We theoretically show that the critical temperature T c of the critical phenomena described within the Gibbs distribu- tion is varying linearly with the energetic parameters of the Hamiltonian. This kind of variation is due to the nature of the competition between the energy of the system and the energy kBT the thermic source. The change of this competition will lead to a different variation of the critical temperature. This linear variation has been reported by well-known theories of the simplest Ising model. In this study, we observe that this linearity concerns all well- known magnetic Hamiltonians formulated through spin-spin interactions. This competition has also a direct effect on the class of universalities in critical phenomena.We theoretically show that the critical temperature T c of the critical phenomena described within the Gibbs distribu- tion is varying linearly with the energetic parameters of the Hamiltonian. This kind of variation is due to the nature of the competition between the energy of the system and the energy kBT the thermic source. The change of this competition will lead to a different variation of the critical temperature. This linear variation has been reported by well-known theories of the simplest Ising model. In this study, we observe that this linearity concerns all well- known magnetic Hamiltonians formulated through spin-spin interactions. This competition has also a direct effect on the class of universalities in critical phenomena.

Jan 1, 20090 citations

Complexity and Criticality

Kim Christensen, Nicholas R. Moloney

Imperial college press advanced physics textOct 1, 2005505 citations

Critical Phenomena

F.J.W. Hahne

Lecture notes in physicsJan 1, 198328 citations

Exotic quantum criticality in one-dimensional coupled dipolar bosons tubes

P. Lecheminant, Heloise Nonne

The competition between intertube hopping processes and density-density interactions is investigated in one-dimensional quantum dipolar bosons systems of $N$ coupled tubes at zero temperature. Using a phenomenological bosonization approach, we show that the resulting competition leads to an exotic quantum phase transition described by a U(1) $\ifmmode\times\else\texttimes\fi{}$ ${\mathbb{Z}}_{N}$ conformal field theory with a fractional central charge. The emerging ${\mathbb{Z}}_{N}$ parafermionic critical degrees of freedom are highly nontrivial in terms of the original atoms or polar molecules of the model. We further determine the main physical properties of the quantum-critical point in a double-tube system which has central charge $c=3/2$ . In triple-tube systems, we show that the competition between the two antagonistic processes is related to the physics of the two-dimensional ${\mathbb{Z}}_{3}$ chiral Potts model. This work opens the possibility to study the exotic properties of the ${\mathbb{Z}}_{N}$ parafermions in the context of ultracold quantum Bose gases.The competition between intertube hopping processes and density-density interactions is investigated in one-dimensional quantum dipolar bosons systems of $N$ coupled tubes at zero temperature. Using a phenomenological bosonization approach, we show that the resulting competition leads to an exotic quantum phase transition described by a U(1) $\ifmmode\times\else\texttimes\fi{}$ ${\mathbb{Z}}_{N}$ conformal field theory with a fractional central charge. The emerging ${\mathbb{Z}}_{N}$ parafermionic critical degrees of freedom are highly nontrivial in terms of the original atoms or polar molecules of the model. We further determine the main physical properties of the quantum-critical point in a double-tube system which has central charge $c=3/2$. In triple-tube systems, we show that the competition between the two antagonistic processes is related to the physics of the two-dimensional ${\mathbb{Z}}_{3}$ chiral Potts model. This work opens the possibility to study the exotic properties of the ${\mathbb{Z}}_{N}$ parafermions in the context of ultracold quantum Bose gases.

