PHYSICAL REVIEW B 83, 214302 (2011)

Fidelity susceptibility and general quench near an anisotropic quantum critical point

Victor Mukherjee* and Amit Dutta†

Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208 016, India(Received 14 January 2011; published 8 June 2011)

We study the scaling behavior of fidelity susceptibility density (χf ) at or close to an anisotropic quantumcritical point (AQCP) characterized by two different correlation length exponents ν|| and ν⊥ along parallel andperpendicular spatial directions, respectively. Our studies show that the response of the system due to a smallchange in the Hamiltonian near an AQCP is different from that seen near an isotropic QCP. In particular, for afinite system with linear dimension L|| (L⊥) in the parallel (perpendicular) directions, the maximum value ofχf is found to increase in a power-law fashion with L|| for small L||, with an exponent depending on both ν||and ν⊥ and eventually crosses over to a scaling with L⊥ for L

1/ν|||| � L

1/ν⊥⊥ . We also propose scaling relations

of heat density and defect density generated following a quench starting from an AQCP and connect them to ageneralized fidelity susceptibility. These predictions are verified exactly both analytically and numerically takingthe example of a Hamiltonian showing a semi-Dirac band-crossing point.

DOI: 10.1103/PhysRevB.83.214302 PACS number(s): 64.70.qj, 64.70.Tg, 03.75.Lm, 67.85.−d

I. INTRODUCTION

Recent studies on fidelity and fidelity susceptibility1–4

(χF ) near a quantum critical point (QCP) have contributedto a deeper understanding of a quantum phase transition5–7

from the viewpoint of quantum information theory. Fidelityis the measure of overlap of two neighboring ground statesof a quantum Hamiltonian in the parameter space. Fidelitysusceptibility provides quantitatively the rate of change of theground state under an infinitesimal variation of the parametersof the Hamiltonian. Since the ground state of a quantum many-body system exhibits different types of symmetries on eitherside of a QCP,5 a sharp drop of fidelity is observed right there.At the same time, the fidelity susceptibility usually diverges ina power-law fashion with the system size where the exponentis given in terms of the quantum critical exponents.2–4,8–16

In recent years, a series of works have been directed tounderstanding the connection between fidelity susceptibilityto quantum phase transition at critical2–4,8–16 and multicriticalpoints.17 Studies of fidelity per site,4,18 reduced fidelity,4,19–21

and geometric phase, which is also closely related to fidelitysusceptibility,2 near quantum critical22,23 and multicritical24

points have also been interesting areas of research.In this paper, we extend the investigation on fidelity

susceptibility to the case of an anisotropic quantum criticalpoint (AQCP) and highlight the marked difference with thecorresponding studies on an isotropic QCP. An interestingrealization of an AQCP is seen in semi-Dirac band-crossingpoints25,26 where the energy gap scales linearly with mo-mentum along one spatial direction but quadratically alongothers unlike Dirac points in graphene where a gap openslinearly along both the directions.27 The possibility of sucha semi-Dirac point has been reported recently25,28 using athree-unit cell slab of VO2 confined within insulating TiO2

and also in liquid He3. A series of works on low-energyproperties of a system with a semi-Dirac point have alreadybeen reported.26,29,30 It is to be noted that the scaling ofdefect density following a slow quench across a QCP, namely,the Kibble Zurek scaling,31–39 has also been generalizedto an AQCP using a semi-Dirac Hamiltonian.40 An AQCPcan also be realized at the edge of the gapless region of a

two-dimensional Kitaev model in a honeycomb lattice41,42 forwhich the Kibble Zurek scaling has also been proposed.43

Let us consider a d-dimensional quantum mechanicalHamiltonian H (λ) designated by a parameter λ. For twoground state wavefunctions, ψ0(λ) and ψ0(λ + δλ), infinites-imally separated in the parameter space (δλ → 0), we candefine fidelity (F ) as1–4

