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

Physical review. B./Physical review. B

Boundary quantum phase transitions in the spin- 12 Heisenberg chain with boundary magnetic fields

Parameshwar R. Pasnoori, Junhyun Lee, J. H. Pixley, Natan Andrei, P. Azaria

Published Jun 12, 2023• 15 citations
Publisher

Abstract

We consider the spin-$1/2$ Heisenberg chain with boundary magnetic fields and analyze it using a combination of Bethe ansatz and density matrix renormalization group (DMRG) techniques. We show that the system exhibits several different ground states, which depend on the orientation of the boundary magnetic fields. When both the boundary fields take equal values greater than a critical field strength, each edge in the ground state accumulates a fractional spin which saturates to spin $1/4$, which is similar to systems exhibiting symmetry protected topological phases (SPT). Unlike in SPT systems, the fractional boundary spin in the Heisenberg spin chain is not a genuine quantum number since the variance of the associated operator does not vanish, this is due to the absence of a bulk gap. The system exhibits high-energy bound states when the boundary fields take values greater than the critical field. All the excitations in the system can be sorted out into towers whose number depends on the number of bound states exhibited by the system. As the boundary fields are varied, in addition to the ground-state phase transition, we find that the system may undergo a boundary eigenstate phase transition (BEPT) where the number of towers of the Hilbert space changes. We further inquire how the EPT reflects itself on local ground-state properties by computing the magnetization profile $\ensuremath{\langle}{S}_{j}^{z}\ensuremath{\rangle}$ using DMRG. We identify a clear qualitative change from low edge fields to high edge fields when crossing the critical field. We though are unable to conclude on the basis of our data that EPT corresponds to a genuine phase transition in the ground state.

DOI
10.1103/physrevb.107.224412

Related papers

Boundary Quantum Phase Transitions in the Spin $\frac{1}{2}$ Heisenberg Chain with Boundary Magnetic Fields

Parameshwar R. Pasnoori, Junhyun Lee, J. H. Pixley +2 more

We consider the spin $\frac{1}{2}$ Heisenberg chain with boundary magnetic fields and analyze it using a combination of Bethe ansatz and density matrix renormalization group (DMRG) techniques. We show that the system exhibits several different ground states which depend on the orientation of the boundary magnetic fields. When both the boundary fields take equal values greater than a critical field strength, each edge in the ground state accumulates a fractional spin which saturates to spin $\frac{1}{4}$, which is similar to systems exhibiting symmetry protected topological phases (SPT). Unlike in SPT systems, the fractional boundary spin in the Heisenberg spin chain is not a genuine quantum number since the variance of the associated operator does not vanish, this is due to the absence of a bulk gap. The system exhibits high energy bound states when the boundary fields take values greater than the critical field. All the excitations in the system can be sorted out into towers whose number depends on the number of bound states exhibited by the system. As the boundary fields are varied, in addition to the ground state phase transition, we find that the system may undergo an eigenstate phase transition (EPT) where the number of towers of the Hilbert space changes. We further inquire how the EPT reflects itself on local ground state properties by computing the magnetization profile $\langle S^z_j \rangle$ using DMRG. We identify a clear qualitative change from low edge fields to high edge fields when crossing the critical field. We though are unable to conclude on the basis of our data that EPT corresponds to a genuine phase transition in the ground state.

arXiv:2212.14832•Dec 30, 2022•1 citation

Complete Boundary Phase Diagram of the Spin-$\frac{1}{2}$ XXZ Chain with Boundary Fields in the Anti-Ferromagnetic Gapped Regime

