arXiv:2103.16203 · Physical review. B./Physical review. B

Topological charge scaling at a quantum phase transition

Piotr Tomczak

Published Jul 12, 2021• 1 citation

Abstract

We reexamine the Kosterlitz-Thouless phase transition in the ground state $|{\mathrm{\ensuremath{\Psi}}}_{0}\ensuremath{\rangle}$ of an antiferromagnetic spin-$\frac{1}{2}$ Heisenberg chain with nearest- and next-nearest-neighbor interactions $\ensuremath{\lambda}$ from a different perspective: After defining winding number (topological charge) $W$ in the basis of resonating valence bond states, the finite-size scaling of $\ensuremath{\langle}{\mathrm{\ensuremath{\Psi}}}_{0}|W|{\mathrm{\ensuremath{\Psi}}}_{0}\ensuremath{\rangle}, \ensuremath{\langle}{\mathrm{\ensuremath{\Psi}}}_{0}|W|{\ensuremath{\partial}}_{\ensuremath{\lambda}}{\mathrm{\ensuremath{\Psi}}}_{0}\ensuremath{\rangle}, \ensuremath{\langle}{\ensuremath{\partial}}_{\ensuremath{\lambda}}{\mathrm{\ensuremath{\Psi}}}_{0}|W|{\ensuremath{\partial}}_{\ensuremath{\lambda}}{\mathrm{\ensuremath{\Psi}}}_{0}\ensuremath{\rangle}$ leads to the accurate value of critical coupling ${\ensuremath{\lambda}}_{c}=0.2412\ifmmode\pm\else\textpm\fi{}0.0007$ and to the value of subleading critical exponent $\ensuremath{\nu}=2.000\ifmmode\pm\else\textpm\fi{}0.001$. This approach should be useful when examining the topological phase transitions in all systems described in the basis of resonating valence bonds.

DOI: 10.1103/physrevb.104.l020405

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arXiv:2103.16203 · Physical review. B./Physical review. B

Topological charge scaling at a quantum phase transition

Piotr Tomczak

Published Jul 12, 2021• 1 citation

arXiv HTML PDF

Abstract

We reexamine the Kosterlitz-Thouless phase transition in the ground state

|{\mathrm{\ensuremath{\Psi}}}_{0}\ensuremath{\rangle}

of an antiferromagnetic spin-

\frac{1}{2}

Heisenberg chain with nearest- and next-nearest-neighbor interactions

\ensuremath{\lambda}

from a different perspective: After defining winding number (topological charge)

W

in the basis of resonating valence bond states, the finite-size scaling of

\ensuremath{\langle}{\mathrm{\ensuremath{\Psi}}}_{0}|W|{\mathrm{\ensuremath{\Psi}}}_{0}\ensuremath{\rangle}, \ensuremath{\langle}{\mathrm{\ensuremath{\Psi}}}_{0}|W|{\ensuremath{\partial}}_{\ensuremath{\lambda}}{\mathrm{\ensuremath{\Psi}}}_{0}\ensuremath{\rangle}, \ensuremath{\langle}{\ensuremath{\partial}}_{\ensuremath{\lambda}}{\mathrm{\ensuremath{\Psi}}}_{0}|W|{\ensuremath{\partial}}_{\ensuremath{\lambda}}{\mathrm{\ensuremath{\Psi}}}_{0}\ensuremath{\rangle}

leads to the accurate value of critical coupling

{\ensuremath{\lambda}}_{c}=0.2412\ifmmode\pm\else\textpm\fi{}0.0007

and to the value of subleading critical exponent

\ensuremath{\nu}=2.000\ifmmode\pm\else\textpm\fi{}0.001

. This approach should be useful when examining the topological phase transitions in all systems described in the basis of resonating valence bonds.

DOI: 10.1103/physrevb.104.l020405

Related papers

Probing the Kosterlitz-Thouless transition in 1D Heisenberg antiferromagnet based on topological properties of its ground state

Tomczak, P., Jablonski, P., M. Tomczak

A Kosterlitz-Thouless phase transition in the ground state of an antiferromagnetic spin- $\frac{1}{2}$ Heisenberg chain with nearest and next-nearest-neighbor interactions is re-investigated from a different perspective: An unequivocal correspondence is found between components of the scalar product $\langle Ψ_0| Ψ_0 \rangle$ and geometrical objects. One can classify these objects according to whether any two of them can be transformed into each other in a continuous way (belong to the same homotopy class). A finite size scaling of the "connection term`` $\langle Ψ_0|\partial Ψ_0 \rangle$ with respect to chain length (16, 18, 20, 22, 24 spins) for each homotopy class of above mentioned objects leads to the critical value of $λ$ with rather high accuracy.A Kosterlitz-Thouless phase transition in the ground state of an antiferromagnetic spin-$\frac{1}{2}$ Heisenberg chain with nearest and next-nearest-neighbor interactions is re-investigated from a different perspective: An unequivocal correspondence is found between components of the scalar product $\langle Ψ_0| Ψ_0 \rangle$ and geometrical objects. One can classify these objects according to whether any two of them can be transformed into each other in a continuous way (belong to the same homotopy class). A finite size scaling of the "connection term`` $\langle Ψ_0|\partial Ψ_0 \rangle$ with respect to chain length (16, 18, 20, 22, 24 spins) for each homotopy class of above mentioned objects leads to the critical value of $λ$ with rather high accuracy.

