arXiv:2004.14669 · Physical review. B./Physical review. B

Kosterlitz-Thouless phase and Zd topological quantum phase

Mohammad Hossein Zarei

Published Jun 8, 2020• 3 citations

Abstract

It has been known that encoding Boltzmann weights of a classical spin model in amplitudes of a many-body wave function can provide quantum models whose phase structure is characterized by using classical phase transitions. In particular, such correspondence can lead to finding new quantum phases corresponding to well-known classical phases. Here, we investigate this problem for the Kosterlitz-Thouless (KT) phase in the $d$-state clock model, where we find a corresponding quantum model constructed by applying a local invertible transformation on a $d$-level version of Kitaev's toric code. In particular, we show the ground-state fidelity in such a quantum model is mapped to the heat capacity of the clock model. Accordingly, we identify an extended topological phase transition in our model in the sense that, for $d\ensuremath{\ge}5$, a KT-like quantum phase emerges between a ${Z}_{d}$ topological phase and a trivial phase. Then, using a mapping to the correlation function in the clock model, we introduce a nonlocal (string) observable for the quantum model which exponentially decays in terms of distance between two end points of the corresponding string in the ${Z}_{d}$ topological phase, while it shows a power law behavior in the KT-like phase. Finally, using well-known transition temperatures for the $d$-state clock model, we give evidence to show that while the stability of both the ${Z}_{d}$ topological phase and the KT-like phase increases by increasing $d$, the KT-like phase is even more stable than the ${Z}_{d}$ topological phase for large $d$.

DOI: 10.1103/physrevb.101.235126

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arXiv:2004.14669 · Physical review. B./Physical review. B

Kosterlitz-Thouless phase and Zd topological quantum phase

Mohammad Hossein Zarei

Published Jun 8, 2020• 3 citations

arXiv HTML PDF

Abstract

d

-state clock model, where we find a corresponding quantum model constructed by applying a local invertible transformation on a

d

-level version of Kitaev's toric code. In particular, we show the ground-state fidelity in such a quantum model is mapped to the heat capacity of the clock model. Accordingly, we identify an extended topological phase transition in our model in the sense that, for

d\ensuremath{\ge}5

, a KT-like quantum phase emerges between a

{Z}_{d}

topological phase and a trivial phase. Then, using a mapping to the correlation function in the clock model, we introduce a nonlocal (string) observable for the quantum model which exponentially decays in terms of distance between two end points of the corresponding string in the

{Z}_{d}

topological phase, while it shows a power law behavior in the KT-like phase. Finally, using well-known transition temperatures for the

d

-state clock model, we give evidence to show that while the stability of both the

{Z}_{d}

topological phase and the KT-like phase increases by increasing

d

, the KT-like phase is even more stable than the

{Z}_{d}

topological phase for large

d

DOI: 10.1103/physrevb.101.235126

Related papers

Conditional global entanglement in a Kosterlitz-Thouless quantum phase transition

Elahe Samimi, Mohammad Hossein Zarei, Afshin Montakhab

Entanglement is known as an important indicator for characterizing different types of quantum phase transitions (QPTs); however, it faces some challenges in the Kosterlitz-Thouless (KT) phase transitions due to an essential singularity which cannot be identified in finite derivatives of the ground-state energy. In this paper, we consider global entanglement (GE) in a KT phase transition and show that while it does not indicate any clear signature of the phase transition, the conditional version of GE is a good indicator with strong signatures of the KT transition. In particular, we study a deformed version of the ${Z}_{d}$ Kitaev model which has an intermediate KT phase which separates a ${Z}_{d}$ topological phase from a magnetized phase at two different KT transition points. Using a mapping to the classical $d$ -state clock model, we consider GE and the generalized GE and show that they do not provide a reliable indicator of transition points. However, their difference, called conditional global entanglement ( $Q$ ), shows a peak at the first KT transition point. Additionally, we show that it can characterize various phases of the model as it behaves substantially differently in each phase. We therefore conclude that $Q$ is a useful measure that can characterize various phases of KT QPTs as well as their related critical points.Entanglement is known as an important indicator for characterizing different types of quantum phase transitions (QPTs); however, it faces some challenges in the Kosterlitz-Thouless (KT) phase transitions due to an essential singularity which cannot be identified in finite derivatives of the ground-state energy. In this paper, we consider global entanglement (GE) in a KT phase transition and show that while it does not indicate any clear signature of the phase transition, the conditional version of GE is a good indicator with strong signatures of the KT transition. In particular, we study a deformed version of the ${Z}_{d}$ Kitaev model which has an intermediate KT phase which separates a ${Z}_{d}$ topological phase from a magnetized phase at two different KT transition points. Using a mapping to the classical $d$-state clock model, we consider GE and the generalized GE and show that they do not provide a reliable indicator of transition points. However, their difference, called conditional global entanglement ($Q$), shows a peak at the first KT transition point. Additionally, we show that it can characterize various phases of the model as it behaves substantially differently in each phase. We therefore conclude that $Q$ is a useful measure that can characterize various phases of KT QPTs as well as their related critical points.