Parameshwar R. Pasnoori, Yicheng Tang, Junhyun Lee +3 more

We consider the spin $\frac{1}{2}$ XXZ chain with diagonal boundary fields and solve it exactly using Bethe ansatz in the gapped anti-ferromagnetic regime and obtain the complete phase boundary diagram. Depending on the values of the boundary fields, the system exhibits several phases which can be categorized based on the ground state exhibited by the system and also based on the number of bound states localized at the boundaries. We show that the Hilbert space is comprised of a certain number of towers whose number depends on the number of boundary bound states exhibited by the system. The system undergoes boundary phase transitions when boundary fields are varied across certain critical values. There exist two types of phase transitions. In the first type the ground state of the system undergoes a change. In the second type, named the `Eigenstate phase transition', the number of towers of the Hilbert space changes, which is again associated with the change in the number of boundary bound states exhibited by the system. We use the DMRG and exact diagonalization techniques to probe the signature of the Eigenstate phase transition and the ground state phase transition by analyzing the spin profiles in each eigenstate.

arXiv:2507.00386•Jul 1, 2025•0 citations

Spin fractionalization and zero modes in the spin-$\frac{1}{2}$ XXZ chain with boundary fields

Parameshwar R. Pasnoori, Yicheng Tang, Junhyun Lee +3 more

In this work we argue that the antiferromagnetic spin $\frac{1}{2}$ XXZ chain in the gapped phase with boundary magnetic fields hosts fractional spin $\frac{1}{4}$ at its edges. Using a combination of Bethe ansatz and the density matrix renormalization group we show that these fractional spins are sharp quantum observables in both the ground and the first excited state as the associated fractional spin operators have zero variance. In the limit of zero edge fields, we argue that these fractional spin operators once projected onto the low energy subspace spanned by the ground state and the first excited state, identify with the strong zero energy mode discovered by P. Fendley \cite{Fendley}.

arXiv:2312.05970•Dec 10, 2023•1 citation

Spin fractionalization and zero modes in the spin- 1 2 XXZ chain with boundary fields

Parameshwar R. Pasnoori, Yicheng Tang, Junhyun Lee +3 more

In this work, we argue that the antiferromagnetic spin-$\frac{1}{2}$ XXZ chain in the gapped phase with boundary magnetic fields hosts fractional spin $\frac{1}{4}$ at its edges. Using a combination of Bethe ansatz and the density matrix renormalization group we show that these fractional spins are sharp quantum observables in both the ground and the first excited state as the associated fractional spin operators have zero variance. In the limit of zero edge fields, we argue that these fractional spin operators, once projected onto the low-energy subspace spanned by the ground state and the first excited state, identify with the strong zero-energy mode discovered by Fendley [J. Phys. A: Math. Theor. 49, 30LT01 (2016)].

Physical review. B./Physical review. B•Jul 17, 2025•4 citations

Breakdown of a Magnetization Plateau in Ferrimagnetic Mixed Spin-(1/2,S) Heisenberg Chains due to a Quantum Phase Transition towards the Luttinger Spin Liquid

Jozef Strečka

Magnetization curves of the ferrimagnetic mixed spin-(1/2,S) Heisenberg chains are calculated with the help of density-matrix renormalization group method for several quantum spin numbers S=1, 3/2, 2 and 5/2.It is shown that the ferrimagnetic mixed spin-(1/2,S) Heisenberg chains exhibit irrespective of the spin value S exactly one intermediate magnetization plateau, which can be identified with the gapped Lieb-Mattis ferrimagnetic ground state.The magnetization plateau due to the Lieb-Mattis ferrimagnetism breaks down at a quantum phase transition towards the Luttinger spin liquid, which is characterized by a continuous change of the magnetization with the magnetic field until another quantum critical point is reached at the saturation field.

arXiv:1607.03617•Acta Physica Polonica A•Apr 1, 2017•13 citations

Topological order parameters of the spin- 12 dimerized Heisenberg ladder in magnetic field

