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Gaussian Critical Line in Anisotropic Mixed Quantum Spin Chains

Rainer Bischof

Published Sep 7, 2012• 0 citations

Abstract

By numerical methods, two models of anisotropic mixed quantum spin chains, consisting of spins of two different sizes, Sa = 1/2 and Sb = 1 as well as Sb = 3/2, are studied with respect to their critical properties at quantum phase transitions in a selected region of parameter space. The quantum spin chains are made up of basecells of four spins, according to the structure Sa − Sa − Sb − Sb. They are described by the XXZ Hamiltonian, that extends the quantum Heisenberg model by a variable anisotropic exchange interaction. As additional control parameter, an alternating exchange constant between nearest-neighbour spins is introduced. Insight gained by complementary application of exact diagonalization and quantum Monte Carlo simulations, as well as appropriate methods of analysis, is embedded in the broad existing knowledge on homogeneous quantum spin chains. In anisotropic homogeneous quantum spin chains, there exist phase boundaries with continuously varying critical exponents, the Gaussian critical lines, along which, in addition to standard scaling relations, further extended scaling relations hold. Reweighting methods, also applied to improved quantum Monte Carlo estimators, and finite-size scaling analysis of simulation data deliver a wealth of numerical results confirming the existence of a Gaussian critical line also in the mixed spin models considered. Extrapolation of exact data offers, apart from confirmation of simulation data, furthermore, insight into the conformal operator content of the model with Sb = 1.

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By numerical methods, two models of anisotropic mixed quantum spin chains, consisting of spins of two different sizes, Sa = 1/2 and Sb = 1 as well as Sb = 3/2, are studied with respect to their critical properties at quantum phase transitions in a selected region of parameter space. The quantum spin chains are made up of basecells of four spins, according to the structure Sa − Sa − Sb − Sb. They are described by the XXZ Hamiltonian, that extends the quantum Heisenberg model by a variable anisotropic exchange interaction. As additional control parameter, an alternating exchange constant between nearest-neighbour spins is introduced. Insight gained by complementary application of exact diagonalization and quantum Monte Carlo simulations, as well as appropriate methods of analysis, is embedded in the broad existing knowledge on homogeneous quantum spin chains. In anisotropic homogeneous quantum spin chains, there exist phase boundaries with continuously varying critical exponents, the Gaussian critical lines, along which, in addition to standard scaling relations, further extended scaling relations hold. Reweighting methods, also applied to improved quantum Monte Carlo estimators, and finite-size scaling analysis of simulation data deliver a wealth of numerical results confirming the existence of a Gaussian critical line also in the mixed spin models considered. Extrapolation of exact data offers, apart from confirmation of simulation data, furthermore, insight into the conformal operator content of the model with Sb = 1.

The critical behaviour of anisotropic Heisenberg models with two kinds of antiferromagnetically exchange-coupled centers are studied numerically by using finite-size calculations and conformal invariance. These models exhibit the interesting property of ferrimagnetism instead of antiferromagnetism. Most of our results are centered in the mixed Heisenberg chain where we have at even (odd) sites a spin-S (S ′) SU(2) operator interacting with a XXZ like interaction (anisotropy ∆). Our results indicate universal properties for all these chains. The whole phase, 1> ∆> −1, where the models change from ferromagnetic ( ∆ = 1) to ferrimagnetic ( ∆ = −1) behaviour is critical. Along this phase the critical fluctuations are ruled by a c = 1 conformal field theory of Gaussian type. The conformal dimensions and critical exponents, along this phase, are calculated by studying these models with several boundary conditions. 1

The critical and hysteresis behaviors of the ferromagnetic mixed-spin system consisting of two sublattices A and B with two types of spins SA=32 and SB=12 described by anisotropic Classical-Heisenberg Model on a square lattice are examined by using Monte Carlo simulations based on metropolis algorithm. By exploiting the thermal variations of the order-parameters and the magnetic susceptibility, the phase diagrams are calculated on the (Dz/J, kBT/J), (Dx/J, kBT/J) and (Δ, kBT/J) planes and the effects of various interaction parameters in the Hamiltonian are explored. It is found that the model yields only second-order phase transitions. We have also estimated the critical exponents ν, β and γ of the model as functions of Dz and Dx by using the finite-size scaling arguments and found that they depend on the details of the model. Finally, the hysteresis behavior of the model is also investigated and it is revealed that the width of hysteresis loop decreases with temperature while it increases for positive Dz values.

The critical behaviour of anisotropic Heisenberg models with two kinds of antiferromagnetically exchange-coupled centres are studied numerically by using finite-size calculations and conformal invariance. These models exhibit the interesting property of ferrimagnetism instead of antiferromagnetism. Most of our results are centred in the mixed Heisenberg chain where we have at even (odd) sites a spin- SU(2) operator interacting with a XXZ-like interaction (anisotropy ). Our results indicate universal properties for all these chains. The whole phase, , where the models change from ferromagnetic to ferrimagnetic behaviour is critical. Along this phase the critical fluctuations are ruled by a c = 1 conformal field theory of Gaussian type. The conformal dimensions and critical exponents, along this phase, are calculated by studying these models with several boundary conditions.

The authors are concerned with the critical properties of the anisotropic Heisenberg chain, or XXZ model, with arbitrary integer or half-integer spin. The eigenspectra of these Hamiltonians, with periodic boundaries, are calculated for finite chains by solving numerically their associated Bethe ansatz equations. The resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified, for lattices with an even and odd number of sites. The results for spin 1 and spin 3/2 indicate that the conformal anomaly c for these models, in the gapless regime, has the value c=3S/(1+S), independent of the anisotropy, and the exponents vary continuously with the anisotropy as in the eight-vertex model. The operator content of these models indicate that the underlying field theory governing these critical spin-S models is described by composite fields formed by the product of Gaussian and Z(N) fields, with N=2S. Finally some of the irrelevant operators which produce the leading finite-size corrections of the eigenenergies are identified.

Scaling aspects of Gaussian topological phase transitions in quantum spin chains are investigated using the prototypical one-dimensional spin-1 XXZ Heisenberg model with uniaxial single-ion anisotropy

D

. This model presents a critical line separating the gapped Haldane and large-

D

phases, with the relevant energy gap closing at the transition point. We show that a proper tangential finite-size scaling analysis is able to accurately locate the Gaussian critical line and to probe the continuously varying set of correlation length critical exponents. The specific features of the tangential scaling are highlighted, in contrast to the standard scaling holding in the Ising-like transition between the gapless antiferromagnetic N\'eel and gapped Haldane phases. Our results are compared with field-theoretic predictions and available high-accuracy data for specific points along the Gaussian line.

We investigate the phase diagram of a quantum mixed-spin

(1/2,1/2,1,1)

Heisenberg tetramer chain featuring bond-alternating antiferromagnetic nearest-neighbor exchange interactions and single-ion anisotropy. This model exhibits a rich phase structure comprising three gapped phases: the antiferromagnetic, topological valence-bond-solid (TVBS), and the trivial nonmagnetic phases (TnM). We identify a multicritical Wess-Zumino-Witten

SU{(2)}_{1}

point where two Ising critical lines merge, forming a higher-symmetry point that realizes conformal embedding on the lattice. Beyond this multicritical point, a Gaussian critical line emerges, determining the boundary between the TVBS and the TnM phases. Utilizing the tangential finite-size scaling method, we demonstrate the precision with which critical points and correlation length critical exponents can be extracted along this Gaussian line. Our results reveal rich critical phenomena induced by the interplay of bond-alternating exchange and anisotropic single-ion interactions in quantum many-body systems.