pith. sign in

arxiv: 2303.01607 · v1 · pith:V23CBYGKnew · submitted 2023-03-02 · ❄️ cond-mat.str-el

Perturbative solution of fermionic sign problem in lattice Quantum Monte Carlo

Sergei Iskakov , Mikhail I. Katsnelson , Alexander I. Lichtenstein This is my paper

Pith reviewed 2026-05-24 10:03 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords Hubbard modelsign problemQuantum Monte Carloperturbation theorycupratesspectral functionpseudogap
0
0 comments X

The pith

First-order perturbation in chemical potential and t' around half-filling gives accurate spectral functions for doped Hubbard models at optimal cuprate parameters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a strong-coupling perturbation scheme for the generic Hubbard model that begins from a half-filled particle-hole-symmetric reference system free of the fermionic sign problem. First-order corrections in the chemical potential shift and the second-neighbor hopping t' are shown to produce reliable electronic spectral functions in the optimally doped regime at temperatures of order T=0.1t, a region otherwise inaccessible to direct lattice QMC. The approach is applied to the strong-coupling case with U equal to the bandwidth and the optimal t' value for cuprates, where it is used to examine pseudogap formation and the nodal-antinodal dichotomy.

Core claim

A strong-coupling perturbation scheme around a half-filled particle-hole-symmetric reference Hubbard model, implemented with lattice determinantal QMC in continuous or discrete time, yields accurate electronic spectral functions when only the first-order terms in the chemical potential shift and second-neighbor hopping are retained; this holds for the parameter range corresponding to optimally doped cuprates at T approximately 0.1t with U equal to the bandwidth.

What carries the argument

The first-order perturbative expansion in chemical potential and t' around the sign-problem-free half-filled particle-hole-symmetric point, evaluated by determinantal QMC on the reference system.

If this is right

  • Spectral functions can be obtained for doped Hubbard systems at temperatures and dopings relevant to cuprates without encountering the sign problem.
  • The pseudogap and nodal-antinodal dichotomy become accessible for study in the strong-coupling regime with optimal t'.
  • The method extends the reach of lattice QMC calculations to the optimally doped regime at T of order 0.1t.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Higher-order terms in the same expansion could extend the method to stronger doping or lower temperatures.
  • The same reference-system approach might be combined with other observables beyond the spectral function.
  • If the nodal-antinodal features survive in the calculation, they could be tested against ARPES data on cuprates at comparable parameters.

Load-bearing premise

That the first-order expansion in chemical potential and t' around half-filling stays quantitatively accurate for the doped regime when U equals the bandwidth.

What would settle it

A comparison of the first-order perturbative spectral function against results from a higher-order expansion or from an independent non-perturbative method at the same U, doping, t', and T=0.1t values would show whether the approximation holds.

Figures

Figures reproduced from arXiv: 2303.01607 by Alexander I. Lichtenstein, Mikhail I. Katsnelson, Sergei Iskakov.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic representation of a half-filled reference system for the doped square lattice. Bellow: calculated density of states (DOS) in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Spectral function [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Spectral function [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Spectral function [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Green’s function [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Spectral function of the square-lattice Hubbard model as a function of momentum at the first Matsubara frequency [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Feynman diagram for the first order dual fermion perturbation for the self-energy [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
read the original abstract

We develop a strong-coupling perturbation scheme for a generic Hubbard model around a half-filled particle-hole-symmetric reference system, which is free from the fermionic sign problem. The approach is based on the lattice determinantal Quantum Monte Carlo (QMC) method in continuous and discrete time versions for large periodic clusters in a fermionic bath. Considering the first-order perturbation in the shift of the chemical potential and of the second-neighbour hopping gives an accurate electronic spectral function for a parameter range corresponding to the optimally doped cuprate system for temperature of the order of $T=0.1t$, the region hardly accessible for the straightforward lattice QMC calculations. We discuss the formation of the pseudogap and the nodal-antinodal dichotomy for a doped Hubbard system in a strong-coupling regime with the interaction parameter $U$ equal to the bandwidth and the optimal value of the next-nearest-neighbor hopping parameter $t'$ for high-temperature superconducting cuprates.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a strong-coupling perturbation scheme for the generic Hubbard model around a half-filled particle-hole-symmetric reference system free from the fermionic sign problem. It employs lattice determinantal QMC (continuous and discrete time) on large periodic clusters and applies a first-order perturbation in the chemical-potential shift δμ and next-nearest-neighbor hopping t' to compute the electronic spectral function, claiming quantitative accuracy for parameters corresponding to optimally doped cuprates at T≈0.1t with U equal to the bandwidth. The work discusses pseudogap formation and nodal-antinodal dichotomy in the doped strong-coupling regime.

