Oscillating fidelity susceptibility near a quantum multicritical point
We study scaling behavior of the geometric tensor and the fidelity susceptibility in the vicinity of a quantum multicritical point (MCP) using the example of a transverse model. We show that the behavior of the geometric tensor (and thus of ) is drastically different from that seen near a critical point. In particular, we find that is highly non-monotonic function of along the generic direction when the system size is bounded between the shorter and longer correlation lengths characterizing the MCP: , where are the two correlation length exponents characterizing the system. We find that the scaling of the maxima of the components of is associated with emergence of quasi-critical points at , related to the proximity to the critical line of finite momentum anisotropic transition. This scaling is different from that in the thermodynamic limit , 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 step-like behavior for small quench amplitudes. Study of heat density and diagonal entropy density also show signatures of quasi-critical points.
I Introduction and main results
Fidelity and fidelity susceptibilityzanardi06 (); venuti07 (); li09 (); gurev08 (); shigu08 () () are information theoretic measures of continuous quantum phase transitions (QPT) . It is well known that a QPT sachdev99 (); dutta96 (), occurring at absolute zero temperature, indicates a change in symmetry of the ground state of a many body quantum system. Fidelity measures the overlap between two neighboring ground states in the parameter space of a quantum Hamiltonian and shows a dip at the quantum critical point (QCP) where the overlap between two neighboring ground states is minimum. At the same time, the fidelity susceptibility, which is a quantitative measure of the rate of change of the ground state of a system under an infinitesimal variation of a parameter of the Hamiltonian, usually attains the maximum value and often diverges with the system size in the thermodynamic limit. The scaling behavior of with the system size at the QCP and with respect to the deviation from the QCP is given in terms of the associated quantum critical exponents.
Following the Kibble Zurek (KZ) predictions zurek05 (); polkovnikov05 (); aeppli10 (); polkovnikov_rmp () on the scaling of the defect density as a function of rate of change of the Hamiltonian following a slow quantum quench (see Refs. [aeppli10, ; dziarmaga_09, ; polkovnikov_rmp, ] for a review), a series of works have been reported addressing the scaling behavior of the defect density for quench across a QCP, gapless phases, gapless lines, etc mukherjee07 (). Of a particular interest is generalization of these results to quenching across a multicritical point (MCP) divakaran09 (); mondal09 (); viola09 (); mukherjee10 (); viola10 (). Such points (both for classical and quantum phase transitions) are characterized by two or more orthogonal directions with different correlation lengths chaikin-lubensky (). They generically appear either as intersection of two critical lines or as a terminating point of one critical line on the other critical line. The second scenario is realized for the transverse field XY model, which is the primary example for this paper (see Fig. 1). In this model one terminating critical direction (horizontal line in Fig. 1) corresponds to a finite momentum anisotropic transition with the critical momentum vanishing at the MCP. In finite size systems the momentum is quantized in units of . In turn this quantization results in emergence of quasi-critical points close to the MCP where the energy gap has a local minimum. As we will show these quasi-critical points dominate the scaling of the fidelity susceptibility along generic directions and in turn determine the scaling of defect density for slow as well as rapid quenches across a QCP as suggested in Refs. gu09 (); gritsev09 (); grandi10 ().
The scaling behavior of at a QCP is well established venuti07 (); gurev08 (); shigu08 (). For a Hamiltonian given by where is the Hamiltonian describing the QCP, is the perturbation not commuting with and measures the deviation from the QCP. For the translationally invariant systems it can be shown that the fidelity susceptibility is related to the connected imaginary time correlation function of the perturbation venuti07 ():
From this representation it is clear that the scaling dimension of is . If this scaling dimension is negative then the fidelity susceptibility diverges at the QCP as , where is the system size. If the scaling dimension is positive then the fidelity susceptibility is still singular but this singularity is a subleading correction on top of a nonuniversal constant (see e.g. Ref. [grandi10long, ]). In case where is a marginal or relevant perturbation (so scales as the energy density) we have additional simplification coming from so that at the critical point schwandt09 (); gritsev09 (); capponi10 (); grandi10 ()
At small but finite this scaling is substituted by
and the two asymptotes match when the correlation length becomes of the order of the system size. These asymptotics are dominant for and subleading for (at there are additional logarithmic singularities grandi10long ()).
