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:math-ph/0509049

SU(2) symmetry breaking in quantum spin chain

Anilesh Mohari

Published Jan 1, 2007• 2 citations
arXivHTMLPDF

Abstract

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

Related papers

Spontaneous $SU(2)$ symmetry breaking in ground state of quantum spin chain

Anilesh Mohari

In this paper we prove that there exists no translation invariant pure state of $\clb=\otimes_{\mathbb{Z}}M_d(\IC)$ that is real, lattice symmetric, reflection positive with a twist and SU(2) invariant when $d$ is an even integer. This in particular says that Heisenberg iso-spin anti-ferromagnetic model with ${1 \over 2}$-odd integer spin degrees of freedom admits spontaneous symmetry breaking at its ground states.

•Sep 21, 2005•2 citations

Detailed balance and split property in quantum spin chain

Anilesh Mohari

In this exposition we investigate further the general methodology proposed in [Mo2] to study properties of the ground states of a translation invariant Hamiltonian for one lattice dimensional quantum spin chain A = ⊗ ZMd, where Md is the matrix of d × d complex matrices. We introduce a notion of quantum detailed balance [Mo1] for a translation invariant state on A and prove that such a pure state is uniformly mixing [BR,Ma2] if and only if the lattice space correlation functions decay exponentially. Furthermore we also prove that a pure lattice symmetric, translation and SU(2) gauge invariant state give rise to a canonical Popescu systems acting on a finite dimensional Hilbert space and thus the lattice space correlation functions of the pure state decay exponentially. As a consequence of these results we conclude that if the ground states for an integer spin SU(2) invariant (2s + 1 = d) detailed balanced Hamiltonian is unique then the state is split. In particular if the ground state for integer

arXiv:math-ph/0505035•May 11, 2005•3 citations

Spontaneous SU2(C) symmetry breaking in the ground states of quantum spin chain

Anilesh Mohari

In this paper, we have proved that there exists no translation invariant pure state of M=⊗k∈Z Md(k)(C) that is real, lattice symmetric with a certain twist and SU2(C) invariant for any even integer d ≥ 2. In particular, this result also shows that the Heisenberg iso-spin anti-ferromagnetic model with 12-odd integer spin degrees of freedom does not admit a unique ground state.

Journal of Mathematical Physics•Nov 1, 2018•3 citations

$SU_2(\mathbb{C})$ symmetry in quantum spin chain ground states and Haldane's conjecture

Anilesh Mohari

In this paper, we prove that any translation and $SU_2(\IC)$-invariant pure state of $\IM=\otimes_{k \in \IZ}\!M^{(k)}_d(\IC)$, that is also real, lattice symmetric and reflection positive with a certain twist $r_0 \in U_d(\IC)$, is finitely correlated and its two-point spatial correlation function decays exponentially whenever $d$ is an odd integer. In particular, the Heisenberg iso-spin anti-ferromagnetic integer spin model admits unique low temperature limiting ground state and its spatial correlation function decays exponentially. The unique low temperature limiting ground state of the Hamiltonian is determined by the unique solution to Clebsch-Gordon inter-twinning isometry between two representations of $SU_2(\IC)$.

arXiv:2112.13976•Dec 28, 2021•0 citations

Translation invariant pure state and its split property

Anilesh Mohari

We prove Haag duality property of any translation invariant pure state on $\clb = \otimes_{\IZ}M_d(C), \;d \ge 2$, where $M_d(C)$ is the set of $d \times d$ dimensional matrices over field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on $\clb$. This result makes it possible to study such a pure state with additional symmetry. We prove that exponentially decaying two point spacial correlation function of a real lattice symmetric reflection positive translation invariant pure state is a split state. Further there exists no translation invariant pure state on $\clb$ that is real, lattice symmetric, refection positive and $su(2)$ invariant when $d$ is an even integer. This in particular says that Heisenberg iso-spin anti-ferromagnets model for 1/2-odd integer spin degrees of freedom admits spontaneous symmetry breaking at it's ground states

arXiv:0904.2104•Apr 14, 2009•0 citations

Translation-symmetry-protected topological orders in quantum spin systems

Su-Peng Kou, Xiao-Gang Wen

In this paper we systematically study a simple class of translation-symmetry protected topological orders in quantum spin systems using slave-particle approach. The lattice spin systems are translation invariant, but may break any other symmetries. We consider topologically ordered ground states that do not spontaneously break any symmetry. Those states can be described by $Z2A$ or $Z2B$ projective symmetry group. We find that the $Z2A$ translation symmetric topological orders can still be divided into 16 subclasses corresponding to 16 translation-symmetry protected topological orders. We introduced four ${Z}_{2}$ topological indices ${\ensuremath{\zeta}}_{\mathbf{k}}=0,1$ at $\mathbf{k}=(0,0)$, $(0,\ensuremath{\pi})$, $(\ensuremath{\pi},0)$, and $(\ensuremath{\pi},\ensuremath{\pi})$ to characterize those 16 topological orders on square lattice. We calculated the topological degeneracies and crystal momenta for those 16 topological phases on even-by-even, even-by-odd, odd-by-even, and odd-by-odd lattices, which allows us to physically measure such topological orders. We predict the appearance of gapless fermionic excitations at the quantum phase transitions between those symmetry protected topological orders. Our result can be generalized to any dimensions. We find 256 translation-symmetry protected $Z2A$ topological orders for a system on three-dimensional cubic lattice.

