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short review & summary
  1. In spin chain model the luttinger liquid theory shows spin correlation $C \sim 1/r$ with odd-even oscillations.
  2. In gapped excitations model shows exponentially decaying of correlation.
  3. Critical scaling behavior of correlation $C \sim r^{-(d-2 + \eta)}$.
  4. Several special transition( Neel order to VBS ) shows bahavior of power-law decaying $\to$ exponentially decaying.

Spin correlation function

As order parameter.

1D Heisenberg chain.

The spin correlation function has power-law decay:

$$ C(r) = \langle \vec{S}_i \cdot \vec{S}_{i+r} \rangle \sim (-1)^{r} \frac{1}{r} $$

for large $r$ , where the sign means odd and even oscillations: $r_{\text{even}} = 2,4,6,…, r_{\text{odd}}= 1,3,5,…$ . And this oscillation should be symmetric about zero.

Since choose plot $(-1)^r \cdot r \cdot C(r)$, it should be converge to a constant $\to 0.25$ when $r \to \infty$. （$0.25 = 1/4$ is contributed by spin $S = 1/2$）. The graph[ See ref.2 absolute value of the spin correlation on Page 100 Fig 34 ] shows the longest-distance correlation. Odd-even grid points still oscillate.[ See ref.2 mentioned about this oscillations on Page 100, reference is also provided: J. Voit, Rep. Prog. Phys. 58 977 (1995) which is about Luttinger liquid ].

If the system has gapped excitations, its ground state correlation is exponentially decaying. [ See ref.2 mentioned: Haldane Chain on Page 14, even number coupled chains on Page 15 ]

Néel(order) - VBS(order) transition[ See ref.2: Neel-VBS transition of J-Q model on Page 22 ]

For the longest distance, while $r_{max} = L/2$ on the periodic $L\times L$ lattice, in the VBS phase($J/Q \to 0$) it should be

$$ C(r=L/2) \to 0, L\to \infty $$

In Neel phase correlation is finite(because of long-range order). Another correlation is mentioned: dimer correlation which should vanish in Neel phase when $L \to \infty$.

This is similar to Dimerization transition.(But different model)

Dimerization Transition. [ See ref.2: next-near-neighbor interation Heibenberg model on Page 101-104 ]

Power-law decaying $C(r)\sim 1/r, 0<g<g_c$ in Heisenberg chain phase $\to$ exponentially decaying with a triplet excitation gap in VBS phase.

The graph[ See ref.2 Fig.37 on Page 104 ] shows directly how Spin correlations exponentially decay in VBS phase and how Dimer correlations $\sim 1/r$ decay in Néel phase. The black line(circles) correspond to the VBS phase.

This is similar to the KT transition.

Kosterliz-Thouless transition[ See ref.2: KT transition of 2D XY model on Page 51 ]

This is a thermodynamic phase transition (order to disorder). Although it has been proven that the ordered phase cannot be achieved as long as the temperature is limited[ References: J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973). J. M. Kosterlitz, J. Phys. C 7, 1046 (1974). provided by ref.2 ], the correlation function shows power-law decaying $C(r)\sim r^{-\eta}, 0<T<T_{KT}$ $\to$ exponentially decaying.

Finite-size scaling and critical exponents. (normal cases)

Be careful about correlation length $\xi$ and correlation function $C$. At the critical point, the behavior of $\xi$ is power law diverge: $\xi \sim |t|^{-\nu}$ where $t$ is reduced temperature or any other parameter about criticality.

The correlation at the critical point is:

$$ C(r) \sim r^{-(d-2 + \eta)} $$

where $d$ is the space dimension in Classic system and $d \to d+1$ in Quantum system.[ See ref.2 mentioned the $d$-dimension lattice with time dimension mapped to $d+1$ spatial dimension, the only thing needed to do for quantum phase transition is directly replacing the $d$ $\to d+1$ of the scaling, on Page 54 ], which means, $C \sim r^{-(1+\eta)}$ in a usually 2D quantum critical point.

Imaginary time correlation

[Hui Shao, A. W. Sandvik. arxiv.org/pdf/2202.09870]

For the real time correlation:

$$ G(t) = \langle S(t)S(0)\rangle $$

Notice, $t = 0$ is right means reacting first. While using Heisenberg picture $O(t) = e^{iHt} O(t=0) e^{-iHt}$, it becomes,

$$ G(t) = \langle e^{iHt} S e^{-iHt}S\rangle $$

to imaginary time $it \to \tau$,

$$ G(\tau = it) = \langle e^{H\tau} S e^{-H\tau}S\rangle $$

be careful, $+\tau$ is on the left and $-\tau$ is on the right.

The dynamic spin structure factor $S(q,\omega)$ can be measured, e.g., by magnetic inelastic neutron scattering as the cross section for momentum $(q)$ and energy $(\omega)$ transfer. So according to,

$$ G(\tau) = \int d\omega S(\omega)e^{-\omega \tau } $$

we can obtain $S(\omega) \to G(\tau)$. QMC also can and only can measure $G(\tau)$. But more valuable is $G(\tau) \to S(\omega)$ which needs Stochastic analytic continuation(SAC).

The imaginary time correlation itself does not seem to have much information and behavior that needs to be discussed.

Correlation & entanglement-entropy

topology

The difference between entanglement entropy(EE) and two-point correlation function is: topological EE.

Although entanglement entropy and correlation properties are similar, and there is also divergence behavior at the phase transition point, using the entropy quantity to measure the entanglement between sub-regions has an advantage compared to the traditional two-point correlation function

  • For example, in spin liquid states, the two-point correlation function decays at large scales, but such states can exhibit topological order, quantified by topological EE.

reference:

[1] n-lab. Correlator. Available at: https://ncatlab.org/nlab/show/correlator

[2] Sandvik. Computational Studies of Quantum Spin Systems.