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:2002.11924 · Physical review. B./Physical review. B

Path integral for spin-1 chain in the fluctuating matrix product state basis

Jintae Kim, Rajarshi Pal, Jin-Hong Park, Jung Hoon Han

Published Jul 6, 2020• 1 citation
arXivHTMLPDF

Abstract

An alternative method of writing down the path integral for spin-1 Heisenberg antiferromagnetic chain is introduced. In place of the conventional coherent-state basis that leads to the nonlinear $\ensuremath{\sigma}$ model, we use a basis called the fluctuating matrix product states which embodies intersite entanglement from the outset. It forms an overcomplete set spanning the entire Hilbert space of the spin-1 chain. Saddle-point analysis performed for the bilinear-biquadratic spin model predicts the Affleck-Kennedy-Lieb-Tasaki (AKLT) state as the ground state in the vicinity of the AKLT Hamiltonian. Quadratic effective action derived by gradient expansion around the saddle point is free from constraints that plagued the nonlinear $\ensuremath{\sigma}$ model and exactly solvable. The obtained excitation modes agree precisely with the single-mode approximation result for the AKLT Hamiltonian. Excitation spectra for other bilinear-biquadratic Hamiltonians are obtained as well by diagonalizing the quadratic action.

DOI
10.1103/physrevb.102.041105

Related papers

Zero-point energy of tensor fluctuations on the matrix product state manifold

Sebastian Leontica, Andrew G. Green

This work presents a method to quantize low energy excitations of systems whose ground state is best represented by a matrix product state (MPS) such as the Affleck-Kennedy-Lieb-Tasaki (AKLT) model. The approach is similar to the spin-wave model of excitations in typical magnetic materials, but includes a more general set of fluctuations in entanglement structure. We argue that the quasilocal action of tensor fluctuations facilitates a description in terms of bosonic modes. We apply this approach to compute fluctuation corrections to the bilinear-biquadratic Heisenberg model, whose ground state we expect to be close to the exact bond dimension 2 AKLT state in a certain parameter range. Our results show significant improvements in energy approximations, highlighting both the qualitative and quantitative potential of this paradigm for studying complex entangled systems. This approach paves the way for new insights into the low-energy physics of correlated materials and the development of effective field theories beyond traditional semiclassical methods.

arXiv:2410.22411•Physical review. B./Physical review. B•May 28, 2025•0 citations

Construction of variational matrix product states for the Heisenberg spin-1 chain

Jintae Kim, Minsoo Kim, Naoki Kawashima +2 more

We propose a simple variational wave function that captures the correct ground-state energy of the spin-1 Heisenberg chain model to within 0.04%. The wave function is written in the matrix product state (MPS) form with the bond dimension $D=8$ and is characterized by three fugacity parameters. The proposed MPS generalizes the Affleck-Kennedy-Lieb-Tasaki state by dressing it with dimers, trimers, and general $q$-mers. The fugacity parameters control the number and the average size of the $q$-mers. Furthermore, the $D=8$ variational MPS state captures the ground states of the entire family of the bilinear-biquadratic Hamiltonian belonging to the Haldane phase to high accuracy. The 2-4-2 degeneracy structure in the entanglement spectrum of our MPS state is found to match well the results of the density matrix renormalization group (DMRG) calculation, which is computationally much heavier. Spin-spin correlation functions also find an excellent fit with those obtained by DMRG.

arXiv:2005.10980•Physical review. B./Physical review. B•Aug 10, 2020•8 citations

Exact Fractionalized Ground States in an Extended Spin-1 Kitaev Chain

Alwyn Jose Raja, R. Ganesh

Inspired by the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, we present exact solutions for a spin-1 chain with Kitaev-like couplings. We consider an expanded Kitaev model with bilinear and biquadratic terms. At an exactly solvable point, the Hamiltonian can be reexpressed as a sum of projection operators. Unlike the AKLT model where projectors act on total spin, we project the component of spin along the bond direction. This leads to exponential ground-state degeneracy, expressed in terms of fractionalized spin-1/2 objects. Each ground state can be expressed concisely as a matrix product state. We construct a phase diagram by varying the relative strength of bilinear and biquadratic terms. The fractionalized states provide a qualitative picture for the spin-1 Kitaev model, yielding approximate forms for the ground state and low-lying excitations.

arXiv:2510.12880•Physical Review Letters•May 27, 2026•0 citations

Entanglement spectra of the q-deformed Affleck-Kennedy-Lieb-Tasaki model and matrix product states

Raul A. Santos, Francis N. C. Paraan, V. E. Korepin +1 more

We exactly calculate the reduced density matrix of matrix product states (MPS). Our compact result enables one to perform analytic studies of entanglement in MPS. In particular, we consider the MPS ground states of two anisotropic spin chains. One is a q -deformed Affleck-Kennedy-Lieb-Tasaki (AKLT) model and the other is a general spin-1 quantum antiferromagnet with nearest-neighbor interactions. Our analysis shows how anisotropy affects entanglement on different continuous parameter manifolds. We also construct an effective boundary spin model that describes a block of spins in the ground state of the q -deformed AKLT Hamiltonian. The temperature of this effective model is given in terms of the deformation parameter q .

