Excited-state quantum phase transitions in constrained systems

Jakub Novotn\'y; Pavel Cejnar; Pavel Str\'ansk\'y

REVIEW 2 major objections 4 minor 67 references

Excited-state quantum phase transitions in constrained systems

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that all excited-state quantum phase transitions in constrained Hamiltonian systems can be found and classified directly from the Hamiltonian plus Lagrange multipliers, without constructing the canonical transformation…

desk verdict A useful, mostly correct extension of ESQPT semiclassics whose advertised generality outruns its proof. read the letter →

arxiv 2412.04240 v2 pith:IOV42H2I submitted 2024-12-05 quant-ph

Jakub Novotný , Pavel Stránský , Pavel Cejnar This is my paper · ORCID

classification quant-ph

keywords excited-statequantumphasetransitionsLagrangemultipliersconstrainedsystemsHolstein-Primakoffmappingleveldensitystationarypointsalgebraicbosonmodelsu(3)model

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper extends the standard semiclassical theory of excited-state quantum phase transitions (ESQPTs) to systems whose dynamics is restricted by conserved quantities. Its central claim is that all stationary points of the reduced classical Hamiltonian—the objects that produce ESQPT singularities in the level density—can be found and classified directly from the original Hamiltonian together with Lagrange multipliers for the constraints. This removes the need to construct the canonical transformation that eliminates the constrained degrees of freedom. The paper proves the correspondence on general grounds and demonstrates it on a u(3) boson model with one and two constraints. It also shows that boundary stationary points that appear in a single Holstein-Primakoff mapping are an artifact of that singular coordinate choice: a complete atlas of mappings reveals them as ordinary ESQPTs.

What carries the argument

The Lagrange function $L(X,\lambda)=H(X)+\sum_\alpha\lambda_\alpha\Phi_\alpha(X)$ is the primary object: solving its full gradient gives stationary points $\{X_{\rm st},\lambda_{\rm st}\}$. The second object is the restricted Hessian $D^2L|_{\Sigma}$, obtained by evaluating the Hessian of $L$ only in directions tangent to the constraint surface $\Sigma$; its negative-eigenvalue count supplies the index $r_\Sigma$. Appendix A shows that a local canonical transformation $X\mapsto(x,\varphi,\Phi)$ separates the constraints into conjugate pairs $(\Phi_\alpha,\varphi_\alpha)$ with cyclic angles, which turns the restricted Hessian into the Hessian of the reduced Hamiltonian plus harmless zero eigenvalues. The Holstein-Primakoff map $M^{(j)}$ is the third object: a singular projection from the constraint sphere $\Sigma$ onto the compact ball $\sigma^{(j)}$, and the atlas $\{M^{(0)},\dots,M^{(f)}\}$ is what guarantees that every stationary point appears in the interior of at least one chart.

What would settle it

Choose a Hamiltonian with two independent, regular constraints whose Poisson brackets do not vanish, compute the quantum level density at large size, and compare the locations and derivative-singularity types of its nonanalytic features with predictions from $\nabla L=0$ and $D^2L|_{\Sigma}$; a mismatch would falsify the claimed generality. Within the u(3) model, a numerical check near $\xi\approx0.42$ should show the derivative of the smoothed level density developing singular kinks exactly at the new stationary energies for $\xi>\xi_e$.

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Extended reading notes

Core claim

For a constrained system with $f+c$ degrees of freedom and $c$ independent constraints $\Phi_\alpha$, define $L(X,\lambda)=H(X)+\sum_\alpha\lambda_\alpha\Phi_\alpha(X)$. The paper's claim is that the solutions of $\nabla_{(X,\lambda)}L=0$ stand in one-to-one correspondence with the stationary points $x_{\rm st}$ of the reduced Hamiltonian $H^{(\sigma)}$ on the physical phase space, with equal energies $E_{\rm st}=H(X_{\rm st})=L(X_{\rm st},\lambda_{\rm st})$ and equal singularity indices $r(x_{\rm st})=r_\Sigma(X_{\rm st})$, the latter read off from the Hessian of $L$ restricted to directions tangent to the constraint surface. This is proven in Appendix A using local action-angle coordinates in which each constraint is a momentum conjugate to a cyclic angle. Applied to the u(3) boson model, the method reproduces the full set of ESQPT energies and indices, including the transition at $E=1-\xi$ that a single Holstein-Primakoff chart misses. The companion claim about the Holstein-Primakoff mapping is that its classical singular boundary is a coordinate artifact: boundary stationary points move to the interior of some chart of a complete atlas of HP mappings, so they are not a separate class of ESQPT.

