We study scaling behavior of the geometric tensor ${\ensuremath{\chi}}_{\ensuremath{\alpha},\ensuremath{\beta}}({\ensuremath{\lambda}}_{1},{\ensuremath{\lambda}}_{2})$ and the fidelity susceptibility ${\ensuremath{\chi}}_{F}$ in the vicinity of a quantum multicritical point (MCP) using the example of a transverse $\mathit{XY}$ model. We show that the behavior of the geometric tensor (and thus of ${\ensuremath{\chi}}_{F}$) is drastically different from that seen near a critical point. In particular, we find that it is a highly nonmonotonic function of $\ensuremath{\lambda}$ along the generic direction ${\ensuremath{\lambda}}_{1}~{\ensuremath{\lambda}}_{2}=\ensuremath{\lambda}$ when the system size $L$ is bounded by the shorter and longer correlation lengths characterizing the MCP: $1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{1}}\ensuremath{\ll}L\ensuremath{\ll}1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{2}}$, where ${\ensuremath{\nu}}_{1}<{\ensuremath{\nu}}_{2}$ are the two correlation-length exponents characterizing the system. We find that the scaling of the maxima of the components of ${\ensuremath{\chi}}_{\ensuremath{\alpha}\ensuremath{\beta}}$ is associated with the emergence of quasicritical points at $\ensuremath{\lambda}~1/{L}^{1/{\ensuremath{\nu}}_{1}}$, related to the proximity to the critical line of the finite-momentum anisotropic transition. This scaling is different from that in the thermodynamic limit $L\ensuremath{\gg}1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{2}}$, which is determined by the conventional critical exponents. We use our results to calculate the defect density following a rapid quench starting from the MCP and show that it exerts a steplike behavior for small quench amplitudes. A study of heat density and diagonal entropy density also shows signatures of quasicritical points.We study scaling behavior of the geometric tensor ${\ensuremath{\chi}}_{\ensuremath{\alpha},\ensuremath{\beta}}({\ensuremath{\lambda}}_{1},{\ensuremath{\lambda}}_{2})$ and the fidelity susceptibility ${\ensuremath{\chi}}_{F}$ in the vicinity of a quantum multicritical point (MCP) using the example of a transverse $\mathit{XY}$ model. We show that the behavior of the geometric tensor (and thus of ${\ensuremath{\chi}}_{F}$) is drastically different from that seen near a critical point. In particular, we find that it is a highly nonmonotonic function of $\ensuremath{\lambda}$ along the generic direction ${\ensuremath{\lambda}}_{1}~{\ensuremath{\lambda}}_{2}=\ensuremath{\lambda}$ when the system size $L$ is bounded by the shorter and longer correlation lengths characterizing the MCP: $1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{1}}\ensuremath{\ll}L\ensuremath{\ll}1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{2}}$, where ${\ensuremath{\nu}}_{1}<{\ensuremath{\nu}}_{2}$ are the two correlation-length exponents characterizing the system. We find that the scaling of the maxima of the components of ${\ensuremath{\chi}}_{\ensuremath{\alpha}\ensuremath{\beta}}$ is associated with the emergence of quasicritical points at $\ensuremath{\lambda}~1/{L}^{1/{\ensuremath{\nu}}_{1}}$, related to the proximity to the critical line of the finite-momentum anisotropic transition. This scaling is different from that in the thermodynamic limit $L\ensuremath{\gg}1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{2}}$, which is determined by the conventional critical exponents. We use our results to calculate the defect density following a rapid quench starting from the MCP and show that it exerts a steplike behavior for small quench amplitudes. A study of heat density and diagonal entropy density also shows signatures of quasicritical points.

arXiv:1010.4446Physical Review BFeb 24, 201146 citations

Density fluctuations at critical points

J. P. Stark

PhysicaJan 1, 19671 citation

Fidelity susceptibility and general quench near an anisotropic quantum critical point

