S=1/2 and S=1 one dimensional quantum magnets

Why are we interested in one-dimensional magnetism? Magnetism in one dimension are of highly theoretical interest as well as there are a variety of highly interesting experimental systems. Most two-dimensional magnets order at T=0, however, in one-dimensional situation, fluctuations are so strong that at T=0 most magnets are disordered. Because of the strong fluctuations, conventional techniques are normally not applicable. So we need to search for new tools to solve problems. One of the representative way is to use numerical methods. An example is, by numerically evaluation, one found a unique, disorder ground state of the nearest-neighbor isotropic anti-ferromagnetic Heisenberg chain for spins S=1/2, with anti-ferromagnetic spin-spin correlations, referred as quasi long-range ordered.

Source: Ulrich Schollwock[1]

Haldane's Conjecture

Introduction to Haldane's Conjecture

Haldane conjectured a fundamental difference between spin chains with S an integer and S a half-integer. Haldane Conjecture is that there is a spin gap of the excitation spectrum in integer Heisenberg spin chains whereas half-integer spin chains have a gapless excitation spectrum. Here we use effective field theories of anti-ferromagnets to show Haldane's Conjecture.

Effective Field Theories of Antiferromagnets

Let's consider an isotropic anti-ferromagnetic Heisenberg model with nearest neighbor interaction between spins of S length.The Hamiltonian can be expressed as

${\mathcal {H}}=J\sum _{i=1}^{N}\mathbf {S} _{i}\mathbf {S} _{j}\qquad J>0$

Assume that local $N{\acute {e}}el$ order on short length scales, we can derive an effective long-wavelength low energy Lagrangian for $S\rightarrow \infty$

${\mathcal {L}}={\frac {v}{2g}}\partial _{\mu }{\overset {\rightarrow }{\phi }}\partial ^{\mu }{\overset {\rightarrow }{\phi }}+{\frac {\Theta }{8\pi }}\epsilon ^{\mu \nu }{\overset {\rightarrow }{\phi }}\left(\partial _{\mu }{\overset {\rightarrow }{\phi }}\times \partial _{\mu }{\overset {\rightarrow }{\phi }}\right)$

Where v=2JS is the spin wave velocity, g=2/S is the coupling constant and $\Theta =2\pi S$ is the topological angle; ${\overset {\rightarrow }{\phi }}$ is a three-component unit vector field. Then we can do a Wick rotation to get the form of Lagrangian as below:

${\mathcal {L}}={\mathcal {L}}_{NL}+i{\mathcal {L}}_{TOP}$

Where ${\mathcal {L}}_{NL}$ is the Lagrangian of an O(3) nonlinear $\sigma$ model and ${\mathcal {L}}_{TOP}={\frac {\Theta }{8\pi }}\epsilon ^{\mu \nu }{\overset {\rightarrow }{\phi }}\left(\partial _{\mu }{\overset {\rightarrow }{\phi }}\times \partial _{\mu }{\overset {\rightarrow }{\phi }}\right)$.With differential geometry, we are able to derive that

$Q[{\overset {\rightarrow }{\phi }}]={\frac {1}{\Theta }}\int {d^{2}x}{\mathcal {L}}_{TOP}[{\overset {\rightarrow }{\phi }}]$

Here Q is an integer for any field ${\overset {\rightarrow }{\phi }}$. It's called winding number.

The partition function is

$Z=\int D[{\overset {\rightarrow }{\phi }}]e^{-\int {d^{2}x}{\mathcal {L}}_{NL}}e^{-i\Theta Q[{\overset {\rightarrow }{\phi }}]}\delta \left(|{\overset {\rightarrow }{\phi }}|^{2}\right)$

$e^{-i\Theta Q[{\overset {\rightarrow }{\phi }}]}={\begin{cases}1&S=interger\\\pm 1&S=half-interger\\\end{cases}}$

For integer S, it's reduced to the nonlinear $\sigma$-model. And for half-integer S,it can be considered by the Bethe ansatz for S=1/2.

Re-normalize group, we can get the below $\beta$ function.

$\beta \left(\mu \right)=a{\frac {du}{da}}=-\epsilon u+{\frac {u^{2}}{2\pi }}+O\left(u^{3}\right)$

This function tells us that for any finite S, the flow is to disordered high-temperature limit. Where we regard $u=ga^{2-d}$,lattice constant a and $\epsilon =d-2$.

The correlations are antiferromagnetic. They decay with the Ornsterin-Zernike law for two-dimensional space-time.

$\left\langle \mathbf {S} _{i}\cdot \mathbf {S} _{j}\right\rangle \propto \left(-1\right)^{i-j}{\frac {e^{-|i-j|/\xi }}{\sqrt[{2}]{|i-j|}}}$

For $k\simeq \pi$, the excitation spectrum is massive-relativistic, with a gap to the ground state and can be depicted as

$E\left(k\right)={\sqrt[{2}]{\Delta ^{2}+v^{2}\left(k-\pi \right)^{2}}}$

In the limit of $S\rightarrow \infty$, the finite gap and the correlation length should have the relation with S, demonstrated like

$\Delta \left(S\right)\propto S^{2}e^{-\pi S}$

$\xi \left(S\right)\propto S^{-1}e^{\pi S}$

In classic limit, at T=0, $\lim _{S\to \infty }\Delta =0$,$\lim _{S\to \infty }\xi =0$

Verify Haldane's Conjecture in S=1/2 case: the Bethe solution for S=1/2

From Reference 2, we can get a view of Bethe solution for S=1/2

The Bethe solution for S=1/2 states that for a rather large class of isotropic, translationally and rotationally invariant Hamiltonians, the spectrum is either critical, or the ground state degenerate with broken translational invariance, for example, by dimerisation, and the spectrum gapped. This indicate that the S=1/2 chain is indeed generic for all half-integer spins.

Verify Haldane's Conjecture in S=1 case

For integer spins, there is no exact analytic result. For S=1, many studies have been carried out to determine the gap, converging to $\Delta _{s=1}=0.41050\left(2\right)$ (从对应文献得到计算的结果，把结果陈述出来) obtained independently by DMRG and exact diagonalization. Experimentally, the gap was first confirmed beyond any doubt by Renard et al.[\Delta_{s=1}=0.41050\left( 2 \right)] on the standard S=1 chain “NENP” ($[Ni\left(C_{2}H_{8}N_{2}\right)_{2}][ClO_{4}]$). The experiments showed isotropic interactions with a weak easy-plane anisotropy, which agreed with theoretical results very well.

The Density Matrix Renormalization Group (DMRG) : Numerical Method to Solve 1-D Quantum Magnets Problem

Various techniques have been introduced or greatly refined: the nonlinear�-model, conformal analysis, bosonization, the matrix product ansatz, the Bethe ansatz, exact diagonalization and quantum Monte Carlo are now standard techniques in the field. Among numerical approaches, certainly the most important recent development has been the introduction of the Density Matrix Renormalization Group (DMRG) by White in 1992[Whit 92b].