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:1310.1876 · Physical Review B

Hierarchy of exactly solvable spin- 12 chains with so(N)1 critical points

Ville Lahtinen, Teresia Månsson, Eddy Ardonne

Published Jan 10, 2014• 14 citations
arXivHTMLPDF

Abstract

We construct a hierarchy of exactly solvable spin-1/2 chains with $so{(N)}_{1}$ critical points. Our construction is based on the framework of condensate-induced transitions between topological phases. We employ this framework to construct a Hamiltonian term that couples $N$ transverse field Ising chains such that the resulting theory is critical and described by the $so{(N)}_{1}$ conformal field theory. By employing spin duality transformations, we then cast these spin chains for arbitrary $N$ into translationally invariant forms that all allow exact solution by the means of a Jordan-Wigner transformation. For odd $N$ our models generalize the phase diagram of the transverse field Ising chain, the simplest model in our hierarchy. For even $N$ the models can be viewed as longer ranger generalizations of the $XY$ chain, the next model in the hierarchy. We also demonstrate that our method of constructing spin chains with given critical points goes beyond exactly solvable models. Applying the same strategy to the Blume-Capel model, a spin-1 generalization of the Ising chain in a generic magnetic field, we construct another critical spin-1 chain with the predicted conformal field theory (CFT) describing the criticality.

DOI
10.1103/physrevb.89.014409

Related papers

$so(N)_1$ criticality in generalized cluster models

Ville Lahtinen, Eddy Ardonne

We show that $so(N)_1$ universality class quantum criticality emerges when one-dimensional generalized cluster models -- the N-cluster models -- are perturbed with Ising or Zeeman terms. Each critical point is described by a low-energy theory of N linearly dispersing fermions, whose spectrum we show to precisely match the prediction by $so(N)_1$ conformal field theory. Furthermore, by an explicit construction we show that the N-cluster models are dual to N non-locally coupled transverse field Ising chains, which enables to identify local representations for the primary fields and shows that the N-cluster models provide the simplest representation of the recently introduced hierarchy of $so(N)_1$ critical spin models. For the experimentally most realistic case of N=3, that corresponds to the original one-dimensional cluster model, our results show that $su(2)_2 \simeq so(3)_1$ Wess-Zumino-Witten model can emerge in a local, translationally invariant and Jordan-Wigner solvable spin-1/2 model.

•Apr 27, 2015•0 citations

Condensate-induced transitions and critical spin chains

Teresia Månsson, Ville Lahtinen, Juha Suorsa +1 more

We show that condensate-induced transitions between two-dimensional topological phases provide a general framework to relate one-dimensional spin models at their critical points. We demonstrate this using two examples. First, we show that two well-known spin chains, namely, the $\mathit{XY}$ chain and the transverse field Ising chain with only next-nearest-neighbor interactions, differ at their critical points only by a nonlocal boundary term and can be related via an exact mapping. The boundary term constrains the set of possible boundary conditions of the transverse field Ising chain, reducing the number of primary fields in the conformal field theory that describes its critical behavior. We argue that the reduction of the field content is equivalent to the confinement of a set of primary fields, in precise analogy to the confinement of quasiparticles resulting from a condensation of a boson in a topological phase. As the second example we show that when a similar confining boundary term is applied to the $\mathit{XY}$ chain with only next-nearest-neighbor interactions, the resulting system can be mapped to a local spin chain with the $u{(1)}_{2}\ifmmode\times\else\texttimes\fi{}u{(1)}_{2}$ critical behavior predicted by the condensation framework.

arXiv:1212.0351•Physical Review B•Jul 16, 2013•22 citations

Symmetry-protected topological phases, conformal criticalities, and duality in exactly solvable SO( n ) spin chains

Sreejith Chulliparambil, Hua-Chen Zhang, Hong-Hao Tu

We introduce a family of $\mathrm{SO}(n$)-symmetric spin chains which generalize the transverse-field Ising chain for $n=1$. These spin chains are defined with gamma matrices and can be exactly solved by mapping to $n$ species of itinerant Majorana fermions coupled to a static ${\mathbb{Z}}_{2}$ gauge field. Their phase diagrams include a critical point described by the $\mathrm{Spin}{(n)}_{1}$ conformal field theory as well as two distinct gapped phases. We show that one of the gapped phases is a trivial phase and the other realizes a symmetry-protected topological phase when $n\ensuremath{\ge}2$. These two gapped phases are proved to be related to each other by a Kramers-Wannier duality. Furthermore, other elegant structures in the transverse-field Ising chain, such as the infinite-dimensional Onsager algebra, also carry over to our models.

