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arxiv: 2606.02013 · v1 · pith:OKCAU4NBnew · submitted 2026-06-01 · ✦ hep-th · cond-mat.str-el· hep-lat

Successes and challenges of using Semidefinite Programming for the study of Spin Chain Hamiltonians

David Berenstein , P. N. Thomas Lloyd This is my paper

Pith reviewed 2026-06-28 13:39 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elhep-lat
keywords semidefinite programmingspin chainsIsing modelPotts modelfree fermionsground state energycorrelation functionsconformal data
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The pith

Semidefinite programming exactly recovers the ground state energy and fermion two-point functions for free-fermion spin chains.

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

This paper applies semidefinite programming to compute ground state energies, excited states, and correlation functions in spin chain Hamiltonians, focusing on the transverse Ising and three-state Potts models. The authors prove that for free-fermion systems the SDP relaxation is exact for both the energy and the two-point functions. They introduce charge constraints to access excited states and test integrability constraints for improvement. At critical points they use approximate Virasoro algebra relations to estimate the central charge and critical exponents. The method works well near free fermions but accuracy declines for correlation functions in interacting cases, and the computational demands grow rapidly with system size.

Core claim

In the special case of free fermions the semidefinite program returns the exact ground state energy and the exact two-point functions. For general models the program still supplies an upper bound on the ground state energy that becomes tight with added constraints, and it produces correlators that qualitatively reproduce the true behavior even when quantitative accuracy is lost.

What carries the argument

The semidefinite programming relaxation of the Hamiltonian minimization problem, augmented by charge and integrability constraints.

If this is right

  • Charge constraints enable computation of the first excited charged states.
  • Integrability constraints tighten the relaxation for integrable models.
  • Approximate Virasoro correlators allow extraction of the central charge and some critical exponents at criticality.
  • The location of the phase transition can be identified using these methods.
  • The approach encounters scaling issues as lattice volume increases.

Where Pith is reading between the lines

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

  • The exactness result for free fermions may extend to other models where similar algebraic constraints can be imposed.
  • Qualitative accuracy in interacting regimes suggests SDP could provide initial estimates in systems lacking exact solutions.
  • The volume scaling problem indicates that symmetry reductions or hierarchical improvements would be needed for larger systems.
  • Linking SDP outputs to conformal data might help benchmark lattice regularizations of conformal field theories.

Load-bearing premise

The semidefinite relaxation stays tight after the addition of charge constraints and any available integrability constraints.

What would settle it

A numerical check on a small free-fermion Ising chain where the SDP energy and two-point functions are compared against exact diagonalization results.

Figures

Figures reproduced from arXiv: 2606.02013 by David Berenstein, P. N. Thomas Lloyd.

Figure 1
Figure 1. Figure 1: FIG. 1. The addition of the additional charge constraints generated from the boost operator have [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Building up the Virasoro generators for the Ising model allowed us to explore if there [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The left figure is the ratio, [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Here we plot [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows the resulting ⟨H† nHn⟩ curves for both choices. The minimal basis fails to produce a clear minimum at the known critical coupling. The enlarged basis is better, exhibiting a localized feature in the vicinity of the critical point, but the feature is shallower than the corresponding Ising signal and shifted by a small but visible amount from the exact location of the transition. Neither basis reproduc… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Two-point correlators [PITH_FULL_IMAGE:figures/full_fig_p043_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Two-point correlators [PITH_FULL_IMAGE:figures/full_fig_p044_7.png] view at source ↗
read the original abstract

We study semidefinite programming (SDP) methods to analyze spin chain Hamiltonians. We examine the ground state energy, the first excited charged states and ground state correlators in two simple models: the Ising model in a transverse magnetic field and the closely related 3-state Potts model. Our goal is to understand precisely what the SDP program is doing and when it works well, why it does so. We focus on the following novel ingredients: using charge constraints to obtain excited states and to see if additional constraints from integrable models are effective at improving the method. At criticality we also explore to what extent we can use approximate Virasoro correlators to extract conformal data: the central charge and some critical exponents of charged states. We also use these to identify the location of the phase transition. In the special case where the system is made of free fermions we prove that the SDP finds the exact energy of the ground state and produces the correct two point functions of the fermions. Away from free fermion theories, the SDP gets progressively worse at estimating data beyond the value of the ground state energy (like correlation functions), although it qualitatively matches these. In order to be effective, the SDP seems to run into scaling issues where the amount of input needed scales poorly with the lattice volume.

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

1 major / 2 minor

Summary. The manuscript examines semidefinite programming (SDP) relaxations for spin-chain Hamiltonians, focusing on the transverse-field Ising and 3-state Potts models. It introduces charge constraints to access excited states and integrability constraints to tighten bounds, proves that the SDP yields the exact ground-state energy and correct two-point fermion correlators in the free-fermion case, reports qualitative agreement for correlators away from free fermions, and explores extraction of central charge and critical exponents from approximate Virasoro correlators at criticality while noting that required input scales poorly with lattice volume.

Significance. The explicit proof of tightness for free fermions supplies a rigorous benchmark that clarifies when SDP relaxations become exact under charge and integrability constraints; this is a concrete strength. The demonstration that charge constraints enable access to charged excited states and that approximate Virasoro data can locate phase transitions adds practical insight, even though accuracy degrades for correlators outside the free-fermion point.

major comments (1)
  1. [free-fermion proof] The central claim that the SDP relaxation remains tight for free fermions once charge (and integrability) constraints are imposed is load-bearing for both the exactness proof and the qualitative claims away from that limit. The manuscript should supply the explicit algebraic steps showing that the SDP bound saturates the known free-fermion ground-state energy and two-point functions without residual gap.
minor comments (2)
  1. [scaling issues] The scaling discussion would be strengthened by a quantitative plot or table showing how the number of constraints or matrix size grows with chain length N.
  2. Notation for the charge and integrability constraints should be defined once in a dedicated subsection before their use in the SDP formulation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and constructive feedback. The single major comment concerns the explicitness of the free-fermion tightness proof. We address it directly below and will revise the manuscript to incorporate the requested algebraic steps.

read point-by-point responses

  1. Referee: The central claim that the SDP relaxation remains tight for free fermions once charge (and integrability) constraints are imposed is load-bearing for both the exactness proof and the qualitative claims away from that limit. The manuscript should supply the explicit algebraic steps showing that the SDP bound saturates the known free-fermion ground-state energy and two-point functions without residual gap.

    Authors: We agree that the current presentation of the proof would benefit from a more expanded derivation. In the revised version we will insert a dedicated subsection that walks through the algebraic steps: starting from the free-fermion Hamiltonian expressed in Majorana operators, imposing the charge and integrability constraints on the moment matrix, and showing explicitly that the SDP objective equals the known ground-state energy while the extracted two-point functions match the exact Wick contractions with no residual gap. This addition will make the saturation argument fully self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents its strongest claim—that the SDP relaxation is tight for free-fermion spin chains, yielding exact ground-state energy and two-point functions—as a mathematical proof once charge and integrability constraints are imposed. This is not obtained by fitting parameters to data, renaming known results, or reducing via self-citation chains; the abstract explicitly distinguishes the exact free-fermion case from the qualitative degradation away from it. No load-bearing step in the provided derivation chain reduces by construction to its own inputs, and the method's scaling limitations are acknowledged rather than hidden. The central result is therefore self-contained and externally verifiable as a proof.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; all such quantities remain unknown.

pith-pipeline@v0.9.1-grok · 5766 in / 1212 out tokens · 23435 ms · 2026-06-28T13:39:50.762310+00:00 · methodology

discussion (0)

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Reference graph

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