Constructing realistic effective spin Hamiltonians with machine learning approaches

Abstract

The effective Hamiltonian method has recently received considerable attention due to its power to deal with finite-temperature problems and large-scale systems. In this work, we put forward a machine learning (ML) approach to generate realistic effective Hamiltonians. In order to find out the important interactions among many possible terms, we propose some new techniques. In particular, we suggest a new criterion to select models with less parameters using a penalty factor instead of the commonly-adopted additional penalty term, and we improve the efficiency of variable selection algorithms by estimating the importance of each possible parameter by its relative uncertainty and the error induced in the parameter reduction. We also employ a testing set and optionally a validation set to help prevent over-fitting problems. To verify the reliability and usefulness of our approach, we take two-dimensional MnO and three-dimensional TbMnO 3 as examples. In the case of TbMnO 3 , our approach not only reproduces the known results that the Heisenberg, biquadratic, and ring exchange interactions are the major spin interactions, but also finds out that the next most important spin interactions are three-body fourth-order interactions. In both cases, we obtain effective spin Hamiltonians with high fitting accuracy. These tests suggest that our ML approach is powerful for identifying the effective spin Hamiltonians. Our ML approach is general so that it can be adopted to construct other effective Hamiltonians.

Spin Hamiltonians in Magnets: Theories and Computations

“…In Ref. [125], a machine learning method for constructing Hamiltonian (MLMCH) is proposed, which is more efficient and more reliable than the traditional forward selection method. Firstly, a testing set is used to avoid over-fitting problems.…”

“…The flow charts of this method of variable selection as well as the forward selection method are shown in Figure 5. [125], where some details are omitted. A testing set is used to check if over-fitting problems occur.…”

“…In the case of TbMnO 3 , besides the biquadratic terms, the four-spin couplings are also found to be important in explaining the non-Heisenberg behaviors [122]; the three-body fourth-order terms are also found to be important in simulating the total energies of different spin configurations [125] (a list of the fitted values of each important interaction parameter in TbMnO 3 is provided in the supplementary material of Ref. [125]). According to Ref.…”

“…as used in Ref. [125]. When using expansions of spin vectors (or directions of the spins), suitable truncations on interaction distances and interaction orders are needed.…”

“…(b) The forward selection method using a testing set to check if over-fitting problems occur. (c) A simplified flow chart of the algorithm used in MLMCH[125], where some details are omitted. A testing set is used to check if over-fitting problems occur.…”