Toshikaze Kariyado, Yasuhiro Hatsugai

Topological properties of the spin-1/2 dimerized Heisenberg ladder are investigated, focusing on the plateau phase in the magnetic field whose magnetization is half of the saturation value. Although the applied magnetic field removes most of the symmetries of the system, there is a symmetry-protected topological phase supported by the spatial inversion symmetry. The ${Z}_{2}$ Berry phase associated with a symmetry-respecting boundary and quantized into 0 and $\ensuremath{\pi}$ is used as a symmetry-protected topological order parameter. Edge states are also analyzed to confirm the bulk-edge correspondence. In addition, a symmetry-breaking boundary is considered. Then, we observe a different type of quantization of the Berry phase, i.e., a quantization into $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$ of the Berry phase. In this case, the bulk-edge correspondence is also different, namely, there emerge ``polarized'' edge states for the case with $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$ quantization. We also evaluate the entanglement entropy by the infinite time-evolving block decimation (iTEBD) to complement the Berry-phase-based arguments. Further, a different type of the topological order parameter is extracted from the matrix product state representation of the ground state given by the iTEBD.

arXiv:1412.7901•Physical Review B•Jun 8, 2015•20 citations
We consider the spin-1/21/21/2 Heisenberg chain with boundary magnetic fields and analyze it using a combination of Bethe ansatz and density matrix renormalization group (DMRG) techniques. We show that the system exhibits several different ground states, which depend on the orientation of the boundary magnetic fields. When both the boundary fields take equal values greater than a critical field strength, each edge in the ground state accumulates a fractional spin which saturates to spin 1/41/41/4, which is similar to systems exhibiting symmetry protected topological phases (SPT). Unlike in SPT systems, the fractional boundary spin in the Heisenberg spin chain is not a genuine quantum number since the variance of the associated operator does not vanish, this is due to the absence of a bulk gap. The system exhibits high-energy bound states when the boundary fields take values greater than the critical field. All the excitations in the system can be sorted out into towers whose number depends on the number of bound states exhibited by the system. As the boundary fields are varied, in addition to the ground-state phase transition, we find that the system may undergo a boundary eigenstate phase transition (BEPT) where the number of towers of the Hilbert space changes. We further inquire how the EPT reflects itself on local ground-state properties by computing the magnetization profile \ensuremath⟨Sjz\ensuremath⟩\ensuremath{\langle}{S}_{j}^{z}\ensuremath{\rangle}\ensuremath⟨Sjz​\ensuremath⟩ using DMRG. We identify a clear qualitative change from low edge fields to high edge fields when crossing the critical field. We though are unable to conclude on the basis of our data that EPT corresponds to a genuine phase transition in the ground state.
Boundary Quantum Phase Transitions in the Spin 12\frac{1}{2}21​ Heisenberg Chain with Boundary Magnetic Fields
We consider the spin 12\frac{1}{2}21​ Heisenberg chain with boundary magnetic fields and analyze it using a combination of Bethe ansatz and density matrix renormalization group (DMRG) techniques. We show that the system exhibits several different ground states which depend on the orientation of the boundary magnetic fields. When both the boundary fields take equal values greater than a critical field strength, each edge in the ground state accumulates a fractional spin which saturates to spin 14\frac{1}{4}41​, which is similar to systems exhibiting symmetry protected topological phases (SPT). Unlike in SPT systems, the fractional boundary spin in the Heisenberg spin chain is not a genuine quantum number since the variance of the associated operator does not vanish, this is due to the absence of a bulk gap. The system exhibits high energy bound states when the boundary fields take values greater than the critical field. All the excitations in the system can be sorted out into towers whose number depends on the number of bound states exhibited by the system. As the boundary fields are varied, in addition to the ground state phase transition, we find that the system may undergo an eigenstate phase transition (EPT) where the number of towers of the Hilbert space changes. We further inquire how the EPT reflects itself on local ground state properties by computing the magnetization profile ⟨Sjz⟩\langle S^z_j \rangle⟨Sjz​⟩ using DMRG. We identify a clear qualitative change from low edge fields to high edge fields when crossing the critical field. We though are unable to conclude on the basis of our data that EPT corresponds to a genuine phase transition in the ground state.
