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Physical review. B, Condensed matter

Phase diagrams and correlation exponents for quantum spin chains of arbitrary spin quantum number

Heinz J. Schulz

Published Nov 1, 1986• 537 citations
Publisher

Abstract

The low-temperature properties of the spin-S quantum spin chain are studied representing a spin-S operator as the sum of 2S spin-1/2 operators. The resulting system of 2S-coupled spin-1/2 systems is studied in the weak-coupling limit, using a continuum representation. It is shown that under scaling, the coupling becomes strong. Under the additional hypothesis (which is shown to be true for S=1) that strong coupling represents correctly the properties of the spin-S system, the following results are obtained: (i) there are, in general, two types of planar (massless) phases XY1 and XY2, separated by an Ising-like transition; (ii) for half-odd-integer S the exponent \ensuremath{\eta} governing the asymptotic power-law decay of transverse spin correlations takes the universal value \ensuremath{\eta}=1 at the boundary between XY1 and the adjacent uniaxial antiferromagnetic phase; (iii) for integer S there is an additional singlet phase between XY1 and the antiferromagnetic state, with \ensuremath{\eta}=(1/4) at the limit of XY1; (iv) a spin-Peierls instability only exists for half-odd-integer S; (v) a magnetic field along the z direction may lead to a transition from the singlet or antiferromagnetic state to a planar phase. Universal scaling relations between exponents for transverse and longitudinal correlations in the phases XY1 and XY2 and an explicit asymptotic expression for correlation functions are derived. Finally, implications of the present results for some generalizations of the spin chain problem are briefly discussed. The above points (ii) and (iii) confirm predictions by Haldane, which were derived using quite different methods.

