Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2105.01307 · Physical review. B./Physical review. B

Disorder lines, modulation, and partition function zeros in free fermion models

P. N. Timonin, Gennady Y. Chitov

Published Jul 6, 2021• 11 citations
arXivHTMLPDF

Abstract

The modulation is analyzed from the analytical properties of zeros of the free fermionic partition function on the complex plane of wave numbers. It is shown how these properties are related to the oscillations of correlation functions. This approach can be used for analysis of phase transitions with local or nonlocal order parameters, as well as for the disorder lines. We find an infinite cascade of disorder lines at finite temperature in the quantum $XY$ chain (equivalent to free fermions). The well-known ground-state factorization on the disorder line, and consequently disentanglement, is shown to follow directly from analytical properties of this model on the complex plane. From the quantum-classical correspondence the results for the chain are used to detect the disorder lines in several frustrated two-dimensional Ising models. The present formalism can be applied to other fermionic models in two and three spatial dimensions. In particular, we find the temperature-dependent Fermi wave vector of oscillations in the degenerate gas of three-dimensional fermions, which naturally leads in the limit $T\ensuremath{\rightarrow}0$ to the definition of the Fermi energy as the surface of quantum criticality. The modulation is a very common phenomenon, and it occurs in a large variety of models. The important point is that all these modulation transitions can be related to the complex zeros of partition functions, as done in the present study.

DOI
10.1103/physrevb.104.045106

Related papers

Disentanglement, disorder lines, and Majorana edge states in a solvable quantum chain

Gennady Y. Chitov, Karun Gadge, P. N. Timonin

We study the exactly solvable one-dimensional model: the dimerized $XY$ chain with uniform and staggered transverse fields, equivalent upon fermionization to the noninteracting dimerized Kitaev-Majorana chain with modulation. The model has three known gapped phases with local and nonlocal (string) orders, along with the gapless incommensurate (IC) phase in the $U(1)$ limit. The criticality is controlled by the properties of zeros of model's partition function, analytically continued onto the complex wave numbers. In the ground state they become complex zeros of the spectrum of the Hamiltonian. The analysis of those roots yields the phase diagram which contains continuous quantum phase transitions and weaker singularities known as disorder lines (DLs) or modulation transitions. The latter are shown to occur in two types: DLs of the first kind with continuous appearance of the IC oscillations, and DLs of the second kind corresponding to a jump of the wave number of oscillations. The salient property of zeros of the spectrum is that the ground state is shown to be separable (factorized), and the model is disentangled on a subset of the DLs. From analysis of those zeros we also find the Majorana edge states and their wave functions.

arXiv:2207.01147•Physical review. B./Physical review. B•Sep 28, 2022•7 citations

Infinite cascades of phase transitions in the classical Ising chain

P. N. Timonin, Gennady Y. Chitov

We report exact results on one of the best studied models in statistical physics: the classical antiferromagnetic Ising chain in a magnetic field. We show that the model possesses an infinite cascade of thermal phase transitions (also known as disorder lines or geometric phase transitions). The phase transition is signaled by a change of asymptotic behavior of the nonlocal string-string correlation functions when their monotonic decay becomes modulated by incommensurate oscillations. The transitions occur for rarefied (m-periodic) strings with arbitrary odd m. We propose a duality transformation which maps the Ising chain onto the m-leg Ising tube with nearest-neighbor couplings along the legs and the plaquette four-spin interactions of adjacent legs. Then the m-string correlation functions of the Ising chain are mapped onto the two-point spin-spin correlation functions along the legs of the m-leg tube. We trace the origin of these cascades of phase transitions to the lines of the Lee-Yang zeros of the Ising chain in m-periodic complex magnetic field, allowing us to relate these zeros to the observable (and potentially measurable) quantities.

arXiv:1707.06365•Physical review. E•Dec 15, 2017•8 citations

Fermi surface as a quantum critical manifold: Gaplessness, order parameter, and scaling in d dimensions