It is natural to expect that the scaling of the fidelity susceptibility can be further extended to understand universal properties of MCPs as well as quantum dynamics through the MCPs. For example, in Refs. [grandi10, ; grandi10long, ] it was shown that the scaling of the generated quasiparticles and excess energy (heat) produced by slow quenches can be understood through the scaling behavior of generalized adiabatic fidelity susceptibilities. To the best of our knowledge such analysis was never performed before. As we mentioned earlier because MCPs are characterized by at least two orthogonal directions (principal axes) characterized by different scaling exponents, the natural object to analyze near the MCP is the so called geometric tensor venuti07 () defined as
where the sum is taken over all excited states. One can also define the fidelity susceptibility, characterizing the drop of the overlap of the ground state wave functions for an infinitesimal change of the tuning parameter . It is clear that now is described by both the magnitude and the direction:
where is the projection of the vector to the direction . If we are interested in the fidelity susceptibility along a generic direction such that all components are of the same order then the most divergent component of the geometric tensor defines the scaling of . It follows from the definition of the geometric tensor that its diagonal components describe the fidelity susceptibility along the principal axes. For the purposes of this work we will focus on the situation of two orthogonal directions . It is straightforward to generalize our results to the situation of multi-component MCPs.
It is natural to expect that both the geometric tensor and fidelity susceptibility will have singular behavior at the MCP. In this work we focus on how this singularity develops as we approach the MCP. We find that this approach to the MCP is highly nontrivial characterized by non-monotonic dependence of the components of the geometric tensor on the coupling . In particular, in the thermodynamic limit , where , we find the that the geometric tensor can be expressed as:
where is the dimensionality, and are the correlation length exponents corresponding to the couplings and respectively, and is a scaling function. For approaching the MCP from a generic direction and with and fixed, the scaling of the geometric tensor is affected by the small argument asymptotic of the scaling function at (we assume that ). While for approach of the critical point along the line the argument of the scaling function is constant of the order of unity and the scaling of the geometric tensor is determined purely by the prefactor in Eq. (6). We point that some asymptotics of the scaling function in Eq. (6) can be constrained by critical exponents characterizing second order phase transitions away from the MCP. Thus if the describes a line of critical points then the asymptotics of at should reduce to the scaling behavior of fidelity susceptibility of the corresponding phase transition (see Eq. (3)). Clearly this requirement determines the asymptotics of at small . One can find similar constraints on other components of at large and small . In the next section we analyze the scaling functions and discuss their asymptotics for a particular one-dimensional transverse field XY-model.
In finite size systems the scaling (6) is replaced by another expression:
If then the scaling above should reduce to Eq. (6). Exactly at the MCP () the geometric tensor should be independent on which leaves only two possibilities or . Thus the nonzero components of the geometric tensor should have the scaling, which is a straightforward generalization of Eq. (2). As it is clear from Eq. (7) there is a third scaling regime where one argument of the scaling function is small and the other is large. For generic direction this regime happens when the system size is bounded between the shorter and longer correlation lengths characterizing the system: with a new scaling emerging when . For the model we analyze we find that all components of the geometric tensor have additional singularity at this (quasi-critical) point leading to the local maximum of with stronger scaling with . We also find that in this intermediate regime all components of the geometric tensor have strongly oscillatory dependence on the magnitude of (or equivalently on the system size). These oscillations, however, maybe an artifact of our model where MCP corresponds to the ending point of a nonzero momentum anisotropic transition. And finally at the two scalings (6) and (7) smoothly connect to each other.
The oscillations of the components of the geometric tensor and the fidelity susceptibility near the MCP in turn translate into a step-like behavior in the number of defects produced during a sudden quench of the magnitude starting from the critical point. The steps are more pronounced in relatively small systems, while in the limit of large system sizes we get a scaling behavior . This result is similar to that observed in the rapid quench from a critical point grandi10 ().
Ii Geometric tensor for the transverse field XY-model
where ’s are the Pauli spin matrices, is the system size, , are the coupling constants along and directions respectively, and denotes the transverse field. Here we consider even boundary condition (, where ) in the odd fermion number sector, i.e., . As we will explain later choice of boundary condition does not affect the results in the thermodynamic limit. The boundary conditions will somewhat modify the finite size scalings (see below) but the exponents will remain unaffected. It is well known that the Hamiltonian (8) can be mapped to a collection of non-interacting quasipartices, labeled by momentum modes (), with an excitation energy of the form lieb61 (); pfeuty70 ()
where and . Clearly the spectrum is gapless along the line provided that and along the two lines . At these boundaries the system undergoes continuous quantum phase transitions. The corresponding phase diagram is plotted in Fig. 1.
