Dynamical mean-field theory for dense spin systems at finite temperature

The gist: This work develops and tests a mean-field theory approach for treating dense spin systems at finite temperature, which it terms dynamical mean-field theory for spin systems (spinDMFT) by utilizing a time-dependent mean field to compute dynamical correlations in imaginary time.

Derivation of the method

The method considers an isotropic Heisenberg model for spins with S = 1/2 in a static, homogeneous magnetic field H = X i<j Ji jS⃗ i · S⃗ j + B⃗ · X iS⃗ i. (2) The lattice geometry is described by the symmetric coupling matrix Ji j, and two energy scales are introduced: the linear and quadratic coupling constants JL:= X i̸=0 J0i, JQ:= J2,0 (3). SpinDMFT is argued to be exact when zi, z′i → ∞. This limit is realized in two cases: either the lattice becoming infinite-dimensional, or the interaction between spins being infinitely long-ranged (4). The method determines imaginary-time spin dynamics for systems at finite temperature by working in thermal equilibrium where expectation values are computed as 〈X〉 = 1/Z Tr e −βHX (5). Imaginary-time evolution is carried out by the evolution operator U(τ) = exp(−τH) acting on operators as X(τ) = U−1 (τ)XU(τ) (6).

Mean-field approximation

The environment fields in Eq. (13a), which are part of the quantum many-body problem, are replaced by a classical, time dependent mean field Hmf(τ) = V⃗(τ) + B⃗ · S⃗ 0 (14). This is justified because as z → ∞, the couplings approach zero, making the influence of each neighbor smaller. The expectation values are computed by performing a quantum average in the single-site model and a classical average over the probability space of the mean fields (15a). The self-consistency condition is expressed in Eq. (17b), which relates spin autocorrelations and expectation values on the right-hand side to unknown quantities on the left-hand side, requiring numerical iteration until convergence is achieved (17).

Implementation and benchmarks

The numerical implementation involves discretizing imaginary time τ ∈ [0,β] into equidistant steps (18) and using a Monte Carlo approach to draw mean-field configurations from the probability distribution p[V⃗] (19). The covariance matrix Mab τ1τ2 is defined as Cov V a(τ1), V b(τ2) (20), and the average vector is U a τ = V a(τ) (21). To efficiently draw samples, the covariance matrix is diagonalized using a Fourier transform to Matsubara frequencies Mab(ωn) = 1/β Z β0 dτ e −iωnτMab τ,0 (A.4). The spin autocorrelations g ab(τ) are computed as an average of C ab(τ)[V⃗]/Z0[V⃗] with the temperature-dependent probability distribution pβ[V⃗] = p[V⃗]Z0[V⃗]/Z0 (A.7).

Comparison to finite-size results

SpinDMFT is benchmarked against results in finite-size systems, including a nearest-neighbor ferromagnet, a nearestneighbor antiferromagnet, and an infinite-range system with random couplings (3.3). At very high temperatures, the agreement is very good in all cases because then the signs of the couplings do not play a role and the single-site approximation is well-justified (10). As temperature decreases, deviations start to become visible for antiferromagnetic systems, which are tentatively attributed to incipient antiferromagnetic order and its spin flop transition (10). The method is unable to describe the antiferromagnetic system quantitatively at low temperatures because it does not consider neighboring spins explicitly (12).

Ferromagnetic phase transition in spinDMFT

The study investigates the dependence of the spin expectation values on an external magnetic field and the possibility of order arising in spinDMFT (3.4). For JL < 0, a clear hallmark of a ferromagnetic phase transition is seen, providing evidence that we can capture the ferromagnetic ordering transition for JL < 0 (16). The determined critical temperature βc JQ ≈ 1.85 indicates that the dynamical fluctuations included in spinDMFT have a non-negligible effect on the critical temperature so that its value differs from what is reached by a static local mean-field treatment (17).

Conclusion

SpinDMFT produces quantitatively reliable results at lower temperatures only for the random-coupling system; in other cases it reproduces the qualitative features of the correlations (4). The algorithm does not converge for low temperatures in the case of an antiferromagnet in an external magnetic field, presumably signaling that the antiferromagnetic order requires an extension beyond the treatment of a single site (11). The approach is promising for effective single-site theory, especially in NMR applications like spin diffusion and dynamic nuclear polarization (18).

Additional details of spinDMFT

The cavity method separates the degrees of freedom of spin S⃗ 0 from the ones of the remaining ensemble, and in the limit z → ∞, this expectation value is identical to an analogous expectation value in the full system (A.2). The numerical implementation uses a preconditioned Crank-Nicolson (pCN) algorithm to sample mean fields from pβ[V⃗] instead of p[V⃗], which drastically reduces the variance at low temperatures (21). The Chebyshev expansion technique is used to approximate the evolution operator by expanding it in terms of Chebyshev polynomials of the Hamiltonian, with a truncation error scaling as εCET ∝ O p aτ Nc Nc (B.6). This method is memory efficient when computing evolution with this approach, as only the action of the Hamiltonian is required, not its matrix form (B.5). The real-time dynamics are obtained from imaginary-time correlations by employing the maximum entropy method to perform stochastic analytic continuation to real time (D).

References

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1. No longer be limited to infinite temperature simulations; spinDMFT can now compute imaginary-time correlations and thermodynamical quantities, enabling study of finite-temperature spin dynamics relevant to solid-state NMR applications.

2. Implement phase transition detection by tracking the spin expectation value as the external magnetic field vanishes, allowing for the determination of a critical temperature, which is shown to be sensitive to dynamical fluctuations included in spinDMFT.

3. Achieve quantitatively reliable results for spin glass systems; when comparing to finite-size methods, the correlations agree with spinDMFT very well and the method is able to precisely calculate correlations for a spin glass in the paramagnetic phase.

4. Perform real-time dynamics simulations by employing the maximum entropy method [53] to perform stochastic analytic continuation of imaginary-time correlations to real time, allowing for the study of non-equilibrium effects.

5. Capture magnetic excitation dispersion; by analyzing the real and imaginary parts of spin correlations as a function of temperature, researchers can determine the dispersion of magnetic excitations such as magnons, triplons and spinons as function of temperature.

In recent years, a method for computing spin dynamics at infinite temperature (spinDMFT) was developed. It utilizes the ideas of dynamical mean-field theory for fermions: single-site approximation and a self-consistency condition to approximate time-dependent spin correlations. In this work, we develop a crucial extension of the method to systems at finite temperature, able to compute imaginary-time correlations and thermodynamical quantities. We benchmark the method by comparison to results in finite-size systems, obtaining very good agreement with correlations in a random-coupling system, good agreement for a ferromagnetic system and large discrepancies in the case of an antiferromagnet. We note the appearance of ferromagnetic order in the method. We discuss possible extensions and potential applications of the approach.