text: Table 3. Structu(cid:20)re of the host ne(cid:21)twork. bedded (random) is very close to that of Not embedded. Group Output size Building block #w × × Table 4 shows the best test errors and embedding losses conv 1 32 32 [3 3, 16] N/A (cid:20) × × (cid:21) E (w) of the host networks with and without embedding. × 3 3, 16 k × × R conv 2 32 32 × × N 144 k We can see that the test errors of Not embedded and ran- 3 3, 16 k (cid:20) × × (cid:21) dom are almost the same while those of direct and diff × 3 3, 32 k × × conv 3 16 16 × × N 288 k are slightly larger. The embedding loss E (w) of ran- 3 3, 32 k R × × dom is extremely low compared with those of direct and × 3 3, 64 k × × conv 4 8 8 × × N 576 k diff. These results indicate that the random approach can 3 3, 64 k × effectively embed a watermark without impairing the per- 1 1 avg-pool, fc, soft-max N/A formance in the original task. 4.2.2 Detecting Watermarks 0.9 101 Figure 2 shows the histogram of the embedded water- Not embedded 0.8 Embedded (direct) mark σ(ΣiXjiwi) (before thresholding) with and without Embedded (diff) watermarks where (a) direct, (b) diff, and (c) random pa- 0.7 Embedded (random) rameters are used in embedding and detection. If we bi- 0.6 100 narize σ(Σ X w ) at a threshold of 0.5, all watermarks i ji i ∀ ≥ 0.5 a⇔re correctly d≥etected because j, σ(ΣiXjiwi) 0.5 ( Σ X w 0) for all embedded cases. Please note 0.4 i ji i ∈ { } that we embedded b = 1 0, 1 T as a watermark as men- 0.3 10-1 tioned before. Although random watermarks will be de- 0.2 tected for the non-embedded cases, it can be easily judged that the watermark is not embedded because the distribution 0.1 of σ(Σ X w ) is quite different from those for embedded i ji i 0.00 50 100 150 20010-2 cases. 4.2.3 Distribution of Model Parameters Figure 1. Training curves for the host network on CIFAR-10 as a function of epochs. Solid lines denote test error (y-axis on the left) We explore how trained model parameters are affected and dashed lines denote training loss E(w) (y-axis on the right). by the embedded watermarks. Figure 3 shows the distribu- tion of model parameters W (not w) with and without wa- Table 4. Test error (%) and embedding loss E (w) with and with- R termarks. These parameters are taken only from the layer out embedding. Test error E (w) in which a watermark was embedded. Note that W is the R parameter before taking the mean over ﬁlters, and thus the Not embedded 8.04 N/A × × × direct 8.21 1.24×10−1 number of parameters is 3 3 64 64. We can see that × − direct and diff signiﬁcantly alter the distribution of param- diff 8.37 6.20 10 2 × − eters while random does not. In direct, many parameters random 7.97 4.76 10 4 became large and a peak appears near 2 so that their mean over ﬁlters becomes a large positive value to reduce the em- mark (T = 256) was embedded into the conv 2 group. bedding loss. In diff, most parameters were pushed in both 4.2.1 Test Error and Training Loss positive and negative directions so that the differences be- tween these parameters became large. In random, a water- Figure 1 shows the training curves for the host network mark is diffused over all parameters with random weights in CIFAR-10 as a function of epochs. Not embedded is and thus does not signiﬁcantly alter the distribution. This is the case that the host network is trained without the embed- one of the desirable properties of watermarking related the ding regularizer. Embedded (direct), Embedded (diff), security requirement; one may be aware of the existence of and Embedded (random) respectively represent training the embedded watermarks for the direct and diff cases. curves with embedding regularizers whose parameters are The results so far indicated that the random approach Xdirect, Xdiff, and Xrandom. We can see that the training seemed to be the best choice among the three, with low em- loss E(w) with a watermark becomes larger than the not- bedding loss, low test error in the original task, and not al- embedded case if the parameters Xdirect and Xdiff are used. tering the parameter distribution. Therefore, in the follow- This large training loss is dominated by the embedding loss ing experiments, we used the random approach in embed- ER(w), which indicates that it is difﬁcult to embed a water- ding watermarks without explicitly indicating it. mark directly into a parameter or even into the difference of 4.2.4 Fine-tune-to-embed and Distill-to-embed two parameters. On the other hand, the training loss of Em- 5 rorre tseT ssol gniniarT ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: Training

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text: as

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text: epochs.

text: Solid

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text: left)

text: We

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text: how

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text: by

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text: embedded.

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text: that

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text: parameter

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text: mean

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text: and

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text: alter

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text: direct,

text: many

text: parameters

text: random

text: became

text: large

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text: a

text: peak

text: appears

text: near

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text: mean

text: over

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text: becomes

text: a

text: large

text: positive

text: value

text: to

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text: mark

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text: group.

text: bedding

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text: one

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text: (diff),

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text: respectively

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text: cases.

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text: experiments,

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text: and

text: Distill-to-embed

text: two

text: parameters.

text: On

text: the

text: other

text: hand,

text: the

text: training

text: loss

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text: rorre

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text: 4 Low-Temperature Series The low-temperature physics of the system can be captured by various dimen- sionless ratios. As independent quantities we deﬁne the parameters m, m and t H as √ ≡ Σ H ≡ H ≡ T s s m , m , t . (4.1) 2πF 3 H 2πF 2 2πF 2 For the eﬀective low-energy expansion to be consistent, the temperature as well as the staggered and magnetic ﬁeld must be small compared to the scale Λ that charac- terizes the microscopic system. The natural scale in the Heisenberg antiferromagnet is the exchange integral J. In the present study, we deﬁne low temperatures and weak ﬁelds by (cid:112) ∝ (cid:46) T, H, M ( H ) 0.3 J . (4.2) II s The factors 2π in Eq. (4.1) were introduced in analogy to the relevant scale in quantum chromodynamics (see Ref. [60]). The point is that for the antiferromagnet – both on the square and honeycomb lattice – the denominator 2πF 2 is of the order of J. The parameters m, m , t hence measure temperature and ﬁeld strength relative H to the underlying microscopic scale. Whereas temperature and magnetic ﬁeld can be arbitrarily small, it should be noted that the staggered ﬁeld can not be switched oﬀ. This is a consequence of the Mermin-Wagner theorem [53] and the fact that the staggered magnetization – unlike the magnetization – represents the order parameter. As we have discussed on previous occasions, the domain where the eﬀective expansion fails due to the smallness of the staggered ﬁeld, is tiny. The interested reader is referred to Figs. 2 and 3 of Ref. [61]. 4.1 Pressure We ﬁrst discuss the pressure, deﬁned by − P = z z . (4.3) 0 The quantity z includes all terms in the energy density that do not depend on 0 temperature. Introducing dimensionless functions h (m, m , t) as i H 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: section

text: Low-Temperature

text: Series

text: The

label: paragraph

text: low-temperature

text: physics

text: of

text: the

text: system

text: can

text: be

text: captured

text: by

text: various

text: dimen-

text: sionless

text: ratios.

