Thermodynamic-limit bootstrap: Solving fractional quantum Hall ground states

arXiv:2610.21317 · cond-mat.str-el, cond-mat.mes-hall · Submitted 2026-10-15 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: Today's paper: "Thermodynamic-limit bootstrap: Solving fractional quantum Hall ground states".

Mira: The gist: The quantum many-body bootstrap provides a universal ground-state solver that determines both the energy and the state directly in the thermodynamic limit for fractional quantum Hall systems,

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So, we've covered what the paper is and how they get these rigorous energy bounds in the thermodynamic limit. Now let's look at the specific titles and authors of this work, "Thermodynamic-limit bootstrap: Solving fractional quantum Hall ground states". Mira I think it’s important to note who wrote it, Qiang Gao and Taige Wang from MIT and Harvard.

Lev: They are working on things that would be very relevant for real hardware because they are dealing with the fundamental physics of these systems <ref:2610.21317#pg2>. When you talk about solving these states directly in the TDL, you’re talking about pushing the limits of what we can model for actual physical devices <ref:2610.21317#pg3>.

Kai: Yeah, and they are tackling systems that are usually computed in finite systems and then extrapolated to the thermodynamic limit <ref:2610.21317#pg1>. That’s the exact challenge this paper is addressing.

Mira: The title itself suggests a direct solution, which is what they aim for by developing this quantum many-body bootstrap method <ref:2610.21317#pg2>. It’s about providing a way to solve fractional quantum Hall ground states directly in the TDL <ref:2610.21317#pg1>.

Lev: And they are specifically focusing on parent Hamiltonians, like Laughlin and Pfaffian ones, where they reproduce the exact ground-state energies and pair correlations almost perfectly <ref:2610.21317#pg2>. That level of accuracy is what makes this work so important for testing our understanding of these states <ref:2610.21317#pg3>.

Kai: It sounds like they’re taking a very specific, well-understood class of systems and showing that their new method can nail the energetics and the structure simultaneously <ref:2610.21317#pg3>.

Mira: And this is important because it shows that even for frustrated systems, where things get more complicated, they still have a path forward using these rigorous constraints <ref:2610.21317#pg3>.

Lev: This paper is setting a benchmark for how we can move from finite system approximations to direct TDL solutions in this field <ref:2610.21317#pg3>.

The paper's summary: Kai: Now let’s go back to the core idea of the paper, "Thermodynamic-limit bootstrap: Solving fractional quantum Hall ground states". It seems like they are proposing this quantum many-body bootstrap as a way to solve these complex ground state problems directly in the TDL <ref:2610.21317#pg1>.

Mira: They do this by replacing a variational wavefunction with universal constraints on local correlations, which turns the ground state problem into something that can be solved through positivity and symmetry conditions on reduced density matrices <ref:2610.21317#pg1>. It’s a shift from guessing the whole state to imposing strong local rules <ref:2610.21317#pg5>.

Lev: And this involves minimizing the energy over window density matrices of the infinite sphere, which is what they do for quantum Hall systems <ref:2610.21317#pg5>. They’re looking at a primal problem that seeks a rigorous lower bound on the TDL ground-state energy per particle, E0 ≡ limN→∞ E0(N)/N <ref:2610.21317#pg5>.

Kai: And those constraints on the window density matrix ρ⋆W are where they get really specific about the local structure of the ground state <ref:2610.21317#pg8>. These constraints include things like trρW = one and tr(ρW na) = ν for all a in W <ref:2610.21317#pg8>.

Mira: For two-particle interactions, they impose homogeneity by specifying the two-particle reduced correlation matrix Γ(two) using bulk pair weights πm ≥ zero <ref:2610.21317#pg8>. And for higher particle numbers, they put constraints on the k-particle correlation matrix Γ(k) T to ensure homogeneity across different center-of-mass sectors <ref:2610.21317#pg8>.