Load-bearing premise

The argument relies on every constraint being pairable with a cyclic coordinate through a local action-angle canonical transformation; if the conserved quantities fail to commute or no such separation exists, the equality of energies and indices is not proven.

Editorial extensions

If this is right

ESQPT energies and singularity types in any constrained system follow from solving $\nabla L=0$ and diagonalizing $D^2L|_{\Sigma}$; the explicit canonical transformation that removes the constraints is never needed.
Boundary stationary points seen in a single Holstein-Primakoff map are ordinary ESQPTs, not a separate class; a complete atlas of HP maps finds every one in the interior of some chart.
Adding extra conserved constraints lowers the effective number of degrees of freedom, moving ESQPT singularities to lower derivatives of the level density and changing the indices.
Additional constraints can also delete ESQPTs: some stationary points live only in subspaces with particular values of the conserved quantities, and at least one singularity in the doubly constrained u(3) model arises from the interplay of all invariant subspaces.
For polynomial algebraic Hamiltonians the Lagrange equations are polynomial, so stationary points can be found analytically and their count bounds the number of ESQPTs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same Lagrange construction should apply to systems whose constraints are not integrals of motion but are imposed by the physical setup, such as gauge constraints, provided the involutive and action-angle conditions hold.
The appearance of complex stationary points near the real axis suggests a general precursor effect: before a pair of real stationary points emerges, the smoothed level density should show a smooth but non-monotonic wiggle, which could serve as an early spectroscopic warning of an approaching ESQPT.
For boson systems with more than two degrees of freedom, the atlas of HP maps grows with the number of boson types, so the Lagrange method likely becomes the only practical route; this could be checked by extending the u(3) analysis to u(4) vibron models.
The u(3) model with two constraints is integrable; applying the Lagrange method to other integrable limits should predict which invariant subspaces carry ESQPTs and which do not.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this paper is worth engaging with. The Lagrange-multiplier method for finding and classifying ESQPT stationary points is a genuine, practical extension of the standard semiclassical machinery, and the Holstein-Primakoff atlas argument is the strongest part: it convincingly shows that boundary ESQPTs in the u(3) vibron model are artifacts of a single singular mapping, not a separate phenomenon. The numerical checks against Gaussian-smoothed quantum spectra are convincing, with stationary-point energies and indices matching the observed nonanalytic structure. No parameters are fitted to the target spectra. The citation pattern is normal; the heavy use of the authors' own prior classification is background, not circularity.

What is genuinely new: the constrained stationary-point and index theorem, the demonstration that a complete HP atlas resolves the boundary singularity, and the multiply-constrained u(3) example showing that additional integrals of motion can change the ESQPT count and character. The paper also gives a useful practical argument that Lagrange multipliers avoid the square-root clutter and boundary issues of HP mappings.

The main soft spot is exactly what the stress-test identifies. Appendix A proves the theorem only under the assumption that the constraints can be paired with cyclic coordinates via a local action-angle canonical transformation, which requires the constraints to Poisson-commute and be independent. The paper never states this involution condition; Section 2.2 only requires nonzero, linearly independent gradients and calls the constraints arbitrary integrals of motion. That overstates the generality. The proof of the level-density equality (16) also relies on the same measure factorization, so the unqualified claim of "arbitrary number of constraints" is not established. The demonstrated examples all use commuting constraints, so the gap does not infect the numerics, but it is a real gap in the general theorem. A second, smaller gap: the claim that a complete HP atlas always suffices is argued and demonstrated for f=2, but not proven for general f or higher-rank algebras. That is minor relative to the main result.

I think the central argument holds for the cases it actually treats. The paper is clearly written and the authors know the ESQPT literature well. The right fix is to make the involution/separability assumption explicit and qualify the generality claim, not to redo the analysis.

Who is this for? Anyone working on ESQPTs in bosonic algebraic models—vibron, IBM, spinor condensates—will get direct value. It deserves a serious referee. I would send it to review with the request that the authors state and prove their theorem under the correct hypotheses, or explicitly restrict the claims to commuting integrals of motion. I would cite it in my own work on ESQPTs.

