gauge

Lattice Gauge Model Client Code

Generalized nematics with polyhedral mesogens can serve as a testbed for the detection and realization of multipolar orders. To this end, a lattice gauge model is employed, first introduced in Phys. Rev. X 6, 041025 (2016). On each site of a cubic lattice, a triad of three orthogonal “spins”, {S^l, S^m, Sⁿ} with S^α × S^β = ±ε_αβγ S^γ, fully determines the orientation (and chirality) of a mesogen. α, β, γ = l, m, n are referred to as “color indices”. The classical Hamiltonian is then given by

where the interaction between mesogens on nearest-neighboring sites is mediated through additional fields U_ij ∈ G ⊂ O(3) which live on the bonds. G is the intended point group of the mesogens. These fields are in fact gauge fields, as the Hamiltonian possesses a local point-group symmetry (Einstein summation implied),

in addition to a global O(3) symmetry,

Since it is impossible to spontaneously break a gauge symmetry, the model eventually orders into a state which breaks O(3) down to G, possibly through one or more intermediate phases with symmetry G₀ such that G ⊂ G₀ ⊂ O(3) [see Phys. Rev. E 95, 022704 (2017)]. The ground-state manifold is hence given by O(3)/G. The phase transitions between these phases have been found to be generically of first order, except in special cases where fine-tuning of the interaction coupling can give second order transitions.

For example, when G = O(2), the gauge theory recovers the Heisenberg model with general exchange interaction while for G = D_∞h it reduces to the Lebwohl-Lasher model [Phys. Rev. A 6, 426–429 (1972)],

A mathematical derivation of these limits is provided in Phys. Rev. E 94, 022701 (2016).

For the purpose of machine-learning these order parameters, raw configurations are sampled from the Monte Carlo simulation of the gauge model and fed to TK-SVM. The three “spins” defining the local triad are thus treated as separate spins in a spin cluster. Their Greek color index takes the place of the sublattice index.

The results of applying TK-SVM to the various orders realized in the gauge model, and analyses thereof, are published in:

• Jonas Greitemann, Ke Liu, and Lode Pollet: Probing hidden spin order with interpretable machine learning, Phys. Rev. B 99, 060404(R) (2019), open access via arXiv:1804.08557;
• Ke Liu, Jonas Greitemann, and Lode Pollet: Learning multiple order parameters with interpretable machines, Phys. Rev. B 99, 104410 (2019), open access via arXiv:1810.05538.

Building and Installation

CMake is used to build this code:

$ cd svm-order-params/gauge
$ mkdir build && cd build
$ cmake ..
$ make -jN

Finally, using make install the compiled executables can be copied to the bin directory at the location configured in CMAKE_INSTALL_PREFIX. This step is optional.

Refer to the top-level README for information on dependencies and cloning of submodules.

Examples & Usage

For the → Basic Usage of the TK-SVM framework, refer to the top-level README.

An assortment of example parameter files are provided in the params directory. Each comes with an explanatory comment and detailed instruction on how to invoke the executables.

Runtime parameters

This sections lists the runtime parameters which are defined by — and exclusive to — this client code. These parameters supplement those lists in the section → Runtime parameters of the top-level README.

Simulation runtime

After an initial thermalization phase of thermalization_sweeps MC sweeps, total_sweeps sweeps are carried out and observables are measured after each. Each sweep consists of sweep_unit unit steps. Each unit step in turn picks a number of random sites corresponding to the number of lattice sites. On each site, hits_R Metropolis attempts are made at updating the local spins and hits_U Metropolis attempts are made at updating the local gauge fields.

Additionally, once per MC sweep, with probability global_gauge_prob, all spins are rotated by a random O(3) transformation. Such a microcanonical update does not reduce autocorrelations on physical observables which are themselves O(3) invariant but can help when "learning" such observables using a machine that does not impose that symmetry explicitly. See J. Greitemann's PhD thesis, Sec. 3.3.1, for further details and motivation in the context of TK-SVM.

Parameter name	Default	Description
total_sweeps	0	Number of MC steps per phase diagram point sampled
thermalization_sweeps	10000	Thermalization steps after each phase point change
sweep_unit	10	Number of unit steps per Monte Carlo sweep
hits_R	1	Number of updates to local spins per MC unit step
hits_U	1	Number of updates to gauge fields per MC unit step
global_gauge_prob	0.05	Probability to perform global gauge transformation

Model parameters

Parameter name	Default	Description
length	required	Linear system size
gauge_group	required	One of: Cinfv, Dinfh, D2h, D2d, D3, D3h, T, Td, Th, O, Oh, I, Ih
group_size	required	Number of elements in the gauge group (has to be chosen accordingly)
O3	0	Whether the group is a subgroup of SO(3) (0) or just O(3) (1) (choose accordingly)
temperature	1	Temperature (rescales J1, J3 couplings, keep at 1)
<phase-diag-point-spec>	optional	Initial phase diagram point (not relevant for TKSVM use case)

The parameters gauge_group, group_size, and O3 have to be selected according to the desired gauge symmetry group, as listed in the table below:

Group symbol	gauge_group	group_size	O3
C∞v	Cinfv	1	1
D∞h	Dinfh	2	1
D2h	D2h	8	1
D2d	D2d	8	1
D3	D3	6	0
D3h	D3h	12	1
T	T	12	0
Td	Td	24	1
Th	Th	24	1
O	O	24	0
Oh	Oh	48	1
I	I	60	0
Ih	Ih	120	1

Tensorial kernel

Parameter name	Default	Description
rank	required	Rank of the monomial mapping
symmetrized	1	Eliminate redundant (symmetric) monomials (1) or not (0)
color	triad	Consider single spin per site (mono) or all three (triad)
cluster	single	Use single spin cluster, bipartite lattice (two-spin cluster), or full spin configurations

Phase diagram point specification

The parameter space of the gauge model is spanned by the temperature as well as the two couplings J₁ = J₂ and J₃ which parametrize the diagonal coupling matrix,

The (inverse) temperature may be absorbed into the couplings, resulting in a two-dimensional parameter space. Points in the phase diagram are described by the type J1J3. The following parameters thus replace mentions of <phase-diag-point-spec> in the specification of the other parameters:

J1J3 phase diagram point	Default	Description
J1	0	β J1 coupling
J3	0	β J3 coupling