Lev: The energy itself is determined by these pair weights πm for two-body Hamiltonians, and for three-body Hamiltonians, the objective function involves the three-body cluster weights γ(three) D <ref:2610.21317#pg8>. That’s how they get a rigorous energy statement from the dual SDP <ref:2610.21317#pg8>.

Kai: So, even though this energy comes from the dual SDP, it doesn't assume that the optimized window density matrix extends to a physical infinite-system state <ref:2610.21317#pg8>.

The paper's improvements: Mira: This leads to some really strong improvements they suggest for this method, and I think these are what make it a practical solver <ref:2610.21317#pg5>. One big improvement is that every finite-K result holds directly in the thermodynamic limit without needing any extrapolation in particle number or window size <ref:2610.21317#pg8>.

Lev: That’s a major win for practical application because it means you don't have to do those tedious extrapolations, which is what I spend half my time dealing with when building real hardware <ref:2610.21317#pg5>.

Kai: And they also use the local structure revealed by ρ⋆W to identify the underlying quantum Hall state directly through OES analysis on an inner cap A ⊂ W <ref:2610.21317#pg8>. This lets them compare their results against known states, like recovering Laughlin and Moore–Read OES count level by level <ref:2610.21317#pg2>.

Mira: They also use this OES as a primary state diagnostic because it sharply distinguishes competing FQH states, resolving ambiguities where candidates share the same Particle Entanglement Spectrum, or PES counting <ref:2610.21317#pg4>. This is useful because sometimes two different candidates can look similar based on their PES alone <ref:2610.21317#pg4>.

Lev: And they can determine both the energetics and the phase identity directly in the thermodynamic limit for both Abelian and non-Abelian topological orders, which is a nice unification <ref:2610.21317#pg5>. That’s a lot of consistency for testing different types of order <ref:2610.21317#pg5>.

Kai: So, it seems the big shift here is moving from just getting an energy number to actually characterizing the state itself using these local diagnostics <ref:2610.21317#pg8>.

Conclusion: Mira: To wrap up on "Thermodynamic-limit bootstrap: Solving fractional quantum Hall ground states", the paper suggests that this method provides a rigorous way to solve these ground state problems in the TDL <ref:2610.21317#pg5>. It’s about using universal constraints on local correlations to build a solver based on positivity and symmetry <ref:2610.21317#pg8>.

Lev: So, for me, the main point is that this framework gives us a way to bypass the usual need for extrapolation and provides a quantitative test because FQH systems have these vast number of thermodynamic benchmarks <ref:2610.21317#pg3>.

Kai: And they’ve shown how they can use diagnostics like OES to identify the underlying topological order directly, which is key for connecting theory to experiment <ref:2610.21317#pg8>. It moves us beyond just numbers and into understanding the actual physical state <ref:2610.21317#pg8>.

Mira: It’s a powerful tool because it lets them determine both energetics and phase identity directly in the thermodynamic limit for both Abelian and non-Abelian topological orders <ref:2610.21317#pg5>. This consistency across different orders is what makes this method so compelling <ref:2610.21317#pg5>.

Lev: I just want to say that the method works across exactly solvable models, established Coulomb states and even unsettled ground states <ref:2610.21317#pg3>. That breadth is what makes it so valuable for testing our current theories <ref:2610.21317#pg3>.

Kai: It’s a lot of rigorous work to get this kind of direct insight, but the ability to solve these problems without relying on finite-size scaling is a huge step forward in this area <ref:2610.21317#pg8>.