Significance. If validated, the approach would enable QMC-based access to the doped Hubbard regime relevant to cuprate physics without the sign problem, by combining a sign-problem-free reference calculation with controlled perturbation. The method targets a temperature and doping window that is otherwise difficult for direct lattice QMC.

major comments (2)
  1. [Abstract] Abstract and results section: the central claim that first-order perturbation in δμ and t' yields quantitatively accurate spectral functions at U equal to bandwidth, optimal t', and T=0.1t is load-bearing, yet no explicit second-order diagrams, truncation-error bounds, or benchmarks against independent methods (e.g., small-cluster ED or sign-problem-free QMC at nearby parameters) are provided to confirm that higher-order terms remain small.
  2. [Results] Method and results: the perturbation is applied to the Green's function or self-energy obtained from the particle-hole symmetric reference; without a concrete demonstration that the first-order correction remains accurate when U equals the bandwidth (where the expansion parameter is not obviously small), the quantitative accuracy asserted for the optimally doped regime cannot be assessed.
minor comments (2)
  1. [Method] Clarify in the equations how the continuous-time and discrete-time QMC implementations are combined with the perturbative correction.
  2. [Method] Add a brief statement on the cluster sizes used and finite-size effects in the reference QMC calculations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We respond to each major comment below, clarifying the nature of the perturbation expansion and agreeing to enhance the discussion of its accuracy.

read point-by-point responses

  1. Referee: [Abstract] Abstract and results section: the central claim that first-order perturbation in δμ and t' yields quantitatively accurate spectral functions at U equal to bandwidth, optimal t', and T=0.1t is load-bearing, yet no explicit second-order diagrams, truncation-error bounds, or benchmarks against independent methods (e.g., small-cluster ED or sign-problem-free QMC at nearby parameters) are provided to confirm that higher-order terms remain small.

    Authors: The expansion is performed with respect to the small parameters δμ and t', which are of order 0.1-0.3t for the doping levels considered. This is distinct from the interaction strength U, which is treated non-perturbatively in the reference system. While the manuscript does not include explicit second-order calculations or direct benchmarks, the results demonstrate consistency with expected physical phenomena in the doped regime. We will revise the manuscript to include a discussion of the expected magnitude of higher-order terms based on the smallness of δμ and t'. revision: partial

  2. Referee: [Results] Method and results: the perturbation is applied to the Green's function or self-energy obtained from the particle-hole symmetric reference; without a concrete demonstration that the first-order correction remains accurate when U equals the bandwidth (where the expansion parameter is not obviously small), the quantitative accuracy asserted for the optimally doped regime cannot be assessed.

    Authors: We emphasize that the expansion parameter is δμ and t', not U. The reference system at half filling with t'=0 is sign-problem-free for arbitrary U, and the perturbation corrects for finite doping and t'. At the parameters studied, δμ and t' are sufficiently small to justify the first-order approximation, as evidenced by the smooth evolution of the spectral functions. We will add a clarification on this point in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: perturbative expansion is independent of target doped spectral function

full rationale

The derivation performs lattice QMC at the half-filled particle-hole symmetric reference point (sign-problem free by construction) and then applies an explicit first-order perturbative correction in δμ and t'. This is a standard expansion whose output is not equivalent to the input by definition; the accuracy for doped parameters at U equal to bandwidth is an empirical claim about truncation error rather than a self-definitional or fitted-input reduction. No equations rename a fit as a prediction, no load-bearing uniqueness theorem is imported via self-citation, and the central result does not reduce to its own inputs. The approach remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract identifies the particle-hole symmetric half-filled point as sign-problem free but supplies no further free parameters, axioms, or invented entities.

axioms (1)
  • domain assumption The half-filled particle-hole-symmetric Hubbard model is free from the fermionic sign problem.
    Explicitly invoked as the reference system for the perturbation expansion.

pith-pipeline@v0.9.0 · 5699 in / 1093 out tokens · 20569 ms · 2026-05-24T10:03:55.694202+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We develop a strong-coupling perturbation scheme for a generic Hubbard model around a half-filled particle-hole-symmetric reference system, which is free from the fermionic sign problem... first-order perturbation in the shift of the chemical potential and of the second-neighbour hopping

  • IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The first order for the vertex in particle-hole (PH) channel is given by the diagram shown in Fig.7 ˜Σ(1)12 = −∑s−QMC ∑3,4 γd1234(s) ˜G034

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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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