arXiv:1112.0517•Europhysics Letters (EPL)•May 1, 2012•18 citations

Variational and density-matrix renormalization-group studies of the frustrated antiferromagnetic Heisenberg S=1 quantum spin chain

Alexei K. Kolezhuk, R. Roth, Ulrich Schollwöck

In this paper we study a frustrated antiferromagnetic isotropic Heisenberg S=1 quantum spin chain H=${\ensuremath{\sum}}_{\mathrm{i}}$${\mathbf{S}}_{\mathrm{i}}$${\mathbf{S}}_{\mathrm{i}+1}$+\ensuremath{\alpha}${\ensuremath{\sum}}_{\mathrm{i}}$${\mathbf{S}}_{\mathrm{i}}$${\mathbf{S}}_{\mathrm{i}+2}$, using a variational ansatz starting from valence bond states and the density-matrix renormalization group. We find both methods to give results in very good qualitative and good quantitative agreement, which clarify the phase diagram as follows: At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{D}}$=0.284(1), there is a disorder point of the first kind, marking the onset of incommensurate spin-spin correlations in the chain. At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{L}}$=0.3725(25) there is a Lifshitz point, at which the excitation spectrum is found to develop a particular doubly degenerate structure. These points are the quantum remnants of the transition from antiferromagnetic to spiral order in the classical frustrated chain. At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{T}}$=0.7444(6) there is a first-order phase transition from an Affleck-Kennedy-Lieb-Tasaki (AKLT) phase characterized by nonvanishing string order to a phase which can be understood as a next-nearest-neighbor generalization of the AKLT model. At the transition, the string order parameter shows (to numerical precision) a discontinuous jump of 0.085 to zero; the correlation length and the gap are both finite at the transition. The problem of edge states in open frustrated chains is discussed at length.

arXiv:cond-mat/9609064•Physical review. B, Condensed matter•Apr 1, 1997•51 citations

Spin and dimer correlations forS=1 bilinear-biquadratic antiferromagnetic Heisenberg chains

J. J. Deisz

Using the world-line quantum Monte Carlo method, we calculate the spin 〈${\mathit{S}}_{\mathit{R}}^{\mathit{z}}$${\mathit{S}}_{0}^{\mathit{z}}$〉 and dimer 〈(${\mathbf{S}}_{0}$\ensuremath{\cdot}${\mathbf{S}}_{1}$)(${\mathbf{S}}_{\mathit{R}}$\ensuremath{\cdot}${\mathbf{S}}_{\mathit{R}+1}$)〉 correlation functions for the S=1 biquadratic Heisenberg antiferromagnetic chain. These results show the relevance of the Affleck-Kennedy-Lieb-Tasaki state for describing the nonclassical behavior of the S=1 antiferromagnetic Heisenberg model. We also include results for S=3/2 and 2, which show the inhibition of staggered dimerization as the classical limit (S\ensuremath{\rightarrow}\ensuremath{\infty}) is approached, where N\'eel order is favored.