arXiv:2305.03398•Physical review. B./Physical review. B•Sep 7, 2023•5 citations

Realizing All so(N)1 Quantum Criticalities in Symmetry Protected Cluster Models

Ville Lahtinen, Eddy Ardonne

We show that all so(N)_{1} universality class quantum criticalities emerge when one-dimensional generalized cluster models are perturbed with Ising or Zeeman terms. Each critical point is described by a low-energy theory of N linearly dispersing fermions, whose spectrum we show to precisely match the prediction by so(N)_{1} conformal field theory. Furthermore, by an explicit construction we show that all the cluster models are dual to nonlocally coupled transverse field Ising chains, with the universality of the so(N)_{1} criticality manifesting itself as N of these chains becoming critical. This duality also reveals that the symmetry protection of cluster models arises from the underlying Ising symmetries and it enables the identification of local representations for the primary fields of the so(N)_{1} conformal field theories. For the simplest and experimentally most realistic case that corresponds to the original one-dimensional cluster model with local three-spin interactions, our results show that the su(2)_{2}≃so(3)_{1} Wess-Zumino-Witten model can emerge in a local, translationally invariant, and Jordan-Wigner solvable spin-1/2 model.

arXiv:1504.07044•Physical Review Letters•Dec 3, 2015•41 citations

Symmetry-protected topological order identified via Gutzwiller-guided density matrix renormalization group: SO ( n ) spin chains

Pei-Yuan Cai, Hui-Ke Jin, Yi Zhou

We present a comprehensive study of topological phases in the SO($n$) spin chains using a combination of analytical parton construction and numerical techniques. For even $n=2l$, we identify a novel SPT$^2$ phase characterized by two distinct topological sectors, exhibiting exact degeneracy at the matrix product state (MPS) exactly solvable point. Through Gutzwiller-projected mean-field theory and density matrix renormalization group (DMRG) calculations, we demonstrate that these sectors remain topologically degenerate in close chains throughout the SPT$^2$ phase, with energy gaps decaying exponentially with system size. For odd $n=2l+1$, we show that the ground state remains unique in close chains. We precisely characterize critical states using entanglement entropy scaling, confirming the central charges predicted by conformal field theories. Our results reveal fundamental differences between even and odd $n$ cases, provide numerical verification of topological protection, and establish reliable methods for studying high-symmetry quantum systems. The Gutzwiller-guided DMRG is demonstrated to be notably efficient in targeting specific topological sectors.

arXiv:2504.10919•Physical review. B./Physical review. B•Jul 7, 2025•3 citations

Quantum-spin-chain realizations of conformal field theories

Diptiman Sen

We present a series of one-dimensional quantum Hamiltonians that, at certain critical points, realize the minimal series of conformal field theories with central charge less than one. The models consist of ferromagnetically coupled SU(2) spins in a transverse magnetic field. The infinite-spin (free-boson) limit is especially studied after performing the Holstein-Primakoff transformation. The analysis can be generalized to SU(3) quantum chains. These realize a different series of conformal field theories at criticality.