Referee Report

2 major / 4 minor

Summary. The paper extends the semiclassical theory of excited-state quantum phase transitions (ESQPTs) to systems with constraints induced by conserved quantities. It proposes to find and classify the stationary points of the constrained classical Hamiltonian by solving the Lagrange equations of L = H + Σ_α λ_α Φ_α, claiming that each stationary point's energy and Hessian index coincide with those obtained after explicit reduction to the unconstrained phase space. The method is demonstrated on a u(3) boson model with one constraint (fixed total boson number N) and with two constraints (additional conserved O(2) Casimir). The paper also analyzes Holstein-Primakoff (HP) mappings for fixed-N bosonic systems, showing that a single HP mapping is singular at the boundary of the reduced phase space and that a complete atlas of HP mappings is needed to reveal all ESQPTs. The Lagrange-multiplier results are verified against numerical diagonalization and against the HP atlas.

Significance. If the claimed correspondence is valid, the paper provides a practical method for identifying ESQPTs in constrained systems without constructing explicit canonical transformations, and it resolves a long-standing issue with boundary stationary points in Holstein-Primakoff mappings by showing they are coordinate singularities rather than a distinct type of ESQPT. The numerical verification in the u(3) model with one and two constraints is convincing, and the method contains no fitted parameters: all semiclassical predictions are derived from the Hamiltonian and constraints. The demonstration that an additional constraint can move ESQPT singularities to lower derivatives, change their indices, or remove them entirely is a useful contribution to the classification of ESQPTs. The construction of a complete atlas of HP mappings is a new and clearly explained technical development.

major comments (2)

[Section 2.2 and Appendix A] The proof of the central correspondence (13)-(14) relies on the existence of a local canonical transformation to coordinates (x, φ_1,...,φ_c, Φ_1,...,Φ_c) in which the Hamiltonian is independent of the cyclic coordinates φ_α (Eq. (43)). Such a simultaneous action-angle separation exists only when the constraint functions are in involution (Poisson-commuting) and define a first-class constraint surface in the sense of Dirac. The manuscript does not state this condition: Section 2.2 requires only nonzero, linearly independent gradients, and the abstract claims 'an arbitrary number of integrals of motion'. For conserved quantities that do not Poisson-commute, no common action-angle representation is guaranteed, the measure factorization in Eq. (16) is not justified, and the index identity (14) is not established by the given proof. The demonstrations in Sections 3.4 and 3.5 use Poisson-commuting constraints (Φ_N and Φ_l), so they provide no evidence for the non-commuting case. Please either restrict the claim to commuting integrals of motion or supply a proof that avoids the action-angle separation, for example by analyzing the Hessian of H restricted to the constraint surface directly.
[Section 2.2, Eq. (7)] The quantum formulation of multiple constraints implicitly requires a common zero eigenspace of the operators ÊˆΦ_α. For constraints of the form (8), ÊˆΦ_α = ÊˆI_α − I_α, this means the conserved operators ÊˆI_α must possess a joint eigenspace, which generally requires the ÊˆI_α to commute (or at least to act as scalars on H_c). The paper does not state this commutativity requirement, despite claiming in the abstract that the method applies to an arbitrary number of integrals of motion. This is not merely a technicality: without it, the physical meaning of imposing several non-commuting constraints is unclear. The authors should state explicitly that the method is intended for commuting (Abelian) integrals of motion, or justify that the non-commuting case is physically meaningful and covered by the proof.

minor comments (4)

[Eq. (40) and Sec. 3.5] The symbol l^2 is used both for the operator Êˆl^2 and for its eigenvalue; the classical counterpart in Eq. (41) uses ℓ^2, which is clearer. Please unify the notation for the eigenvalue throughout Section 3.5.
[Eq. (47)] The block decomposition of the Hessian would be easier to follow if the rows and columns were labeled (x, φ, Φ) explicitly; the current display has three rows of blocks but the second row contains only zero blocks, which is initially confusing.
[Appendix B, Eq. (57)] The expression 'j ≠ k ≠ j′' is ambiguous; it should read 'k ≠ j and k ≠ j′'.
[Section 3.5, final paragraph] The statement that the ESQPT at E = −0.2 'can be considered as a result of an interplay of all the subspaces' is vague; a brief explanation of how the finite-dimensional subspaces H_l conspire to produce the semiclassical singularity would strengthen the argument.

