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.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:1212.0351Physical Review BJul 16, 201322 citations

Critical Properties of Spin Chains

Satoshi Takada

Springer series in solid-state sciencesJan 1, 19873 citations

Bose condensation in quasi-one-dimensional antiferromagnets in strong fields

Ian K. Affleck

Recent experiments show that axially symmetric integer-spin antiferromagnetic chains undergo a phase transition at a critical applied magnetic field. It was argued, using Landau-Ginzburg theory, that this is one-dimensional Bose condensation. The theory is further analyzed at the mean-field and Gaussian level. Then some exact results concerning the critical behavior are determined using known results on one-dimensional Bose fluids. These are shown to be consistent with recent numerical simulations on spin chains. Breaking of axial symmetry produces crossover to a two-dimensional Ising transition.Recent experiments show that axially symmetric integer-spin antiferromagnetic chains undergo a phase transition at a critical applied magnetic field. It was argued, using Landau-Ginzburg theory, that this is one-dimensional Bose condensation. The theory is further analyzed at the mean-field and Gaussian level. Then some exact results concerning the critical behavior are determined using known results on one-dimensional Bose fluids. These are shown to be consistent with recent numerical simulations on spin chains. Breaking of axial symmetry produces crossover to a two-dimensional Ising transition.

Physical review. B, Condensed matterFeb 1, 1991274 citations

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

Ville Lahtinen, Teresia Månsson, Eddy Ardonne

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.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.

arXiv:1310.1876Physical Review BJan 10, 201414 citations

Phase Transitions In Spin Systems

Johannes Richter, S E Kreuger, DAMIAN J. J. FARNELL +1 more

Jan 1, 20010 citations

\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.

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.