Recently it has been shown that the fidelity of the ground state of a quantum many-body system can be used to detect its quantum critical points (QCPs). If g denotes the parameter in the Hamiltonian with respect to which the fidelity is computed, we find that for one-dimensional models with large but finite size, the fidelity susceptibility χF can detect a QCP provided that the correlation length exponent satisfies ν< 2. We then show that χF can be used to locate a QCP even if ν 2 if we introduce boundary conditions labeled by a twist angle Nθ ,w hereN is the system size. If the QCP lies at g = 0, we find that if N is kept constant, χF has a scaling form given by χF ∼ θ −2/ν f(g/θ 1/ν )i fθ � 2π/N. We illustrate this both in a tight-binding model of fermions with a spatially varying chemical potential with amplitude h and period 2q in which ν = q, and in a XY spin-1/2 chain in which ν = 2. Finally we show that when q is very large, the model has two additional QCPs at h =± 2 which cannot be detected by studying the energy spectrum but are clearly detected by χF. The peak value and width of χF seem to scale as nontrivial powers of q at these QCPs. We argue that these QCPs mark a transition between extended and localized states at the Fermi energy.
Recently it has been shown that the fidelity of the ground state of a quantum many-body system can be used to detect its quantum critical points (QCPs). If
g denotes the parameter in the Hamiltonian with respect to which the fidelity is computed, we find that for one-dimensional models with large but finite size, the fidelity susceptibility ${\ensuremath{\chi}}_{F}$ can detect a QCP provided that the correlation length exponent satisfies $\ensuremath{\nu}<2$. We then show that ${\ensuremath{\chi}}_{F}$ can be used to locate a QCP even if $\ensuremath{\nu}\ensuremath{\ge}2$ if we introduce boundary conditions labeled by a twist angle $N\ensuremath{\theta}$, where
N is the system size. If the QCP lies at
g=0, we find that if
N is kept constant, ${\ensuremath{\chi}}_{F}$ has a scaling form given by ${\ensuremath{\chi}}_{F}\ensuremath{\sim}{\ensuremath{\theta}}^{\ensuremath{-}2/\ensuremath{\nu}}f(g/{\ensuremath{\theta}}^{1/\ensuremath{\nu}})$ if $\ensuremath{\theta}\ensuremath{\ll}2\ensuremath{\pi}/N$. We illustrate this both in a tight-binding model of fermions with a spatially varying chemical potential with amplitude
h and period
2q in which $\ensuremath{\nu}=q$, and in a
XY spin-1/2 chain in which $\ensuremath{\nu}=2$. Finally we show that when
q is very large, the model has two additional QCPs at $h=\ifmmode\pm\else\textpm\fi{}2$ which cannot be detected by studying the energy spectrum but are clearly detected by ${\ensuremath{\chi}}_{F}$. The peak value and width of ${\ensuremath{\chi}}_{F}$ seem to scale as nontrivial powers of
q at these QCPs. We argue that these QCPs mark a transition between extended and localized states at the Fermi energy.
We study scaling of the ground-state fidelity in neighborhoods of quantum critical points in a model of interacting spinfull fermions - a BCS-like model. Due to the exact diagonalizability of the model, in one and higher dimensions, scaling of the ground-state fidelity can be analyzed numerically with great accuracy, not only for small systems but also for macroscopic ones, together with the crossover region between them. Additionally, in one-dimensional case we have been able to derive a number of analytical formulae for fidelity and show that they accurately fit our numerical results; these results are reported in the article. Besides regular critical points and their neighborhoods, where well-known scaling laws are obeyed, there is the multi-critical point and critical points in its proximity where anomalous scaling behavior is found. We consider also scaling of fidelity in neighborhoods of critical points where fidelity oscillates strongly as the system size or the chemical potential is varied. Our results for a one-dimensional version of a BCS-like model are compared with those obtained by Rams and Damski in similar studies of a quantum spin chain - an anisotropic XY model in transverse magnetic field.
The deconfined quantum critical point was proposed as a second-order quantum phase transition between two broken-symmetry phases beyond the Landau-Ginzburg-Wilson paradigm. However, numerical studies cannot completely rule out a weakly-first-order transition because of strong violations of finite-size scaling. We demonstrate that the fidelity is a simple probe to study the deconfined quantum critical point. We study the ground-state fidelity susceptibility close to the deconfined quantum critical point in a spin chain using the large-scale finite-size density-matrix renormalization-group method. We find that the finite-size scaling of the fidelity susceptibility obeys the conventional scaling behavior for continuous phase transitions, supporting the idea that the deconfined quantum phase transition is continuous. We numerically determine the deconfined quantum critical point and the associated correlation length critical exponent from the finite-size-scaling theory of the fidelity susceptibility. Our results are consistent with recent results obtained directly from the matrix product states for infinite-size lattices using others observables. Our work provides a useful probe to study critical behaviors at the deconfined quantum critical point from the concept of quantum information.
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 scaling of the ground-state fidelity in neighborhoods of quantum critical points in a model of interacting spinful fermions -a BCS-like model.Due to the exact diagonalizability of the model, in one and higher dimensions, scaling of the groundstate fidelity can be analyzed numerically with great accuracy, not only for small systems but also for macroscopic ones, together with the crossover region between them.Additionally, in one-dimensional case we have been able to derive a number of analytical formulae for fidelity and show that they accurately fit our numerical results; these results are reported in the article.Besides regular critical points and their neighborhoods, where well-known scaling laws are obeyed, there is the multicritical point and critical points in its proximity where anomalous scaling behavior is found.We consider also scaling of fidelity in neighborhoods of critical points where fidelity oscillates strongly as the system size or the chemical potential is varied.Our results for a one-dimensional version of a BCS-like model are compared with those obtained by Rams and Damski [11] in similar studies of a quantum spin chain -an anisotropic XY model in transverse magnetic field.
We consider a spin ladder model which is known to have matrix product states as exact ground states with spin liquid characteristics. The model has two critical-point transitions at the parameter values
u=0 and $\ensuremath{\infty}$. We study the variation of entanglement and fidelity measures in the ground states as a function of
u and specially look for signatures of quantum phase transitions at
u=0 and $\ensuremath{\infty}$. The two different entanglement measures used are
S(i) (the single-site von Neumann entropy) and
S(i,j) (the two-body entanglement). At the quantum critical point (QCP) $u=\ensuremath{\infty}$, the entanglement measure
E [=S(i),S(i,j)] vanishes but remains nonzero at the other QCP
u=0. The first and second derivatives of
E with respect to the parameter
u and the entanglement length associated with
S(i,j) are further calculated to identify special features, if any, near the QCPs. We further determine the GS fidelity
F and a quantity
ln∣D∣ related to the second derivative of
F and show that these quantities calculated for finite-sized systems are good indicators of QPTs occurring in the infinite system.