The points where the two critical lines meet are multicritical. We will focus on the MCP where and (denoted by MC1 on Fig. 1). Then the vanishing gap corresponds to the momentum therefore it is convenient to shift momentum . Then the spectrum (9) linearized near the MCP simplifies to
The ground state of the Hamiltonian (8) is expressed as
where the angles satisfy the relationgu09 ()
Here the states , and are respectively the empty and doubly occupied states of Jordan-Wigner fermions with momentum and .
ii.1 Scaling analysis of the spectrum near the MCP
As we discussed earlier multicritical points are characterized by two (or more) tuning parameters chaikin-lubensky (). This affects the scaling form of various observables. The natural tuning parameters in the present model are and . It is convenient to fix the overall energy scale . Then dispersion of the low energy excitations (10) can be rewritten in the scaling form
where is the dynamical exponent at the multicritical point, is the exponent assosciated with the divergence of the correlation length when one approaches the MCP and is the correlation length exponent governing the cross-over from linear to quadratic dispersion at (see also Ref. [damle96, ]). Physically the exponent governs the scaling of the healing length determining the decay of the longitudinal correlationsbunder99 ().
The scaling function is explicitly written as follows:
Because the MCP in our case is the intersection of two critical lines one of which terminates at the MCP (the horizontal line on Fig. 1), the scaling function should satisfy some constraints. In particular, at , the spectrum of excitation should reduce to that for the Ising critical point characterized by the exponent : . This implies that
This is clearly the case since for , , and we have at large . Likewise in the opposite limit of large the spectrum should be characteristic of the anisotropic transition occurring at finite value of with a finite gap at . This implies that for
which is also the case. One can also infer the information about the scaling of the minimum gap near the anisotropic transition which occurs at . Considering the limit and and noting the gap in the spectrum scales as from Eq. (13) we find that near the MCP the gap scales as
Using the explicit exponents for our model we find that . For the generic direction where this equation reduces to . Similar analysis can reveal relations between correlation length exponents and the asymptotics of the scaling function.
ii.2 Scaling of the geometric tensor
Next we will analyze the scaling of the different components of the geometric tensor. We will first perform the analysis specific for the transverse field XY model and then discuss generalizations to generic MCPs. Following eq. (4) and the definitions of , , we can rewrite Eq. (12)
It is easy to see that the components of the geometric tensor in our model take the form
where is the ground state wavefunction in the state and is the excited state of the pair of fermions with momenta :
Using explicit properties of the wavefunction, this geometric tensor can be rewritten as
With help of Eq. (18) we can compute all three independent components of this tensor and find their scaling:
Now, let us study each term of the tensor separately. First we focus on the MCP where . In this case clearly the only nonzero component of the geometric tensor is :
Note that this scaling is exactly the same as across the Ising quantum critical points (vertical lines in Fig. 1). In this respect the geometric tensor evaluated right at the MCP does not contain any additional information about the additional singularity in the system compared to the standard critical point.
The situation changes dramatically if we consider the scaling of close to the MCP. In particular, if we approach the MCP from a generic direction with (to be specific we assume ) we see that the other diagonal component of the geometric tensor becomes singular at the quasi-critical points occurring at , with , where only one momentum mode dominates the sum in Eq. (22). There we find
The lowest possible value of the dominant momentum for the quasi-critical point is (corresponding to ). At this coupling we find the maximum value for the component of the geometric tensor
The next local maximum for will occur at the second quasi-critical point corresponding to and it is easy to see that it will be down by a factor of compared to the first maximum. Once we go sufficiently far from the MCP all momenta will start contributing to the value of in Eq. (22) so that
It is interesting that if we naively connect the scalings (25) with (26) we will get a mismatch. Indeed one can expect that when becomes comparable to the system size dependence in should disappear and the geometric tensor should become a function of only with no dependence. This is indeed the case for usual critical points. However, here the situation is very different. At this value of (corresponding to the shorter correlation length becoming comparable to the system size) the component of the geometric tensor starts to oscillate as a function of and these oscillations persist until the moment when the contribution of the single dominant momentum corresponding to the quasi-critical point becomes comparable with that of all other momentum modes. Comparing Eqs. (25) with (26) we see that this happens when the second, longer, correlation length reaches the system size. So we conclude that highly oscillatory behavior of occurs in the intermediate values of the coupling .