Complete Boundary Phase Diagram of the Spin-12\frac{1}{2}21​ XXZ Chain with Boundary Fields in the Anti-Ferromagnetic Gapped Regime
We consider the spin 12\frac{1}{2}21​ XXZ chain with diagonal boundary fields and solve it exactly using Bethe ansatz in the gapped anti-ferromagnetic regime and obtain the complete phase boundary diagram. Depending on the values of the boundary fields, the system exhibits several phases which can be categorized based on the ground state exhibited by the system and also based on the number of bound states localized at the boundaries. We show that the Hilbert space is comprised of a certain number of towers whose number depends on the number of boundary bound states exhibited by the system. The system undergoes boundary phase transitions when boundary fields are varied across certain critical values. There exist two types of phase transitions. In the first type the ground state of the system undergoes a change. In the second type, named the `Eigenstate phase transition', the number of towers of the Hilbert space changes, which is again associated with the change in the number of boundary bound states exhibited by the system. We use the DMRG and exact diagonalization techniques to probe the signature of the Eigenstate phase transition and the ground state phase transition by analyzing the spin profiles in each eigenstate.
Spin fractionalization and zero modes in the spin-12\frac{1}{2}21​ XXZ chain with boundary fields
In this work we argue that the antiferromagnetic spin 12\frac{1}{2}21​ XXZ chain in the gapped phase with boundary magnetic fields hosts fractional spin 14\frac{1}{4}41​ at its edges. Using a combination of Bethe ansatz and the density matrix renormalization group we show that these fractional spins are sharp quantum observables in both the ground and the first excited state as the associated fractional spin operators have zero variance. In the limit of zero edge fields, we argue that these fractional spin operators once projected onto the low energy subspace spanned by the ground state and the first excited state, identify with the strong zero energy mode discovered by P. Fendley \cite{Fendley}.
In this work, we argue that the antiferromagnetic spin-12\frac{1}{2}21​ XXZ chain in the gapped phase with boundary magnetic fields hosts fractional spin 14\frac{1}{4}41​ at its edges. Using a combination of Bethe ansatz and the density matrix renormalization group we show that these fractional spins are sharp quantum observables in both the ground and the first excited state as the associated fractional spin operators have zero variance. In the limit of zero edge fields, we argue that these fractional spin operators, once projected onto the low-energy subspace spanned by the ground state and the first excited state, identify with the strong zero-energy mode discovered by Fendley [J. Phys. A: Math. Theor. 49, 30LT01 (2016)].
Magnetization curves of the ferrimagnetic mixed spin-(1/2,S) Heisenberg chains are calculated with the help of density-matrix renormalization group method for several quantum spin numbers S=1, 3/2, 2 and 5/2.It is shown that the ferrimagnetic mixed spin-(1/2,S) Heisenberg chains exhibit irrespective of the spin value S exactly one intermediate magnetization plateau, which can be identified with the gapped Lieb-Mattis ferrimagnetic ground state.The magnetization plateau due to the Lieb-Mattis ferrimagnetism breaks down at a quantum phase transition towards the Luttinger spin liquid, which is characterized by a continuous change of the magnetization with the magnetic field until another quantum critical point is reached at the saturation field.
Topological properties of the spin-1/2 dimerized Heisenberg ladder are investigated, focusing on the plateau phase in the magnetic field whose magnetization is half of the saturation value. Although the applied magnetic field removes most of the symmetries of the system, there is a symmetry-protected topological phase supported by the spatial inversion symmetry. The Z2{Z}_{2}Z2​ Berry phase associated with a symmetry-respecting boundary and quantized into 0 and $\ensuremath{\pi}$ is used as a symmetry-protected topological order parameter. Edge states are also analyzed to confirm the bulk-edge correspondence. In addition, a symmetry-breaking boundary is considered. Then, we observe a different type of quantization of the Berry phase, i.e., a quantization into $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$ of the Berry phase. In this case, the bulk-edge correspondence is also different, namely, there emerge ``polarized'' edge states for the case with $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$ quantization. We also evaluate the entanglement entropy by the infinite time-evolving block decimation (iTEBD) to complement the Berry-phase-based arguments. Further, a different type of the topological order parameter is extracted from the matrix product state representation of the ground state given by the iTEBD.