text: As

text: independent

text: quantities

text: we

text: deﬁne

text: the

text: parameters

text: m,

text: m

text: and

text: t

text: H

text: as

label: equation

text: Σ

text: H

text: H

text: T

text: s

text: s

text: m

text: m

text: t

text: H

text: For

text: the

text: eﬀective

text: low-energy

text: expansion

text: to

text: be

text: consistent,

text: the

text: temperature

text: as

text: well

text: as

text: the

text: staggered

text: and

text: magnetic

text: ﬁeld

text: must

text: be

text: small

text: compared

text: to

text: the

text: scale

text: Λ

text: that

text: charac-

text: terizes

text: the

text: microscopic

text: system.

text: The

text: natural

text: scale

text: in

text: the

text: Heisenberg

text: antiferromagnet

text: is

text: the

text: exchange

text: integral

text: J.

text: In

text: the

text: present

text: study,

text: we

text: deﬁne

text: low

text: temperatures

text: and

text: weak

text: ﬁelds

text: by

text: T,

text: H,

text: M

text: H

text: J

text: II

text: s

text: The

text: factors

text: 2π

text: in

text: Eq.

text: were

text: introduced

text: in

text: analogy

text: to

text: the

text: relevant

text: scale

text: in

text: quantum

text: chromodynamics

text: (see

text: Ref.

text: The

text: point

text: is

text: that

text: for

text: the

text: antiferromagnet

text: both

text: on

text: the

text: square

text: and

text: honeycomb

text: lattice

text: the

text: denominator

text: is

text: of

text: the

text: order

text: of

text: J.

text: The

text: parameters

text: m,

text: m

text: t

text: hence

text: measure

text: temperature

text: and

text: ﬁeld

text: strength

text: relative

text: H

text: to

text: the

text: underlying

text: microscopic

text: scale.

text: Whereas

text: temperature

text: and

text: magnetic

text: ﬁeld

text: can

text: be

text: arbitrarily

text: small,

text: it

text: should

text: be

text: noted

text: that

text: the

text: staggered

text: ﬁeld

text: can

text: not

text: be

text: switched

text: oﬀ.

text: This

text: is

text: a

text: consequence

text: of

text: the

text: Mermin-Wagner

text: theorem

text: and

text: the

text: fact

text: that

text: the

text: staggered

text: magnetization

text: unlike

text: the

text: magnetization

text: represents

text: the

text: order

text: parameter.

text: As

text: we

text: have

text: discussed

text: on

text: previous

text: occasions,

text: the

text: domain

text: where

text: the

text: eﬀective

text: expansion

text: fails

text: due

text: to

text: the

text: smallness

text: of

text: the

text: staggered

text: ﬁeld,

text: is

text: tiny.

text: The

text: interested

text: reader

text: is

text: referred

text: to

text: Figs.

text: and

text: of

text: Ref.

text: Pressure

text: We

text: ﬁrst

text: discuss

text: the

text: pressure,

text: deﬁned

text: by

text: P

text: z

text: z

text: The

text: quantity

text: z

text: includes

text: all

text: terms

text: in

text: the

text: energy

text: density

text: that

text: do

text: not

text: depend

text: on

text: temperature.

text: Introducing

text: dimensionless

text: functions

text: h

text: (m,

text: m

text: t)

text: as

text: i

text: H

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: While the order parameter is indeed expected to increase when the staggered ﬁeld becomes stronger, the behavior with respect to the magnetic ﬁeld comes rather un- expectedly: in the series (4.14), the term linear in H is small, but positive. The order parameter thus increases when a weak perpendicular magnetic ﬁeld is applied. Notice that the subleading correction (order H2) involves the NLO eﬀective constant k whose sign remains open. Still, the behavior of the order parameter in weak mag- 1 netic ﬁelds is dominated by the leading term that is strictly positive. We emphasize that this result is universal in the sense that the term of order H is the same for any bipartite lattice: the only diﬀerence between, e.g., the square and honeycomb lattice, concerns the actual values of the eﬀective constants Σ and F . s The phenomenon that the order parameter is enhanced by an external magnetic ﬁeld when the order parameter is already present in zero magnetic ﬁeld, is called magnetic catalysis according to Ref. [73]. It has been observed in quantum chro- modynamics, where the quark condensate – the order parameter of the spontaneously broken chiral symmetry – increases at T =0 in presence of a magnetic ﬁeld [73–76]. Magnetic catalysis has also been reported in condensed matter systems like graphene [77] and three-dimensional topological insulators [78]. The fact that the staggered magnetization is enhanced at T =0 in square lattice antiferromagnets subjected to a magnetic ﬁeld perpendicular to the order parameter, has been reported in Ref. [46]. It should be pointed out that the phenomenon of magnetic catalysis – as it is discussed in the context of QCD or, e.g., graphene – involves charged particles and Landau levels. The mechanism of magnetic catalysis is thus rather subtle and diﬀerent from the mechanism leading to magnetic catalysis in d=2+1 antiferromagnets where no charged particles are involved in its low-energy description. The fact that the staggered magnetization grows in presence of a weak perpendicular magnetic ﬁeld, is simply due to the suppression of quantum ﬂuctuations of the order parameter vector by the magnetic ﬁeld. Still, according to the deﬁnition given in Ref. [73], we are dealing with magnetic catalysis. 4.3 Magnetization The low-temperature expansion of the magnetization, −∂z(T, H , H) s Σ(T, H , H) = , (4.15) s ∂H takes the form11 11The coeﬃcient σˆ can trivially be obtained from z given in Sec. . 2 4b 17 ##LTLine## ##LTLine##

text: While

text: the

text: order

text: parameter

text: is

text: indeed

text: expected

text: to

text: increase

text: when

text: the

text: staggered

text: ﬁeld

text: becomes

text: stronger,

text: the

text: behavior

text: with

text: respect

text: to

text: the

text: magnetic

text: ﬁeld

text: comes

text: rather

text: un-

text: expectedly:

text: in

text: the

text: series

text: the

text: term

text: linear

text: in

text: H

text: is

text: small,

text: but

text: positive.

text: The

text: order

text: parameter

text: thus

text: increases

text: when

text: a

text: weak

text: perpendicular

text: magnetic

text: ﬁeld

text: is

text: applied.

text: Notice

text: that

text: the

text: subleading

text: correction

text: (order

text: involves

text: the

text: NLO

text: eﬀective

text: constant

text: k

text: whose

text: sign

text: remains

text: open.

text: Still,

text: the

text: behavior

text: of

text: the

text: order

text: parameter

text: in

text: weak

text: mag-

text: netic

text: ﬁelds

text: is

text: dominated

text: by

text: the

text: leading

text: term

text: that

text: is

text: strictly

text: positive.

text: We

text: emphasize

text: that

text: this

text: result

text: is

text: universal

text: in

text: the

text: sense

text: that

text: the

text: term

text: of

text: order

text: H

text: is

text: the

text: same

text: for

text: any

text: bipartite

text: lattice:

text: the

text: only

text: diﬀerence

text: between,

text: e.g.,

text: the

text: square

text: and

text: honeycomb

text: lattice,

text: concerns

text: the

text: actual

text: values

text: of

text: the

text: eﬀective

text: constants

text: Σ

text: and

text: F

text: s

text: The

text: phenomenon

text: that

text: the

text: order

text: parameter

text: is

text: enhanced

text: by

text: an

text: external

text: magnetic

text: ﬁeld

text: when

text: the

text: order

text: parameter

text: is

text: already

text: present

text: in

text: zero

text: magnetic

text: ﬁeld,

text: is

text: called

text: magnetic

text: catalysis

text: according

text: to

text: Ref.

