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:1607.08594

Inversion symmetry breaking and criticality in free fermionic lattices

Zoltán Kádár

Published Jul 28, 2016• 0 citations
arXivHTMLPDF

Abstract

We describe the connection between inversion symmetry breaking and criticality in free fermionic lattice models. It is shown that for translation-invariant spinless fermions, the breaking of this symmetry in the ground state implies criticality, i.e., the existence of long-range correlations and the vanishing of the spectral gap; while for models with spin, only the asymmetry of the spin-averaged covariance matrix implies a similar conclusion. Our results are proved by introducing invariants under global translation-invariant free fermion quenches. Using this result, we identify a set of models where the generalized Hartree-Fock approximation must break down.

DOI
10.48550/arxiv.1607.08594

Related papers

On Spectral Gap,U(1) Symmetry and Split Property in Quantum Spin Chains

Taku Matsui

We consider a class of ground states for quantum spin chains on an integer lattice. First we show that presence of the spectral gap between the ground state energy and the rest of spectrum implies the split property of certain subsystems.As a corollary, we show that gapless excitation exists for spinless Fermion if the pure gauge invariant ground state is non-trivial and translationally invariant.

arXiv:0808.1537•Aug 11, 2008•1 citation

Spectral gap and exponential decay of correlations

Matthew B. Hastings, Tohru Koma

We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short-range interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, then the nonvanishing spectral gap implies exponential decay of the corresponding correlation. When two observables commute with each other at large distance, the connected correlation function decays exponentially under the gap assumption. If the observables behave as a vector under the U(1) rotation of a global symmetry of the system, we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself, rather than just the connected correlation function, decays exponentially under the gap assumption on a lattice with a certain self-similarity in (fractal) dimensions D < 2. In particular, if the system is translationally invariant in one of the spatial directions, then this self-similarity condition is automatically satisfied. We also treat systems with long-range, power-law decaying interactions. Spectral Gap and Exponential Decay of Correlations 2 1

•Sep 13, 2013•333 citations

The Spectral Gap for Some Spin Chains With Discrete Symmetry Breaking

Bruno Nachtergaele

We prove that for any finite set of generalized valence bond solid (GVBS) states of a quantum spin chain there exists a translation invariant finite-range Hamiltonian for which this set is the set of ground states. This result implies that there are GVBS models with arbitrary broken discrete symmetries that are described as combinations of lattice translations, lattice reflections, and local unitary or anti-unitary transformations. We also show that all GVBS models that satisfy some natural conditions have a spectral gap. The existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics. This general scheme is interesting in its own right and therefore, although the basic idea is not new, we present it in a system-independent setting. The results are illustrated with an number of examples. Keywords: quantum spin chains, Heisenberg model, spectral gap, symmetr...

•Jan 1, 1994•80 citations

Universal construction of order parameters for translation-invariant quantum lattice systems with symmetry-breaking order

Jinhua Liu, Qian‐Qian Shi, Honglei Wang +2 more

For any translation-invariant quantum lattice system with a symmetry group G, we propose a practical and universal construction of order parameters which identify quantum phase transitions with symmetry-breaking order. They are defined in terms of the fidelity between a ground state and its symmetry-transformed counterpart, and are computed through tensor network representations of the ground-state wave function. To illustrate our scheme, we consider three quantum systems on an infinite lattice in one spatial dimension, namely, the quantum Ising model in a transverse magnetic field, the quantum spin-1/2XYX model in an external magnetic field, and the quantum spin-1 XXZ model with single-ion anisotropy. All these models have symmetry group Z(2) and exhibit broken-symmetry phases. We also discuss the role of the order parameters in identifying factorized states.

arXiv:0909.3031•Physical Review E•Aug 15, 2012•4 citations

SU(2) symmetry breaking in quantum spin chain

Anilesh Mohari

Though our primary motivation is to study properties of the ground states of a Hamiltonian for quantum spin chain, in this exposition we propose a general methodology valid for a translation invariant state. To that end we study the associated Popescu systems [Po,BJKW] representing the translation invariant state and find an useful criteria for the state to be pure. As a simple consequence of this criteria we prove a surprising result which says that there exists no translation invariant lattice symmetric and SU(2)-gauge invariant pure state for half-odd integer spin chain. As a consequence of this result we prove that ground state for half-odd integer spin chain is not unique for any SU(2)-gauge invariant translation and lattice symmetric Hamiltonian. In other words this exhibits that there is a phase transition in the ground state for anti-ferromagnetic isotropic Heisenberg half-odd integer spin chain. 2 It has been known in physics literature that the spin 1

