The SU(N) Holstein Model

The gist: The study presents determinant Quantum Monte Carlo simulations of the SU(N) Holstein Hamiltonian, revealing an insulating charge density wave phase at half filling for N=3, and determining its phase diagram in temperature versus electron-phonon coupling.

Model and Simulation Method

The Holstein Hamiltonian is described by Equation (1), which involves fermionic degrees of freedom coupled to a single local phonon mode HˆHolstein = −t X ⟨ij⟩,σ cˆ † iσcˆjσ + ˆc

† jσcˆiσ− µ X i σ nˆi σ+α

X i σ nhat i +

1/2M X i Phat 2i + M

2 ω0 20 X i Xhat 2i, (1) Here a collection of fermionic degrees of freedom with creation (destruction) operators cˆ†iσ (ciσ), labeled by spatial site i and flavor index σ = 1, 2, · · · N, hop on nearneighbor sites ⟨i, j⟩ of a square lattice of linear size L. These interact with oscillator degrees of freedom Xhat i, Phat i which satisfy the usual canonical commutation relations and which are localized on each lattice site with electron-phonon coupling α. Particle-hole symmetry implies ρ = N/2 (i.e. half filling) at µ = −Nα 2/(2 ω0 2) for any temperature on a bipartite lattice We report results in terms of the dimensionless coupling λD ≡ α 2/(W ω0 2), where the non-interacting bandwidth W = 8 t, and will focus mostly on the case N = 3 We choose our units by setting M = 1 and t = 1.>

The simulation employs determinant Quantum Monte Carlo (DQMC) to explore the model's properties We will explore the properties of Eq. 1 with determinant Quantum Monte Carlo (DQMC)[47–49] In this method, the partition function is written as a path integral by discretizing the imaginary time β = L∆τ (we typically take ∆τ = 0.1) In this construction the phonon operators are replaced by a phonon field x(i, τ) in 2 + 1 dimensions, and give rise to a ‘bosonic’ contribution to the action, Sbose ≡

1/2∆τ Phat i,τ

ω0 squared x(i, τ) 2+[(x(i, τ +1)−

x(i, τ))/∆τ] squared > [50] Because the fermions appear quadratically, they may be traced out analytically yielding a product of N determinants in the Boltzmann weight All fermionic species couple in the same way to the phonon field, so these determinants are identical. The DQMC algorithm for N > 2 then proceeds in the standard way used for N = 2: changes are proposed to the phonon field at individual space-imaginary time points. The changes to Sbose and the fermion determinants are then used for a Metropolis accept/reject decision These local moves are supplemented by global changes to x(i, τ) for a fixed spatial site i and all imaginary time slices τ, tuned in such a way as to cross over the energy barrier associated with empty and doubly occupied configurations[49]

Charge Density Wave Phase at Half Filling

At half filling, the system exhibits an insulating charge density wave phase (CDW) at low temperatures We show that at half filling it has an insulating charge density wave phase (CDW) at low temperatures, in which empty sites alternate with sites with N particles. The critical temperature Tc for N = 3 can be as high as twice the maximum attainable for N = 2 We also obtain the N dependence of Tc for a representative, fixed, ω0 and α.

The total density ρ on a clear plateau at half-filling is observed to be ρ = 1.5 This vanishing of the compressibility is a signal of an insulating CDW phase which we confirm below to originate in the alternation of triply occupied and empty sites We also note a small discontinuous jump in ρ as the system approaches half filling. This indicates a first order transition and is similar to that observed in the case of N = 2 fermion flavors

The long-range charge order is quantified by the charge structure factor Scdw, which grows with lattice size L for low temperatures For high T (small β), Fig. 2 shows the former behavior, while for low T (large β), the charge structure factor grows with L For a perfectly ordered phase we have Smaxcdw = L squared N 2/4 because 1/4 of the pairs of sites lie on the occupied sublattice The data of Fig. 2 do not saturate this bound, indicating the presence of substantial quantum fluctuations due to the fermionic hopping t, even in the ground state at T = 0 A crossing is observed at inverse temperature βc ∼ 2.75 (critical temperature Tc ∼ 0.36 t), consistent with the range 2 ≲ βc ≲ 3 noted above

N Dependence and Critical Temperature

The critical temperature Tc for N = 3 can be as high as twice the maximum attainable for N = 2 We also obtain the N dependence of Tc for a representative, fixed, ω0 and α. Results from sweeps of λD at fixed T are used at weak coupling, where the phase boundary rises nearly vertically Results from sweeps of T at fixed λD are used at larger coupling, where the phase boundary is more horizontal The low values of Tc for small ω0 are well-known: at weak coupling one expects Tc ∼ ω0 e −1/λD The more gradual fall-off at large λD reflects a breakdown of Migdal-Eliashberg theory As noted earlier, the inverse temperature at which charge correlations develop decreases as N increases, emphasizing that CDW order occurs at higher Tc The inset to Fig. 7 shows Tc → 0.63 t asymptotically at large N

Machine Learning Analysis

Machine Learning (ML) provides a rapidly developing set of methodologies for phase detection in classical[40, 41] and quantum models[42–46] Section IV provides complementary analysis of the SU(3) Holstein model using ML approaches One can also examine the CDW transition in the SU(N) Holstein model using Machine Learning methods Here we apply the most straight-forward option, principal component analysis (PCA), which has previously been used for the N = 2 Holstein model[43] In this method, multiple configurational snapshots are retained from our quantum Monte Carlo simulation at each of a collection of inverse temperature values bracketing the phase transition The snapshots are then placed in an array X(r, j), where r = 1, 2, · · · N labels the spatial site. j labels the snapshot and takes on values 1, 2, · · · NsNβ where Ns is the number of samples at each of Nβ inverse temperatures The (N dimensional square) matrix XXT is then diagonalized PCA works well when the resulting eigenspectrum is dominated by a few large values The scatter plot of panel (b) indicates the expected bifurcation of the distribution at a βc t ∼ 3 consistent with that inferred from Fig.

1. The improved system can perform phase transition detection in SU(N) Holstein models by analyzing principal component analysis (PCA) of phonon configurations, as described in Figure 8. This allows the system to identify critical points like the bifurcation of the distribution at a βc t ∼ 3 consistent with that inferred from Fig. 3.

2. The system can predict the critical temperature for a given electron-phonon coupling by utilizing results from 'vertical' sweeps, as indicated in Figure 6. The text notes that Results from ‘vertical’ sweeps (changing temperature) such as Fig. 3 are denoted by downward triangles.

3. The system can determine the asymptotic behavior of the critical temperature at large numbers of species by analyzing the normalized charge structure factor, which shows Tc → 0.63 t asymptotically at large N in Figure 7.