text: It

text: has

text: been

text: observed

text: in

text: quantum

text: chro-

text: modynamics,

text: where

text: the

text: quark

text: condensate

text: the

text: order

text: parameter

text: of

text: the

text: spontaneously

text: broken

text: chiral

text: symmetry

text: increases

text: at

text: T

text: in

text: presence

text: of

text: a

text: magnetic

text: ﬁeld

text: Magnetic

text: catalysis

text: has

text: also

text: been

text: reported

text: in

text: condensed

text: matter

text: systems

text: like

text: graphene

text: and

text: three-dimensional

text: topological

text: insulators

text: The

text: fact

text: that

text: the

text: staggered

text: magnetization

text: is

text: enhanced

text: at

text: T

text: in

text: square

text: lattice

text: antiferromagnets

text: subjected

text: to

text: a

text: magnetic

text: ﬁeld

text: perpendicular

text: to

text: the

text: order

text: parameter,

text: has

text: been

text: reported

text: in

text: Ref.

text: It

text: should

text: be

text: pointed

text: out

text: that

text: the

text: phenomenon

text: of

text: magnetic

text: catalysis

text: as

text: it

text: is

text: discussed

text: in

text: the

text: context

text: of

text: QCD

text: or,

text: e.g.,

text: graphene

text: involves

text: charged

text: particles

text: and

text: Landau

text: levels.

text: The

text: mechanism

text: of

text: magnetic

text: catalysis

text: is

text: thus

text: rather

text: subtle

text: and

text: diﬀerent

text: from

text: the

text: mechanism

text: leading

text: to

text: magnetic

text: catalysis

text: in

text: antiferromagnets

text: where

text: no

text: charged

text: particles

text: are

text: involved

text: in

text: its

text: low-energy

text: description.

text: The

text: fact

text: that

text: the

text: staggered

text: magnetization

text: grows

text: in

text: presence

text: of

text: a

text: weak

text: perpendicular

text: magnetic

text: ﬁeld,

text: is

text: simply

text: due

text: to

text: the

text: suppression

text: of

text: quantum

text: ﬂuctuations

text: of

text: the

text: order

text: parameter

text: vector

text: by

text: the

text: magnetic

text: ﬁeld.

text: Still,

text: according

text: to

text: the

text: deﬁnition

text: given

text: in

text: Ref.

text: we

text: are

text: dealing

text: with

text: magnetic

text: catalysis.

text: Magnetization

text: The

text: low-temperature

text: expansion

text: of

text: the

text: magnetization,

text: −∂z(T,

text: H

text: H)

text: s

text: Σ(T,

text: H

text: H)

text: s

text: ∂H

text: takes

text: the

text: coeﬃcient

text: σˆ

text: can

text: trivially

text: be

text: obtained

text: from

text: z

text: given

text: in

text: Sec.

text: ##LTLine##

text: ##LTLine##


text: over the coordinates of the ﬁrst vertex, i.e., integration over the coordinates x , x , x 0 1 2 of the one-magnon vertex, is identically zero. One concludes that the relevant new diagrams must involve vertices with at least three magnon lines. This then leads to the two-loop diagrams 4c and 4d of Fig. 2. Any other diagram that involves vertices with an odd number of magnon lines is at least of order p5, i.e., beyond the scope of the present study. Remarkably, the explicit evaluation of diagram 4c yields zero, z = 0 , (A.4) 4c while the sunset diagram con(cid:90)tributes with 2H2 z = ddx GI(x) ∂ GI(x) ∂ GII(x) . (A.5) 4d F 2 T 0 0 T R × − ≤ This integral over the torus = ds S1, with circle S1 deﬁned as β/2 ≤ x β/2, is divergent in the ultraviolet. In the subsequent appendix we show how to 0 isolate the singularities and how to evaluate the ﬁnite pieces. B Evaluation of the Sunset Diagram In order to process the integral (A.5), we decompose the thermal propagators GI,II(x) as GI,II(x) = ∆I,II(x) + GI,II(x) , (B.1) where the ∆I,II(x) are the zero-temperature propagators deﬁned in Eq. (2.12). The integral then takes the form (cid:90) (cid:16) (cid:17) ddx GI ∂ GI ∂ GII + ∆I ∂ GI ∂ GII + GI ∂ ∆I ∂ GII + GI ∂ GI ∂ ∆II 0 0 0 0 0 0 0 0 T +∆I ∂ GI ∂ ∆II + ∆I ∂ ∆I ∂ GII + GI ∂ ∆I ∂ ∆II + ∆I ∂ ∆I ∂ ∆II (.B.2) 0 0 0 0 0 0 0 0 → The ﬁrst four integrals over the torus are convergent in d 3. The four remaining integrals that involve two or three zero-temperature propagators, however, are singular 22 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: over

text: the

text: coordinates

text: of

text: the

text: ﬁrst

text: vertex,

text: i.e.,

text: integration

text: over

text: the

text: coordinates

text: x

text: x

text: x

text: of

text: the

text: one-magnon

text: vertex,

text: is

text: identically

text: zero.

text: One

text: concludes

text: that

text: the

text: relevant

text: new

text: diagrams

text: must

text: involve

text: vertices

text: with

text: at

text: least

text: three

text: magnon

text: lines.

text: This

text: then

text: leads

text: to

text: the

text: two-loop

text: diagrams

text: and

text: of

text: Fig.

text: Any

text: other

text: diagram

text: that

text: involves

text: vertices

text: with

text: an

text: odd

text: number

text: of

text: magnon

text: lines

text: is

text: at

text: least

text: of

text: order

text: i.e.,

text: beyond

text: the

text: scope

text: of

text: the

text: present

text: study.

text: Remarkably,

text: the

text: explicit

text: evaluation

text: of

text: diagram

text: yields

text: zero,

text: z

text: while

text: the

text: sunset

text: diagram

text: with

text: z

text: ddx

text: GI(x)

text: GI(x)

text: GII(x)

text: F

text: T

text: T

text: R

text: This

text: integral

text: over

text: the

text: torus

text: ds

text: with

text: circle

text: deﬁned

text: as

text: x

text: is

text: divergent

text: in

text: the

text: ultraviolet.

text: In

text: the

text: subsequent

text: appendix

text: we

text: show

text: how

text: to

text: isolate

text: the

text: singularities

text: and

text: how

text: to

text: evaluate

text: the

text: ﬁnite

text: pieces.

text: B

text: Evaluation

text: of

text: the

text: Sunset

text: Diagram

text: In

text: order

text: to

text: process

text: the

text: integral

text: we

text: decompose

text: the

text: thermal

text: propagators

text: GI,II(x)

text: as

text: GI,II(x)

text: ∆I,II(x)

text: GI,II(x)

text: where

text: the

text: ∆I,II(x)

text: are

text: the

text: zero-temperature

text: propagators

text: deﬁned

text: in

text: Eq.