Physical review. B, Condensed matter•Aug 1, 1992•4 citations
An alternative method of writing down the path integral for spin-1 Heisenberg antiferromagnetic chain is introduced. In place of the conventional coherent-state basis that leads to the nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ model, we use a basis called the fluctuating matrix product states which embodies intersite entanglement from the outset. It forms an overcomplete set spanning the entire Hilbert space of the spin-1 chain. Saddle-point analysis performed for the bilinear-biquadratic spin model predicts the Affleck-Kennedy-Lieb-Tasaki (AKLT) state as the ground state in the vicinity of the AKLT Hamiltonian. Quadratic effective action derived by gradient expansion around the saddle point is free from constraints that plagued the nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ model and exactly solvable. The obtained excitation modes agree precisely with the single-mode approximation result for the AKLT Hamiltonian. Excitation spectra for other bilinear-biquadratic Hamiltonians are obtained as well by diagonalizing the quadratic action.
This work presents a method to quantize low energy excitations of systems whose ground state is best represented by a matrix product state (MPS) such as the Affleck-Kennedy-Lieb-Tasaki (AKLT) model. The approach is similar to the spin-wave model of excitations in typical magnetic materials, but includes a more general set of fluctuations in entanglement structure. We argue that the quasilocal action of tensor fluctuations facilitates a description in terms of bosonic modes. We apply this approach to compute fluctuation corrections to the bilinear-biquadratic Heisenberg model, whose ground state we expect to be close to the exact bond dimension 2 AKLT state in a certain parameter range. Our results show significant improvements in energy approximations, highlighting both the qualitative and quantitative potential of this paradigm for studying complex entangled systems. This approach paves the way for new insights into the low-energy physics of correlated materials and the development of effective field theories beyond traditional semiclassical methods.
We propose a simple variational wave function that captures the correct ground-state energy of the spin-1 Heisenberg chain model to within 0.04%. The wave function is written in the matrix product state (MPS) form with the bond dimension D=8D=8D=8 and is characterized by three fugacity parameters. The proposed MPS generalizes the Affleck-Kennedy-Lieb-Tasaki state by dressing it with dimers, trimers, and general qqq-mers. The fugacity parameters control the number and the average size of the qqq-mers. Furthermore, the D=8D=8D=8 variational MPS state captures the ground states of the entire family of the bilinear-biquadratic Hamiltonian belonging to the Haldane phase to high accuracy. The 2-4-2 degeneracy structure in the entanglement spectrum of our MPS state is found to match well the results of the density matrix renormalization group (DMRG) calculation, which is computationally much heavier. Spin-spin correlation functions also find an excellent fit with those obtained by DMRG.
Inspired by the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, we present exact solutions for a spin-1 chain with Kitaev-like couplings. We consider an expanded Kitaev model with bilinear and biquadratic terms. At an exactly solvable point, the Hamiltonian can be reexpressed as a sum of projection operators. Unlike the AKLT model where projectors act on total spin, we project the component of spin along the bond direction. This leads to exponential ground-state degeneracy, expressed in terms of fractionalized spin-1/2 objects. Each ground state can be expressed concisely as a matrix product state. We construct a phase diagram by varying the relative strength of bilinear and biquadratic terms. The fractionalized states provide a qualitative picture for the spin-1 Kitaev model, yielding approximate forms for the ground state and low-lying excitations.
We exactly calculate the reduced density matrix of matrix product states (MPS). Our compact result enables one to perform analytic studies of entanglement in MPS. In particular, we consider the MPS ground states of two anisotropic spin chains. One is a q -deformed Affleck-Kennedy-Lieb-Tasaki (AKLT) model and the other is a general spin-1 quantum antiferromagnet with nearest-neighbor interactions. Our analysis shows how anisotropy affects entanglement on different continuous parameter manifolds. We also construct an effective boundary spin model that describes a block of spins in the ground state of the q -deformed AKLT Hamiltonian. The temperature of this effective model is given in terms of the deformation parameter q .
In this paper we study a frustrated antiferromagnetic isotropic Heisenberg S=1 quantum spin chain H=${\ensuremath{\sum}}_{\mathrm{i}}$Si{\mathbf{S}}_{\mathrm{i}}Si​Si+1{\mathbf{S}}_{\mathrm{i}+1}Si+1​+\ensuremath{\alpha}${\ensuremath{\sum}}_{\mathrm{i}}$Si{\mathbf{S}}_{\mathrm{i}}Si​Si+2{\mathbf{S}}_{\mathrm{i}+2}Si+2​, using a variational ansatz starting from valence bond states and the density-matrix renormalization group. We find both methods to give results in very good qualitative and good quantitative agreement, which clarify the phase diagram as follows: At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{D}}$=0.284(1), there is a disorder point of the first kind, marking the onset of incommensurate spin-spin correlations in the chain. At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{L}}$=0.3725(25) there is a Lifshitz point, at which the excitation spectrum is found to develop a particular doubly degenerate structure. These points are the quantum remnants of the transition from antiferromagnetic to spiral order in the classical frustrated chain. At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{T}}$=0.7444(6) there is a first-order phase transition from an Affleck-Kennedy-Lieb-Tasaki (AKLT) phase characterized by nonvanishing string order to a phase which can be understood as a next-nearest-neighbor generalization of the AKLT model. At the transition, the string order parameter shows (to numerical precision) a discontinuous jump of 0.085 to zero; the correlation length and the gap are both finite at the transition. The problem of edge states in open frustrated chains is discussed at length.
Using the world-line quantum Monte Carlo method, we calculate the spin 〈SRz{\mathit{S}}_{\mathit{R}}^{\mathit{z}}SRz​S0z{\mathit{S}}_{0}^{\mathit{z}}S0z​〉 and dimer 〈(S0{\mathbf{S}}_{0}S0​\ensuremath{\cdot}S1{\mathbf{S}}_{1}S1​)(SR{\mathbf{S}}_{\mathit{R}}SR​\ensuremath{\cdot}SR+1{\mathbf{S}}_{\mathit{R}+1}SR+1​)〉 correlation functions for the S=1 biquadratic Heisenberg antiferromagnetic chain. These results show the relevance of the Affleck-Kennedy-Lieb-Tasaki state for describing the nonclassical behavior of the S=1 antiferromagnetic Heisenberg model. We also include results for S=3/2 and 2, which show the inhibition of staggered dimerization as the classical limit (S\ensuremath{\rightarrow}\ensuremath{\infty}) is approached, where N\'eel order is favored.