Physical Review A

Berezinskii-Kosterlitz-Thouless transition uncovered by the fidelity susceptibility in the XXZ model

Bo Wang, Mang Feng, Zeqian Chen

Published Jun 2, 2010• 37 citations

Abstract

We investigate the one-dimensional Heisenberg $\mathit{XXZ}$ model involving the Dzyaloshinskii-Moriya (DM) interaction from the viewpoint of quantum information. By using the density-matrix--renormalization-group technique, we study the Berezinskii-Kosterlitz-Thouless (BKT) phase transition using the ground-state fidelity, the fidelity susceptibility, and the entanglement entropy. We argue that the BKT phase transition can be well characterized in our model, and the critical line induced by the DM interaction can also be detected.

DOI: 10.1103/physreva.81.064301

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Physical Review A

Berezinskii-Kosterlitz-Thouless transition uncovered by the fidelity susceptibility in the XXZ model

Bo Wang, Mang Feng, Zeqian Chen

Published Jun 2, 2010• 37 citations

Publisher

Abstract

We investigate the one-dimensional Heisenberg

\mathit{XXZ}

model involving the Dzyaloshinskii-Moriya (DM) interaction from the viewpoint of quantum information. By using the density-matrix--renormalization-group technique, we study the Berezinskii-Kosterlitz-Thouless (BKT) phase transition using the ground-state fidelity, the fidelity susceptibility, and the entanglement entropy. We argue that the BKT phase transition can be well characterized in our model, and the critical line induced by the DM interaction can also be detected.

DOI: 10.1103/physreva.81.064301

Related papers

Fidelity study of the quantum phase transition in the one-dimensional spin-1 XXZ model in the thermodynamic limit

Shanhu Yang, Jun-Hui Zheng, Tao Yang

The potential of fidelity in accurately detecting unconventional quantum phase transitions (QPTs) remains an open question. Specific attention is directed towards high-order phase transitions, including the infinite-order Berezinskii-Kosterlitz-Thouless (BKT) transition and the Gaussian transition. In this study, we extend the concept of fidelity susceptibility to fixed-point fidelity susceptibility in the thermodynamic limit and assess this measure using tensor-network algorithms. Our results demonstrate that the fixed-point fidelity susceptibility effectively identifies the BKT and Gaussian phase transitions, a capability lacking in the traditional finite-fidelity approach. Furthermore, we establish a correlation between the fixed-point fidelity method and the entanglement method, demonstrating their equivalence. This research not only confirms the efficacy of fidelity in detecting QPTs but also enhances the understanding of fidelity-related methodologies.The potential of fidelity in accurately detecting unconventional quantum phase transitions (QPTs) remains an open question. Specific attention is directed towards high-order phase transitions, including the infinite-order Berezinskii-Kosterlitz-Thouless (BKT) transition and the Gaussian transition. In this study, we extend the concept of fidelity susceptibility to fixed-point fidelity susceptibility in the thermodynamic limit and assess this measure using tensor-network algorithms. Our results demonstrate that the fixed-point fidelity susceptibility effectively identifies the BKT and Gaussian phase transitions, a capability lacking in the traditional finite-fidelity approach. Furthermore, we establish a correlation between the fixed-point fidelity method and the entanglement method, demonstrating their equivalence. This research not only confirms the efficacy of fidelity in detecting QPTs but also enhances the understanding of fidelity-related methodologies.

