Quantum criticality of the Lipkin-Meshkov-Glick Model in terms of fidelity susceptibility
We study the critical properties of the Lipkin-Meshkov-Glick Model in terms of the fidelity susceptibility. By using the Holstein-Primakoff transformation, we obtain explicitly the critical exponent of the fidelity susceptibility around the second-order quantum phase transition point. Our results provide a rare analytical case for the fidelity susceptibility in describing the universality class in quantum critical behavior. The different critical exponents in two phases are non-trivial results, indicating the fidelity susceptibility is not always extensive.
pacs:64.60.-i, 05.70.Fh, 75.10.-b
The Lipkin-Meshkov-Glick (LMG) model LMG () was introduced in nuclear physics. It describes a cluster of mutually interacting spins in a transverse magnetic field. In condensed matter physics, this model is associated with a system of infinite coordination number. In earlier time, scaling behaviors of critical observables have been studied by mean field analysis infcoor (), while recently the finite-size scaling of this model was studied by the expansion in the Holstein-Primakoff single boson representation HP () and by the continuous unitary transformations (CUT) CUT1 (); CUT2 (); JVidalStat (). Meanwhile, a rich structure of four different regions is revealed in the parameter space through a careful scrutiny on the spectrum JVidalSpectra (). Besides, the quantum criticality has been investigated by studying its entanglement properties QPT1st (); QPT2nd (); EntE1 (); EntE2 (); nilesen (). Both the first- and second-order quantum phase transitions (QPTs) sachdev () have been revealed, in the antiferromagnetic and the ferromagnetic cases respectively QPT1st (); QPT2nd ().
Regarding the QPT itself, the ground state of a system would undergo a significant structural change at certain critical point. This primary observation suggests a new description of QPTs in terms of fidelity HTQuan06 (); PZanardi06 (); Buonsante07 (); PZanardi0606130 (); PZanardi07 (); WLYou07 (); HQZhou07 (); LCVenuti07 (); SJGu07 (); SChen07 (); WQNing07 (); MFYang07 (); NPaunkovic07 (), a concept introduced in quantum information theory nilesen (). Mathematically it is the overlap between two ground states in which their driving parameters deviate by a small amount. However, the fidelity depends computationally on an arbitrarily small yet finite change of the driving parameter. For this, Zanardi et. al. introduced the Riemannian metric tensor PZanardi07 (), while You et. al. suggested the fidelity susceptibility WLYou07 (), both focus on the leading term of the fidelity, in order to explain singularities in QPTs. In addition, scaling analysis of these quantities has been informative: it helps understanding their divergence and the criticality of the system LCVenuti07 (), and it also reveals the intrinsic relation between the critical exponent of some physical quantities and that of the fidelity susceptibility SJGu07 ().
In this paper, we explicitly compute the ground-state fidelity susceptibility and its critical exponent of the LMG model. Numerical analysis is also performed to check with our analytic calculations. We show that, the expansion in the Holstein-Primakoff transformation is sufficient to determine the critical exponent of the fidelity susceptibility . In addition, we revealed two distinct critical exponents in two phases which is not a general feature. Therefore, our findings not only suggest another route on understanding the quantum criticality of the LMG model, but also show the fidelity susceptibility is not always extensive in describing the universality class of a quantum many-body system.
This paper consists of five sections. In Sec. II, we review the Hamiltonian, symmetry, and conserved quantities of the LMG model. In Sec. III, we diagonalize the model Hamiltonian and compute the fidelity susceptibility in the anisotropic model. In Sec. IV, we perform finite size scaling analysis and discuss the scaling relation between different exponents. Finally, we give a brief summary in Sec. V.
Ii The model Hamiltonian
The Hamiltonian of the LMG model reads
where are the Pauli matrices, , and . The prefactor is necessary to ensure finite energy per spin in the thermodynamic limit. It is understood that the total spin and the parity are the conserved quantities, i.e.,
In addition, in the isotropic case , one has and simultaneous eigenstates can be found. In the main context, the following parameter space is considered: . We take as the spectrum is invariant under the transformation . In addition, as a common practice we only consider the maximum spin sector which contains the lowest energy state.
Iii Critical behavior of the fidelity susceptibility
We briefly review of the concept of the fidelity susceptibility here. Suppose there is a Hamiltonian of a general form as
for is defined as the driving term of the system, which simply does not commute with . The function coupled to is often considered as the linear external field . Then the fidelity susceptibility is defined as PZanardi07 (); WLYou07 ()
where and stand for the eigenenergies and eigenstates of the (whole) Hamiltonian respectively.
The fidelity susceptibility is well-defined for a non-degenerate ground state of the continuous variable , but it is not suitable to deal with states with good quantum numbers. The LMG model undergoes ground state level crossing when , the ground states are assigned the magnetization as the quantum numbers.
We put our focus on the fidelity susceptibility for an arbitrary isotropy . One resolution is to use the Bethe-Ansatz solution FPan99 (); JLinks03 (), which is rather complicated. So we adopt the expansion method which was used extensively by Dusuel and Vidal CUT1 (); CUT2 (), that corresponds to the large limit.