F = |〈ψ0(λ)|ψ0(λ + δλ)〉| ≈ 1 − δλ2

2χF (λ) + · · · , (1)

where the fidelity susceptibility χF is the first nonvanishingterm in the expansion of fidelity. The scaling behavior ofχF at a QCP is well established.2,4,13 Let us choose theHamiltonian to be of the form H = H0 + λHI . Here H0 is theHamiltonian describing a QCP at λ = 0 while HI ≡ ∂λH |λ=0

is the perturbation not commuting with H0. One can relate thefidelity susceptibility density (χf = 1/LdχF) to the connectedimaginary time (τ ) correlation function of the perturbationHI (τ ) using the relation2

χf(λ) = 1

LdχF = 1

Ld

∫ ∞

0τ 〈HI (τ )HI (0)〉cdτ. (2)

Using dimensional analysis in Eq. (2), we get that the scalingdimension of χf is given by dim[χf] = 2HI

− 2z + d, wherez is the dynamical exponent associated with the QCP andHI

is the scaling dimension of the operator HI . Clearly anegative value of the scaling dimension leads to a fidelitysusceptibility diverging with the system size L at the QCP asχf(λ = 0) ∼ L2z−d−2HI . A positive value, on the other hand,implies a singular χf though the singular behavior appears as asubleading correction to a nonuniversal constant.11 A marginalor relevant perturbation HI (so that λHI scales as the energy)allows us to make an additional simplification coming fromHI

= z − 1/ν so that at the critical point8–11

χf ∼ L2/ν−d . (3)

Further we get a crossover from system size dependence to λ

dependence when the correlation length ξ ∼ λ−ν becomes ofthe order of system size:

χf ∼ |λ|νd−2. (4)

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VICTOR MUKHERJEE AND AMIT DUTTA PHYSICAL REVIEW B 83, 214302 (2011)

These asymptotics are dominant for dν < 2 and subleadingfor dν > 2, while at dν = 2 there are additional logarithmicsingularities.8,11

In the following analysis, we show that the general scalingof fidelity susceptibility valid near an isotropic QCP getsmodified due to the anisotropy in critical behavior. Thechanged scaling form naturally includes the correlation lengthexponents along the different spatial directions, namely, ν||and ν⊥. In addition, for a finite system with linear dimensionL|| (L⊥) in the parallel (perpendicular) directions, the max-imum value of χf increases with L|| in the limit of small

L|| (L1/ν|||| L

1/ν⊥⊥ ) only. In contrast, for higher values of

L|| (L1/ν|||| � L

1/ν⊥⊥ ), we observe a crossover and χf scales

with L⊥. We also study the defect density and heat densityfollowing a rapid quantum quench starting from an AQCP andrelate them through a generalized fidelity susceptibility.11 Wealso highlight the connection to the Kibble-Zurek scaling forthe defect density following a slow quench through an AQCPand retrieve the scaling relations derived previously.40,43

The paper is organized as follows: Sec. II provides a generalscaling relation of χf associated with an AQCP. We do alsopropose the same for the heat density and the defect densityfollowing a general quench starting from the AQCP, and relatethem through a generalized fidelity susceptibility density. InSec. III, we have taken a model Hamiltonian which shows anAQCP occurring in the physical systems described above andconfirm our scaling predictions using exact analytical and nu-merical methods. Concluding remarks are presented in Sec. IV.

II. GENERAL SCALING RELATIONS

A. Fidelity susceptibility

Let us consider a d-dimensional quantum Hamiltonianshowing an AQCP at λ = 0. The correlations length exponentis ν = ν|| along m spatial directions and ν = ν⊥ along rest ofthe (d − m) directions, called the parallel and perpendiculardirections, respectively. The fidelity susceptibility as obtainedfrom adiabatic perturbation theory32,44 is of the form4

χF =∑n�=0

∣∣〈ψn

∣∣ ∂H∂λ

∣∣ψ0⟩∣∣2

(En − E0)2, (5)

so that for a finite system with linear dimension L|| (L⊥) in theparallel (perpendicular) directions, the corresponding fidelitysusceptibility density (χf) can be written as

χf = 1

Lm|| L

d−m⊥

χF = 1

Lm|| L

d−m⊥

∑n�=0

∣∣⟨ψn

∣∣ ∂H∂λ

∣∣ψ0⟩∣∣2

(En − E0)2. (6)