It is worthwhile to mention that in case of the Hamiltonian (8) with even boundary condition, the quantization of changes to in the sector henkel (). This shifts the position of the quasi-critical points by a factor of , but does not change the scaling of . In the thermodynamic limit when the components of the geometric tensor are determined by the contributions from many momentum modes the boundary conditions become unimportant.
Similar analysis can be performed on the other two components of the geometric tensor. In particular, one can check that the off-diagonal component also has a very sharp singularity near the quasi-critical point rapidly changing from large positive to large negative value both of which scale as . And finally the component also has two (positive) maxima on the left and on the right of the quasi-critical point which scale as . Weaker singularities are observed in the vicinity of other quasi-critical points. As in the case of they persist until both correlation lengths become smaller than the system size.
Next we will attempt to generalize our scaling results to the case of generic MCP which has two different correlation length exponents and . We will always assume that (when the scaling results reduce to those of a conventional QCP). For this purpose we will make the assumptions similar to those used in Refs. [polkovnikov05, ; grandi101, ]. Namely, we will still rely on the assumption that elementary excitations are created in pairs with opposite momenta. In Refs. [gritsev09, ; grandi10long, ] it was shown that this assumption is not affecting the universal scaling results, which alternatively can be obtained through the analysis of the correlation functions, but significantly simplifies the derivation. Besides in gapped systems lowest energy excitations generically have quasi-particle nature sachdev99 (). Then
Here denotes the state with two excited quasi-particles having momentum and respectively. Note that
For relevant or marginal operators the scaling dimension of the perturbation is the same as the scaling dimension of energy (alternatively this perturbation is irrelevant). Therefore the scaling dimension of the matrix element (28) is the same as the scaling dimension of . This can be expressed introducing the dimensionless scaling function
From this we can immediately infer the scaling of the geometric tensor in the thermodynamic limit when the summation over momenta can be substituted by integration:
Note that we implicitly assumed that the integral over momenta converges in the infrared limit. Otherwise one can expect nonuniversal cutoff dependent corrections to the geometric tensor coming from regularization grandi10long (). For our particular XY model the diagonal components of the scaling function are given by the following integrals:
If we would consider the curve then the argument of the scaling function would be of the order of one and the scaling of the components of the geometric tensor will be given by the direct generalization of that for a QCP:
However, this setup does not describe a generic situation. Because and are natural tuning parameters of the system it is more appropriate to analyze the scaling of the geometric tensor along the lines and , where is some fixed slope. In this case it is easy to see that the argument of the scaling function goes to zero as because . In turn this can affect the scaling of . In our case this is indeed the case and at small and the integral in Eq. (31) is determined by the interval , i.e. by the vicinity of the finite momentum critical points at and . The asymptotics of the functions are then set by the properties of these finite momentum critical points for the anisotropic transition (see Fig. 1). E.g. the asymptotics of is set from requiring that . Combining this with Eq. (30) we find that at small , which is indeed the case. Similarly from the fact that the fidelity susceptibility along direction diverges as at the anisotropic critical point we find that at small . While these results are quite specific for the XY model the fact that they can be obtained from combining Eq. (30) with the critical properties of the transition occurring at and finite is generic.
For the path parametrized by and with there are no singularities occurring at because the critical line at terminates at the MCP. In this case the scaling functions do not have any singularities at and we recover the regular scaling (34).
And finally at very small values of such that the sum in Eq. (27) is dominated by the single momentum corresponding to the quasi-critical point we find that the scaling of the geometric tensor is determined by the scaling of the minimum gap at . Combining the scaling analysis with this observation we find that the maxima of are determined by the scaling properties of the functions evaluated at . In general using Eq. (29) and for simplicity assuming one can write the expression for the geometric tensor in the form (7) with
For the system size smaller than the longer correlation length this sum can be dominated by a single most divergent term which determines the scaling of the geometric tensor. As we saw this is indeed the case for the transverse field XY model.
Finally, we consider a path characterized by for a general (see appendix A). It can be shown mukherjee10 () that quasi-critical points appear only for where diverges as . In contrast, for , the scaling relation saturates to .
ii.3 Analysis of the geometric tensor through correlation functions.
