Majorana-Based Pfaffian QMC

• Majorana-based Pfaffian QMC is a numerical approach that uses explicit Pfaffian wavefunctions to simulate non-Abelian paired states in fractional quantum Hall systems.
• The method efficiently computes key observables and correlation lengths, distinguishing gapped phases (Moore–Read and anti-Pfaffian) from gapless PH-Pfaffian behavior.
• Its scalable O(N^3) implementation enables large-scale simulations, guiding theoretical insights into composite fermion pairing and topological phase transitions.

Majorana-Based Pfaffian QMC refers to the application of quantum Monte Carlo (QMC) methods, with explicit use of Pfaffian wavefunctions built on Majorana (i.e., non-Abelian) topological order, to the simulation and understanding of strongly correlated fermionic systems, particularly in the context of fractional quantum Hall (FQH) states at even-denominator filling factors such as ν=5/2. This approach enables large-scale, numerically exact studies of paired composite fermion (CF) states, drawing on the special structure of Pfaffian forms—most notably, those appearing in the Moore–Read, anti-Pfaffian, and particle-hole-symmetric (PH-Pfaffian) states.

1. Pfaffian Trial Wavefunctions: Structure and Physical Context

The essential form of a paired CF Pfaffian state at filling fraction ν=1/M is

$$\Psi(z_1, \ldots, z_N) = P_{\mathrm{LLL}}\left\{\mathrm{Pf}[g(z_i-z_j)] \prod_{i<j} (z_i-z_j)^M\right\},$$

where $z_i$ is the complex coordinate of the $i$-th particle, $g(z)$ is a pairing function, the Jastrow factor $\prod_{i<j} (z_i-z_j)^M$ attaches M vortices to each electron, and $P_{\mathrm{LLL}}$ denotes projection to the lowest Landau level (LLL). The Pfaffian denotes an antisymmetrized product over pairs: $$\mathrm{Pf}[g(z_i-z_j)] = \mathcal{A}\{g(z_1-z_2)g(z_3-z_4)\cdots\}$$.

Moore–Read Pfaffian (MR, $\ell=-1$ pairing): $g_{\mathrm{MR}}(z)=1/z$, corresponding to weak ($p_x-ip_y$) pairing of CFs; $z_i$0 for $z_i$1.

Anti-Pfaffian (aPf, $z_i$2 pairing): $z_i$3 (on the sphere: $z_i$4 with $z_i$5).

PH-Pfaffian (PH, $z_i$6 pairing): $z_i$7 (sphere: $z_i$8).

These paired states are key candidates for the even-denominator FQH plateaus, especially at $z_i$9, where both experiment and theory highlight the role of non-Abelian topological order (Yutushui et al., 2020).

2. Majorana Topological Order and Physical Implications

The non-Abelian FQH phases described by such Pfaffian wavefunctions are fundamentally associated with the presence of Majorana zero modes localized to topological defects (e.g., quasiholes). The Ising (Majorana) anyon sector is characterized by fusion rules

$i$0

where $i$1 is a quasihole operator and $i$2 is a neutral fermion. Braiding two $i$3's yields a non-commuting unitary operation on the degenerate ground state manifold, supporting topological quantum computation and quantum statistics beyond Abelian anyons (Milovanovic et al., 2021).

3. Efficient QMC with Pfaffian Structure

The Pfaffian structure greatly facilitates QMC sampling by allowing the electronic (or composite fermion) part of the wavefunction to be efficiently evaluated for large system sizes. For the Moore–Read and related states, the necessity of lowest-Landau-level projection is typically handled using either direct derivatives ($i$4) or approximate but scalable projection schemes ("single-CF" or "pairwise-CF" projections) (Yutushui et al., 2020). The resulting wavefunctions, while incorporating long-range quantum correlations and non-Abelian statistics, become tractable for large-scale Monte Carlo evaluation, enabling direct computation of pair, density, and correlation functions for $i$5 up to 56 or more.

4. Two-Point Correlation Function and Extraction of Correlation Length

A central observable computed in Pfaffian QMC studies is the composite-fermion two-point correlator

$i$6

which, at long distance, decays exponentially in a gapped phase: $i$7 for $i$8, with $i$9 the magnetic length and $g(z)$0 the correlation length. This is extracted by fitting the MC data for $g(z)$1 on a semi-log plot (Yutushui et al., 2020). In practice, the exponential decay and the magnitude of $g(z)$2 quantify the strength and range of the topological order; absence of exponential falloff suggests gaplessness.

5. Large-Scale Numerical Results and Physical Interpretation at ν=5/2

Simulations reveal that both the MR and anti-Pfaffian trial states yield finite correlation lengths (in units of $g(z)$3): $g(z)$4 and $g(z)$5, confirming they correspond to gapped non-Abelian phases. In stark contrast, the fully projected PH-Pfaffian trial state shows no evidence of a finite correlation length up to $g(z)$6—the two-point correlator decays algebraically (or, if exponential, with a length scale exceeding all accessible sizes), indicating compressible, gapless behavior that is more consistent with a composite Fermi liquid state than with a topological order (Yutushui et al., 2020).

State	Correlation Length $g(z)$7	Phase Character
Moore-Read	1.30(5)	Gapped, non-Abelian
Anti-Pfaffian	1.38(14)	Gapped, non-Abelian
PH-Pfaffian	None resolved ($g(z)$8)	Gapless, compressible

The concluding interpretation is that, for clean systems at $g(z)$9, only the MR or anti-Pfaffian paired phases realize fully gapped non-Abelian topological order. The particle-hole symmetric PH-Pfaffian—though possessing the correct quantized thermal Hall effect in edge theory—does not emerge as a robust gapped ground state in these numerically accessible regimes (Yutushui et al., 2020, Milovanovic et al., 2021).

6. Theoretical Frameworks: Field-Theory, FQHE, and Extensions

Field-theoretical analyses using composite-fermion Chern-Simons theory and Dirac composite fermion approaches further elucidate the role of Landau-level mixing and particle-hole symmetry breaking. These frameworks predict that the MR and anti-Pfaffian are stabilized for weak LL mixing, and the PH-Pfaffian only arises when nonperturbative effects or strong LL mixing are included (Milovanovic et al., 2021).

QMC with Pfaffian-based trial states complements these theoretical predictions, providing quantitative estimates for correlation lengths, validating the stability/instability of proposed topological orders, and clarifying the true ground-state phase diagram of the half-filled Landau level.

7. Broader Implications and Computational Methodology

The Majorana-based Pfaffian QMC approach establishes a rigorous numerical paradigm for evaluating the fate of non-Abelian quantum fluids in regimes that are analytically and experimentally challenging. The method exploits the antisymmetric Pfaffian, which not only encodes the appropriate pairing and statistics but also supports scaling to large $\prod_{i<j} (z_i-z_j)^M$0 due to efficient algorithmic implementations (O($\prod_{i<j} (z_i-z_j)^M$1) scaling for Pfaffian evaluation in QMC).

The same techniques inform the design of parent Hamiltonians in engineered platforms (e.g., circuit QED (Hafezi et al., 2013)), guide the search for new topological phases, and serve as a template for incorporating nontrivial pairing and projection into variational QMC for broader classes of strongly correlated fermion systems.

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References:

• Yutushui & Mross, "Large-scale simulations of particle-hole-symmetric Pfaffian trial wavefunctions" (Yutushui et al., 2020)
• Milovanović et al., "Pfaffian paired states for half-integer fractional quantum Hall effect" (Milovanovic et al., 2021)
• Deng et al., "Engineering three-body interaction and Pfaffian states in circuit QED systems" (Hafezi et al., 2013)