Here, E0 and En denote the energy of the ground state and nthenergy level, respectively. We can contrast the above Eq. (6)with the specific heat density [the second derivative of theground state energy density (E0/L

m|| L

d−m⊥ )] given by9

χE = − 1

Lm|| L

d−m⊥

∂2E0

∂λ2

∼ 1

Lm|| L

d−m⊥

∑n�=0

∣∣⟨ψn

∣∣ ∂H∂λ

∣∣ψ0⟩∣∣2

En − E0. (7)

Comparison of Eqs. (6) and (7) indicates that near an AQCP, amuch stronger divergence of χf as compared to χE is expected;this is due to the higher power of energy difference term in thedenominator of χf .

We note that near an AQCP, the specific heat χE ∼ |λ|−α

where below the upper critical dimension, the exponentα satisfies a modified hyperscaling relation7,45,46 2 − α =ν||m + ν⊥(d − m) + ν||z||. In the limit of large |λ| (|λ| �L

−1/ν|||| ,L

−1/ν⊥⊥ ), χE scales as

χE ∼ |λ|−α ∼ |λ|ν||m+ν⊥(d−m)+ν||z||−2. (8)

Now, in the same limit, the scaling of the fidelity susceptibilitydensity is given by9

χf = 1

Lm|| L

d−m⊥

∑n�=0

∣∣⟨ψn

∣∣ ∂H∂λ

∣∣ψ0⟩∣∣2

(En − E0)2∼ χE

|En − E0|

∼ |λ|ν||m+ν⊥(d−m)−2(|λ| � L−1/ν|||| ,L

−1/ν⊥⊥ ). (9)

In deriving Eq. (9), we have used Eq. (8) and the fact thatnear the AQCP, En − E0 ∼ |λ|ν||z|| = |λ|ν⊥z⊥ . In the specialcase of ν|| = ν⊥ = ν, we retrieve the expected scaling relationχf ∼ λνd−2 valid near an isotropic QCP (Refs. 8 and 11) [seeEq. (4)].

On the other hand, right at the AQCP (λ = 0) and in thelimit L

1/ν|||| L

1/ν⊥⊥ , χf(λ = 0) scales with the system size

L|| as

χf(λ = 0) ∼ L

2ν|| −

ν⊥ν|| (d−m)−m

|| (L1/ν|||| L

1/ν⊥⊥ ). (10)

However, in the opposite limit L1/ν|||| � L

1/ν⊥⊥ , χf(λ =

0) instead starts scaling with L⊥, and Eq. (10) getsmodified to

χf(λ = 0) ∼ L2

ν⊥ − ν||ν⊥ m−(d−m)

⊥ (L1/ν|||| � L

1/ν⊥⊥ ). (11)

Clearly, the special condition L1/ν|||| ∼ L

1/ν⊥⊥ yields

χf (λ = 0) ∼ L

2ν|| −

ν⊥ν|| (d−m)−m

|| ∼ L2

ν⊥ − ν||ν⊥ m−(d−m)

⊥ . (12)

The above scalings in Eqs. (10)–(12) suggest χf(λ = 0)initially increases with L|| until L

1/ν|||| ∼ L

1/ν⊥⊥ . Beyond which,

χf(λ = 0) becomes independent of L|| and saturates to aconstant value. However, in this limit, the fidelity susceptibilitydensity scales with L⊥, as shown in Eq. (11).

An alternative way of arriving at the above scalings is bythe use of correlation functions:2–4,8–10,12

χf = 1

Lm|| L

d−m⊥

∫ ∞

0τ 〈HI (τ )HI (0)〉cdτ, (13)

where we define

HI (τ ) = eHτHI e−Hτ

and

〈HI (τ )HI (0)〉c = 〈HI (τ )HI (0)〉 − 〈HI (τ )〉〈HI (0)〉,with τ being the imaginary time. For a relevant perturbation,λHI should scale as the energy, so that HI ∼ λν||z||−1. Using

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FIDELITY SUSCEPTIBILITY AND GENERAL QUENCH . . . PHYSICAL REVIEW B 83, 214302 (2011)

the relation τ ∼ Lz|||| and L||,⊥ ∼ λ−ν||,⊥ , we get the scaling of

χf from Eq. (13) given by

χf ∼ |λ|ν||m+ν⊥(d−m)−2, (14)

which is identical to Eq. (9).