Physical review. B./Physical review. BApr 22, 20254 citations

Universal shift of the fidelity susceptibility peak away from the critical point of the Berezinskii-Kosterlitz-Thouless quantum phase transition

Łukasz Cincio, Marek M. Rams, Jacek Dziarmaga +1 more

We show that the peak which can be observed in fidelity susceptibility around the Berezinskii-Kosterlitz-Thouless transition is shifted from the quantum critical point (QCP) at ${J}_{c}$ to ${J}^{*}$ in the gapped phase by a value $|{J}^{*}\ensuremath{-}{J}_{c}|={B}^{2}/36$, where ${B}^{2}$ is a transition width controlling the asymptotic form of the correlation length $\ensuremath{\xi}\ensuremath{\sim}exp(\ensuremath{-}B/\sqrt{|J\ensuremath{-}{J}_{c}|})$ in that phase. This is in contrast to the conventional continuous QCP where the maximum is an indicator of the position of the critical point. The shape of the peak is universal, emphasizing the close connection between fidelity susceptibility and the correlation length. We support those arguments with numerical matrix product state simulations of the one-dimensional Bose-Hubbard model in the thermodynamic limit, where the broad peak is located at ${J}^{*}=0.212$ that is significantly different from ${J}_{c}=0.3048(3)$ . In the spin- $3/2$ XXZ model the shift from ${J}_{c}=1$ to ${J}^{*}=1.0021$ is small but the narrow universal peak is much more pronounced over the nonuniversal background.We show that the peak which can be observed in fidelity susceptibility around the Berezinskii-Kosterlitz-Thouless transition is shifted from the quantum critical point (QCP) at ${J}_{c}$ to ${J}^{*}$ in the gapped phase by a value $|{J}^{*}\ensuremath{-}{J}_{c}|={B}^{2}/36$, where ${B}^{2}$ is a transition width controlling the asymptotic form of the correlation length $\ensuremath{\xi}\ensuremath{\sim}exp(\ensuremath{-}B/\sqrt{|J\ensuremath{-}{J}_{c}|})$ in that phase. This is in contrast to the conventional continuous QCP where the maximum is an indicator of the position of the critical point. The shape of the peak is universal, emphasizing the close connection between fidelity susceptibility and the correlation length. We support those arguments with numerical matrix product state simulations of the one-dimensional Bose-Hubbard model in the thermodynamic limit, where the broad peak is located at ${J}^{*}=0.212$ that is significantly different from ${J}_{c}=0.3048(3)$. In the spin-$3/2$ XXZ model the shift from ${J}_{c}=1$ to ${J}^{*}=1.0021$ is small but the narrow universal peak is much more pronounced over the nonuniversal background.

arXiv:1906.05307Physical review. B./Physical review. BAug 19, 201913 citations

Berezinskii-Kosterlitz-Thouless Renormalization Group Flow at a Quantum Phase Transition

Matthias Thamm, Harini Radhakrishnan, Hatem Barghathi +4 more

We present a controlled numerical study of the Berezinskii-Kosterlitz-Thouless (BKT) transition in the one-dimensional Bose-Hubbard model at unit filling, providing evidence of the characteristic logarithmic finite-size scaling of the BKT transition. Employing density matrix renormalization group and quantum Monte Carlo simulations under periodic boundary conditions, together with a systematic finite-size scaling analysis of bipartite particle number fluctuations, we resolve boundary-induced complications that previously obscured critical scaling. We demonstrate that a suitably chosen central region under open boundaries reproduces universal renormalization group signatures, reconciling earlier discrepancies. Finally, leveraging a nonparametric Bayesian analysis, we determine the critical interaction strength with high precision to be U_{c}/J=3.275(2), establishing a benchmark for BKT physics in one-dimensional quantum models.We present a controlled numerical study of the Berezinskii-Kosterlitz-Thouless (BKT) transition in the one-dimensional Bose-Hubbard model at unit filling, providing evidence of the characteristic logarithmic finite-size scaling of the BKT transition. Employing density matrix renormalization group and quantum Monte Carlo simulations under periodic boundary conditions, together with a systematic finite-size scaling analysis of bipartite particle number fluctuations, we resolve boundary-induced complications that previously obscured critical scaling. We demonstrate that a suitably chosen central region under open boundaries reproduces universal renormalization group signatures, reconciling earlier discrepancies. Finally, leveraging a nonparametric Bayesian analysis, we determine the critical interaction strength with high precision to be U_{c}/J=3.275(2), establishing a benchmark for BKT physics in one-dimensional quantum models.