arXiv:2301.09168Physical Review AMay 17, 20233 citations

Non-local order parameter for topological phase transition in the quantum toric code

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It has been known that by encoding the Boltzmann weights of a classical Ising model in the amplitudes of the wave function of the ground state of the toric code model, the classical phase transition in Ising model is mapped to a topological phase transition in a perturbed toric code model. Since such topological phase transitions cannot be characterized by any local order parameter, it will be an important challenge to find an order parameter which describes the above topological phase transition. In this paper, using a simple technic based on mapping between classical Ising model and the ground state of the toric code model, we find a non-local order parameter which well reveals the topological nature of the above phase transition. We show that such an order parameter is, in fact, a kind of string order parameter which has been recently introduced for some topological phase transitions.It has been known that by encoding the Boltzmann weights of a classical Ising model in the amplitudes of the wave function of the ground state of the toric code model, the classical phase transition in Ising model is mapped to a topological phase transition in a perturbed toric code model. Since such topological phase transitions cannot be characterized by any local order parameter, it will be an important challenge to find an order parameter which describes the above topological phase transition. In this paper, using a simple technic based on mapping between classical Ising model and the ground state of the toric code model, we find a non-local order parameter which well reveals the topological nature of the above phase transition. We show that such an order parameter is, in fact, a kind of string order parameter which has been recently introduced for some topological phase transitions.

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Quantum Kosterlitz-Thouless phase transition of the self-dual toric code phase

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In the tensor network representation, a deformed $Z_{2}$ topological ground state wave function is proposed and its norm can be exactly mapped to the two-dimensional solvable Ashkin-Teller (AT) model. Then the topological (toric code) phase with anyonic excitations corresponds to the partial order phase of the AT model, and possible topological phase transitions are precisely determined. With the electric-magnetic self-duality, a novel gapless Coulomb state with quasi-long-range order is obtained via a quantum Kosterlitz-Thouless phase transition. The corresponding ground state is a condensate of pairs of logarithmically confined electric charges and magnetic fluxes, and the scaling behavior of various anyon correlations can be exactly derived, revealing the effective interaction between anyons and their condensation. Deformations away from the self-duality drive the Coulomb state into either the gapped Higgs phase or confining phase.In the tensor network representation, a deformed $Z_{2}$ topological ground state wave function is proposed and its norm can be exactly mapped to the two-dimensional solvable Ashkin-Teller (AT) model. Then the topological (toric code) phase with anyonic excitations corresponds to the partial order phase of the AT model, and possible topological phase transitions are precisely determined. With the electric-magnetic self-duality, a novel gapless Coulomb state with quasi-long-range order is obtained via a quantum Kosterlitz-Thouless phase transition. The corresponding ground state is a condensate of pairs of logarithmically confined electric charges and magnetic fluxes, and the scaling behavior of various anyon correlations can be exactly derived, revealing the effective interaction between anyons and their condensation. Deformations away from the self-duality drive the Coulomb state into either the gapped Higgs phase or confining phase.