Circularity Check

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No significant circularity: the Lagrange-multiplier ESQPT method is derived from standard constrained optimization and independently benchmarked against numerical spectra.

full rationale

The central derivation chain is: (i) define the classical Hamiltonian H and constraints Phi on the full phase space; (ii) form the Lagrange function L = H + sum lambda Phi and solve grad L = 0; (iii) prove in Appendix A that each such solution corresponds to a stationary point of the reduced Hamiltonian H^(sigma) with the same energy and Hessian index, using a local action-angle canonical transformation (Eq. 43); and (iv) compare the resulting ESQPT energies and indices against independent numerical diagonalization in Figs. 4 and 6. None of these steps fits a parameter to the target quantum spectra, and identities (13)-(14) follow from the standard constrained-optimization theorem plus the explicit restricted-Hessian construction, not from assuming the conclusion. Citations to the authors' earlier classification [27] and HP-mapping work [39] provide background results that are parameter-free and externally validated in unconstrained settings; they are not used to force the constrained-system result. The only genuine caveat is that the Appendix A proof requires each constraint to be paired with a cyclic action-angle coordinate (Eq. 43), i.e., local Liouville-Arnold separability; for non-commuting conserved quantities the level-density identity (16) and index relation (14) are not established. That is an assumption limiting the generality claim, not a circular reduction: the paper's equations do not define the target result into existence, and the u(3) demonstrations use commuting constraints and are checked against exact diagonalization. Therefore no circular step is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted constants appear in the central derivation. The parameters xi, epsilon, and l are external physical parameters of the u(3) example, and the smoothing width is a visualization tool for finite-N spectra. The central claim rests on the domain assumptions listed above, especially the canonical separability of the constraints.

assumptions (4)

domain assumption The smooth level density is given by the Weyl formula and nondegenerate stationary points of the classical Hamiltonian classify ESQPT singularities via Eq. (6).
Borrowed from prior ESQPT semiclassical theory [27]; this anchors the whole method and is not derived in this paper.
domain assumption For every constraint there exists a canonical transformation to local action-angle coordinates separating (Phi_alpha, phi_alpha), with phi_alpha cyclic.
Invoked in Section 2.2 before Eq. (16) and in Appendix A Eq. (43). Requires the constraints to be in involution, which is not stated as an assumption.
domain assumption Constraints are regular: gradients are nonzero and linearly independent on the constraint surface.
Stated in Section 2.2; needed for the constraint surface to be a smooth submanifold and for the Lagrange method to apply.
domain assumption The classical limit of the fully-connected boson system is obtained by a large-N position-momentum substitution, and the constraint becomes a sphere of radius sqrt(2).
Standard large-N limit for algebraic boson models, cited to [52]; used throughout Section 3.

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Pith. "Pith review of Excited-state quantum phase transitions in constrained systems." pith.science (2026). https://pith.science/paper/IOV42H2I

@misc{pith2026241204240,
  author       = {Pith},
  title        = {Pith review of: Excited-state quantum phase transitions in constrained systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IOV42H2I}},
  note         = {Machine review of arXiv:2412.04240}
}

read the original abstract

We extend the standard semiclassical theory of Excited-State Quantum Phase Transitions (ESQPTs), based on a classification of stationary points in the classical Hamiltonian, to constrained systems. We adopt the method of Lagrange multipliers to find all stationary points and their properties directly from the Hamiltonian constrained by an arbitrary number of integrals of motion, and demonstrate the procedure on an algebraic u(3) boson model with two independent constraints. We also elaborate the Holstein-Primakoff (HP) mapping, used to eliminate one degree of freedom in bosonic systems constrained by a conserved number of excitations, and address the fact that this mapping leads, in the classical limit, to a compact phase space with singular behaviour that conceals some stationary points at the phase space boundary. It is shown that the HP method reveals all ESQPTs only after constructing a complete atlas of different HP mappings.

Figures

Figures reproduced from arXiv: 2412.04240 by the authors.