text: The

text: integral

text: then

text: takes

text: the

text: form

text: ddx

text: GI

text: GI

text: GII

text: ∆I

text: GI

text: GII

text: GI

text: ∆I

text: GII

text: GI

text: GI

text: ∆II

text: T

text: +∆I

text: GI

text: ∆II

text: ∆I

text: ∆I

text: GII

text: GI

text: ∆I

text: ∆II

text: ∆I

text: ∆I

text: ∆II

text: The

text: ﬁrst

text: four

text: integrals

text: over

text: the

text: torus

text: are

text: convergent

text: in

text: d

text: The

text: four

text: remaining

text: integrals

text: that

text: involve

text: two

text: or

text: three

text: zero-temperature

text: propagators,

text: however,

text: are

text: singular

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Finally, the last integral in Eq. (B.2) that contains three zero-temperature propa- gators, is decomposed as (cid:90) (cid:90) (cid:90) (cid:90) ddx ∆I ∂ ∆I ∂ ∆II = ddx ∆I ∂ ∆I ∂ ∆II + ddx ∆I ∂ ∆I ∂ ∆II 0 0 0 0 0 0 T T \S R − ddx ∆I ∂ ∆I ∂ ∆II . (B.10) 0 0 R\S T \ S R \ S The integrals over and are ﬁnite, but the integral over all Euclidean → C space is singular in d 3. The co(cid:90)rresponding counterterm , C = ddx ∆I ∂ ∆I ∂ ∆II , (B.11) 0 0 R can be absorbed by NLO eﬀective constants in z , Eq. (3.2). 4a In conclusion, the ﬁrst four integrals in the sunset contribution, Eq. (B.2), are well-deﬁned and can be evaluated numerically in a straightforward manner, using the fact that the integrals are two-dimensional, d3x = 2πrdrdt . (B.12) The evaluation of the remaining four integrals in Eq. (B.2) is more subtle, but can be handled within dimensional regularization using the method established in → Ref. [79]. In the limit d 3, the ﬁnal – and ﬁnite – representation for the free energy density originating from the sunset diagram 4d reads 25

text: Finally,

text: the

text: last

text: integral

text: in

text: Eq.

text: that

text: contains

text: three

text: zero-temperature

text: propa-

text: gators,

text: is

text: decomposed

text: as

text: ddx

text: ∆I

text: ∆I

text: ∆II

text: ddx

text: ∆I

text: ∆I

text: ∆II

text: ddx

text: ∆I

text: ∆I

text: ∆II

text: T

text: T

text: \S

text: R

text: ddx

text: ∆I

text: ∆I

text: ∆II

text: R\S

text: T

text: S

text: R

text: S

text: The

text: integrals

text: over

text: and

text: are

text: ﬁnite,

text: but

text: the

text: integral

text: over

text: all

text: Euclidean

text: C

text: space

text: is

text: singular

text: in

text: d

text: The

text: counterterm

text: C

text: ddx

text: ∆I

text: ∆I

text: ∆II

text: R

text: can

text: be

text: absorbed

text: by

text: NLO

text: eﬀective

text: constants

text: in

text: z

text: Eq.

text: In

text: conclusion,

text: the

text: ﬁrst

text: four

text: integrals

text: in

text: the

text: sunset

text: contribution,

text: Eq.

text: are

text: well-deﬁned

text: and

text: can

text: be

text: evaluated

text: numerically

text: in

text: a

text: straightforward

text: manner,

text: using

text: the

text: fact

text: that

text: the

text: integrals

text: are

text: two-dimensional,

text: The

text: evaluation

text: of

text: the

text: remaining

text: four

text: integrals

text: in

text: Eq.

text: is

text: more

text: subtle,

text: but

text: can

text: be

text: handled

text: within

text: dimensional

text: regularization

text: using

text: the

text: method

text: established

text: in

text: Ref.

text: In

text: the

text: limit

text: d

text: the

text: ﬁnal

text: and

text: ﬁnite

text: representation

text: for

text: the

text: free

text: energy

text: density

text: originating

text: from

text: the

text: sunset

text: diagram

text: reads


text: 4 2 3 4a 4b Figure 1: Low-temperature expansion of the partition function for the d=2+1 an- tiferromagnet: Feynman diagrams up to two-loop order T 4. Filled circles refer to L L 2 , while the vertex associated with the subleading piece 4 is represented by the eff eff number 4. Each loop is suppressed by one power of T . two, but three Goldstone ﬁelds emerge in the low-energy spectrum. This scenario, i.e., the low-temperature physics of canted phases, we postpone for future studies – in the present investigation we consider weak magnetic ﬁelds where only two spin-wave branches are relevant. 3 Evaluation of the Partition Function We now evaluate the partition function for the d=2+1 antiferromagnet subjected to the magnetic and staggered ﬁelds deﬁned in Eq. (2.9). The relevant Feynman diagrams up to two-loop order are shown in Fig. 1.5 The crucial point is that we are dealing with a systematic low-temperature expansion of the partition function where each magnon loop is suppressed by one power of temperature. The free Bose gas contribution is given by the one-loop graph 3 (order T 3), while the two-loop graph 4b is of order T 4. The incorporation of a perpendicular magnetic ﬁeld generates extra vertices that L involve an odd number of magnon lines. With respect to 2 , the explicit terms are eff proportional to one time derivative(cid:16)and read (cid:17) − iF 2H U0∂ U2 U2∂ U0 . (3.1) 0 0 L These contributions, along with those originating from 4 , create vertices with eff 1, 3, 5, . . . magnon lines: in presence of a perpendicular magnetic ﬁeld, the set of 5The perturbative evaluation of the partition function is described in more detail in section 2 of Ref. [60] and in appendix A of Ref. [65]. Regarding the eﬀective Lagrangian technique in general, the interested reader may consult Refs. [66–68]. 7 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine##

font: default

text: Figure

text: Low-temperature

label: caption

text: expansion

text: of

text: the

text: partition

text: function

text: for

text: the

text: an-

text: tiferromagnet:

text: Feynman

text: diagrams

text: up

text: to

text: two-loop

text: order

text: T

text: Filled

text: circles

text: refer

text: to

text: L

text: L

text: while

text: the

text: vertex

text: associated

text: with

text: the

text: subleading

text: piece

text: is

text: represented

text: by

text: the

text: eff

text: eff

text: number

text: Each

text: loop

text: is

text: suppressed

text: by

text: one

text: power

text: of

text: T

text: two,

text: but

text: three

text: Goldstone

text: ﬁelds

text: emerge

text: in

text: the

text: low-energy

text: spectrum.

text: This

text: scenario,

text: i.e.,

text: the

text: low-temperature

text: physics

text: of

text: canted

text: phases,

text: we

text: postpone

text: for

text: future

text: studies

text: in

text: the

text: present

text: investigation

text: we

text: consider

text: weak

text: magnetic

text: ﬁelds

text: where

text: only

text: two

text: spin-wave

text: branches

text: are

text: relevant.

