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International Journal of Modern Physics Conference Series

CRITICAL EXPONENTS FOR ANTIFERROMAGNETIC SPIN CHAINS OBTAINED FROM BOSONISATION

Marcel Kossow, Peter Schupp, Stefan Kettemann

Published Jan 1, 2012• 2 citations
Publisher

Abstract

The Heisenberg spin 1/2 chain is revisited in the perturbative RG approach with special focus on the transition of the critical exponents. We give a compact review that first order RG in the couplings is sufficient to derive the exact transition from ν = 1 to ν = 2/3, if the boson radius obtained in the bosonization procedure is replaced by the exact radius obtained in the Bethe approach. We explain the fact, that from the bosonization procedure alone, the critical exponent can not be derived correctly in the isotropic limit Jz → J. We further state that this fact is important if we consider to bosonize the antiferromagnetic super spin chain for the quantum Hall effect.

DOI
10.1142/s2010194512006101

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The Heisenberg spin 1/2 chain is revisited in the perturbative RG approach with special focus on the transition of the critical exponents. We give a compact review that first order RG in the couplings is sufficient to derive the exact transition from ν = 1 to ν = 2/3, if the boson radius obtained in the bosonization procedure is replaced by the exact radius obtained in the Bethe approach. We explain the fact, that from the bosonization procedure alone, the critical exponent can not be derived correctly in the isotropic limit Jz → J. We further state that this fact is important if we consider to bosonize the antiferromagnetic super spin chain for the quantum Hall effect.
The thermodynamic Bethe-Ansatz equations for the isotropic ferromagnetic S=12S=\frac{1}{2}S=21​ Heisenberg chain have been solved numerically. At low temperatures, I find a power-law dependence on TTT for the specific heat and the susceptibility with critical exponents $\ensuremath{\alpha}=\ensuremath{-}0.49\ifmmode\pm\else\textpm\fi{}0.02$, $\ensuremath{\gamma}=2.00\ifmmode\pm\else\textpm\fi{}0.02$, and $\ensuremath{\Delta}\ensuremath{\simeq}\ensuremath{\gamma}$. The exponent $\ensuremath{\alpha}$ is compatible with effective spin waves and $\ensuremath{\gamma}$ and $\ensuremath{\Delta}$ are the exponents of the classical chain. Amplitudes and corrections to scaling are obtained and differences with previous results are discussed.
The antiferromagnetic spin-1/2 Heisenberg model, with axial anisotropy and applied magnetic field parallel to the axis, is solved numerically by Lanczos diagonalization of finite systems. We devise an extremely accurate method that allows for a reliable extrapolation to the infinite size limit. In the above limit, our results are compared with exact analytical solutions obtained through the Bethe ansatz. As an important example, we probe the behavior of the Critical Magnetic Field in the whole anisotropic region. We find indications that suggest the presence of two different regimes for the scaling of size effects as a function of the anisotropy.
A recently developed numerical method, entanglement perturbation theory (EPT), is used to study the antiferromagnetic Heisenberg spin chains with z-axis anisotropy λλλ and magnetic field B. To demonstrate the accuracy, we first apply EPT to the isotropic spin-1/2 antiferromagnetic Heisenberg model, and find that EPT successfully reproduces the exact Bethe Ansatz results for the ground state energy, the local magnetization, and the spin correlation functions (Bethe ansatz result is available for the first 7 lattice separations). In particular, EPT confirms for the first time the asymptotic behavior of the spin correlation functions predicted by the conformal field theory, which realizes only for lattice separations larger than 1000. Next, turning on the z-axis anisotropy and the magnetic field, the 2-spin and 4-spin correlation functions are calculated, and the results are compared with those obtained by Bosonization and density matrix renormalization group methods. Finally, for the spin-1 antiferromagnetic Heisenberg model, the ground state phase diagram in λλλ space is determined with help of the Roomany-Wyld RG finite-size-scaling. The results are in good agreement with those obtained by the level-spectroscopy method.
We study the 1D quantum Heisenberg chain with randomly ferromagnetic or antiferromagnetic couplings [a model previously studied by approximate strong-disorder renormalization group (RG)]. We find that, at least for sufficiently large spin S , the ground state has “spin glass” order. The spin waves on top of this state have the dynamical exponent z = 3 / 2 , intermediate between the values z = 1 of the antiferromagnet and z = 2 of the ferromagnet. Density matrix renormalization group (DMRG) simulations are in good agreement with the analytical results for spins S = 1 and S = 3 / 2 . The case S = 1 / 2 shows large finite size effects: We suggest that this case is also ordered, but with a small ordered moment.
We perform a renormalization group analysis of some important effective field theoretic models for deconfined spinons. We show that deconfined spinons are critical for an isotropic SU(N)\mathrm{SU}(N)SU(N) Heisenberg antiferromagnet, if NNN is large enough. We argue that nonperturbatively this result should persist down to N=2N=2N=2 and provide further evidence for the so-called deconfined quantum criticality scenario. Deconfined spinons are also shown to be critical for the case describing a transition between quantum spin nematic and dimerized phases. On the other hand, the deconfined quantum criticality scenario is shown to fail for a class of easy-plane models. For the cases where deconfined quantum criticality occurs, we calculate the critical exponent $\ensuremath{\eta}$ for the decay of the two-spin correlation function to first-order in $ϵ=4\ensuremath{-}d$. We also note the scaling relation $\ensuremath{\eta}=d+2(1\ensuremath{-}\ensuremath{\varphi}∕\ensuremath{\nu})$ connecting the exponent $\ensuremath{\eta}$ for the decay to the correlation length exponent $\ensuremath{\nu}$ and the crossover exponent $\ensuremath{\varphi}$.
The properties of spin-1/2 antiferromagnetic chains with various types of random exchange coupling are studied via an asymptotically exact decimation renormalization-group transformation, which is a generalization of that introduced by Dasgupta and Ma. Random-singlet phases occur in which each spin is paired with one other spin that may be very far away; more exotic phases also occur. The behavior of typical and mean correlation functions is analyzed and found to be very different, with very small sets of spins dominating the latter at long distances as well as the low-temperature thermodynamics. Some of the phase transitions that occur between antiferromagnetically ordered phases and random singlet or other antiferromagnetic phases are also analyzed. For example, if a small uniaxial anisotropy perturbation is added to a random Heisenberg antiferromagnetic chain, a transition occurs from a random-singlet phase to an Ising antiferromagnetic phase, as the anisotropy changes sign from easy plane to easy axis. The staggered magnetization vanishes at the transition with critical exponent \ensuremath{\beta}=8/(1+ \ensuremath{\surd}7 ). Possible implications for the properties of random quantum magnetic systems in higher dimensions are briefly discussed.