Victor Mukherjee, Amit Dutta

We study the scaling behavior of fidelity susceptibility density $({\ensuremath{\chi}}_{\mathrm{f}})$ at or close to an anisotropic quantum critical point (AQCP) characterized by two different correlation length exponents ${\ensuremath{\nu}}_{||}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ along parallel and perpendicular spatial directions, respectively. Our studies show that the response of the system due to a small change in the Hamiltonian near an AQCP is different from that seen near an isotropic QCP. In particular, for a finite system with linear dimension ${L}_{||}$ (${L}_{\ensuremath{\perp}}$) in the parallel (perpendicular) directions, the maximum value of ${\ensuremath{\chi}}_{\mathrm{f}}$ is found to increase in a power-law fashion with ${L}_{||}$ for small ${L}_{||}$ , with an exponent depending on both ${\ensuremath{\nu}}_{||}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ and eventually crosses over to a scaling with ${L}_{\ensuremath{\perp}}$ for ${L}_{||}^{1/{\ensuremath{\nu}}_{||}}\ensuremath{\gtrsim}{L}_{\ensuremath{\perp}}^{1/{\ensuremath{\nu}}_{\ensuremath{\perp}}}$. We also propose scaling relations of heat density and defect density generated following a quench starting from an AQCP and connect them to a generalized fidelity susceptibility. These predictions are verified exactly both analytically and numerically taking the example of a Hamiltonian showing a semi-Dirac band-crossing point.We study the scaling behavior of fidelity susceptibility density $({\ensuremath{\chi}}_{\mathrm{f}})$ at or close to an anisotropic quantum critical point (AQCP) characterized by two different correlation length exponents ${\ensuremath{\nu}}_{||}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ along parallel and perpendicular spatial directions, respectively. Our studies show that the response of the system due to a small change in the Hamiltonian near an AQCP is different from that seen near an isotropic QCP. In particular, for a finite system with linear dimension ${L}_{||}$ (${L}_{\ensuremath{\perp}}$) in the parallel (perpendicular) directions, the maximum value of ${\ensuremath{\chi}}_{\mathrm{f}}$ is found to increase in a power-law fashion with ${L}_{||}$ for small ${L}_{||}$, with an exponent depending on both ${\ensuremath{\nu}}_{||}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ and eventually crosses over to a scaling with ${L}_{\ensuremath{\perp}}$ for ${L}_{||}^{1/{\ensuremath{\nu}}_{||}}\ensuremath{\gtrsim}{L}_{\ensuremath{\perp}}^{1/{\ensuremath{\nu}}_{\ensuremath{\perp}}}$. We also propose scaling relations of heat density and defect density generated following a quench starting from an AQCP and connect them to a generalized fidelity susceptibility. These predictions are verified exactly both analytically and numerically taking the example of a Hamiltonian showing a semi-Dirac band-crossing point.

arXiv:1101.1713Physical Review BJun 8, 201120 citations

\mathit{XY}

model. We show that the behavior of the geometric tensor (and thus of ${\ensuremath{\chi}}_{F}$) is drastically different from that seen near a critical point. In particular, we find that it is a highly nonmonotonic function of $\ensuremath{\lambda}$ along the generic direction ${\ensuremath{\lambda}}_{1}~{\ensuremath{\lambda}}_{2}=\ensuremath{\lambda}$ when the system size

L

is bounded by the shorter and longer correlation lengths characterizing the MCP: $1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{1}}\ensuremath{\ll}L\ensuremath{\ll}1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{2}}$, where ${\ensuremath{\nu}}_{1}<{\ensuremath{\nu}}_{2}$ are the two correlation-length exponents characterizing the system. We find that the scaling of the maxima of the components of ${\ensuremath{\chi}}_{\ensuremath{\alpha}\ensuremath{\beta}}$ is associated with the emergence of quasicritical points at $\ensuremath{\lambda}~1/{L}^{1/{\ensuremath{\nu}}_{1}}$, related to the proximity to the critical line of the finite-momentum anisotropic transition. This scaling is different from that in the thermodynamic limit $L\ensuremath{\gg}1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{2}}$, which is determined by the conventional critical exponents. We use our results to calculate the defect density following a rapid quench starting from the MCP and show that it exerts a steplike behavior for small quench amplitudes. A study of heat density and diagonal entropy density also shows signatures of quasicritical points.

{L}_{||}

(${L}_{\ensuremath{\perp}}$) in the parallel (perpendicular) directions, the maximum value of ${\ensuremath{\chi}}_{\mathrm{f}}$ is found to increase in a power-law fashion with

{L}_{||}

for small

{L}_{||}

, with an exponent depending on both ${\ensuremath{\nu}}_{||}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ and eventually crosses over to a scaling with ${L}_{\ensuremath{\perp}}$ for ${L}_{||}^{1/{\ensuremath{\nu}}_{||}}\ensuremath{\gtrsim}{L}_{\ensuremath{\perp}}^{1/{\ensuremath{\nu}}_{\ensuremath{\perp}}}$. We also propose scaling relations of heat density and defect density generated following a quench starting from an AQCP and connect them to a generalized fidelity susceptibility. These predictions are verified exactly both analytically and numerically taking the example of a Hamiltonian showing a semi-Dirac band-crossing point.