text: Evaluation

text: of

text: the

text: Partition

text: Function

text: We

text: now

text: evaluate

text: the

text: partition

text: function

text: for

text: the

text: antiferromagnet

text: subjected

text: to

text: the

text: magnetic

text: and

text: staggered

text: ﬁelds

text: deﬁned

text: in

text: Eq.

text: The

text: relevant

text: Feynman

text: diagrams

text: up

text: to

text: two-loop

text: order

text: are

text: shown

text: in

text: Fig.

text: The

text: crucial

text: point

text: is

text: that

text: we

text: are

text: dealing

text: with

text: a

text: systematic

text: low-temperature

text: expansion

text: of

text: the

text: partition

text: function

text: where

text: each

text: magnon

text: loop

text: is

text: suppressed

text: by

text: one

text: power

text: of

text: temperature.

text: The

text: free

text: Bose

text: gas

text: contribution

text: is

text: given

text: by

text: the

text: one-loop

text: graph

text: (order

text: T

text: while

text: the

text: two-loop

text: graph

text: is

text: of

text: order

text: T

text: The

text: incorporation

text: of

text: a

text: perpendicular

text: magnetic

text: ﬁeld

text: generates

text: extra

text: vertices

text: that

text: L

text: involve

text: an

text: odd

text: number

text: of

text: magnon

text: lines.

text: With

text: respect

text: to

text: the

text: explicit

text: terms

text: are

text: eff

text: proportional

text: to

text: one

text: time

text: read

text: iF

text: L

text: These

text: contributions,

text: along

text: with

text: those

text: originating

text: from

text: create

text: vertices

text: with

text: eff

text: magnon

text: lines:

text: in

text: presence

text: of

text: a

text: perpendicular

text: magnetic

text: ﬁeld,

text: the

text: set

text: of

text: perturbative

text: evaluation

text: of

text: the

text: partition

text: function

text: is

text: described

text: in

text: more

text: detail

text: in

text: section

text: of

text: Ref.

text: and

text: in

text: appendix

text: A

text: of

text: Ref.

text: Regarding

text: the

text: eﬀective

text: Lagrangian

text: technique

text: in

text: general,

text: the

text: interested

text: reader

text: may

text: consult

text: Refs.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: X EOP, PDEM Scarf I potential and its coherent states revivals 11 m where γ = ωt. Here e(m) are the dimensionless non-degenerate ener(cid:81)gy eigenvalues n · · · ≥ (2.16), satisfying e(m) > e(m) > e(m−) > > e(m) (2=.16) 0, (m 1), and the parameter n+1 n n 1 0 ρ(m) denotes the moments of probability distribution deﬁned by ρ(m) = n e(m), with n n i=1 n ρ(m) = 1. The last param(cid:32)eter appe(cid:33)aring in (4.1) is the normalization constant given by 0 (cid:88) ∞ 1/2 N Jn (J) = , (4.2) m ρ(m) n=0 n (cid:113) (cid:104) | (cid:105) which is deduced from the normalization condition ξ(m)(x; J, γ) ξ(m)(x; J, γ) = 1, where 0 < J < R = lim → ∞ sup n ρ(m) and R denotes the radius of convergence. n + n Under these considerations, the moments ρ(m) and the squared of normalization constant N n (J) are given by m n! Γ(n + 2σ + 1) ρ(m) = , (4.3) n 2n Γ(2σ + 1) √ N − 2 (J) = (2J) σΓ(2σ + 1)I (2 2J), (4.4) m 2σ ∞ R (cid:51) and applying the Stirling’s approximation to (4.3) we get R = . Here σ = − · s m and I ( ) are the modiﬁed Bessel functions of the ﬁrst kind [57]. 2σ So that, the Gazeau-Klauder CS (4.1) ar(cid:110)e reduced to (cid:111) (cid:88) (cid:113) (cid:112) − ∞ (2J)n/2 exp i ω n(n+2σ) t |ξ(m)(x; J, t)(cid:105) = (2J)σ√/2 2 |ψ(m)(x)(cid:105), (4.5) n n! Γ(n + 2σ + 1) I2σ(2 2J) n=0 where ψ(m)(x) are given by (2.17). It is well known that the concept of quantum n | | revivals arises from the weighting probabilities c(m) 2 for the general wave-packet, i.e., (cid:88) n ∞ | (cid:105) | (cid:105) (cid:80) Ψ(m)(x, t) = c(m) ψ(m)(x) , (4.6) n n n n=0 ∞ | | | (cid:105) where c(m) 2 = 1. So, when Ψ(m)(x, t) in (4.6) play the role of our Gazeau- n=0 n n Klauder CS (4.5), then the weighting distribution depends on J as | | ≡ Jn (2J)n+σ √ c(m)(J) 2 N = . (4.7) n 2 (J)ρ(m) n! Γ(n + 2σ + 1)I (2 2J) m n 2σ | | In Figure 2 we display the curves of c(m)(J) 2 as a function of quantum number n n for various values of J and m. It is clear that all frames show a Gaussian-shaped ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: X

text: EOP,

text: PDEM

text: Scarf

text: I

text: potential

text: and

text: its

text: coherent

text: states

text: revivals

text: m

text: where

text: γ

text: ωt.

text: Here

text: e(m)

text: are

text: the

text: dimensionless

text: non-degenerate

text: eigenvalues

text: n

text: satisfying

text: e(m)

text: e(m)

text: e(m−)

text: e(m)

text: (m

text: and

text: the

text: parameter

text: n

text: n

text: ρ(m)

text: denotes

text: the

text: moments

text: of

text: probability

text: distribution

text: deﬁned

text: by

text: ρ(m)

text: n

text: e(m),

text: with

text: n

text: n

text: n

text: ρ(m)

text: The

text: last

text: in

text: is

text: the

text: normalization

text: constant

text: given

text: by

text: N

text: Jn

text: (J)

text: m

text: ρ(m)

text: n

text: which

text: is

text: deduced

text: from

text: the

text: normalization

text: condition

text: ξ(m)(x;

text: J,

text: γ)

text: ξ(m)(x;

text: J,

text: γ)

text: where

text: J

text: R

text: lim

text: sup

text: n

text: ρ(m)

text: and

text: R

text: denotes

text: the

text: radius

text: of

text: convergence.

text: n

text: n

text: Under

text: these

text: considerations,

text: the

text: moments

text: ρ(m)

text: and

text: the

text: squared

text: of

text: normalization

text: constant

text: N

text: n

text: (J)

text: are

text: given

text: by

text: m

text: n!

text: Γ(n

text: 2σ

text: ρ(m)

text: n

text: Γ(2σ

text: N

text: (J)

text: σΓ(2σ

text: m

text: 2σ

text: R

text: and

text: applying

text: the

text: Stirling’s

text: approximation

text: to

text: we

text: get

text: R

text: Here

text: σ

text: s

text: m

text: and

text: I

text: are

text: the

text: modiﬁed

text: Bessel

text: functions

text: of

text: the

text: ﬁrst

text: kind

text: 2σ

text: So

text: that,

text: the

text: Gazeau-Klauder

text: CS

text: reduced

text: to

text: exp

text: i

text: ω

text: t

text: |ξ(m)(x;

text: J,

text: n

text: n!