Qiang Gao, *, Taige Wang2, 3, †

Massachusetts Institute of Technology · Harvard University

cond-mat.str-el, cond-mat.mes-hall

Submitted: 2026-10-15

Updated: 2026-10-15

Comments: 17 pages, 4 main figures, 5 extended figures

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 83/100

The gist: The gist: The quantum many-body bootstrap provides a universal ground-state solver that determines both the energy and the state directly in the thermodynamic limit for fractional quantum Hall

Key concepts

Quantum Many-Body Bootstrap: This is a universal method that replaces guessing a wavefunction with imposing strict mathematical constraints on local correlations. By enforcing positivity, symmetry, and consistency conditions on reduced density matrices, it allows for the rigorous determination of the ground state properties without needing an initial guess.
Window Density Matrix ($ ho_W$): This matrix is central to the method; it captures the local structure of the ground state by exchanging particles with an infinite surrounding fluid. Constraints on $ ho_W$, such as ensuring particle number conservation and homogeneity across different sectors, define how particles are organized locally.
Orbital Entanglement Spectrum (OES): The OES is a diagnostic tool derived from the local structure revealed by $ ho_W$. It is used to sharply distinguish between competing fractional quantum Hall states, helping researchers identify specific topological orders like composite Fermi liquids or RR3 phases based on their unique entanglement patterns.

Summary

The gist: The quantum many-body bootstrap provides a universal ground-state solver that determines both the energy and the state directly in the thermodynamic limit for fractional quantum Hall systems, identifying composite Fermi liquid and RR3 topological orders.

How it works

The method replaces a variational wavefunction with universal constraints on local correlations by turning the ground-state problem into a convex optimization via positivity, symmetry, and consistency conditions on reduced density matrices (Page 1). For quantum Hall systems, this involves minimizing the energy over window density matrices of the infinite sphere (Page 5). The primal problem seeks a rigorous lower bound on the TDL ground-state energy E0 which is bounded by E⋆ (Page 5).

The constraints imposed on the window density matrix ρW are crucial for defining the local structure of the ground state (Page 8). These constraints include:

**)&tr ρW = 1, tr(ρW na) = ν, a ∈ W (Page 8). The window contains no fixed number of particles because it exchanges particles with the surrounding infinite fluid (Page 8). For two-particle interactions, homogeneity requires that the two-particle reduced correlation matrix Γ(2) is specified by bulk pair weights πm ≥ 0 (Page 8). For higher particle numbers, constraints are imposed on the k-particle correlation matrix Γ(k) T, ensuring homogeneity across different center-of-mass sectors (Page 8). The energy is determined by pair weights πm for two-body Hamiltonians (Page 8). For three-body Hamiltonians, the objective function involves the three-body cluster weights γ(3) D (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8). The rigorous energy statement follows from the dual SDP and does not assume that the optimized window density matrix extends to a physical infinite-system state (Page 8). Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size (Page 8).

Improvements for AI systems

No finite-size extrapolation required for thermodynamic limit energy bounds: The bootstrap determines rigorous lower bounds on ground-state energy directly in the TDL, stating that Every finite-K result therefore holds directly in the thermodynamic limit, without extrapolation in particle number or window size.
Direct identification of topological order via Orbital Entanglement Spectrum (OES): The method reveals the local structure of the ground state through window density matrix ρ⋆W, which allows for a direct comparison with known states, such as recovering the Laughlin levels match nearly one by one or identifying RR3 order at LL1 ν = 2/5.
Precise characterization of real-space correlations: The bootstrap reconstructs the pair distribution function g(r) = ⟨: ρ(r)ρ(0):⟩/n2, resolving the correlation hole and revealing how particles organize in real space, with short-distance behavior being controlled entirely by the lowest relative-angular-momentum channels.
Distinction between competing phases using entanglement diagnostics: The OES is used as a primary state diagnostic because it sharply distinguishes competing FQH states, resolving ambiguities where candidates share the same Particle Entanglement Spectrum (PES) counting, such as identifying a composite Fermi liquid rather than a paired Pfaffian state at LLL ν = 1/4.
Robust phase identification across different Landau levels: The framework allows for the determination of both energetics and phase identity directly in the thermodynamic limit for both Abelian and non-Abelian topological orders, ensuring that The same framework determines both the energetics and phase identity of Abelian and non-Abelian topological orders.