Jan 29, 20190 citations

Perturbative study of the one-dimensional quantum clock model

Bingnan Zhang

We calculate the ground-state energy density $\ensuremath{\epsilon}(g)$ for the one-dimensional $N$ -state quantum clock model up to order 18, where $g$ is the coupling and $N=3,4,5,...,10,20$ . Using methods based on the Pad\'e approximation, we extract the singular structure of ${\ensuremath{\epsilon}}^{\ensuremath{''}}(g)$ or $\ensuremath{\epsilon}(g)$. They correspond to the specific heat and free energy of the classical two-dimensional (2D) clock model. We find that, for $N=3,4$ , there is a single critical point at ${g}_{c}=1$ . The heat capacity exponent of the corresponding 2D classical model is $\ensuremath{\alpha}=0.34\ifmmode\pm\else\textpm\fi{}0.01$ for $N=3$ , and $\ensuremath{\alpha}=\ensuremath{-}0.01\ifmmode\pm\else\textpm\fi{}0.01$ for $N=4$ . For $N>4$ , there are two exponential singularities related by ${g}_{c1}=1/{g}_{c2}$ , and $\ensuremath{\epsilon}(g)$ behaves as $A{e}^{\ensuremath{-}\frac{c}{|{g}_{c}{\ensuremath{-}g|}^{\ensuremath{\sigma}}}}+\text{analytic}\phantom{\rule{4.pt}{0ex}}\text{terms}$ near ${g}_{c}$ . The exponent $\ensuremath{\sigma}$ gradually grows from 0.2 to 0.5 as $N$ increases from 5 to 9, and it stabilizes at 0.5 when $N>9$ . The phase transitions exhibited in these models should be generalizations of the Kosterlitz-Thouless transition, which has $\ensuremath{\sigma}=0.5$.We calculate the ground-state energy density $\ensuremath{\epsilon}(g)$ for the one-dimensional $N$-state quantum clock model up to order 18, where $g$ is the coupling and $N=3,4,5,...,10,20$. Using methods based on the Pad\'e approximation, we extract the singular structure of ${\ensuremath{\epsilon}}^{\ensuremath{''}}(g)$ or $\ensuremath{\epsilon}(g)$. They correspond to the specific heat and free energy of the classical two-dimensional (2D) clock model. We find that, for $N=3,4$, there is a single critical point at ${g}_{c}=1$. The heat capacity exponent of the corresponding 2D classical model is $\ensuremath{\alpha}=0.34\ifmmode\pm\else\textpm\fi{}0.01$ for $N=3$, and $\ensuremath{\alpha}=\ensuremath{-}0.01\ifmmode\pm\else\textpm\fi{}0.01$ for $N=4$. For $N>4$, there are two exponential singularities related by ${g}_{c1}=1/{g}_{c2}$, and $\ensuremath{\epsilon}(g)$ behaves as $A{e}^{\ensuremath{-}\frac{c}{|{g}_{c}{\ensuremath{-}g|}^{\ensuremath{\sigma}}}}+\text{analytic}\phantom{\rule{4.pt}{0ex}}\text{terms}$ near ${g}_{c}$. The exponent $\ensuremath{\sigma}$ gradually grows from 0.2 to 0.5 as $N$ increases from 5 to 9, and it stabilizes at 0.5 when $N>9$. The phase transitions exhibited in these models should be generalizations of the Kosterlitz-Thouless transition, which has $\ensuremath{\sigma}=0.5$.

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Guido Giachetti, Nicolò Defenu, Stefano Ruffo +1 more

The Berezinskii-Kosterlitz-Thouless (BKT) transition is the paradigmatic example of a topological phase transition without symmetry breaking, where a quasiordered phase, characterized by a power-law scaling of the correlation functions at low temperature, is disrupted by the proliferation of topological excitations above the critical temperature T_{BKT}. In this Letter, we consider the effect of long-range decaying couplings ∼r^{-2-σ} on the BKT transition. After pointing out the relevance of this nontrivial problem, we discuss the phase diagram, which is far richer than the corresponding short-range one. It features-for 7/4<σ<2-a quasiordered phase in a finite temperature range T_{c}<T<T_{BKT}, which occurs between a symmetry broken phase for T<T_{c} and a disordered phase for T>T_{BKT}. The transition temperature T_{c} displays unique universal features quite different from those of the traditional, short-range XY model. Given the universal nature of our findings, they may be observed in current experimental realizations in 2D atomic, molecular, and optical quantum systems.The Berezinskii-Kosterlitz-Thouless (BKT) transition is the paradigmatic example of a topological phase transition without symmetry breaking, where a quasiordered phase, characterized by a power-law scaling of the correlation functions at low temperature, is disrupted by the proliferation of topological excitations above the critical temperature T_{BKT}. In this Letter, we consider the effect of long-range decaying couplings ∼r^{-2-σ} on the BKT transition. After pointing out the relevance of this nontrivial problem, we discuss the phase diagram, which is far richer than the corresponding short-range one. It features-for 7/4<σ<2-a quasiordered phase in a finite temperature range T_{c}<T<T_{BKT}, which occurs between a symmetry broken phase for T<T_{c} and a disordered phase for T>T_{BKT}. The transition temperature T_{c} displays unique universal features quite different from those of the traditional, short-range XY model. Given the universal nature of our findings, they may be observed in current experimental realizations in 2D atomic, molecular, and optical quantum systems.