arXiv:math-ph/0509049•Jan 1, 2007•2 citations

Off-diagonal long-range order implies vanishing charge gap

Hal Tasaki, Haruki Watanabe

For a large class of quantum many-body systems with U(1) symmetry, we prove a general inequality that relates the (off-diagonal) long-range order with the charge gap. For a system of bosons or fermions on a lattice or in the continuum, the inequality implies that a ground state with off-diagonal long-range order inevitably has a vanishing charge gap, and hence is characterized by nonzero charge susceptibility. For a quantum spin system, the inequality implies that a ground state within a magnetization plateau cannot have transverse long-range order.

arXiv:2105.10692•Physical review. B./Physical review. B•Nov 8, 2021•1 citation
We describe the connection between inversion symmetry breaking and criticality in free fermionic lattice models. It is shown that for translation-invariant spinless fermions, the breaking of this symmetry in the ground state implies criticality, i.e., the existence of long-range correlations and the vanishing of the spectral gap; while for models with spin, only the asymmetry of the spin-averaged covariance matrix implies a similar conclusion. Our results are proved by introducing invariants under global translation-invariant free fermion quenches. Using this result, we identify a set of models where the generalized Hartree-Fock approximation must break down.
We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short-range interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, then the nonvanishing spectral gap implies exponential decay of the corresponding correlation. When two observables commute with each other at large distance, the connected correlation function decays exponentially under the gap assumption. If the observables behave as a vector under the U(1) rotation of a global symmetry of the system, we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself, rather than just the connected correlation function, decays exponentially under the gap assumption on a lattice with a certain self-similarity in (fractal) dimensions D < 2. In particular, if the system is translationally invariant in one of the spatial directions, then this self-similarity condition is automatically satisfied. We also treat systems with long-range, power-law decaying interactions. Spectral Gap and Exponential Decay of Correlations 2 1
We prove that for any finite set of generalized valence bond solid (GVBS) states of a quantum spin chain there exists a translation invariant finite-range Hamiltonian for which this set is the set of ground states. This result implies that there are GVBS models with arbitrary broken discrete symmetries that are described as combinations of lattice translations, lattice reflections, and local unitary or anti-unitary transformations. We also show that all GVBS models that satisfy some natural conditions have a spectral gap. The existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics. This general scheme is interesting in its own right and therefore, although the basic idea is not new, we present it in a system-independent setting. The results are illustrated with an number of examples. Keywords: quantum spin chains, Heisenberg model, spectral gap, symmetr...
For any translation-invariant quantum lattice system with a symmetry group G, we propose a practical and universal construction of order parameters which identify quantum phase transitions with symmetry-breaking order. They are defined in terms of the fidelity between a ground state and its symmetry-transformed counterpart, and are computed through tensor network representations of the ground-state wave function. To illustrate our scheme, we consider three quantum systems on an infinite lattice in one spatial dimension, namely, the quantum Ising model in a transverse magnetic field, the quantum spin-1/2XYX model in an external magnetic field, and the quantum spin-1 XXZ model with single-ion anisotropy. All these models have symmetry group Z(2) and exhibit broken-symmetry phases. We also discuss the role of the order parameters in identifying factorized states.
Though our primary motivation is to study properties of the ground states of a Hamiltonian for quantum spin chain, in this exposition we propose a general methodology valid for a translation invariant state. To that end we study the associated Popescu systems [Po,BJKW] representing the translation invariant state and find an useful criteria for the state to be pure. As a simple consequence of this criteria we prove a surprising result which says that there exists no translation invariant lattice symmetric and SU(2)-gauge invariant pure state for half-odd integer spin chain. As a consequence of this result we prove that ground state for half-odd integer spin chain is not unique for any SU(2)-gauge invariant translation and lattice symmetric Hamiltonian. In other words this exhibits that there is a phase transition in the ground state for anti-ferromagnetic isotropic Heisenberg half-odd integer spin chain. 2 It has been known in physics literature that the spin 1
For a large class of quantum many-body systems with U(1) symmetry, we prove a general inequality that relates the (off-diagonal) long-range order with the charge gap. For a system of bosons or fermions on a lattice or in the continuum, the inequality implies that a ground state with off-diagonal long-range order inevitably has a vanishing charge gap, and hence is characterized by nonzero charge susceptibility. For a quantum spin system, the inequality implies that a ground state within a magnetization plateau cannot have transverse long-range order.