B. Heat and defect density following a sudden quench

In this section, we study a sudden quench47,48 of a quantumsystem of amplitude λ, starting from the AQCP. The quantitiesof interest are defect density37–39 (nex) and heat density11 (Q)generated in the process. Advantage of using heat density, orthe excess energy above the new ground state, is that it canbe defined even for nonintegrable systems. On the other hand,for an integrable system with noninteracting quasiparticles, itis useful to define defect density, which is a measure of thedensity of excited quasiparticles generated in the system.

As λ is suddenly increased from λ = 0 to its final value λ,all the momentum modes k|| � λν|| and k⊥ � λν⊥ get excitedwith excitation energy ∼ λν||z|| = λν⊥z⊥ for each mode. Thisgives an excitation energy density or heat density of the form

Q ∼ λν||m+ν⊥(d−m)+ν||z|| . (15)

Defect density is related to the probability of exci-tation, which in turn can be expressed in terms of fi-delity susceptibility.8,11,49 Following the above argument onefinds that

nex ∼ λ2χf ∼ λν||m+ν⊥(d−m). (16)

Equation (16) can also be derived by noticing that for a suddenquench of amplitude λ, all the momentum modes k1 � λν||

and k2 � λν⊥ get excited with unit probability, giving nex ∼λν||m+ν⊥(d−m).

C. Generalized fidelity susceptibility density

In this section, we deal with a generic quench from anAQCP at time t = 0 given by

λ(t) = δtr

r!�(t), (17)

where δ is a small parameter, and � is the step function.11

The case r = 0 denotes a rapid quench of amplitude δ; thecase r = 1 implies a slow linear quench with a rate δ and soon. In all these cases, the limit δ → 0 is considered to signifya slow adiabatic time evolution. If the system is initially inthe ground state, the transition probability to the instantaneousexcited state as obtained from the adiabatic perturbation theoryis given by

Pex = δ2∑n�=0

∣∣⟨ψn

∣∣ ∂H∂λ

∣∣ψ0⟩∣∣2

(En − E0)2r+2

= δ2Lm|| L

d−m⊥ χ2r+2, (18)

which leads to a density of defect of the form

nex = 1

Lm|| L

d−m⊥

Pex = δ2χ2r+2. (19)

In the above, we have used the definition of a generalizedfidelity susceptibility density χl given by11

χl = 1

Lm||

1

Ld−m⊥

∑n�=0

∣∣⟨ψn

∣∣ ∂H∂λ

∣∣ψ0⟩∣∣2

(En − E0)l. (20)

From Eq. (20), one finds that χ1 stands for the specific densityχE while χ2 is the fidelity susceptibility density χf ; χ4, on theother hand, yields the excitation probability following a slowlinear quench starting from an AQCP.

In the same spirit as in Eq. (2), a general χl can also beexpressed in terms of time-dependent connected correlationfunctions given by

χl = 1

Lm|| L

d−m⊥ (l − 1)!

∫ ∞

0τ l−1〈HI (τ )HI (0)〉cdτ . (21)

Now, using λ ∼ L−1/ν|||| ∼ L

−1/ν⊥⊥ and t ∼ L

z|||| ∼ L

z⊥⊥ in

Eq. (17) leads to the scaling relations L|| ∼ δ− ν||

1+ν||z||r , L⊥ ∼δ

− ν⊥1+ν⊥z⊥r = δ

− ν⊥1+ν||z||r . These suggest that one can further

conclude HI ∼ λν||z||−1 ∼ δν||z||−1

1+rν||z|| , and τ ∼ Lz|||| ∼ δ

− ν||z||1+ν||z||r .