text: Γ(n

text: 2σ

text: where

text: ψ(m)(x)

text: are

text: given

text: by

text: It

text: is

text: well

text: known

text: that

text: the

text: concept

text: of

text: quantum

text: n

text: revivals

text: arises

text: from

text: the

text: weighting

text: probabilities

text: c(m)

text: for

text: the

text: general

text: wave-packet,

text: i.e.,

text: n

text: Ψ(m)(x,

text: t)

text: c(m)

text: ψ(m)(x)

text: n

text: n

text: n

text: where

text: c(m)

text: So,

text: when

text: Ψ(m)(x,

text: t)

text: in

text: play

text: the

text: role

text: of

text: our

text: Gazeau-

text: n

text: n

text: Klauder

text: CS

text: then

text: the

text: weighting

text: distribution

text: depends

text: on

text: J

text: as

text: Jn

text: c(m)(J)

text: N

text: n

text: (J)ρ(m)

text: n!

text: Γ(n

text: 2σ

text: m

text: n

text: 2σ

text: In

text: Figure

text: we

text: display

text: the

text: curves

text: of

text: c(m)(J)

text: as

text: a

text: function

text: of

text: quantum

text: number

text: n

text: n

text: for

text: various

text: values

text: of

text: J

text: and

text: m.

text: It

text: is

text: clear

text: that

text: all

text: frames

text: show

text: a

text: Gaussian-shaped

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: X EOP, PDEM Scarf I potential and its coherent states revivals 17 m exceptional (m = 1, 2) PDEM Scarf I potential. The timescales are T (m) = 4π and rev T (m) = 2π/(n+σ). As we can see, the usual CM involves a perfect full revivals as it was cl expected above and the sharp peaks arise due to the fractional revivals which become more apparent as J increases. However some permanent peaks still exist, no matter the values attributed to J and m, like those of T /3 and 2T /3, for all m = 0, 1, 2, with rev rev a change in the width and magnitude. We also observe that as J increases all curves merge into a single one, as can be seen in Figure 4(d). | | Figure 5. Plot of ξ(m)(x; J, t) 2 against x for J = 20, plotted for α = 1, β = 2, and n = 50. As for Figure 4, the parameter m refers to: m = 0 (red line) and m = 1 max (blue line). On the other hand we plot in Figure 5 the probability density of a coherent state | | wave-packet ξ(m)(x; J, t) 2 for J = 20, α = 1, β = 2 and m = 0, 1. It is evident that the probability density is reconstructed after the time revival T (m) as it was illustrated rev in the frame (d) compared to (a). In the ﬁrst frame, t = 0, we observe two principal (cid:39) (cid:39) and symmetrical peaks for both cases, (m = 0, 1), centered at x 1.25 and x 2, 1 2 followed by a secondary ripple in the case m = 0 before and after dominant peaks. All these phenomena are periodic in the whole real line with a period δx = 2π, being the behavior for the case where the mass is constant. With time, our numerical simulations show us that these peaks oscillate back and forth between the walls of the well with relative change in shape. 4.2. The case of position-dependent eﬀective mass Even as (4.14) and (4.15) are applicable for the exceptional X PDEM Scarf I potential, m two diﬀerent proﬁles of the mass function, introduced in section 2, without and with singularities were chosen 1 1 M (x) = , and M (x) = − , (4.16) wos 1 + (λx)2 ws (1 (λx)2)2 respectively. These mass functions allow us to construct a coherent state wave- packet with the proper behavior near the boundaries and have been used in many studies, see for instance [59]. We take the proﬁle of the mass function to be of the form of M (x). This proﬁle wos R is without singularities and is a bounded function deﬁned in the whole real line , where ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: X

text: EOP,

text: PDEM

text: Scarf

text: I

text: potential

text: and

text: its

text: coherent

text: states

text: revivals

text: m

text: exceptional

text: (m

text: PDEM

text: Scarf

text: I

text: potential.

text: The

text: timescales

text: are

text: T

text: (m)

text: 4π

text: and

text: rev

text: T

text: (m)

text: As

text: we

text: can

text: see,

text: the

text: usual

text: CM

text: involves

text: a

text: perfect

text: full

text: revivals

text: as

text: it

text: was

text: cl

text: expected

text: above

text: and

text: the

text: sharp

text: peaks

text: arise

text: due

text: to

text: the

text: fractional

text: revivals

text: which

text: become

text: more

text: apparent

text: as

text: J

text: increases.

text: However

text: some

text: permanent

text: peaks

text: still

text: exist,

text: no

text: matter

text: the

text: values

text: attributed

text: to

text: J

text: and

text: m,

text: like

text: those

text: of

text: T

text: and

text: for

text: all

text: m

text: with

text: rev

text: rev

text: a

text: change

text: in

text: the

text: width

text: and

text: magnitude.

text: We

text: also

text: observe

text: that

text: as

text: J

text: increases

text: all

text: curves

text: merge

text: into

text: a

text: single

text: one,

text: as

text: can

text: be

text: seen

text: in

text: Figure

text: Figure

text: Plot

text: of

text: ξ(m)(x;

text: J,

text: t)

text: against

text: x

text: for

text: J

text: plotted

text: for

text: α

text: β

text: and

text: n

text: As

text: for

text: Figure

text: the

text: parameter

text: m

text: refers

text: to:

text: m

text: (red

text: line)

text: and

text: m

text: max

text: (blue

text: line).

text: On

text: the

text: other

text: hand

text: we

text: plot

text: in

text: Figure

text: the

text: probability

text: density

text: of

text: a

text: coherent

text: state

text: wave-packet

text: ξ(m)(x;

text: J,

text: t)

text: for

text: J

text: α

text: β

text: and

text: m

text: It

text: is

text: evident

text: that

text: the

text: probability

text: density

text: is

text: reconstructed

text: after

text: the

text: time

text: revival

text: T

text: (m)

text: as

text: it

text: was

text: illustrated

text: rev

text: in

text: the

text: frame

text: (d)

text: compared

text: to

text: (a).

text: In

text: the

text: ﬁrst

text: frame,

text: t

text: we

text: observe

text: two

text: principal

text: and

text: symmetrical

text: peaks

text: for

text: both

text: cases,

text: (m

text: centered

text: at

text: x

text: and

text: x

text: followed

text: by

text: a

text: secondary

text: ripple

text: in

text: the

text: case

text: m

text: before

text: and

text: after

text: dominant

text: peaks.

text: All

text: these

text: phenomena

text: are

text: periodic

text: in

text: the

text: whole

text: real

text: line

text: with

text: a

text: period

text: δx

text: being

text: the

text: behavior

text: for

text: the

text: case

text: where

text: the

text: mass

text: is

text: constant.

text: With

text: time,

text: our

text: numerical

text: simulations

text: show

text: us

text: that

text: these

text: peaks

text: oscillate

text: back

text: and

text: forth

text: between

text: the

text: walls

text: of

text: the

text: well

text: with

text: relative

text: change

text: in

text: shape.