As we mentioned earlier the same scaling analysis can be repeated without making any assumptions about the nature of quasi-particles. Alternatively one can analyze the scaling of imaginary time correlation functions of the quench operator venuti07 (); gurev08 (); li09 ():
is the imaginary time Heisenberg representation of the operator . Using our notations and translational invariance of the Hamiltonian we find
where the subscript “c” means that only the connected part of the correlation function should be taken into account. One can check that all the scaling results of the previous section can be reproduced through the language of the correlation functions. For the XY model we give the details of the derivation of the correlation functions in the Appendix (B). In particular, we find that at the quasi-critical point
which again yields
ii.4 Numerical analysis of the fidelity susceptibility along the fixed path.
Having discussed the scaling of the geometric tensor, it is instructive to explicitly show the obtained dependencies. To be specific we will focus on the fidelity susceptibility along a particular path given by fixed and being varied, which is equivalent to setting . The system crosses the MCP ( as shown in Fig.1). As we discussed earlier the fidelity susceptibility is just the convolution of the geometric tensor with the derivatives (see Eq. (5)) so that in our case
Clearly the fidelity susceptibility is dominated by the most divergent component of the geometric tensor.
Combining Eqs. (22) we find
Right at the MCP the scaling of coincides with that of (see Eq. (23)). As increases the fidelity susceptibility is becoming dominated by the component of the geometric tensor having very pronounced singularity at the first quasi-critical point . In Figs. 2 and 3 we show the numerically computed fidelity susceptibility for the XY-model. In Fig. 2 we focus on a larger range of which contains both the Ising critical point occurring at (with the expected linear scaling of with the system size) and the MCP occurring at . While Fig. 3 focuses on the vicinity of the MCP. We see that indeed has very pronounced non-monotonic behavior on the one side of the transition corresponding to . In Fig. 4 we show the scaling of the maxima with the system size and confirm asymptotics of the maximum of the fidelity susceptibility together with linear in scaling of the minima of . As we explained earlier the oscillatory behavior is due to proximity to the critical line corresponding to the anisotropic transition () where critical points occur at finite momentum. Because the generic direction is effectively pushed towards the anisotropic transition, indeed . In turn this results in emergence of the quasi-critical points characterized by anomalously small gap scaling as occurring when the allowed value of the momentum crosses the gapless point at the anisotropic transition line () (see Fig.(5)).
ii.5 Defect density following the instantaneous quantum quench from the MCP.
Recently it was understood that the density of defects (which is equivalent to the density of excited quasi-particles in the model) produced during the quantum quench is closely related to the probability of exciting the system per unit volume, which in turn is related to fidelity gritsev09 (); grandi10 (). In the limit of large transverse field in the final Hamiltonian, when the ground state is all spins up (or down), the defect density becomes equivalent to the number of wrongly oriented spins. In the same spirit, we next look at the defect density () for a fast quench starting from the MCP. For a usual critical point the defect density following a fast quench of magnitude starting exactly at the QCP is given by gritsev09 (); grandi10 () (see also Fig. 6)
This scaling can be obtained from e.g. the adiabatic perturbation theory which states that (see Ref. [grandi10long, ])
Then the scaling (44) immediately follows from that of . In our case the integral of the fidelity susceptibility diverges at small since therefore we need to analyze the scaling of separately. Let us observe that as in the case of the excitations are dominated by the matrix element which scales as . The second observation we make is that the sum over momenta is determined by the proximity to the anisotropic transition with shorter correlation length, i.e. . Combining these considerations we find that
The other simpler way to get this scaling is to note again that for the generic quench where the spectrum is dominated by the direction with smaller value of (since then . Thus the characteristic momentum above which transitions are suppressed is approximately equal to the inverse of the shorter correlation length . This characteristic momentum (in power of dimensionality) determines the scaling of the defect density. The situation qualitatively changes for a quench with . In this case the spectrum is dominated by the critical line at and we expect . Numerical analysis of fast quench into the paramagnetic region shows for large (). (see Fig. 6). The anisotropy of the MCP reveals itself, however, for the quenches with very small amplitude , where only for the positive sign of we observe step-like structure. These steps appear once a new quasi-critical point emerges when , where . As a result, the momentum mode gets excited with high probability close to one leading to a sharp rise of the order in the defect density. In contrast, between any two consecutive quasi-critical points there is no significant change in the total number of defects (see the inset in Fig. 6) and the density does not appreciably change with . Same oscillations can be revealed by fixing the quench amplitude at a very small value and varying the system size (see Fig. 7). There the non-monotonicity is related to the fact that the defect density decreases as with the system size between the jumps where the total number of excitations is approximately fixed.