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arXiv:2605.19011May 18, 20260 citations

We reexamine the Kosterlitz-Thouless phase transition in the ground state

|{\mathrm{\ensuremath{\Psi}}}_{0}\ensuremath{\rangle}

of an antiferromagnetic spin-

\frac{1}{2}

Heisenberg chain with nearest- and next-nearest-neighbor interactions

\ensuremath{\lambda}

from a different perspective: After defining winding number (topological charge)

W

in the basis of resonating valence bond states, the finite-size scaling of

\ensuremath{\langle}{\mathrm{\ensuremath{\Psi}}}_{0}|W|{\mathrm{\ensuremath{\Psi}}}_{0}\ensuremath{\rangle}, \ensuremath{\langle}{\mathrm{\ensuremath{\Psi}}}_{0}|W|{\ensuremath{\partial}}_{\ensuremath{\lambda}}{\mathrm{\ensuremath{\Psi}}}_{0}\ensuremath{\rangle}, \ensuremath{\langle}{\ensuremath{\partial}}_{\ensuremath{\lambda}}{\mathrm{\ensuremath{\Psi}}}_{0}|W|{\ensuremath{\partial}}_{\ensuremath{\lambda}}{\mathrm{\ensuremath{\Psi}}}_{0}\ensuremath{\rangle}

leads to the accurate value of critical coupling

{\ensuremath{\lambda}}_{c}=0.2412\ifmmode\pm\else\textpm\fi{}0.0007

and to the value of subleading critical exponent

\ensuremath{\nu}=2.000\ifmmode\pm\else\textpm\fi{}0.001

. This approach should be useful when examining the topological phase transitions in all systems described in the basis of resonating valence bonds.

A Kosterlitz-Thouless phase transition in the ground state of an antiferromagnetic spin-

\frac{1}{2}

Heisenberg chain with nearest and next-nearest-neighbor interactions is re-investigated from a different perspective: An unequivocal correspondence is found between components of the scalar product

\langle Ψ_0| Ψ_0 \rangle

and geometrical objects. One can classify these objects according to whether any two of them can be transformed into each other in a continuous way (belong to the same homotopy class). A finite size scaling of the "connection term``

\langle Ψ_0|\partial Ψ_0 \rangle

with respect to chain length (16, 18, 20, 22, 24 spins) for each homotopy class of above mentioned objects leads to the critical value of

λ

with rather high accuracy.

We present three different methods of finding the critical spin value

S_c

of the quantum phase transition between the spin liquid and the Néel order phase on the mean field triangular Heisenberg antiferromagnet at

T=0

K. These methods are: using the sublattice magnetization as an order parameter, looking at the scaling of the energy difference of topologically degenerate states in different phases, and the quantum fidelity approach. All of these methods are able to pick up the signal of the phase transition, and their estimated numerical values of

S_c

agree for systems where the relative strength of next nearest neighbour interaction compared to the nearest neighbour interaction is small. For the relative interaction strength

j=0

and

j=0.1

, all three methods predict

S_c\\approx0.21

and

S_c\\approx0.25

respectively, while

j=0.2

gives deviating results. This deviation might be caused by the influence of other magnetically ordered states that we have not accounted for in our calculations, and so we restrict the validity of our results to

j<0.125

We study the ground-state properties of antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chains with half-integer spins ranging from

S=\frac{3}{2}

S=\frac{11}{2}

using the density-matrix renormalization group method. We map out the ground-state phase diagrams as a function of

\frac{{J}_{2}}{{J}_{1}}

containing topological phases with alternating $\frac{2S\ensuremath{-}1}{2}$ and

\frac{2S+1}{2}

valence bonds. We identify these topological phases and their boundaries by calculating the string order parameter, the dimer order parameter, and the spin gap for those $\mathrm{high}\text{\ensuremath{-}}S$ systems in the thermodynamic limit (finite-size scaling). We find that these topological regions narrow down inversely with

S

and converge to a single point at

\frac{{J}_{2}}{{J}_{1}}=\frac{1}{4}

in the classical limit---a critical threshold between commensurate and incommensurate orders. In addition, we extend the discussion of the Majumder-Ghosh state, previously noted only for

S=\frac{1}{2}

, and speculate about its possible presence as a ground state in half-integer high-spin systems over a substantial range of

\frac{{J}_{2}}{{J}_{1}}

values.

We investigate the boundary phases of a (2+1)-dimensional quantum critical Heisenberg model with a dangling spin chain. By introducing a multispin

Q

-term along the boundary, we drive a continuous boundary transition from an antiferromagnetic (AF) order to a valence-bond solid (VBS) order. Using large-scale quantum Monte Carlo simulations, we locate the critical point at

Q_{c}=0.310(11)

, and obtain the critical exponents at

Q_{c}

, including

y_{s}=0.81(4)

and the scaling dimensions of AF and VBS order parameters

Δ_{s}=0.660(15)

and

Δ_{v}=0.204(14)

. The weak long-range AF order for $Q