QHAF has been intensely debated on the theoretical front, albeit without any definitive conclusions. Different approximate numerical techniques have claimed a variety of ground states. On the one hand, densitymatrix renormalization group (DMRG) calculations have been claimed for a fully gapped (nonchiral) Z 2 topological spin-liquid ground state that does not break any point group symmetry.13,14 On the other hand, an algebraic and fully symmetric U (1) Dirac spin liquid has been proposed as the ground state, by using projected fermionic wave functions and the variational Monte Carlo (VMC) approach. [15][16][17][18][19][20] In addition, valence bond crystals have been suggested from many other techniques. In particular, a 36-site unit cell valence-bond crystal [21][22][23] was proposed using quantum dimer models, 24-28 series expansion 29,30 and multiscale entanglement renormalization ansatz (MERA) 31 techniques. Finally, a recent coupled cluster method (CCM) suggested a q = 0 (uniform) state. 32On general theoretical grounds, the Z 2 spin liquids in two spatial dimensions are known to be stable phases, 33-35 as compared to algebraic U (1) spin liquids, which are known to be only marginally stable.36 However, explicit numerical calculations using projected wave functions have shown the U (1) Dirac spin liquid to be stable (locally and globally) with respect to dimerizing into all known valence-bond crystal phases. 15,17,18,20 Furthermore, it was shown that, within this class of Gutzwiller projected wave functions, all the fully symmetric, gapped Z 2 spin ...

The phase diagram of the ferromagnetic Kondo model for manganites is investigated using computational techniques. In clusters of dimensions 1 and 2, Monte Carlo simulations in the limit where the localized spins are classical show a rich low temperature phase diagram with three dominant regions: (i) a ferromagnetic phase, (ii) phase separation between hole-poor antiferromagnetic and hole-rich ferromagnetic domains, and (iii) a phase with incommensurate spin correlations. Possible experimental consequences of the regime of phase separation are discussed. Studies using the Lanczos algorithm and the Density Matrix Renormalization Group method applied to chains with localized spin 1/2 (with and without Coulombic repulsion for the mobile electrons) and spin 3/2 degrees of freedom give results in excellent agreement with those in the spin localized classical limit. The Dynamical Mean Field (D = ∞) approximation was also applied to the same model. At large Hund coupling phase separation and ferromagnetism were identified, again in good agreement with results in low dimensions. In addition, a Monte Carlo study of spin correlations allowed us to estimate the critical temperature for ferromagnetism T F M c in 3 dimensional clusters. It is concluded that T F M c is compatible with current experimental results.Actually, the phase diagram of La 1−x Ca x MnO 3 is very rich with not only ferromagnetic phases, but also regions with charge-ordering and antiferromagnetic correlations at x > 0.5 [6], and a poorly understood "normal" state above the critical temperature for ferromagnetism, T F M c , which has insulating characteristics at x ∼ 0.33. Finding an insulator above T F M c is a surprising result since it would have been more natural to have a standard metallic phase in that regime which could smoothly become a ferromagnetic metal as the temperature is reduced. Some theories for manganites propose that the insulating regime above T F M c is caused by a strong correlation between electronic and phononic degrees of freedom [7]. Other proposals include the presence of Berry phases in the DE model that may lead to electronic localization [8]. On the other hand, the regime of charge ordering has received little theoretical attention and its features remain mostly unexplored. To complicate matters further, recent experiments testing the dynamical response of manganites have reported anomalous results in the ferromagnetic phase using neutron scattering [9], while in photoemission experiments [10] the possible existence of a pseudogap above the critical temperature was reported.

We have performed a numerical investigation of the ground state properties of the frustrated quantum Heisenberg antiferromagnet on the square lattice ("J 1 − J 2 model"), using exact diagonalization of finite clusters with 16, 20, 32, and 36 sites. Using a finite-size scaling analysis we obtain results for a number of physical properties: magnetic order parameters, ground state energy, and magnetic susceptibility (at q = 0). For the unfrustrated case these results agree with series expansions and quantum Monte Carlo calculations to within a percent or better. In order to assess the reliability of our calculations, we also investigate regions of parameter space with well-established magnetic order, in particular the non-frustrated case J 2 < 0. We find that in many cases, in particular for the intermediate region 0.3 < J 2 /J 1 < 0.7, the 16 site cluster shows anomalous finite size effects. Omitting this cluster from the analysis, our principal result is that there is Néel type order for J 2 /J 1 < 0.34 and collinear magnetic order (wavevector Q = (0, π)) for J 2 /J 1 > 0.68. There thus is a region in parameter space without any form of magnetic order. Including the 16 site cluster, or analyzing the independently calculated magnetic susceptibility we arrive at the same conclusion, but with modified values for the range of existence of the nonmagnetic region. We also find numerical values for the spin-wave velocity and the spin stiffness. The spin-wave velocity remains finite at the magnetic-nonmagnetic transition, as expected from the nonlinear sigma model analogy. 75.10.Jm, 75.40.Mg