DOI
10.1103/physrevb.34.6372

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The low-temperature properties of the spin-S quantum spin chain are studied representing a spin-S operator as the sum of 2S spin-1/2 operators. The resulting system of 2S-coupled spin-1/2 systems is studied in the weak-coupling limit, using a continuum representation. It is shown that under scaling, the coupling becomes strong. Under the additional hypothesis (which is shown to be true for S=1) that strong coupling represents correctly the properties of the spin-S system, the following results are obtained: (i) there are, in general, two types of planar (massless) phases XY1 and XY2, separated by an Ising-like transition; (ii) for half-odd-integer S the exponent \ensuremath{\eta} governing the asymptotic power-law decay of transverse spin correlations takes the universal value \ensuremath{\eta}=1 at the boundary between XY1 and the adjacent uniaxial antiferromagnetic phase; (iii) for integer S there is an additional singlet phase between XY1 and the antiferromagnetic state, with \ensuremath{\eta}=(1/4) at the limit of XY1; (iv) a spin-Peierls instability only exists for half-odd-integer S; (v) a magnetic field along the z direction may lead to a transition from the singlet or antiferromagnetic state to a planar phase. Universal scaling relations between exponents for transverse and longitudinal correlations in the phases XY1 and XY2 and an explicit asymptotic expression for correlation functions are derived. Finally, implications of the present results for some generalizations of the spin chain problem are briefly discussed. The above points (ii) and (iii) confirm predictions by Haldane, which were derived using quite different methods.
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.
We study excited states and finite temperature properties of chains of randomly placed spins with long range interactions. To this end we extend the recently introduced strong disorder renormalization group method in real space, well suited to study bond disordered antiferromagnetic power law coupled quantum spin chains. For bond disordered, short range coupled spin chains, defined by keeping only interactions between adjacent spins, we show that the distribution of the absolute value of the couplings is the infinite randomness fixed point distribution also for the excited states. Moreover, we derive the distribution function of the sign of the couplings and find that the number of negative couplings increases with temperature T . For power law long range interaction with power exponent α we derive and solve the Master equation for the distribution function of couplings for α > 2 . The coupling amplitude is found to have the strong disorder distribution with finite width 2 α , with small corrections. We derive resulting finite temperature properties of both short and power law long ranged spin systems for α > 2 . The magnetic susceptibility is dominated by paramagnetic S = 1 / 2 -spin, resulting in a Curie law susceptibility. The distribution function of pair lengths, and the spin correlation function are found be the same as in the ground state. We find that the entanglement entropy increases logarithmically with the partition length n and has the same functional form as for the ground state of critical quantum spin chains with temperature dependent effective central charge c ¯ ( T ) , which converges at high temperature to 1 2 ln 2 , half the value it has in the ground state. We confirm that the entanglement entropy grows with time t after a quantum quench logarithmically as S ( t ) ∼ ln ( t ) / ( 2 α ) .
A recent remarkable prediction by Haldane is the 1D integer spin antiferromagnets of XXZ type should show strikingly different T=0 phase behavior from their counterparts with half-integer spin. The consensus of a wide variety of numerical evidence is in support of the Haldane prediction. However, one aspect which has been particularly difficult to confirm has been the behavior in the vicinity of the critical point Δ=Δ2. The point Δ2 is predicted to be a second-order transition in the universality class of the transverse Ising model at its critical field. It has been numerically established that at Δ=Δ2∼1.18–1.20, the Haldane gap disappears and an excited SzT=0 state becomes degenerate with the SzT=0 ground state for Δ≥Δ2. The mapping to the transverse Ising model implies the existence in the limit N→∞ of an infinite continuum of scaling states quasi-degenerate with the ground state(s) at Δ=Δ2. Numerical calculations to determine the presence of these scaling states have been performed up to N=12 spins for the spin-1 XXZ model. The development of this scaling continuum is only apparent for large N, when a class of k=0, SzT=0 high-lying spectral excitations develop a minimum in the vicinity of Δ∼1.18 which intensifies with increasing N. These excitations extrapolate well below the lower edge of the triplet continuum, and we conclude these are the Haldane scaling states. This conclusion is reinforced by a detailed study of the corresponding excitations for the spin-1/2 transverse Ising model. However, we also find a class of k=0, ‖SzT‖=1 excitations which show similar scaling behavior in the vicinity of Δ2. These states were not included in the Haldane prediction. The implications for the behavior of the correlation functions at Δ2 are discussed.
Low-temperature properties of S=1 antiferromagnetic Heisenberg chains with alternating interaction, scrH=Jtsumi{\mathit{Jtsum}}_{\mathit{i}}Jtsumi​[1-(-1)i{)}^{\mathit{i}})i\ensuremath{\delta}]Si{\mathrm{S}}_{\mathit{i}}Si​\ensuremath{\cdot}Si+1{\mathrm{S}}_{\mathit{i}+1}Si+1​, are studied by a quantum Monte Carlo method. Spatial distributions of the spin correlation function 〈S1z{\mathit{S}}_{1}^{\mathit{z}}S1z​Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 and the local magnetic moment 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 are visualized as a function of \ensuremath{\delta}. The continuous phase transition at \ensuremath{\delta}=0.25, which was indicated by several authors, is confirmed by the \ensuremath{\delta} dependence of the low-lying level structure, the divergence of the correlation length, and also the collapse of the edge states. The critical exponent \ensuremath{\eta}, governing the power-law decay of the spin correlation function, is estimated to be 0.99\ifmmode\pm\else\textpm\fi{}0.05. The present result raises a possibility that the variety of quantum spin chains including not only the half-odd-integer-S ones but also the integer-S ones may be described in terms of the generic critical theory.
We study XY and dimerized XX spin-1/2 chains with random exchange couplings by analytical and numerical methods and scaling considerations. We extend previous investigations to dynamical properties, to surface quantities, and operator profiles, and give a detailed analysis of the Griffiths phase. We present a phenomenological scaling theory of average quantities based on the scaling properties of rare regions, in which the distribution of the couplings follows a surviving random-walk character. Using this theory we have obtained the complete set of critical decay exponents of the random XY and XX models, both in the volume and at the surface. The scaling results are confirmed by numerical calculations based on a mapping to free fermions, which then lead to an exact correspondence with directed walks. The numerically calculated critical operator profiles on large finite systems (L< 512)(L<~512)(L< 512) are found to follow conformal predictions with the decay exponents of the phenomenological scaling theory. Dynamical correlations in the critical state are in average logarithmically slow and their distribution shows multiscaling character. In the Griffiths phase, which is an extended part of the off-critical region, average autocorrelations have a power-law form with a nonuniversal decay exponent, which is analytically calculated. We note on extensions of our work to the random antiferromagnetic XXZ chain and to higher dimensions.
We study the long-range behavior of spin correlations in low-dimensional spin-1/2 antiferromagnets with both Ising anisotropy and spatial anisotropy. We represent the spin operators in terms of spin-density operators and spin-phase operators. Then for spin-1/2 antiferromagnets, within a simple harmonic approximation, we obtain expressions for the spin correlations in the ground state. For the one-dimensional linear chain, we find that the two leading terms in spin correlation are (-1)r{)}^{\mathit{r}})rc/${\mathit{r}}^{\mathrm{\ensuremath{\eta}}}$-c1{\mathit{c}}_{1}c1​/r2{\mathit{r}}^{2}r2 for large distance r. The power \ensuremath{\eta} depends on the Ising anisotropy, and \ensuremath{\eta}=0.405 at XY point and \ensuremath{\eta}=1.062 at Heisenberg point. For the two-dimensional square lattice, there is long-range staggered magnetization, and the dependence of magnetization on Ising anisotropy is given. For the crossover from one-dimensional linear chain to two-dimensional square lattice, we find that the disorder-order transition occurs as long as the interchain coupling is introduced. We believe that our method gives a unified approach to the long-range behavior of spin correlations in low-dimensional spin-1/2 antiferromagnets.