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Ranjith Kumar R, S Rahul, Surya Narayan +1 more

We present extensive calculation and the results of quantum Berezinskii-Kosterlitz-Thouless (BKT) transition for interacting helical liquid system. This system shows the quantum BKT transition for the different physical situation which we present by two model Hamiltonians. We derive the renormalization group (RG) equations explicitly and present the flow lines behavior. We also present RG flow lines based on the exact solution. We observe that the Majorana fermion zero modes physics and the gapped Ising phase of the system. We also observe that the evidence of gapless Luttinger liquid phase as a common phase for both quantum BKT transitions.We present extensive calculation and the results of quantum Berezinskii-Kosterlitz-Thouless (BKT) transition for interacting helical liquid system. This system shows the quantum BKT transition for the different physical situation which we present by two model Hamiltonians. We derive the renormalization group (RG) equations explicitly and present the flow lines behavior. We also present RG flow lines based on the exact solution. We observe that the Majorana fermion zero modes physics and the gapped Ising phase of the system. We also observe that the evidence of gapless Luttinger liquid phase as a common phase for both quantum BKT transitions.

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Berezinskii-Kosterlitz-Thouless Quantum Supercriticality in XXZ Heisenberg Spin Chain

Haoshun Chen, Enze Lv, Ning Xi +2 more

Quantum fluctuations can give rise to a singular quantum critical point (QCP) in the ground state, whose influence extends to finite temperatures, forming a quantum critical regime (QCR). Recently, it has been shown that in the quantum Ising model, the symmetry-breaking, longitudinal field can induce a quantum supercritical regime (QSR) emanating from the QCP, which hosts a universally enhanced quantum supercritical magnetocaloric effect (MCE). In this paper, we show that the QSR also emerges in the spin-1/2 XXZ model, in both the form of Ising and Berezinskii-Kosterlitz-Thouless (BKT) supercriticality. Using ground-state and finite-temperature tensor-network methods, we investigate quantum supercritical phenomena near a BKT QCP. We reveal a quantum supercritical crossover scaling $T \propto h^{2/3}$ and a Grüneisen ratio scaling $Γ_h \propto T^{-3/2}$ for the BKT QCP, which differ from the corresponding Ising supercritical scalings. Nevertheless, we find that the scaling function $ϕ_Γ(x)$ of the singular Grüneisen ratio for both BKT and Ising cases can be approximately described by the same expression $ϕ_Γ(x) \approx x/(1+x^2)$ . Our work extends the study of quantum supercritical phenomena from the Ising to the XXZ Heisenberg model, thereby revealing the presence of BKT quantum supercriticality and broadening the scope of quantum supercritical physics.Quantum fluctuations can give rise to a singular quantum critical point (QCP) in the ground state, whose influence extends to finite temperatures, forming a quantum critical regime (QCR). Recently, it has been shown that in the quantum Ising model, the symmetry-breaking, longitudinal field can induce a quantum supercritical regime (QSR) emanating from the QCP, which hosts a universally enhanced quantum supercritical magnetocaloric effect (MCE). In this paper, we show that the QSR also emerges in the spin-1/2 XXZ model, in both the form of Ising and Berezinskii-Kosterlitz-Thouless (BKT) supercriticality. Using ground-state and finite-temperature tensor-network methods, we investigate quantum supercritical phenomena near a BKT QCP. We reveal a quantum supercritical crossover scaling $T \propto h^{2/3}$ and a Grüneisen ratio scaling $Γ_h \propto T^{-3/2}$ for the BKT QCP, which differ from the corresponding Ising supercritical scalings. Nevertheless, we find that the scaling function $ϕ_Γ(x)$ of the singular Grüneisen ratio for both BKT and Ising cases can be approximately described by the same expression $ϕ_Γ(x) \approx x/(1+x^2)$. Our work extends the study of quantum supercritical phenomena from the Ising to the XXZ Heisenberg model, thereby revealing the presence of BKT quantum supercriticality and broadening the scope of quantum supercritical physics.

