arXiv:1407.3924 · Communications in Mathematical Physics

C 1-Classification of Gapped Parent Hamiltonians of Quantum Spin Chains

Sven Bachmann, Yoshiko Ogata

Published Apr 9, 2015• 14 citations

arXiv HTML PDF

DOI: 10.1007/s00220-015-2350-8

Related papers

A Class of Asymmetric Gapped Hamiltonians on Quantum Spin Chains and its Characterization I

Yoshiko Ogata

arXiv:1510.07753Communications in Mathematical PhysicsJun 20, 201622 citations

A Class of Asymmetric Gapped Hamiltonians on Quantum Spin Chains and its Characterization II

Yoshiko Ogata

arXiv:1510.07751Communications in Mathematical PhysicsJun 29, 201613 citations

A Class of Asymmetric Gapped Hamiltonians on Quantum Spin Chains and its Characterization III

Yoshiko Ogata

arXiv:1606.05508Communications in Mathematical PhysicsJan 23, 201711 citations

Parent Hamiltonians of Ergodic Matrix Product States

Owen Ekblad, Eloy Moreno-Nadales, Eric B. Roon +1 more

math-phCross-listedarXiv:2607.01446Jul 3, 20260 citations

C^1-Classification of gapped parent Hamiltonians of quantum spin chains with local symmetry

Yoshiko Ogata

arXiv:1602.08839Feb 29, 20161 citation

Exact results for SU(3) spin chains: Trimer states, valence bond solids, and their parent Hamiltonians

Martin Greiter, Stephan Rachel, Dirk Schuricht

arXiv:cond-mat/0701354Physical Review BFeb 1, 200760 citations

arXiv:1407.3924 · Communications in Mathematical Physics

C 1-Classification of Gapped Parent Hamiltonians of Quantum Spin Chains

Sven Bachmann, Yoshiko Ogata

Published Apr 9, 2015• 14 citations

arXiv HTML PDF

DOI: 10.1007/s00220-015-2350-8

Related papers

A Class of Asymmetric Gapped Hamiltonians on Quantum Spin Chains and its Characterization I

Yoshiko Ogata

arXiv:1510.07753Communications in Mathematical PhysicsJun 20, 201622 citations

A Class of Asymmetric Gapped Hamiltonians on Quantum Spin Chains and its Characterization II

Yoshiko Ogata

arXiv:1510.07751Communications in Mathematical PhysicsJun 29, 201613 citations

A Class of Asymmetric Gapped Hamiltonians on Quantum Spin Chains and its Characterization III

Yoshiko Ogata

arXiv:1606.05508Communications in Mathematical PhysicsJan 23, 201711 citations

Parent Hamiltonians of Ergodic Matrix Product States

Owen Ekblad, Eloy Moreno-Nadales, Eric B. Roon +1 more

Matrix product states (MPS) are quintessential examples of frustration-free gapped ground states of local interactions called parent Hamiltonians. In this work, we investigate parent Hamiltonians for a class of ergodic matrix product states (EMPS), which are MPS defined by site-dependent random tensors $\{X_j^{[k]}\}_{j=1}^D$ which are homogeneously distributed at every site $k$ in the spin chain. Here, the EMPS are not translation-invariant but rather statistically translation-invariant. Under a mild injectivity assumption, we show the thermodynamic limit of an EMPS is the unique frustration-free ground state of a parent Hamiltonian on the whole spin chain, which, depending on the statistical properties of the EMPS, may or may not be finite-range. In contrast to the translation-invariant regime, these Hamiltonians need not be gapped. Nevertheless, applying the martingale method while keeping track of local statistics gives conditions for a gap, in addition to pointing towards why there need not be a gap in general. We include examples of EMPS both with and without spectral gaps to illustrate our results.Matrix product states (MPS) are quintessential examples of frustration-free gapped ground states of local interactions called parent Hamiltonians. In this work, we investigate parent Hamiltonians for a class of ergodic matrix product states (EMPS), which are MPS defined by site-dependent random tensors $\{X_j^{[k]}\}_{j=1}^D$ which are homogeneously distributed at every site $k$ in the spin chain. Here, the EMPS are not translation-invariant but rather statistically translation-invariant. Under a mild injectivity assumption, we show the thermodynamic limit of an EMPS is the unique frustration-free ground state of a parent Hamiltonian on the whole spin chain, which, depending on the statistical properties of the EMPS, may or may not be finite-range. In contrast to the translation-invariant regime, these Hamiltonians need not be gapped. Nevertheless, applying the martingale method while keeping track of local statistics gives conditions for a gap, in addition to pointing towards why there need not be a gap in general. We include examples of EMPS both with and without spectral gaps to illustrate our results.

math-phCross-listedarXiv:2607.01446Jul 3, 20260 citations

C^1-Classification of gapped parent Hamiltonians of quantum spin chains with local symmetry