Substituting for L||, L⊥, HI , and τ in Eq. (21) with l = 2r + 2,

one gets in the limit δ � L− 1

ν|| −z||r

|| ,L− 1

ν⊥ −z⊥r

⊥

χ2r+2 ∼ δν||m+ν⊥(d−m)−2−2ν||z||r

1+ν||z||r . (22)

Therefore, for a generic quench from an AQCP scaling ofdefect density gets modified to

nex ∼ δν||m+ν⊥(d−m)

ν||z||r+1(δ � L

− 1ν|| −z||r

|| ,L− 1

ν⊥ −z⊥r

⊥), (23)

while the corresponding heat density scales as

Q ∼ λν||z||nex ∼ δν||m+ν⊥(d−m)+ν||z||

ν||z||r+1 . (24)

The expressions for nex and Q match exactly with the samefor fast quench [Eqs. (15) and (16)] if we put r = 0 and δ = λ,whereas, the case r = 1 correctly reproduces the values for aslow linear quench starting from the AQCP.40,43

The scaling relations presented above are valid as longas the corresponding exponents do not exceed 2. Otherwisecontributions from short-wavelength modes become dominantand hence the low-energy singularities associated with thecritical point become subleading.8,11

III. MODEL AND HAMILTONIAN

We illustrate the above analytical predictions using therepresentative case of a semi-Dirac point in spatial dimensiond = 2. In this case, ν|| = 1/2, ν⊥ = 1, and d = 2, m = 1. Inthe momentum k space, the Hamiltonian near a semi-Diracpoint can be written as the direct product of 2 × 2 Hamiltoniansgiven by26,40,43

Hk =[

λ k2|| + ik⊥

k2|| − ik⊥ −λ

]. (25)

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VICTOR MUKHERJEE AND AMIT DUTTA PHYSICAL REVIEW B 83, 214302 (2011)

fχ

λ 0

0.5

1

1.5

2

2.5

3

3.5

4

−0.6 −0.4 −0.2 0 0.2 0.6 0.4

FIG. 1. Variation of χf with λ, as obtained numerically for L|| =10 000, L⊥ = 1000, ν|| = 1/2, ν⊥ = 1, d = 2, and m = 1. χf peaksat the AQCP and falls as |λ|−1/2, as predicted in Eq. (9).

The fidelity susceptibility density near the semi-Dirac point(λ = 0) can be written as

χf = 1

π2

∫ π

π/L||

∫ π

π/L⊥

k4|| + k2

⊥(λ2 + k4

|| + k2⊥)2

dk||dk⊥. (26)

Rescaling k||/√

λ = x1, k⊥/λ = x2 and taking the limit λ �L−2

|| ,L−1⊥ , we get

χf = 1

|λ|1/2π2

∫ π/√

λ

π/√

λL||

∫ π/λ

π/λL⊥

x41 + x2

2(1 + x4

1 + x22

)2 dx1dx2

≈ 1

|λ|1/2π2

∫ ∞

0

∫ ∞

0

x41 + x2

2(1 + x4

1 + x22

)2 dx1dx2

∼ |λ|−1/2, (27)

which shows that divergence of χf at λ → 0 [see Fig. (1)] andexponent (1/2), and are in complete agreement with Eq. (9)for d = 2, m = 1, ν|| = 1/2, and ν⊥ = 1.

L ||

fχ

5

6

7

8

9

10

100 120 140 160 180 200

FIG. 2. Variation of χf (λ = 0) with L|| as obtained numericallyfor L⊥ = 100 000, ν|| = 1/2, ν⊥ = 1, d = 2, and m = 1. χf divergesas χf ∼ L||, in agreement with the scaling given in Eq. (10).

χf

L 3.8

4

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

1000 1200 1400 1600 1800 2000

FIG. 3. Variation of χf (λ = 0) with L⊥ as obtained numericallyfor L|| = 10 000, ν|| = 1/2, ν⊥ = 1, d = 2, and m = 1. χf divergesas χf ∼ L

1/2⊥ as expected from the scaling Eq. (11).

Right at the AQCP (λ = 0), we have.