Gennady Y. Chitov

We study several models of $d$-dimensional fermions ($d=1,2,3$) with an emphasis on the properties of their gapless (metallic) phase. It occurs at $T=0$ as a continuous transition when zeros of the partition function reach the real range of parameters. Those zeros define the $(d\ensuremath{-}1)$-manifold of quantum criticality (Fermi surface). Its appearance or restructuring correspond to the Lifshitz transition. Such $(d\ensuremath{-}1)$-membrane breaks the symmetry of the momentum space, leading to gapless excitations, a hallmark of metallic phase. To probe quantitatively the gapless phase we introduce the geometric order parameter as $d$-volume of the Fermi sea. From analysis of the chain, ladder, and free fermions with different spectra, this proposal is shown to be consistent with scaling near the Lifshitz points of other quantities: correlation length, oscillation wavelength, susceptibilities, and entanglement. All the (hyper)scaling relations are satisfied. Two interacting cases of the Tomonaga-Luttinger ($d=1$) and the Fermi ($d=2,3$) liquids are analyzed, yielding the same universality classes as free fermions.

arXiv:2504.07014•Physical review. B./Physical review. B•Aug 9, 2025•0 citations

Finite Temperature Dynamical Correlations in Massive Integrable Quantum Field Theories

Fabian H. L. Eßler, Robert Konik

We consider the finite-temperature frequency and momentum dependent two-point functions of local operators in integrable quantum field theories. We focus on the case where the zero temperature correlation function is dominated by a delta-function line arising from the coherent propagation of single particle modes. Our specific examples are the two-point function of spin fields in the disordered phase of the quantum Ising and the O(3) nonlinear sigma models. We employ a Lehmann representation in terms of the known exact zero-temperature form factors to carry out a low-temperature expansion of two-point functions. We present two different but equivalent methods of regularizing the divergences present in the Lehmann expansion: one directly regulates the integral expressions of the squares of matrix elements in the infinite volume whereas the other operates through subtracting divergences in a large, finite volume. Our central results are that the temperature broadening of the line shape exhibits a pronounced asymmetry and a shift of the maximum upwards in energy ("temperature dependent gap"). The field theory results presented here describe the scaling limits of the dynamical structure factor in the quantum Ising and integer spin Heisenberg chains. We discuss the relevance of our results for the analysis of inelastic neutron scattering experiments on gapped spin chain systems such as CsNiCl3 and YBaNiO5.

•Aug 15, 2016•88 citations

Finite-temperature dynamical correlations in massive integrable quantum field theories

Fabian H L Essler, Robert M Konik

We consider the finite-temperature frequency and momentum dependent two-point functions of local operators in integrable quantum field theories. We focus on the case where the zero temperature correlation function is dominated by a delta-function line arising from the coherent propagation of single particle modes. Our specific examples are the two-point function of spin fields in the disordered phase of the quantum Ising and the O(3) nonlinear sigma models. We employ a Lehmann representation in terms of the known exact zero-temperature form factors to carry out a low-temperature expansion of two-point functions. We present two different but equivalent methods of regularizing the divergences present in the Lehmann expansion: one directly regulates the integral expressions of the squares of matrix elements in the infinite volume whereas the other operates through subtracting divergences in a large, finite volume. Our central results are that the temperature broadening of the line shape exhibits a pronounced asymmetry and a shift of the maximum upwards in energy ("temperature dependent gap"). The field theory results presented here describe the scaling limits of the dynamical structure factor in the quantum Ising and integer spin Heisenberg chains. We discuss the relevance of our results for the analysis of inelastic neutron scattering experiments on gapped spin chain systems such as CsNiCl3 and YBaNiO5.

arXiv:0907.0779•Journal of Statistical Mechanics Theory and Experiment•Sep 30, 2009•62 citations

Exactly Solvable Disorder-free Quantum Breakdown Model: Spectrum, Thermodynamics, and Dynamics