arXiv:0907.4537•Physical Review B•Dec 4, 2009•55 citations
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
Spontaneous SU(2)SU(2)SU(2) symmetry breaking in ground state of quantum spin chain
In this paper we prove that there exists no translation invariant pure state of $\clb=\otimes_{\mathbb{Z}}M_d(\IC)$ that is real, lattice symmetric, reflection positive with a twist and SU(2) invariant when ddd is an even integer. This in particular says that Heisenberg iso-spin anti-ferromagnetic model with 12{1 \over 2}21​-odd integer spin degrees of freedom admits spontaneous symmetry breaking at its ground states.
In this exposition we investigate further the general methodology proposed in [Mo2] to study properties of the ground states of a translation invariant Hamiltonian for one lattice dimensional quantum spin chain A = ⊗ ZMd, where Md is the matrix of d × d complex matrices. We introduce a notion of quantum detailed balance [Mo1] for a translation invariant state on A and prove that such a pure state is uniformly mixing [BR,Ma2] if and only if the lattice space correlation functions decay exponentially. Furthermore we also prove that a pure lattice symmetric, translation and SU(2) gauge invariant state give rise to a canonical Popescu systems acting on a finite dimensional Hilbert space and thus the lattice space correlation functions of the pure state decay exponentially. As a consequence of these results we conclude that if the ground states for an integer spin SU(2) invariant (2s + 1 = d) detailed balanced Hamiltonian is unique then the state is split. In particular if the ground state for integer
SU2(C)SU_2(\mathbb{C})SU2​(C) symmetry in quantum spin chain ground states and Haldane's conjecture
In this paper, we prove that any translation and $SU_2(\IC)$-invariant pure state of $\IM=\otimes_{k \in \IZ}\!M^{(k)}_d(\IC)$, that is also real, lattice symmetric and reflection positive with a certain twist $r_0 \in U_d(\IC)$, is finitely correlated and its two-point spatial correlation function decays exponentially whenever ddd is an odd integer. In particular, the Heisenberg iso-spin anti-ferromagnetic integer spin model admits unique low temperature limiting ground state and its spatial correlation function decays exponentially. The unique low temperature limiting ground state of the Hamiltonian is determined by the unique solution to Clebsch-Gordon inter-twinning isometry between two representations of $SU_2(\IC)$.
We prove Haag duality property of any translation invariant pure state on $\clb = \otimes_{\IZ}M_d(C), \;d \ge 2$, where Md(C)M_d(C)Md​(C) is the set of d×dd \times dd×d dimensional matrices over field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on $\clb$. This result makes it possible to study such a pure state with additional symmetry. We prove that exponentially decaying two point spacial correlation function of a real lattice symmetric reflection positive translation invariant pure state is a split state. Further there exists no translation invariant pure state on $\clb$ that is real, lattice symmetric, refection positive and su(2)su(2)su(2) invariant when ddd is an even integer. This in particular says that Heisenberg iso-spin anti-ferromagnets model for 1/2-odd integer spin degrees of freedom admits spontaneous symmetry breaking at it's ground states
In this paper we systematically study a simple class of translation-symmetry protected topological orders in quantum spin systems using slave-particle approach. The lattice spin systems are translation invariant, but may break any other symmetries. We consider topologically ordered ground states that do not spontaneously break any symmetry. Those states can be described by Z2AZ2AZ2A or Z2BZ2BZ2B projective symmetry group. We find that the Z2AZ2AZ2A translation symmetric topological orders can still be divided into 16 subclasses corresponding to 16 translation-symmetry protected topological orders. We introduced four Z2{Z}_{2}Z2​ topological indices ${\ensuremath{\zeta}}_{\mathbf{k}}=0,1$ at k=(0,0)\mathbf{k}=(0,0)k=(0,0), $(0,\ensuremath{\pi})$, $(\ensuremath{\pi},0)$, and $(\ensuremath{\pi},\ensuremath{\pi})$ to characterize those 16 topological orders on square lattice. We calculated the topological degeneracies and crystal momenta for those 16 topological phases on even-by-even, even-by-odd, odd-by-even, and odd-by-odd lattices, which allows us to physically measure such topological orders. We predict the appearance of gapless fermionic excitations at the quantum phase transitions between those symmetry protected topological orders. Our result can be generalized to any dimensions. We find 256 translation-symmetry protected Z2AZ2AZ2A topological orders for a system on three-dimensional cubic lattice.