χf(λ = 0) ≈ 1

π2

∫ ∞

π/L||

∫ ∞

π/L⊥

1

k4|| + k2

⊥dk||dk⊥. (28)

The scalings k⊥ = pk2|| or k|| = q

√k⊥ simplify the above

integral to

χf = 1

π2

∫ ∞

π/L||

dk||k2||

∫ ∞

π/L⊥k2||

dp

p2 + 1

= 1

π2

∫ ∞

π/L⊥

dk⊥k

3/2⊥

∫ ∞

π/L||√

k⊥

dq

q4 + 1. (29)

In the limit L2|| L⊥ (or L2

|| � L⊥), we can approximateπ/L⊥k2

|| (or π/L||√

k⊥) to zero, so that the scalings in Eq. (29)depend on one of the length scales, and we get

χf(λ = 0) ∼ L||(for L2|| L⊥),

∼ L1/2⊥ (for L2

|| � L⊥). (30)

L

χf

fχ

L ||

1

1.5

2

2.5

3

3.5

4

4.5

5

0 50000 100000 150000 200000 250000 300000

4

5

6

7

8

9

10

11

12

13

4000 0 8000 12000 16000 20000

FIG. 4. Variation of χf (λ = 0) with L|| as obtained numericallyfor ν|| = 1/2, ν⊥ = 1, d = 2, m = 1, and L⊥ = 10 000. χf saturates atL2

|| � L⊥, as expected from Eqs. (29) and (30). Inset shows variationof χf with L⊥ when L|| kept fixed at L|| = 100. χf saturates atL⊥ � L2

||.

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FIDELITY SUSCEPTIBILITY AND GENERAL QUENCH . . . PHYSICAL REVIEW B 83, 214302 (2011)

nex

λ

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0

FIG. 5. Kink density nex as a function of λ as obtained nu-merically for L|| = L⊥ = 1000, ν|| = 1/2, ν⊥ = 1, d = 2, m = 1,

and λ � L−1/ν||1 , L

−1/ν⊥⊥ . nex varies as nex ∼ λ3/2, as predicted in

Eq. (16).

Numerical verifications for the scalings of χf with L|| and L⊥discussed in Eq. (30) are provided in Figs. 2–4. Extending ouranalysis of χf to find the defect density following a fast quenchstarting from the AQCP (λ = 0), we arrive at the scalingnex ∼ λ2χf ∼ λ3/2. This relation is in perfect agreement withEq. (16) and is verified numerically as shown in Fig. 5.However, scaling analysis of heat density Eq. (15) predictsQ ∼ |λ|2.5, which is subleading to the quadratic form Q ∼ λ2

arising from contributions of short-wavelength modes. Thisleads to the scaling relation Q ∼ λ2. This quadratic scaling isalso checked numerically in Fig. 6.

IV. CONCLUSIONS

We have studied the scaling behavior of fidelity suscepti-bility near an AQCP. Anisotropic critical behavior modifiesthe general scaling form of χf . In particular, both ν|| and ν⊥appear in the scaling. In addition, even though the maximum

λ

Q

0

0.0002

0.0004

0.0006

0.0008

0.001

0 0.02 0.04 0.06 0.08 0.1

FIG. 6. Heat density Q as a function of λ as obtained numericallyfor L|| = L⊥ = 1000, ν|| = 1/2, ν⊥ = 1, d = 2, m = 1, and λ �L

−1/ν||1 ,L

−1/ν⊥⊥ . Q follows the perturbative scaling law Q ∼ λ2, as

discussed in the text.

value of χf scales with L|| in the limit of small L||, at highervalues of the same, a crossover is observed and χf starts scalingwith L⊥ instead. We also propose the scaling relations for thedefect density and heat density following a generic quantumquench starting from an AQCP and relate them through ageneralized fidelity susceptibility. We have verified our generalscaling predictions both numerically and analytically using theillustrative example of a Hamiltonian showing a semi-Diracpoint. Interestingly, we show that the heat density following arapid quench starting from a two-dimensional semi-Dirac pointvaries quadratically with the amplitude and the scaling arisingdue to low-energy critical modes appear only as a subleadingcorrection.

ACKNOWLEDGMENTS

The authors acknowledge U. Divakaran, A. Polkovnikov,and R. R. P. Singh for collaboration in related works. A.D.acknowledges CSIR, New Delhi, for partial financial support.