Physical review. B, Condensed matter•Aug 1, 1991•5 citations
We construct a hierarchy of exactly solvable spin-1/2 chains with so(N)1so{(N)}_{1}so(N)1​ critical points. Our construction is based on the framework of condensate-induced transitions between topological phases. We employ this framework to construct a Hamiltonian term that couples NNN transverse field Ising chains such that the resulting theory is critical and described by the so(N)1so{(N)}_{1}so(N)1​ conformal field theory. By employing spin duality transformations, we then cast these spin chains for arbitrary NNN into translationally invariant forms that all allow exact solution by the means of a Jordan-Wigner transformation. For odd NNN our models generalize the phase diagram of the transverse field Ising chain, the simplest model in our hierarchy. For even NNN the models can be viewed as longer ranger generalizations of the XYXYXY chain, the next model in the hierarchy. We also demonstrate that our method of constructing spin chains with given critical points goes beyond exactly solvable models. Applying the same strategy to the Blume-Capel model, a spin-1 generalization of the Ising chain in a generic magnetic field, we construct another critical spin-1 chain with the predicted conformal field theory (CFT) describing the criticality.
so(N)1so(N)_1so(N)1​ criticality in generalized cluster models
We show that so(N)1so(N)_1so(N)1​ universality class quantum criticality emerges when one-dimensional generalized cluster models -- the N-cluster models -- are perturbed with Ising or Zeeman terms. Each critical point is described by a low-energy theory of N linearly dispersing fermions, whose spectrum we show to precisely match the prediction by so(N)1so(N)_1so(N)1​ conformal field theory. Furthermore, by an explicit construction we show that the N-cluster models are dual to N non-locally coupled transverse field Ising chains, which enables to identify local representations for the primary fields and shows that the N-cluster models provide the simplest representation of the recently introduced hierarchy of so(N)1so(N)_1so(N)1​ critical spin models. For the experimentally most realistic case of N=3, that corresponds to the original one-dimensional cluster model, our results show that su(2)2≃so(3)1su(2)_2 \simeq so(3)_1su(2)2​≃so(3)1​ Wess-Zumino-Witten model can emerge in a local, translationally invariant and Jordan-Wigner solvable spin-1/2 model.
We show that condensate-induced transitions between two-dimensional topological phases provide a general framework to relate one-dimensional spin models at their critical points. We demonstrate this using two examples. First, we show that two well-known spin chains, namely, the XY\mathit{XY}XY chain and the transverse field Ising chain with only next-nearest-neighbor interactions, differ at their critical points only by a nonlocal boundary term and can be related via an exact mapping. The boundary term constrains the set of possible boundary conditions of the transverse field Ising chain, reducing the number of primary fields in the conformal field theory that describes its critical behavior. We argue that the reduction of the field content is equivalent to the confinement of a set of primary fields, in precise analogy to the confinement of quasiparticles resulting from a condensation of a boson in a topological phase. As the second example we show that when a similar confining boundary term is applied to the XY\mathit{XY}XY chain with only next-nearest-neighbor interactions, the resulting system can be mapped to a local spin chain with the $u{(1)}_{2}\ifmmode\times\else\texttimes\fi{}u{(1)}_{2}$ critical behavior predicted by the condensation framework.
We introduce a family of SO(n\mathrm{SO}(nSO(n)-symmetric spin chains which generalize the transverse-field Ising chain for n=1n=1n=1. These spin chains are defined with gamma matrices and can be exactly solved by mapping to nnn species of itinerant Majorana fermions coupled to a static Z2{\mathbb{Z}}_{2}Z2​ gauge field. Their phase diagrams include a critical point described by the Spin(n)1\mathrm{Spin}{(n)}_{1}Spin(n)1​ conformal field theory as well as two distinct gapped phases. We show that one of the gapped phases is a trivial phase and the other realizes a symmetry-protected topological phase when $n\ensuremath{\ge}2$. These two gapped phases are proved to be related to each other by a Kramers-Wannier duality. Furthermore, other elegant structures in the transverse-field Ising chain, such as the infinite-dimensional Onsager algebra, also carry over to our models.
We show that all so(N)_{1} universality class quantum criticalities emerge when one-dimensional generalized cluster models are perturbed with Ising or Zeeman terms. Each critical point is described by a low-energy theory of N linearly dispersing fermions, whose spectrum we show to precisely match the prediction by so(N)_{1} conformal field theory. Furthermore, by an explicit construction we show that all the cluster models are dual to nonlocally coupled transverse field Ising chains, with the universality of the so(N)_{1} criticality manifesting itself as N of these chains becoming critical. This duality also reveals that the symmetry protection of cluster models arises from the underlying Ising symmetries and it enables the identification of local representations for the primary fields of the so(N)_{1} conformal field theories. For the simplest and experimentally most realistic case that corresponds to the original one-dimensional cluster model with local three-spin interactions, our results show that the su(2)_{2}≃so(3)_{1} Wess-Zumino-Witten model can emerge in a local, translationally invariant, and Jordan-Wigner solvable spin-1/2 model.
We present a comprehensive study of topological phases in the SO(nnn) spin chains using a combination of analytical parton construction and numerical techniques. For even n=2ln=2ln=2l, we identify a novel SPT2^22 phase characterized by two distinct topological sectors, exhibiting exact degeneracy at the matrix product state (MPS) exactly solvable point. Through Gutzwiller-projected mean-field theory and density matrix renormalization group (DMRG) calculations, we demonstrate that these sectors remain topologically degenerate in close chains throughout the SPT2^22 phase, with energy gaps decaying exponentially with system size. For odd n=2l+1n=2l+1n=2l+1, we show that the ground state remains unique in close chains. We precisely characterize critical states using entanglement entropy scaling, confirming the central charges predicted by conformal field theories. Our results reveal fundamental differences between even and odd nnn cases, provide numerical verification of topological protection, and establish reliable methods for studying high-symmetry quantum systems. The Gutzwiller-guided DMRG is demonstrated to be notably efficient in targeting specific topological sectors.
We present a series of one-dimensional quantum Hamiltonians that, at certain critical points, realize the minimal series of conformal field theories with central charge less than one. The models consist of ferromagnetically coupled SU(2) spins in a transverse magnetic field. The infinite-spin (free-boson) limit is especially studied after performing the Holstein-Primakoff transformation. The analysis can be generalized to SU(3) quantum chains. These realize a different series of conformal field theories at criticality.