arXiv:1202.6541Physical Review BMay 10, 201226 citations

Nonlocality-induced critical-length hierarchy from non-Hermitian competition

Mengjie Yang, Alexander N. Poddubny, Ching Hua Lee

Spectral transitions in non-Hermitian lattices often arise from the competition between non-reciprocal skin accumulation and inter-component hybridization. In short-range systems formed by two coupled chains, this competition conventionally leads to the logarithmic critical-length law $N_c\sim\ln D$ , where $D$ is the transverse separation between the chains. Here we show that long-range hoppings fundamentally reorganizes this critical behavior, producing a hierarchy of distinct scaling laws. When only the hybridization couplings are power-law decaying with exponent $α$ , the onset becomes algebraic, $N_c\sim D^{α/3}$ . When the hoppings within each chain are themselves also power-law decaying, in addition to the hybridization couplings, the system enters a scale-covariant regime for $α<2$ , in which the criticality threshold equation depends only on the system aspect ratio $N_c/D$ . At $α=2$ and beyond, this regime is followed by a marginal logarithmically corrected and algebraically corrected regimes, respectively. We identify two new non-local mechanisms that enable this unconventional critical hierarchy: a nonanalytic band-edge dispersion from long-range intra-chain hoppings, and parity-mixing hybridization induced by non-reciprocity. Our results show that nonlocality systematically removes the physical length scales i.e. skin depth underlying conventional critical non-Hermitian skin behavior, offering a platform-independent framework testable in programmable topoelectrical circuits, photonic lattices and digital quantum simulators.Spectral transitions in non-Hermitian lattices often arise from the competition between non-reciprocal skin accumulation and inter-component hybridization. In short-range systems formed by two coupled chains, this competition conventionally leads to the logarithmic critical-length law $N_c\sim\ln D$, where $D$ is the transverse separation between the chains. Here we show that long-range hoppings fundamentally reorganizes this critical behavior, producing a hierarchy of distinct scaling laws. When only the hybridization couplings are power-law decaying with exponent $α$, the onset becomes algebraic, $N_c\sim D^{α/3}$. When the hoppings within each chain are themselves also power-law decaying, in addition to the hybridization couplings, the system enters a scale-covariant regime for $α<2$, in which the criticality threshold equation depends only on the system aspect ratio $N_c/D$. At $α=2$ and beyond, this regime is followed by a marginal logarithmically corrected and algebraically corrected regimes, respectively. We identify two new non-local mechanisms that enable this unconventional critical hierarchy: a nonanalytic band-edge dispersion from long-range intra-chain hoppings, and parity-mixing hybridization induced by non-reciprocity. Our results show that nonlocality systematically removes the physical length scales i.e. skin depth underlying conventional critical non-Hermitian skin behavior, offering a platform-independent framework testable in programmable topoelectrical circuits, photonic lattices and digital quantum simulators.

cond-mat.mes-hallCross-listedarXiv:2608.02746Aug 5, 2026

Zero Curvature Condition for Quantum Criticality

Chaoming Song

Quantum criticality often lies beyond the scope of the conventional Landau paradigm, and a unifying framework has yet to emerge, due in part to the wide variety of quantum orders. We propose a geometric approach to quantum phase transitions that shifts focus from microscopic order to the competition between noncommuting operators. This competition is encoded in the boundary geometry of their expectation values, defining a quantum observable space. We show that quantum phase transitions occur at zero-curvature points on the quantum observable space boundary, signaling maximal commutativity and suggesting an underlying integrable structure at criticality.Quantum criticality often lies beyond the scope of the conventional Landau paradigm, and a unifying framework has yet to emerge, due in part to the wide variety of quantum orders. We propose a geometric approach to quantum phase transitions that shifts focus from microscopic order to the competition between noncommuting operators. This competition is encoded in the boundary geometry of their expectation values, defining a quantum observable space. We show that quantum phase transitions occur at zero-curvature points on the quantum observable space boundary, signaling maximal commutativity and suggesting an underlying integrable structure at criticality.

arXiv:2303.09591Physical Review LettersJun 2, 20252 citations

The competition between intertube hopping processes and density-density interactions is investigated in one-dimensional quantum dipolar bosons systems of

N

coupled tubes at zero temperature. Using a phenomenological bosonization approach, we show that the resulting competition leads to an exotic quantum phase transition described by a U(1) $\ifmmode\times\else\texttimes\fi{}$

{\mathbb{Z}}_{N}

conformal field theory with a fractional central charge. The emerging

{\mathbb{Z}}_{N}

parafermionic critical degrees of freedom are highly nontrivial in terms of the original atoms or polar molecules of the model. We further determine the main physical properties of the quantum-critical point in a double-tube system which has central charge

c=3/2

. In triple-tube systems, we show that the competition between the two antagonistic processes is related to the physics of the two-dimensional

{\mathbb{Z}}_{3}

chiral Potts model. This work opens the possibility to study the exotic properties of the

{\mathbb{Z}}_{N}

parafermions in the context of ultracold quantum Bose gases.

N_c\sim\ln D

, where

D

is the transverse separation between the chains. Here we show that long-range hoppings fundamentally reorganizes this critical behavior, producing a hierarchy of distinct scaling laws. When only the hybridization couplings are power-law decaying with exponent

α

, the onset becomes algebraic,

N_c\sim D^{α/3}

. When the hoppings within each chain are themselves also power-law decaying, in addition to the hybridization couplings, the system enters a scale-covariant regime for

α<2

, in which the criticality threshold equation depends only on the system aspect ratio

N_c/D

. At

α=2

and beyond, this regime is followed by a marginal logarithmically corrected and algebraically corrected regimes, respectively. We identify two new non-local mechanisms that enable this unconventional critical hierarchy: a nonanalytic band-edge dispersion from long-range intra-chain hoppings, and parity-mixing hybridization induced by non-reciprocity. Our results show that nonlocality systematically removes the physical length scales i.e. skin depth underlying conventional critical non-Hermitian skin behavior, offering a platform-independent framework testable in programmable topoelectrical circuits, photonic lattices and digital quantum simulators.