text: The

text: case

text: of

text: position-dependent

text: eﬀective

text: mass

text: Even

text: as

text: and

text: are

text: applicable

text: for

text: the

text: exceptional

text: X

text: PDEM

text: Scarf

text: I

text: potential,

text: m

text: two

text: diﬀerent

text: proﬁles

text: of

text: the

text: mass

text: function,

text: introduced

text: in

text: section

text: without

text: and

text: with

text: singularities

text: were

text: chosen

text: M

text: (x)

text: and

text: M

text: (x)

text: wos

text: ws

text: respectively.

text: These

text: mass

text: functions

text: allow

text: us

text: to

text: construct

text: a

text: coherent

text: state

text: wave-

text: packet

text: with

text: the

text: proper

text: behavior

text: near

text: the

text: boundaries

text: and

text: have

text: been

text: used

text: in

text: many

text: studies,

text: see

text: for

text: instance

text: We

text: take

text: the

text: proﬁle

text: of

text: the

text: mass

text: function

text: to

text: be

text: of

text: the

text: form

text: of

text: M

text: (x).

text: This

text: proﬁle

text: wos

text: R

text: is

text: without

text: singularities

text: and

text: is

text: a

text: bounded

text: function

text: deﬁned

text: in

text: the

text: whole

text: real

text: line

text: where

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: X EOP, PDEM Scarf I potential and its coherent states revivals 4 m (cid:48)(cid:48) − (cid:48) where S(z) = z /z 3/2 (z /z)2 is the Schwartz derivative of the function z(x) and the prime denotes the derivative with respect to x. It follows that the PDEM SE can be solved if the forms of Q and R are given for a mass function M(x). In order to obtain the eﬀective potential in the above equation, we impose that there must be a constant on the right-hand side of (2.6) representing the bound-state energy spectrum E on the n left-hand side. From (2.5) the solu(cid:115)tion of the e(cid:40)igenfunctions ψ ((cid:41)x) are given by (cid:90) n ∼ M(x) 1 g(x) ψn(x) (cid:48) exp Q(g) dg Fn(g(x)), (2.7) g (x) 2 up to a normalization constant. It is worth to note that all expressions reduce to the well known ones if the mass is taken t(cid:98)o be constant, i.e., M(x) = 1. In the remainder of the paper, we choose to work under the special function F (m)(g) n to be the PDEM X Jacobi polynomials P (α,β,m)(g) studied in more details in [29, 55], ≥ m n where n m. This new family of orthogonal polynomials is orthonormal with respect (cid:99) to the weight function [55] − (1 g)α(1 + g)β ≡ (cid:99) W (m)(g) = − − − , (g g(x)) , P ( α 1,(cid:16)β 1)(g) (cid:17) m (cid:99) where W (0)(g) = W (g) is the weight function for the classical Jacobi polynomials and − ≤ ≤ 1 g(x) +1 in order that L2 g, W (m)(g)dg -orthonormality holds. Moreover these polynomials are related to the classical Jacobi orthogonal polynomials P (α,β)(g) by the following relations n (cid:98) (cid:20) (cid:98) P (α,β,0)(g) = P (α,β)(g), (2.8) n n (cid:21) − α + β + j + 1 − − − − P (α,β,m)(g) = ( 1)m (g 1)P ( α 1,β 1)(g)P (−α+2,β)(g) n 2(α + j + 1) m j 1 − α m + 1 − − − − ≥ + P ( α 2,β)(g)P (α+1,β 1)(g) , (j = n m 0) (2.9) α + j + 1 m j if and only if the following restrictions hold simultaneously [22, 55] (cid:54) − − (cid:54)∈ { · · · − } (R1) β = 0, α, and α β m + 1 0, 1, 2, , m 1 , (2.10a) − − (R2) α > m 2, and sgn(α β + 1) = sgn(β), (2.10b) · where sgn( ) is the signum function. Under these conditions, the scalar product of the exceptional X Jacobi polynomials leads to orthogonality relation, m ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: X

text: EOP,

text: PDEM

text: Scarf

text: I

text: potential

text: and

text: its

text: coherent

text: states

text: revivals

text: m

text: where

text: S(z)

text: z

text: /z

text: (z

text: is

text: the

text: Schwartz

text: derivative

text: of

text: the

text: function

text: z(x)

text: and

text: the

text: prime

text: denotes

text: the

text: derivative

text: with

text: respect

text: to

text: x.

text: It

text: follows

text: that

text: the

text: PDEM

text: SE

text: can

text: be

text: solved

text: if

text: the

text: forms

text: of

text: Q

text: and

text: R

text: are

text: given

text: for

text: a

text: mass

text: function

text: M(x).

text: In

text: order

text: to

text: obtain

text: the

text: eﬀective

text: potential

text: in

text: the

text: above

text: equation,

text: we

text: impose

text: that

text: there

text: must

text: be

text: a

text: constant

text: on

text: the

text: right-hand

text: side

text: of

text: representing

text: the

text: bound-state

text: energy

text: spectrum

text: E

text: on

text: the

text: n

text: left-hand

text: side.

text: From

text: the

text: of

text: the

text: ψ

text: are

text: given

text: by

text: n

text: M(x)

text: g(x)

text: ψn(x)

text: exp

text: Q(g)

text: dg

text: Fn(g(x)),

text: g

text: (x)

text: up

text: to

text: a

text: normalization

text: constant.

text: It

text: is

text: worth

text: to

text: note

text: that

text: all

text: expressions

text: reduce

text: to

text: the

text: well

text: known

text: ones

text: if

text: the

text: mass

text: is

text: taken

text: be

text: constant,

text: i.e.,

text: M(x)

text: In

text: the

text: remainder

text: of

text: the

text: paper,

text: we

text: choose

text: to

text: work

text: under

text: the

text: special

text: function

text: F

text: (m)(g)

text: n

text: to

text: be

text: the

text: PDEM

text: X

text: Jacobi

text: polynomials

text: P

text: (α,β,m)(g)

text: studied

text: in

text: more

text: details

text: in

text: m

text: n

text: where

text: n

text: m.

text: This

text: new

text: family

text: of

text: orthogonal

text: polynomials

text: is

text: orthonormal

text: with

text: respect

text: to

text: the

text: weight

text: function

text: g)β

text: W

text: (m)(g)

text: (g

text: g(x))

text: P

text: α

text: m

text: where

text: W

text: W

text: (g)

text: is

text: the

text: weight

text: function

text: for

text: the

text: classical

text: Jacobi

text: polynomials

text: and

text: g(x)

text: in

text: order

text: that

text: g,

text: W

text: (m)(g)dg

text: -orthonormality

text: holds.

text: Moreover

text: these

text: polynomials

text: are

text: related

text: to

text: the

text: classical

text: Jacobi

text: orthogonal

text: polynomials

text: P

text: (α,β)(g)

text: by

text: the

text: following

text: relations

text: n

text: P

text: P

text: (α,β)(g),

text: n

text: n

text: α

text: β

text: j

text: P

text: (α,β,m)(g)

text: (g

text: α

text: n

text: j

text: m

text: j

text: α

text: m

text: P

text: α

text: (j

text: n

text: m

text: α

text: j

text: m

text: j

text: if

text: and

text: only

text: if

text: the

text: following

text: restrictions

text: hold

text: simultaneously

text: β

text: α,

text: and

text: α

text: β

text: m

text: m

text: α

text: m

text: and

text: sgn(α

text: β

text: sgn(β),

text: where

text: sgn(

text: is

text: the

text: signum

text: function.