*victor@iitk.ac.in†dutta@iitk.ac.in1P. Zanardi and N. Paunkovic, Phys. Rev. E 74, 031123 (2006).2L. CamposVenuti and P. Zanardi, Phys. Rev. Lett. 99, 095701(2007).

3P. Zanardi, P. Giorda, and M. Cozzini, Phys. Rev. Lett. 99, 100603(2007).

4S.-J. Gu, Int. J. Mod. Phys. B 24, 4371 (2010).5S. Sachdev, Quantum Phase Transitions (Cambridge UniversityPress, Cambridge, 1999).

6B. K. Chakrabarti, A. Dutta, and P. Sen, Quantum Ising Phasesand Transitions in Transverse Ising Models (Springer, Heidelberg,1996), Vol. m41.

7M. A. Continentino, Quantum Scaling in Many-Body Systems(World Scientific, Singapore, 2001).

8V. Gritsev and A. Polkovnikov, Developments in Quantum PhaseTransitions, edited by Lincoln D. Carr (Taylor and Francis, BocaRaton, 2010).

9A. F. Albuquerque, F. Alet, C. Sire, and S. Capponi, Phys. Rev. B81, 064418 (2010).

10D. Schwandt, F. Alet, and S. Capponi, Phys. Rev. Lett. 103, 170501(2009).

11C. De Grandi, V. Gritsev, and A. Polkovnikov, Phys. Rev. B 81,012303 (2010).

12S.-J. Gu and H.-Q. Lin, Europhys. Lett. 87, 10003 (2009).13W.-L. You, Y.-W. Li, and S.-J. Gu, Phys. Rev. E 76, 022101 (2007);

S. Chen, L. Wang, Y. Hao, and Y. Wang, ibid. 76, 061108 (2007);S.-J. Gu, H.-M. Kwok, W.-Q. Ning, and H.-Q. Lin, Phys. Rev. B77, 245109 (2008); S. Chen, L. Wang, S.-J. Gu, and Y. Wang, Phys.Rev. A 77, 032111 (2008); S. Yang, S.-J. Gu, C.-P. Sun, and H.-Q.Lin, ibid. 78, 012304 (2008); W.-C. Yu, H.-M. Kwok, J. Cao, andS.-J. Gu, Phys. Rev. E 80, 021108 (2009); Z. Wang, T. Ma, S.-J.Gu, and H.-Q. Lin, Phys. Rev. A 81, 062350 (2010).

14N. T. Jacobson, S. Garnerone, S. Haas, and P. Zanardi, Phys. Rev.B 79, 184427 (2009); A. T. Rezakhani, D. F. Abasto, D. A. Lidar,and P. Zanardi, Phys. Rev. A 82, 012321 (2010); S. Garnerone,

214302-5

VICTOR MUKHERJEE AND AMIT DUTTA PHYSICAL REVIEW B 83, 214302 (2011)

N. T. Jacobson, S. Haas, and P. Zanardi, ibid. 79, 032302 (2009);S. Garnerone, D. Abasto, S. Haas, and P. Zanardi, Phys. Rev.Lett. 102, 057205 (2009); X.-M. Lu, Z. Sun, X. Wang, andP. Zanardi, Phys. Rev. A 78, 032309 (2008); L. CamposVenuti,H. Saleur, and P. Zanardi, Phys. Rev. B 79, 092405 (2009); D. F.Abasto, A. Hamma, and P. Zanardi, Phys. Rev. A 78, 010301(R)(2008); M. Cozzini, R. Ionicioiu, and P. Zanardi, Phys. Rev. B76, 104420 (2007); M. Cozzini, P. Giorda, and P. Zanardi, ibid.75, 014439 (2007); L. Campos Venuti, M. Cozzini, P. Buonsante,F. Massel, N. Bray-Ali, and P. Zanardi, ibid. 78, 115410 (2008).