arXiv:2104.13217Physical Review LettersOct 5, 202146 citations

Phase transitions in the Zp and U(1) clock models

Gaoyong Sun, T. Vekua, Emilio Cobanera +1 more

Quantum phase transitions are studied in the nonchiral $p$ -clock chain, and a new explicitly U(1)-symmetric clock model, by monitoring the ground-state fidelity susceptibility. For $p\ensuremath{\ge}5$, the self-dual ${\mathbb{Z}}_{p}$ -symmetric chain displays a double-hump structure in the fidelity susceptibility with both peak positions and heights scaling logarithmically to their corresponding thermodynamic values. This scaling is precisely as expected for two Beresinskii-Kosterlitz-Thouless (BKT) transitions located symmetrically about the self-dual point, and so confirms numerically the theoretical scenario that sets $p=5$ as the lowest $p$ supporting BKT transitions in ${\mathbb{Z}}_{p}$ -symmetric clock models. For our U(1)-symmetric, non-self-dual minimal modification of the $p$ -clock model we find that the phase diagram depends strongly on the parity of $p$ and only one BKT transition survives for $p\ensuremath{\ge}5$. Using asymptotic calculus we map the self-dual clock model exactly, in the large $p$ limit, to the quantum $O(2)$ rotor chain. Finally, using bond-algebraic dualities we estimate the critical BKT transition temperatures of the classical planar $p$ -clock models defined on square lattices, in the limit of extreme spatial anisotropy. Our values agree remarkably well with those determined via classical Monte Carlo for isotropic lattices. This work highlights the power of the fidelity susceptibility as a tool for diagnosing the BKT transitions even when only discrete symmetries are present.Quantum phase transitions are studied in the nonchiral $p$-clock chain, and a new explicitly U(1)-symmetric clock model, by monitoring the ground-state fidelity susceptibility. For $p\ensuremath{\ge}5$, the self-dual ${\mathbb{Z}}_{p}$-symmetric chain displays a double-hump structure in the fidelity susceptibility with both peak positions and heights scaling logarithmically to their corresponding thermodynamic values. This scaling is precisely as expected for two Beresinskii-Kosterlitz-Thouless (BKT) transitions located symmetrically about the self-dual point, and so confirms numerically the theoretical scenario that sets $p=5$ as the lowest $p$ supporting BKT transitions in ${\mathbb{Z}}_{p}$-symmetric clock models. For our U(1)-symmetric, non-self-dual minimal modification of the $p$-clock model we find that the phase diagram depends strongly on the parity of $p$ and only one BKT transition survives for $p\ensuremath{\ge}5$. Using asymptotic calculus we map the self-dual clock model exactly, in the large $p$ limit, to the quantum $O(2)$ rotor chain. Finally, using bond-algebraic dualities we estimate the critical BKT transition temperatures of the classical planar $p$-clock models defined on square lattices, in the limit of extreme spatial anisotropy. Our values agree remarkably well with those determined via classical Monte Carlo for isotropic lattices. This work highlights the power of the fidelity susceptibility as a tool for diagnosing the BKT transitions even when only discrete symmetries are present.

arXiv:1907.09082Physical review. B./Physical review. BSep 17, 201927 citations

d

-state clock model, where we find a corresponding quantum model constructed by applying a local invertible transformation on a

d

d\ensuremath{\ge}5

, a KT-like quantum phase emerges between a

{Z}_{d}

{Z}_{d}

topological phase, while it shows a power law behavior in the KT-like phase. Finally, using well-known transition temperatures for the

d

-state clock model, we give evidence to show that while the stability of both the

{Z}_{d}

topological phase and the KT-like phase increases by increasing

d

, the KT-like phase is even more stable than the

{Z}_{d}

topological phase for large

d

{Z}_{d}

Kitaev model which has an intermediate KT phase which separates a

{Z}_{d}

topological phase from a magnetized phase at two different KT transition points. Using a mapping to the classical

d

-state clock model, we consider GE and the generalized GE and show that they do not provide a reliable indicator of transition points. However, their difference, called conditional global entanglement (

Q

), shows a peak at the first KT transition point. Additionally, we show that it can characterize various phases of the model as it behaves substantially differently in each phase. We therefore conclude that

Q

is a useful measure that can characterize various phases of KT QPTs as well as their related critical points.