**Figure 2.** (a) A schematic example of a continuous path (green solid oriented line) [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Stationary points of the u(3) Hamiltonians H(j) in their corresponding phase spaces σ (j) , j = 0, 1, 2, parametrised by the coordinates ηk (33). Model parameters are ϵ = 0.3 and ξ ∈ [0, 1]. Varying ξ creates paths of stationary points in the phase space; their directions for increasing ξ are marked by arrows. Line colours indicate the indexes r of the stationary points, see the legend on the right. Roman numerals e… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: (a) Energies of all the stationary points of the [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: Stationary points of Lagrange function ( [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: Smooth level density ρ(E) obtained from a numerical diagonalization of the u(3) Hamiltonian with ξ = 0.6, ϵ = 0. (a) First derivative dρ/dE (ρ(E) is plotted in the inset) when the constraint Φl is switched off. (b) Constraint Φl switched on and ℓ = {0, 0.2, 0.4, 0.6, 0…

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Introduction An Excited-State Quantum Phase Transition (ESQPT) is an extension of a Quantum Phase Transition (QPT), a phenomenon that manifests itself as nonanalytic properties of the ground state of a quantum system in the infinite-size limit when the system’s external or internal coupling strengths are varied [1,2], to the domain of excited energy state...

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ESQPTs in fully-connected bosonic systems This section introduces algebraically formulated bosonic systems with a natural constraint originating in the conservation of the total number of boson excitations, and focuses on the HP mapping—an explicit prescription to eliminate the constrained degree of freedom. We recall the construction of the HP mapping an...

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As shown in Appendix B, the classical HP mapping preserves the relative number of excitations of each boson type, nk = Nk N = Q2 k + P 2 k 2 = q(j) k 2 + p(j) k 2 2 , j̸= k, (32) Excited-state quantum phase transitions in constrained systems 10 where Nk is an eigenvalue of the number operator ˆNk = ˆB† k ˆBk = ˆb(j)† k ˆb(j) k . This fact helps discuss an...

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Using the Lagrange method, we have found a complete set of stationary points and explained in full detail the rich ESQPT structure

Conclusions We have presented a straightforward and easy-to-implement semiclassical method of finding and classifying the ESQPTs in the spectrum of a quantum system with any number of constraints induced by conserving quantities, based on the Lagrange multipliers, and demonstrated it on a case study of the algebraicu(3) model. Using the Lagrange method, w...

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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boundary

Introduction An Excited-State Quantum Phase Transition (ESQPT) is an extension of a Quantum Phase Transition (QPT), a phenomenon that manifests itself as nonanalytic properties of the ground state of a quantum system in the infinite-size limit when the system’s external or internal coupling strengths are varied [1,2], to the domain of excited energy state...

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The ESQPT analysis is then extended to systems with an arbitrary number of constraints or integrals of motion by adopting the method of Lagrange multipliers

Semiclassical level density in general constrained systems This section reviews the principal concepts of the ESQPT theory and summarises the connection between the ESQPTs and the stationary points of the corresponding classical Hamiltonian; more details on ESQPTs can be found in the recently published review [22]. The ESQPT analysis is then extended to s...

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ESQPTs in fully-connected bosonic systems This section introduces algebraically formulated bosonic systems with a natural constraint originating in the conservation of the total number of boson excitations, and focuses on the HP mapping—an explicit prescription to eliminate the constrained degree of freedom. We recall the construction of the HP mapping an...

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projections

As shown in Appendix B, the classical HP mapping preserves the relative number of excitations of each boson type, nk = Nk N = Q2 k + P 2 k 2 = q(j) k 2 + p(j) k 2 2 , j̸= k, (32) Excited-state quantum phase transitions in constrained systems 10 where Nk is an eigenvalue of the number operator ˆNk = ˆB† k ˆBk = ˆb(j)† k ˆb(j) k . This fact helps discuss an...

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[5] [5]

Using the Lagrange method, we have found a complete set of stationary points and explained in full detail the rich ESQPT structure

Conclusions We have presented a straightforward and easy-to-implement semiclassical method of finding and classifying the ESQPTs in the spectrum of a quantum system with any number of constraints induced by conserving quantities, based on the Lagrange multipliers, and demonstrated it on a case study of the algebraicu(3) model. Using the Lagrange method, w...

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Carr L 2011 Understanding Quantum Phase Transitions (CRC Press)

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