15M. M. Rams and B. Damski, Phys. Rev. Lett. 106, 055701 (2011).16H.-Q. Zhou and J. P. Barjaktarevic, J. Phys. A: Math. Theor. 41,

412001 (2008); H.-Q. Zhou, R. Orus, and G. Vidal, Phys. Rev.Lett. 100, 080601 (2008); B. Li, S.-H. Li, and H.-Q. Zhou, Phys.Rev. E 79, 060101(R) (2009); J.-H. Zhao and H.-Q. Zhou, Phys.Rev. B 80, 014403 (2009); H.-Q. Zhou, J.-H. Zhao, and B. Li,J. Phys. A: Math. Theor. 41, 492002 (2008).

17V. Mukherjee, A. Polkovnikov, and A. Dutta, Phys. Rev. B 83,075118 (2011).

18H.-Q. Zhou, J. H. Zhao, and B. Li, J. Phys. A: Math. Theor. 41,492002 (2008); J.-H. Zhao and H.-Q. Zhou, Phys. Rev. B 80, 014403(2009).

19M. Znidaric and T. Prosen, J. Phys. A: Math. Gen. 36, 2463 (2003).20J. Ma, L. Xu, H.-N. Xiong, and X. Wang, Phys. Rev. E 78, 051126

(2008).21E. Eriksson and H. Johannesson, Phys. Rev. A 79, 060301(R)

(2009).22A. C. M. Carollo and J. K. Pachos, Phys. Rev. Lett. 95, 157203

(2005).23S.-L. Zhu, Phys. Rev. Lett. 96, 077206 (2006).24A. Patra, V. Mukherjee, and A. Dutta, J. Stat. Mech. (2011) P03026.25V. Pardo and W. E. Pickett, Phys. Rev. Lett. 102, 166803 (2009).26S. Banerjee, R. R. P. Singh, V. Pardo, and W. E. Pickett, Phys. Rev.

Lett. 103, 016402 (2009).27A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and

A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).

28G. I. Volovik, JETP Lett. 73, 162 (2001).29V. Pardo and W. E. Pickett, Phys. Rev. B 81, 035111 (2010).30P. Delplace and G. Montambaux, Phys. Rev. B 82, 035438 (2010).31W. H. Zurek, U. Dorner, and P. Zoller, Phys. Rev. Lett. 95, 105701

(2005).32A. Polkovnikov, Phys. Rev. B 72, 161201(R) (2005).33J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005).34R. W. Cherng and L. S. Levitov, Phys. Rev. A 73, 043614

(2006).35V. Mukherjee, U. Divakaran, A. Dutta, and D. Sen, Phys. Rev. B

76, 174303 (2007).36D. Sen, K. Sengupta, and S. Mondal, Phys. Rev. Lett. 101, 016806

(2008).37J. Dziarmaga, Adv. Phys. 59, 1063 (2010).38A. Dutta, U. Divakaran, D. Sen, B. K. Chakrabarti, T. F. Rosenbaum,

and G. Aeppli, Transverse field spin models: From statisticalphysics to quantum information, e-print arXiv:1012.0653.

39A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, e-printarXiv:1007.5331.

40A. Dutta, R. R. P. Singh, and U. Divakaran, Europhys. Lett. 89,67001 (2010).

41A. Kitaev, Ann. Phys. (NY) 303, 2 (2003).42K. Sengupta, D. Sen, and S. Mondal, Phys. Rev. Lett. 100, 077204

(2008).43T. Hikichi, S. Suzuki, and K. Sengupta, Phys. Rev. B 82, 174305

(2010).44G. Rigolin, G. Ortiz, and V. H. Ponce, Phys. Rev. A 78, 052508

(2008).45K. Binder and J.-S. Wang, J. Stat. Phys. 55, 87 (1989).46A. Dutta, B. K. Chakrabarti, and J. K. Bhattacharjee, Phys. Rev. B

55, 5619 (1997).47D. Rossini, A. Silva, G. Mussardo, and G. E. Santoro, Phys. Rev.

Lett. 102, 127204 (2009).48E. Canovi, D. Rossini, R. Fazio, G. E. Santoro, and A. Silva, Phys.

Rev. B 83, 094431 (2011).49S.-J. Gu, Phys. Rev. E 79, 061125 (2009).

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