Yoshiko Ogata

We consider the family of gapped Hamiltonians introduced in [FNW] on the quantum spin chains $\bigotimes_{\mathbb Z}\mathop{\mathrm{M}}\nolimits_n$ , with local symmetry given by a group $G$ . The $G$ -symmetric gapped Hamiltonians are given by triples $(k,u,V)$ , where $u$ is a projective unitary representation of $G$ on a finite dimensional space ${\mathbb C}^k$ , and $V$ is an isometry from ${\mathbb C}^k$ to ${\mathbb C}^n\otimes {\mathbb C}^k$ . We show that Hamiltonians $H_0,H_1$ , given by the triples $(k^{(0)}, u^{(0)}, V^{(0)})$ and $(k^{(1)}, u^{(1)}, V^{(1)})$ are $C^1$ -equivalent if the projective representations $u^{(0)}$ and $u^{(1)}$ are unitary equivalent.We consider the family of gapped Hamiltonians introduced in [FNW] on the quantum spin chains $\bigotimes_{\mathbb Z}\mathop{\mathrm{M}}\nolimits_n$, with local symmetry given by a group $G$. The $G$-symmetric gapped Hamiltonians are given by triples $(k,u,V)$, where $u$ is a projective unitary representation of $G$ on a finite dimensional space ${\mathbb C}^k$, and $V$ is an isometry from ${\mathbb C}^k$ to ${\mathbb C}^n\otimes {\mathbb C}^k$. We show that Hamiltonians $H_0,H_1$, given by the triples $(k^{(0)}, u^{(0)}, V^{(0)})$ and $(k^{(1)}, u^{(1)}, V^{(1)})$ are $C^1$-equivalent if the projective representations $u^{(0)}$ and $u^{(1)}$ are unitary equivalent.

arXiv:1602.08839Feb 29, 20161 citation

Exact results for SU(3) spin chains: Trimer states, valence bond solids, and their parent Hamiltonians

Martin Greiter, Stephan Rachel, Dirk Schuricht

We introduce several exact models for SU(3) spin chains: (1) a translationally invariant parent Hamiltonian involving four-site interactions for the trimer chain, with a threefold degenerate ground state. We provide numerical evidence that the elementary excitations of this model transform under representation $\overline{\mathbf{3}}$ of SU(3) if the original spins of the model transform under representation 3. (2) a family of parent Hamiltonians for valence bond solids of SU(3) chains with spin representations 6, 10, and 8 on each lattice site. We argue that of these three models, only the latter two exhibit spinon confinement, and a Haldane gap in the excitation spectrum.We introduce several exact models for SU(3) spin chains: (1) a translationally invariant parent Hamiltonian involving four-site interactions for the trimer chain, with a threefold degenerate ground state. We provide numerical evidence that the elementary excitations of this model transform under representation $\overline{\mathbf{3}}$ of SU(3) if the original spins of the model transform under representation 3. (2) a family of parent Hamiltonians for valence bond solids of SU(3) chains with spin representations 6, 10, and 8 on each lattice site. We argue that of these three models, only the latter two exhibit spinon confinement, and a Haldane gap in the excitation spectrum.

arXiv:cond-mat/0701354Physical Review BFeb 1, 200760 citations

\{X_j^{[k]}\}_{j=1}^D

which are homogeneously distributed at every site

k

in the spin chain. Here, the EMPS are not translation-invariant but rather statistically translation-invariant. Under a mild injectivity assumption, we show the thermodynamic limit of an EMPS is the unique frustration-free ground state of a parent Hamiltonian on the whole spin chain, which, depending on the statistical properties of the EMPS, may or may not be finite-range. In contrast to the translation-invariant regime, these Hamiltonians need not be gapped. Nevertheless, applying the martingale method while keeping track of local statistics gives conditions for a gap, in addition to pointing towards why there need not be a gap in general. We include examples of EMPS both with and without spectral gaps to illustrate our results.

We consider the family of gapped Hamiltonians introduced in [FNW] on the quantum spin chains

\bigotimes_{\mathbb Z}\mathop{\mathrm{M}}\nolimits_n

, with local symmetry given by a group

G

. The

G

-symmetric gapped Hamiltonians are given by triples

(k,u,V)

, where

u

is a projective unitary representation of

G

on a finite dimensional space

{\mathbb C}^k

, and

V

is an isometry from

{\mathbb C}^k

{\mathbb C}^n\otimes {\mathbb C}^k

. We show that Hamiltonians

H_0,H_1

, given by the triples

(k^{(0)}, u^{(0)}, V^{(0)})

and

(k^{(1)}, u^{(1)}, V^{(1)})

are

C^1

-equivalent if the projective representations

u^{(0)}

and

u^{(1)}

are unitary equivalent.

\overline{\mathbf{3}}

of SU(3) if the original spins of the model transform under representation 3. (2) a family of parent Hamiltonians for valence bond solids of SU(3) chains with spin representations 6, 10, and 8 on each lattice site. We argue that of these three models, only the latter two exhibit spinon confinement, and a Haldane gap in the excitation spectrum.