In the tensor network representation, a deformed

Z_{2}

topological ground state wave function is proposed and its norm can be exactly mapped to the two-dimensional solvable Ashkin-Teller (AT) model. Then the topological (toric code) phase with anyonic excitations corresponds to the partial order phase of the AT model, and possible topological phase transitions are precisely determined. With the electric-magnetic self-duality, a novel gapless Coulomb state with quasi-long-range order is obtained via a quantum Kosterlitz-Thouless phase transition. The corresponding ground state is a condensate of pairs of logarithmically confined electric charges and magnetic fluxes, and the scaling behavior of various anyon correlations can be exactly derived, revealing the effective interaction between anyons and their condensation. Deformations away from the self-duality drive the Coulomb state into either the gapped Higgs phase or confining phase.

We calculate the ground-state energy density $\ensuremath{\epsilon}(g)$ for the one-dimensional

N

-state quantum clock model up to order 18, where

g

is the coupling and

N=3,4,5,...,10,20

. Using methods based on the Pad\'e approximation, we extract the singular structure of ${\ensuremath{\epsilon}}^{\ensuremath{''}}(g)$ or $\ensuremath{\epsilon}(g)$. They correspond to the specific heat and free energy of the classical two-dimensional (2D) clock model. We find that, for

N=3,4

, there is a single critical point at

{g}_{c}=1

. The heat capacity exponent of the corresponding 2D classical model is $\ensuremath{\alpha}=0.34\ifmmode\pm\else\textpm\fi{}0.01$ for

N=3

, and $\ensuremath{\alpha}=\ensuremath{-}0.01\ifmmode\pm\else\textpm\fi{}0.01$ for

N=4

. For

N>4

, there are two exponential singularities related by

{g}_{c1}=1/{g}_{c2}

, and $\ensuremath{\epsilon}(g)$ behaves as $A{e}^{\ensuremath{-}\frac{c}{|{g}_{c}{\ensuremath{-}g|}^{\ensuremath{\sigma}}}}+\text{analytic}\phantom{\rule{4.pt}{0ex}}\text{terms}$ near

{g}_{c}

. The exponent $\ensuremath{\sigma}$ gradually grows from 0.2 to 0.5 as

N

increases from 5 to 9, and it stabilizes at 0.5 when

N>9

. The phase transitions exhibited in these models should be generalizations of the Kosterlitz-Thouless transition, which has $\ensuremath{\sigma}=0.5$.

Quantum phase transitions are studied in the nonchiral

p

-clock chain, and a new explicitly U(1)-symmetric clock model, by monitoring the ground-state fidelity susceptibility. For $p\ensuremath{\ge}5$, the self-dual

{\mathbb{Z}}_{p}

-symmetric chain displays a double-hump structure in the fidelity susceptibility with both peak positions and heights scaling logarithmically to their corresponding thermodynamic values. This scaling is precisely as expected for two Beresinskii-Kosterlitz-Thouless (BKT) transitions located symmetrically about the self-dual point, and so confirms numerically the theoretical scenario that sets

p=5

as the lowest

p

supporting BKT transitions in

{\mathbb{Z}}_{p}

-symmetric clock models. For our U(1)-symmetric, non-self-dual minimal modification of the

p

-clock model we find that the phase diagram depends strongly on the parity of

p

and only one BKT transition survives for $p\ensuremath{\ge}5$. Using asymptotic calculus we map the self-dual clock model exactly, in the large

p

limit, to the quantum

O(2)

rotor chain. Finally, using bond-algebraic dualities we estimate the critical BKT transition temperatures of the classical planar

p

-clock models defined on square lattices, in the limit of extreme spatial anisotropy. Our values agree remarkably well with those determined via classical Monte Carlo for isotropic lattices. This work highlights the power of the fidelity susceptibility as a tool for diagnosing the BKT transitions even when only discrete symmetries are present.