We investigate the spin-1 2 Heisenberg model on the triangular lattice in the presence of nearestneighbor J1 and next-nearest-neighbor J2 antiferromagnetic couplings. Motivated by recent findings from density-matrix renormalization group (DMRG) claiming the existence of a gapped spin liquid with signatures of spontaneously broken lattice point group symmetry [Zhu and White, Phys. Rev. B 92, 041105 (2015); Hu, Gong, Zhu, and Sheng, Phys. Rev. B 92, 140403 (2015)], we employ the variational Monte Carlo (VMC) approach to analyze the model from an alternative perspective that considers both magnetically ordered and paramagnetic trial states. We find a quantum paramagnet in the regime 0.08 J2/J1 0.16, framed by 120°coplanar (stripe collinear) antiferromagnetic order for smaller (larger) J2/J1. By considering the optimization of spin-liquid wave functions of different gauge group and lattice point group content as derived from Abrikosov mean field theory, we obtain the gapless U (1) Dirac spin liquid as the energetically most preferable state in comparison to all symmetric or nematic gapped Z2 spin liquids so far advocated by DMRG. Moreover, by the application of few Lanczos iterations, we find the energy to be the same as the DMRG result within error-bars. To further resolve the intriguing disagreement between VMC and DMRG, we complement our methodological approach by pseudofermion functional renormalization group (PFFRG) to compare the spin structure factors for the paramagnetic regime calculated by VMC, DMRG, and PFFRG. This model promises to be an ideal test-bed for future numerical refinements in tracking the long-range correlations in frustrated magnets.

11 pages, 17 figuresInternational audienceUsing both exact diagonalizations and diagonalizations in a subset of short-range valence bond singlets, we address the nature of the groundstate of the Heisenberg spin-1/2 antiferromagnet on the square lattice with competing next-nearest and next-next-nearest neighbor antiferromagnetic couplings (J1-J2-J3 model). A detailed comparison of the two approaches reveals a region along the line (J2+J3)/J1=1/2, where the description in terms of nearest-neighbor singlet coverings is excellent, therefore providing evidence for a magnetically disordered region. Furthermore a careful analysis of dimer-dimer correlation functions, dimer structure factors and plaquette-plaquette correlation functions provides striking evidence for the presence of a plaquette valence bond solid order in part of the magnetically disordered region

Bipartite entanglement measures are fantastic tools to investigate quantum phases of correlated electrons. Here, I analyze the entanglement spectrum of gapped two-leg quantum Heisenberg ladders on a periodic ribbon partitioned into two identical periodic chains. Comparison of various entanglement entropies proposed in the literature is given. The entanglement spectrum is shown to closely reflect the low-energy gapless spectrum of each individual edge, for any sign of the exchange coupling constants. This extends the conjecture initially drawn for Fractional Quantum Hall systems to the field of quantum magnetism, stating a direct correspondence between the low-energy entanglement spectrum of a partitioned system and the true spectrum of the virtual edges. A mapping of the reduced density matrix to a thermodynamic density matrix is also proposed via the introduction of an effective temperature. Introduction -The recent application of quantum information concepts to several domains of condensed matter [1] has proven to be extremely successful, giving new type of physical insights on exotic quantum phases. Upon partitioning a many-body quantum system into two parts A and B, quantum entanglement can be characterized by the properties of the groundstate reduced density matrix of either one of the two parts, ρ A or ρ B . For example, entanglement entropies such as the Von Neumann entropy −Tr{ρ A ln ρ A } or the family of Rényi entropies offer an extraordinary tool to identify a onedimensional conformal invariant system [2] and provides e.g. a direct (numerical) calculation of its central charge [3].Furthermore, the entanglement spectrum (ES) defined by the eigenvalues of a fictitious Hamiltonian H, where ρ A is written as exp (−H), has been shown to provide much more complete information on the system. In one dimension, underlying conformal field theory (CFT) leads to universal scalings of the ES (Ref. 4) and topological properties of the groundstate (GS) can be reflected by specific degeneracies [5]. Choosing a partition corresponding to a very non-local realspace cut, the ES has also been used to define non-local order in gapless spin chains [6].Many-particle quantum entanglement is also a powerful tool to characterize topological features of two-dimensional GS (Ref. 7) as e.g. in dimer liquids on a cylinder geometry [8]. Also, bipartite ES have been shown to provide valuable informations on the edge states of fractional quantum Hall states on spherical [9] and torus geometries [11] upon partition into two (identical) subsystems. Interestingly, the ES of the incompressible GS of a generic Landau-level-projected Coulomb Hamiltonian arranges into a low-energy CFT spectrum, a fingerprint of topological order, separated by an 'entanglement gap' from the high energy levels [9,10].Such advanced insightful analysis of the ES has not however been fully exploited in low dimensional quantum magnets. In particular, the conjecture by Haldane of a precise correspondence between the entanglement spectrum and the true spectrum in r...