A version-1 arXiv preprint proposes a general mathematical way to show that quantum phase transitions exist in one-parameter lattice Hamiltonians and to place the critical value of the control parameter inside analytic bounds. In the paper’s setup, the Hamiltonian is H = K + gV, with K and V noncommuting Hermitian operators. The central theorem is conditional: if K and V scale linearly with system size and V is self-averaging, the thermodynamic-limit system—the idealized limit of an ever-larger system—has at least one transition, with its critical g between a lower and an upper bound.
To make that idea precise, the authors split the Hilbert space—the mathematical space of possible quantum states—into a condensed sector and a normal sector. If the condensed sector accounts for a vanishing fraction of the full space in the thermodynamic limit, the full rescaled ground-state energy per particle equals the lower of the two energies restricted to those sectors. The paper uses that lower-envelope result to interpret the transition as a condensation in state space.
How the bounds are built
The proposed critical point comes from a finite-size crossing: the parameter where the restricted energies meet is followed as the system grows, and its thermodynamic limit is taken as the critical value, assuming smooth limiting behavior. The proof also uses the variational principle and concavity, showing that the full and restricted ground-state energies, including their thermodynamic limits, are concave functions of g.
The bounds are built from the minimum kinetic energies and the extreme or threshold values of the potential term. When the condensed-sector kinetic minimum k_cond is zero, the lower and upper estimates coincide in a simple approximation for the critical point; when k_cond is nonzero, that simplification does not apply.
Examples put numbers on the idea
Two model examples show that the framework can cover different transition types. In Grover’s model, the bounds coincide at the exact critical point g_c = 1, and the paper reports a first-order transition. In the one-dimensional transverse-field Ising example, the exact critical point is also g_c = 1 and the transition is second-order; the theorem-based estimate is about 1, with a lower bound of 1/2 and an approximate upper bound of 2.
The interval is wider in a one-dimensional fermion model with a heterogeneous field. At half filling, the simple estimate is about 2.55, compared with a reported numerical critical point of about 4.0; the reported lower and upper bounds are about 1.28 and 5.09.
At quarter filling, the reported lower bound is about 0.53, while the upper estimates are about 5.09 and 3.60; the reported numerical critical point is about 2.0. Here k_cond is nonzero, and the analysis notes that the upper bounds rely on approximations and are broad.
A framework, not a classification
The theorem does not settle the classification problem. By itself, it does not determine how many critical points a model has or whether a particular transition is first- or second-order; the paper allows for multiple critical points of differing nature and says more information about K and V is needed.
The authors nevertheless give the framework a broad interpretation: g-associated quantum phase transitions in lattice systems with noncommuting operators that scale with system size can be viewed as state-space condensations when the stated conditions hold. The evidence here is entirely theoretical—the objects analyzed are Hamiltonians, Hilbert-space sectors and thermodynamic-limit energies, not human, animal or cell samples.
For researchers applying the method to other models, the main tests remain model-specific: self-averaging must be established, sector quantities such as v_cond and k_cond may be difficult to obtain, and further information is needed to count and classify the transitions. The preprint therefore offers a conditional theorem and a way to narrow the search for a critical point, rather than a universally exact answer for every lattice system.
Paper data and sources
Original title: Rigorous existence and location of quantum phase transitions in lattice Hamiltonian systems
Authors: Massimo Ostilli, Carlo Presilla
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text