The expansion method is done under the Holstein-Primakoff boson representation HP () framework. In low energy spectrum the spin operators in the subspace are mapped into boson operators:
where () is the standard bosonic annihilation (creation) operator satisfying . The above transformation is valid when , but when it can also be used through semi-classical treatment CUT1 (); CUT2 (). This representation is also known as the spin-wave theory. It is well adapted to the computation of the low-energy physics when . After inserting these expressions of the spin operators in Eq. (3), one can approximate the square roots as one and express the result in normal ordered form with respect to the boson vacuum state. Keeping terms of order , and for (in which the approximation is justified), the Hamiltonian becomes
The above Hamiltonian can be diagonalized by a standard Bogoliubov transformation
where is the quasi-bosonic annihilation (creation) operator, and
then the Hamiltonian is diagonalized as
Thus the low-energy spectrum of the model is mapped to the spectrum of a simple harmonic oscillator. The eigenstates are just , where . We consider the driving Hamiltonian responsible for the QPT,
By transforming them into combinations of and operators, the fidelity susceptibility is calculated as
The derivation above is only valid for , for the calculation is actually similar to the above case of , provided that one first rotates the axis to bring it along the classical spin direction. We do not show it explicitly here, but interested readers are recommended to refer to Ref. CUT1 (); CUT2 (). We simply quote the main result, after all the procedures the Hamiltonian becomes:
The driving Hamiltonians also takes a different form:
for the HP transformation is done on the operators. The fidelity susceptibilities are then obtained accordingly:
Thus we obtained of the anisotropic LMG model in large limit. We first see the effect of isotropy to the fidelity susceptibility. It dominates when , but fades out for large . Especially in the isotropic limit, when , diverges when , but tends to zero when . This is the effect of the level-crossing points in the thermodynamic limit. They together form a region of criticality, and the system undergoes continuous level crossing. The fidelity susceptibility responds drastically while moving along . But when , there are no further critical points, naturally measures zero when moving along because we have .
An interesting observation is behaves extensively when even in the large limit. When discarding the extensive part of Eq. (17), we arrive a zero point at , which does not fit with numerical analysis [Fig. 1]. This discrepancy may be eliminated by adopting other transformations of the driving Hamiltonian. Particularly, the flow of operators in the LMG model haven been studied by the continuous unitary transformation (CUT) method CUT1 (); CUT2 (). However, such discrepancy would not hinder us from getting the correct critical exponent of the fidelity susceptibility.
Let us emphasize the intensive property of the fidelity susceptibility, which measures the average response to some driving Hamiltonians. Its divergence should correspond to a critical point of a second-order QPT rather than to the increasing system size. In order to predict the critical exponent correctly, we should average the fidelity susceptibility whenever necessary. To the leading order, Eq. (17) becomes
Then it comes to a key result of our paper: bears different critical exponents across the critical point. It diverges as when , when . It is unlike the Ising model in a transverse field PZanardi06 () nor the one-dimensional asymmetric Hubbard model SJGu07 (), where the critical exponent is a single number over the phases.
Iv Finite size scaling analysis
To illustrate the scaling behavior of the fidelity susceptibility, we perform the exact diagonalization (ED) to solve the spectrum of and then calculate the corresponding fidelity susceptibility numerically.
Let us recall the fidelity susceptibility scaling analysis performed in the asymmetric Hubbard model SJGu07 (). According to the scaling ansatz MAContinentinob () and the obvious power-law divergence observed in Fig. 1, the rescaled fidelity susceptibility around its maximum point at is a simple function of a scaling variable, i.e.
where is the scaling function and is the correlation length critical exponent. This function is universal and does not depend on the system size, as shown in Fig. 2 for cases of and . Remarkably, the critical exponent for three cases are very close. This observation strongly implies that is a universal constant and does not depend on the parameters and .
around the critical point. In the last section, we have already obtained
which is also a universal constant. Then if the maximum point of the intensive fidelity susceptibility scales like
the scaling ansatz also implies another important relation, i.e.
We try to confirm this equality in numerically. In Fig. 2, Eq. (19) is best fitted with . The case to determine is more subtle. It is because Eq. (19) remains the same form even for averaged , but the maximum of does not. To resolve this problem, we first determine from the “bare” . By using least square fit method, we evaluated “bare” for at different . The numerical details are shown in table 1. However, the exponent does not converge perfectly. We compare the obtained in a range of , and those from the range . The results converge better for larger scaling regions. According to the trend of in larger system sizes, we roughly estimate with three effective digits [Fig. 3].
Thus , this will give the relation . These two values of are consistent with our analytic calculation in the last section.
The exponent , can also be discussed from the scaling ansatz at the critical point rather than the maximum point of a finite system, as shown by Vidal, Dusuel, and Barthel CUT2 (); JVidalStat (). Based on their approach, the critical exponent takes the value of 1/3, and is independent of magnitude of . Our results on the maximum simply agree with this value and can be generalized to other models where the precise critical point is not known.
Another scaling analysis is to examine how tends to the critical point . It should scale like
in the large limit. We find with two effective digits for various .
In short, we can confirm that the exponents , and of the fidelity susceptibility do not depend on the value of and . They are universal constants for the LMG model and are related to the critical exponent of the fidelity susceptibility .
In summary, we computed explicitly the fidelity susceptibility and its critical exponent of the LMG model at different isotropy. We confirmed the different critical exponents in two phases numerically by ED, which is a rather non-trivial result. Several scaling exponents are also found in consistence with previous studies. Since the fidelity susceptibility is believed to be able to characterize the universality class of quantum phenomena, our results therefore provide a rare explicit case for the study of fidelity susceptibility.
Acknowledgements.We are very grateful to J. Vidal for many fruitful comments. S. J. G. thanks X. Wang and J. P. Cao for helpful discussions. This work was partially supported RGG Grant CUHK 400906, 401504, and MOE B06011.
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