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arXiv:1102.3545 · Physical Review B

Spin-supersolid phase in Heisenberg chains: A characterization via matrix product states with periodic boundary conditions

Davide Rossini, Vittorio Giovannetti, Rosario Fazio

Published Apr 25, 2011• 14 citations
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Abstract

By means of a variational calculation using matrix product states with periodic boundary conditions, we accurately determine the extension of the spin-supersolid phase predicted to exist in the spin-1 anisotropic Heisenberg chain. We compute both the structure factor and the superfluid stiffness and extract the critical exponents of the supersolid-to-solid phase transition.

DOI
10.1103/physrevb.83.140411

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By means of a variational calculation using matrix product states with periodic boundary conditions, we accurately determine the extension of the spin-supersolid phase predicted to exist in the spin-1 anisotropic Heisenberg chain. We compute both the structure factor and the superfluid stiffness and extract the critical exponents of the supersolid-to-solid phase transition.
We study the spin-3/23/23/2 Heisenberg model including easy-plane and exchange anisotropies in one and two dimensions. In the Ising limit, when the off-diagonal exchange interaction JJJ is zero, the phase diagram in magnetic field is characterized by magnetization plateaus that are either translationally invariant or have a two-sublattice order, with phase boundaries that are macroscopically degenerate. Using a site-factorized variational wave function and perturbational expansion around the Ising limit, we find that superfluid and supersolid phases emerge between the plateaus for small finite values of JJJ. The variational approach is complemented by a density matrix renormalization group study of a one-dimensional chain and exact diagonalization calculations on small clusters of a square lattice. The studied model may serve as a minimal model for the layered Ba2{}_{2}2​CoGe2{}_{2}2​O7{}_{7}7​ material compound, and we believe that the vicinity of the uniform 1/3 plateau in the model parameter space can be observed as an anomaly in the measured magnetization curve.
Using a Lanczos method, we study the spectra of Heisenberg chains with spin S=1, 3/2, 2, 5/2, and 3, as a function of the anisotropy parameter \ensuremath{\lambda} and the number of sites of the chain. We use periodic, twisted, and open boundary conditions. We found that for all values of the spin when -1\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}0, these models are massless with critical behavior described by a conformal theory with central charge c=1. We discuss the whole operator content. In particular, we found that the critical exponent ${\mathrm{\ensuremath{\eta}}}_{\mathit{x}}$ follows the behavior ${\mathrm{\ensuremath{\eta}}}_{\mathit{x}}$=(\ensuremath{\pi}-\ensuremath{\gamma})/2\ensuremath{\pi}S and ${\mathrm{\ensuremath{\eta}}}_{\mathit{z}}$=2 where \ensuremath{\gamma}=${\mathrm{cos}}^{\mathrm{\ensuremath{-}}1}$(\ensuremath{\lambda}). For \ensuremath{\lambda}>0, our results indicate that the S=1 model is massive, while for S>1, the massless behavior characterized by the critical exponents described above continues up to \ensuremath{\lambda}=${\ensuremath{\lambda}}^{\mathrm{*}}$(S). For ${\ensuremath{\lambda}}^{\mathrm{*}}$\ensuremath{\le}1, our results are consistent with a massive phase for the integer-spin models and a massless phase with a rapid variation of the critical exponents for the half-integer-spin models.
Using (infinite) density-matrix renormalization-group techniques, ground-state properties of antiferromagnetic S=1S=1S=1 Heisenberg spin chains with exchange and single-site anisotropies in an external field are studied. The phase diagram is known to display a plenitude of interesting phases. We elucidate quantum phase transitions between the supersolid and spin-liquid phases as well as the spin-liquid and ferromagnetic phases. Analyzing spin-correlation functions in the spin-liquid phase, commensurate and (two distinct) incommensurate regions are identified.
Recent studies have suggested a new phase in the extended Bose-Hubbard model in one dimension at integer filling. In this work, we show that this new phase is phase-separated into a supersolid and superfluid part, generated by mechanical instability. Numerical simulations are performed by means of the density matrix renormalization group algorithm in terms of matrix product states. In the phase-separated phase and the adjacent homogeneous superfluid and supersolid phases, we find peculiar spatial patterns in the entanglement spectrum and string-order correlation functions and show that they survive in the thermodynamic limit. In particular, we demonstrate that the elementary excitations of the homogeneous superfluid with enhanced periodic modulations are phonons, find the central charge to be c=1c=1c=1, and show that the velocity of sound, extracted from the intrinsic level splitting for finite systems, matches with the propagation velocity of local excitations in dynamical simulations. This suggests that the low-energy spectrum of the phase under investigation is effectively captured by a spinless Luttinger liquid, for which we find consistent results between the Luttinger parameter obtained from the linear dependence of the structure factor and the algebraic decay of the one-body density matrix.
Using density-matrix renormalization group calculations, ground-state properties of the spin-1 Heisenberg chain with exchange and single-ion anisotropies in an external field are studied. Our findings confirm and refine recent numerical and analytic results by Sengupta and Batista [Phys. Rev. Lett. 99, 217205 (2007)] on the same model. In particular, we present evidence for two types of biconical (or supersolid) and for two types of spin-flop (or superfluid) structures for chains of finite length. Basic features of the quantum phase diagram may be interpreted qualitatively in the framework of classical spin models.
We discuss in detail a modified variational matrix product state algorithm for periodic boundary conditions, based on a recent work by Pippan et al (2010 Phys. Rev. B 81 081103(R)), which enables one to study large systems on a ring (composed of N ∼ 10 2 sites). In particular, we introduce a couple of improvements allowing us to enhance the algorithm in terms of stability and reliability. We employ such a method to compute the stiffness of one-dimensional strongly correlated quantum lattice systems. The accuracy of our calculations is tested in the exactly solvable spin-1/2 Heisenberg chain.