We can extend the above analysis to the case of heat density gritsev09 (); grandi10 () (or the excess energy above the new ground state) which, unlike , can be defined even for non-integrable systems not describable in terms of independent quasi-particles. For a fast quench through MCP into the ferromagnetic region, is expected to vary as
as is verified numerically (see Fig. 8). Further, in contrast to , does not show abrupt rise at the quasi-critical points very close to the MCP, even though its slope with respect to increases slightly at these points. This is a consequence of additional suppression of the energy by a factor of . A similar quench into the paramagnetic region gives
which is expected if we use the exponents relevant in this case.
where is the probability to be in excited state of momentum . Unlike , the diagonal entropy density is defined even if we break integrability:
where is the probability to occupy -th eigenstate. The diagonal entropy is equivalent to von Neumann entropy of the time-averaged density matrix. For any mode, is close to zero for (i.e., for ) or (i.e., for ), and assumes a finite value of order unity for . This results in oscillatory behavior of entropy density inside the ferromagnetic near the quasicritical points (see inset of Fig. 9). On the other hand, coarse grained diagonal entropy density inside the ferromagnetic region as well as the diagonal entropy density inside the paramagnetic region are dominated by , which gives
In our present model, numerical results suggest for quench into the ferromagnetic phase, whereas for a fast quench into the paramagnetic region. (See Fig. 9).
On a related note, a recent study mukherjee10 () concerning slow non-linear quench across the MCP following a path parametrized by for a general has shown defect density to continuously vary with as () in the transverse field model, where varies with time as , . A similar behavior of fidelity susceptibility is also observed when we consider paths parametrized by , where diverges as for (see appendix A).
We have studied the scaling of the geometric tensor and the fidelity susceptibility near a quantum MCP using the example of a spin- chain in a transverse field. Our analysis shows that oscillates near the MCP, with the peaks, given by , occurring at the quasi-critical points. In turn these points are associated with finite momentum anisotropic transition terminating at the MCP. The oscillations occur in the intermediate region of system sizes bounded between shorter and longer correlation lengths characterizing the vicinity of the MCP. Associated with these jumps we found a step-like behavior of generated defect density for sudden quenches of small amplitude starting from the MCP into the ferromagnetic region with a new jump occurring once a new quasi-critical point is crossed. In the thermodynamic limit these oscillations disappear and the defect density scales as , where is a smaller correlation length exponent characterizing the MCP. A similar analysis of heat density gives a scaling of , whereas diagonal entropy density shows oscillations near the quasi-critical points, finally scaling as in the limit of large . In contrast, behavior inside the paramagnetic region is always dominated by the Ising critical line, with .
Using the adiabatic perturbation theory we generalized some of our findings to the situations of generic MCPs and showed how various scaling relations follow from the analysis of the scaling functions describing the transition matrix elements or equivalently the non-equal time correlation functions of the perturbation along the critical line characterized by a smaller correlation length exponent.
Acknowledgements.The authors acknowledge helpful discussions with V. Gritsev, M. Tomka, D. Sen and U. Divakaran. The work of A.P. was supported by NSF (DMR-0907039), AFOSR FA9550-10-1-0110, and Sloan Foundation. A.D. acknowledges CSIR, New Delhi, for financial support. The authors also acknowledge Abdus Salam ICTP and SISSA, Trieste, for hospitality.
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Appendix A Appendix: scaling of the geometric tensor for a non-generic path
In the main text we focused on the scaling of the geometric tensor along the generic direction . While this setup is indeed most natural, it is instructive to analyze the scaling along other directions, in particular . For simplicity we choose . This scheme lets us approach the multicritical point following a generalized path mukherjee10 (), parametrized by the exponent . In Refs. [barankov08, ; sen08, ] it was shown that for a nonlinear quench in time near a usual critical point where the exponent associated with the amplitude gets replaced by the product . A similar scenario occurs in our case. Indeed the scaling of gap can be written as
where in the last step we have taken . Similar transformations apply to the scaling of matrix elements.
For our setup of the model in one dimension the spectrum gets modified tomukherjee10 ()
which givesmukherjee10 (), for , owing to the presence of quasi-critical points at , . Whereas, for , quasi-critical points are absent, and the scaling saturates to . Repeating the same analysis as in the main text we find that the maxima of say component of the geometric tensor near the quasi-critical point now scale as
for . In the case of we get back our previous result . However, for the exponent in the -dependence of saturates, oscillation disappear and we get
which describes the scaling of the maximum value of the fidelity susceptibility for the Ising transition:
occurring at . This critical power of