Kinya Guan, Hosho Katsura

We introduce and study a disorder-free version of the quantum breakdown model with all-to-all interactions. The Hamiltonian factorizes into the product of the zero-momentum-mode occupation number and a quadratic Hamiltonian including only pairing terms. This structure makes the model exactly solvable and produces an extensive zero-energy degeneracy. We obtain the spectrum and partition function in closed form and analyze the thermodynamics, spectral form factor, two-point functions, and a regulated out-of-time-ordered correlator (OTOC). The OTOC is time independent at leading order in the large-$N$ limit, while its time-dependent contributions first appear at order $1/N$. At infinite temperature, the time-dependent correction reduces to a two-point function arising from the zero-momentum component of the fermion operators. The model therefore provides a controlled setting for isolating the consequences of the breakdown interaction in the absence of disorder, spatial structure, and environmental coupling.

arXiv:2603.17379•Mar 18, 2026•0 citations
The modulation is analyzed from the analytical properties of zeros of the free fermionic partition function on the complex plane of wave numbers. It is shown how these properties are related to the oscillations of correlation functions. This approach can be used for analysis of phase transitions with local or nonlocal order parameters, as well as for the disorder lines. We find an infinite cascade of disorder lines at finite temperature in the quantum XYXYXY chain (equivalent to free fermions). The well-known ground-state factorization on the disorder line, and consequently disentanglement, is shown to follow directly from analytical properties of this model on the complex plane. From the quantum-classical correspondence the results for the chain are used to detect the disorder lines in several frustrated two-dimensional Ising models. The present formalism can be applied to other fermionic models in two and three spatial dimensions. In particular, we find the temperature-dependent Fermi wave vector of oscillations in the degenerate gas of three-dimensional fermions, which naturally leads in the limit T\ensuremath→0T\ensuremath{\rightarrow}0T\ensuremath→0 to the definition of the Fermi energy as the surface of quantum criticality. The modulation is a very common phenomenon, and it occurs in a large variety of models. The important point is that all these modulation transitions can be related to the complex zeros of partition functions, as done in the present study.
We study the exactly solvable one-dimensional model: the dimerized XYXYXY chain with uniform and staggered transverse fields, equivalent upon fermionization to the noninteracting dimerized Kitaev-Majorana chain with modulation. The model has three known gapped phases with local and nonlocal (string) orders, along with the gapless incommensurate (IC) phase in the U(1)U(1)U(1) limit. The criticality is controlled by the properties of zeros of model's partition function, analytically continued onto the complex wave numbers. In the ground state they become complex zeros of the spectrum of the Hamiltonian. The analysis of those roots yields the phase diagram which contains continuous quantum phase transitions and weaker singularities known as disorder lines (DLs) or modulation transitions. The latter are shown to occur in two types: DLs of the first kind with continuous appearance of the IC oscillations, and DLs of the second kind corresponding to a jump of the wave number of oscillations. The salient property of zeros of the spectrum is that the ground state is shown to be separable (factorized), and the model is disentangled on a subset of the DLs. From analysis of those zeros we also find the Majorana edge states and their wave functions.
We report exact results on one of the best studied models in statistical physics: the classical antiferromagnetic Ising chain in a magnetic field. We show that the model possesses an infinite cascade of thermal phase transitions (also known as disorder lines or geometric phase transitions). The phase transition is signaled by a change of asymptotic behavior of the nonlocal string-string correlation functions when their monotonic decay becomes modulated by incommensurate oscillations. The transitions occur for rarefied (m-periodic) strings with arbitrary odd m. We propose a duality transformation which maps the Ising chain onto the m-leg Ising tube with nearest-neighbor couplings along the legs and the plaquette four-spin interactions of adjacent legs. Then the m-string correlation functions of the Ising chain are mapped onto the two-point spin-spin correlation functions along the legs of the m-leg tube. We trace the origin of these cascades of phase transitions to the lines of the Lee-Yang zeros of the Ising chain in m-periodic complex magnetic field, allowing us to relate these zeros to the observable (and potentially measurable) quantities.