arXiv:2603.27789Mar 29, 20260 citations

Berezinskii-Kosterlitz-Thouless transition close to zero temperature

Mikhail Gennag'evich Vasin

The Berezinskii-Kosterlitz-Thouless (BKT) phase transition is considered in the condition of lowest temperatures, when thermal fluctuations give place to quantum ones. For this goal, the critical dynamic of the Sine-Gordon model near the quantum critical point is considered. The approach based on the Keldysh-Schwinger technique of non-equilibrium dynamics description is used, as well as the method of taking into account the crossover from the thermal fluctuation regime to the quantum one in the renormalization group technique. For the system with low vortex concentration, it is shown that the BKT transition unavoidably occurs at a temperature above the crossover temperature from the thermal fluctuation regime to quantum one. As a result at small temperatures the critical exponent of the BKT transition, $η=1/4$ , remains unchanged, however, the universal relation of the superfluid density jump adjusts a little from the well-known relation by Nelson and Kosterlitz.The Berezinskii-Kosterlitz-Thouless (BKT) phase transition is considered in the condition of lowest temperatures, when thermal fluctuations give place to quantum ones. For this goal, the critical dynamic of the Sine-Gordon model near the quantum critical point is considered. The approach based on the Keldysh-Schwinger technique of non-equilibrium dynamics description is used, as well as the method of taking into account the crossover from the thermal fluctuation regime to the quantum one in the renormalization group technique. For the system with low vortex concentration, it is shown that the BKT transition unavoidably occurs at a temperature above the crossover temperature from the thermal fluctuation regime to quantum one. As a result at small temperatures the critical exponent of the BKT transition, $η=1/4$, remains unchanged, however, the universal relation of the superfluid density jump adjusts a little from the well-known relation by Nelson and Kosterlitz.

arXiv:2104.10983Apr 22, 20210 citations

We investigate the one-dimensional Heisenberg

\mathit{XXZ}

We show that the peak which can be observed in fidelity susceptibility around the Berezinskii-Kosterlitz-Thouless transition is shifted from the quantum critical point (QCP) at

{J}_{c}

{J}^{*}

in the gapped phase by a value $|{J}^{*}\ensuremath{-}{J}_{c}|={B}^{2}/36$, where

{B}^{2}

is a transition width controlling the asymptotic form of the correlation length $\ensuremath{\xi}\ensuremath{\sim}exp(\ensuremath{-}B/\sqrt{|J\ensuremath{-}{J}_{c}|})$ in that phase. This is in contrast to the conventional continuous QCP where the maximum is an indicator of the position of the critical point. The shape of the peak is universal, emphasizing the close connection between fidelity susceptibility and the correlation length. We support those arguments with numerical matrix product state simulations of the one-dimensional Bose-Hubbard model in the thermodynamic limit, where the broad peak is located at

{J}^{*}=0.212

that is significantly different from

{J}_{c}=0.3048(3)

. In the spin-

3/2

XXZ model the shift from

{J}_{c}=1

{J}^{*}=1.0021

is small but the narrow universal peak is much more pronounced over the nonuniversal background.

T \propto h^{2/3}

and a Grüneisen ratio scaling

Γ_h \propto T^{-3/2}

for the BKT QCP, which differ from the corresponding Ising supercritical scalings. Nevertheless, we find that the scaling function

ϕ_Γ(x)

of the singular Grüneisen ratio for both BKT and Ising cases can be approximately described by the same expression

ϕ_Γ(x) \approx x/(1+x^2)

. Our work extends the study of quantum supercritical phenomena from the Ising to the XXZ Heisenberg model, thereby revealing the presence of BKT quantum supercriticality and broadening the scope of quantum supercritical physics.

η=1/4

, remains unchanged, however, the universal relation of the superfluid density jump adjusts a little from the well-known relation by Nelson and Kosterlitz.