We study several models of ddd-dimensional fermions (d=1,2,3d=1,2,3d=1,2,3) with an emphasis on the properties of their gapless (metallic) phase. It occurs at T=0T=0T=0 as a continuous transition when zeros of the partition function reach the real range of parameters. Those zeros define the $(d\ensuremath{-}1)$-manifold of quantum criticality (Fermi surface). Its appearance or restructuring correspond to the Lifshitz transition. Such $(d\ensuremath{-}1)$-membrane breaks the symmetry of the momentum space, leading to gapless excitations, a hallmark of metallic phase. To probe quantitatively the gapless phase we introduce the geometric order parameter as ddd-volume of the Fermi sea. From analysis of the chain, ladder, and free fermions with different spectra, this proposal is shown to be consistent with scaling near the Lifshitz points of other quantities: correlation length, oscillation wavelength, susceptibilities, and entanglement. All the (hyper)scaling relations are satisfied. Two interacting cases of the Tomonaga-Luttinger (d=1d=1d=1) and the Fermi (d=2,3d=2,3d=2,3) liquids are analyzed, yielding the same universality classes as free fermions.
We consider the finite-temperature frequency and momentum dependent two-point functions of local operators in integrable quantum field theories. We focus on the case where the zero temperature correlation function is dominated by a delta-function line arising from the coherent propagation of single particle modes. Our specific examples are the two-point function of spin fields in the disordered phase of the quantum Ising and the O(3) nonlinear sigma models. We employ a Lehmann representation in terms of the known exact zero-temperature form factors to carry out a low-temperature expansion of two-point functions. We present two different but equivalent methods of regularizing the divergences present in the Lehmann expansion: one directly regulates the integral expressions of the squares of matrix elements in the infinite volume whereas the other operates through subtracting divergences in a large, finite volume. Our central results are that the temperature broadening of the line shape exhibits a pronounced asymmetry and a shift of the maximum upwards in energy ("temperature dependent gap"). The field theory results presented here describe the scaling limits of the dynamical structure factor in the quantum Ising and integer spin Heisenberg chains. We discuss the relevance of our results for the analysis of inelastic neutron scattering experiments on gapped spin chain systems such as CsNiCl3 and YBaNiO5.
We consider the finite-temperature frequency and momentum dependent two-point functions of local operators in integrable quantum field theories. We focus on the case where the zero temperature correlation function is dominated by a delta-function line arising from the coherent propagation of single particle modes. Our specific examples are the two-point function of spin fields in the disordered phase of the quantum Ising and the O(3) nonlinear sigma models. We employ a Lehmann representation in terms of the known exact zero-temperature form factors to carry out a low-temperature expansion of two-point functions. We present two different but equivalent methods of regularizing the divergences present in the Lehmann expansion: one directly regulates the integral expressions of the squares of matrix elements in the infinite volume whereas the other operates through subtracting divergences in a large, finite volume. Our central results are that the temperature broadening of the line shape exhibits a pronounced asymmetry and a shift of the maximum upwards in energy ("temperature dependent gap"). The field theory results presented here describe the scaling limits of the dynamical structure factor in the quantum Ising and integer spin Heisenberg chains. We discuss the relevance of our results for the analysis of inelastic neutron scattering experiments on gapped spin chain systems such as CsNiCl3 and YBaNiO5.
We introduce and study a disorder-free version of the quantum breakdown model with all-to-all interactions. The Hamiltonian factorizes into the product of the zero-momentum-mode occupation number and a quadratic Hamiltonian including only pairing terms. This structure makes the model exactly solvable and produces an extensive zero-energy degeneracy. We obtain the spectrum and partition function in closed form and analyze the thermodynamics, spectral form factor, two-point functions, and a regulated out-of-time-ordered correlator (OTOC). The OTOC is time independent at leading order in the large-NNN limit, while its time-dependent contributions first appear at order 1/N1/N1/N. At infinite temperature, the time-dependent correction reduces to a two-point function arising from the zero-momentum component of the fermion operators. The model therefore provides a controlled setting for isolating the consequences of the breakdown interaction in the absence of disorder, spatial structure, and environmental coupling.