text: Under

text: these

text: conditions,

text: the

text: scalar

text: product

text: of

text: the

text: exceptional

text: X

text: Jacobi

text: polynomials

text: leads

text: to

text: orthogonality

text: relation,

text: m

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: X EOP, PDEM Scarf I potential and its coherent states revivals 6 m (cid:0) (cid:1) (cid:0) (cid:1) k2 − − k2 − V (m)(x) = 2α2 + 2β2 1 sec2 ϑ(x) β2 α2 sec ϑ(x) tan ϑ(x) eﬀ 8 4 − − k2 − − (cid:34) − (cid:35)P ( −α,β)(sin ϑ(x)) (α β m + 1) (α + β + (α β + 1) sin ϑ(x)) −m−1 − 2 P ( α 1,β 1)(sin ϑ(x)) m − 2 k2 − − P ( −α,β)(sin ϑ(x)) + (α β m + 1)2 cos2 ϑ(cid:2) (x) −m−1 − (cid:3) 4 P ( α 1,β 1)(sin ϑ(x)) m (cid:48)(cid:48)(cid:48) (cid:48)(cid:48) 1 µ (x) − 5 µ 2(x) − k2 − − + (cid:48) (cid:48) (α + β + 1)2 + 4m(α 3β m + 1) , (2.15) 4 µ 3(x) 8 µ 4(x) 8 ≡ k2 − E(m) k2e(m) = n(n 2m + α + β + 1), (cid:98) (2.16) n n 2 − (1 sin ϑ(x))α+1 (1 + sin ϑ(x))β +1 ψ(m)(x) = N(m)M1/4(x) − − 2 −4 2 4 P (α,β,m)(sin ϑ(x)), (2.17) n n P ( α 1,β 1)(sin ϑ(x)) n m (cid:112) where ϑ(x) = kµ(x), N(m) is the normalization constant and where we introduce n (cid:48) the auxiliary mass function µ (x) = M(x). We expressly chose to deduce the eﬀective potential in its shape given by (2.15) and its associated energy spectra (2.16), so that the ground-state energy E(m) is chosen to be zero, in order to construct their associated 0 coherent states `a la Gazeau-Klauder in the next section. (cid:112) It is worth noting that the bound-state wavefunctions (2.17) are physically acceptable if and only if the square-integrability condition fulﬁlls the restriction | | → ψ(m)(x) 2/ M(x) 0 at the end points x and x of the interval of the eﬀective n 1 2 potential (2.15).Then in order to normalize (2.17), as in the case of usual CM, we must require that the auxiliary mass function µ(x) can be restricted to the region − ≤ ≤ ⇒ − π ≤ ≤ π 1 g(x) +1 µ(x) , (2.18) 2k 2k which will(cid:115)be used later once we choose the proﬁle of the mass function M(x). Thus with the help of (2.11), the normalized constant N(m) is given by n − − − − k (n m + s) (n m + α + 1)2 Γ(n m + 1) Γ(n m + 2s) N(m) = − − − , n 2s 1/2 (n 2m + α + 1) Γ(n m + α + 2) Γ(n + β + 1) − where s = (α + β + 1)/2 and α, β > 1. The new PDEM potentials (2.15) are inﬁnite, ≥ since each m 1 gives rise to new exactly solvable PDEM potentials which are all singular in the interval (2.18), due to the properties of the Jacobi polynomials. For m = 0, we recognize the well-known PDEM Scarf I potential associated to the classical Jacobi polynomials [20] (see for instance, equation (23b) therein), namely (cid:0) (cid:1) (cid:0) (cid:1) k2 − − k2 − V (0)(x) = 2α2 + 2β2 1 sec2 ϑ(x) β2 α2 sec ϑ(x) tan ϑ(x) eﬀ 8 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: X

text: EOP,

text: PDEM

text: Scarf

text: I

text: potential

text: and

text: its

text: coherent

text: states

text: revivals

text: m

text: V

text: (m)(x)

text: ϑ(x)

text: β2

text: α2

text: sec

text: ϑ(x)

text: tan

text: ϑ(x)

text: eﬀ

text: −α,β)(sin

text: ϑ(x))

text: (α

text: β

text: m

text: (α

text: β

text: (α

text: β

text: sin

text: ϑ(x))

text: −m−1

text: P

text: α

text: ϑ(x))

text: m

text: P

text: −α,β)(sin

text: ϑ(x))

text: (α

text: β

text: m

text: (x)

text: −m−1

text: P

text: α

text: ϑ(x))

text: m

text: µ

text: (x)

text: µ

text: (α

text: β

text: 3β

text: m

text: µ

text: µ

text: E(m)

text: n(n

text: α

text: β

text: n

text: n

text: sin

text: sin

text: ϑ(x))β

text: ψ(m)(x)

text: −4

text: P

text: (α,β,m)(sin

text: ϑ(x)),

text: n

text: n

text: P

text: α

text: ϑ(x))

text: n

text: m

text: where

text: ϑ(x)

text: kµ(x),

text: N(m)

text: is

text: the

text: normalization

text: constant

text: and

text: where

text: we

text: introduce

text: n

text: the

text: auxiliary

text: mass

text: function

text: µ

text: (x)

text: M(x).

text: We

text: expressly

text: chose

text: to

text: deduce

text: the

text: eﬀective

text: potential

text: in

text: its

text: shape

text: given

text: by

text: and

text: its

text: associated

text: energy

text: spectra

text: so

text: that

text: the

text: ground-state

text: energy

text: E(m)

text: is

text: chosen

text: to

text: be

text: zero,

text: in

text: order

text: to

text: construct

text: their

text: associated

text: coherent

text: states

text: `a

text: la

text: Gazeau-Klauder

text: in

text: the

text: next

text: section.

text: It

text: is

text: worth

text: noting

text: that

text: the

text: bound-state

text: wavefunctions

text: are

text: physically

text: acceptable

text: if

text: and

text: only

text: if

text: the

text: square-integrability

text: condition

text: fulﬁlls

text: the

text: restriction

text: ψ(m)(x)

text: M(x)

text: at

text: the

text: end

text: points

text: x

text: and

text: x

text: of

text: the

text: interval

text: of

text: the

text: eﬀective

text: n

text: potential

text: in

text: order

text: to

text: normalize

text: as

text: in

text: the

text: case

text: of

text: usual

text: CM,

text: we

text: must

text: require

text: that

text: the

text: auxiliary

text: mass

text: function

text: µ(x)

text: can

text: be

text: restricted

text: to

text: the

text: region

text: π

text: π

text: g(x)

text: µ(x)

text: which

text: used

text: later

text: once

text: we

text: choose

text: the

text: proﬁle

text: of

text: the

text: mass

text: function

text: M(x).

text: Thus

text: with

text: the

text: help

text: of

text: the

text: normalized

text: constant

text: N(m)

text: is

text: given

text: by

text: n

text: k

text: (n

text: m

text: s)

text: (n

text: m

text: α

text: Γ(n

text: m

text: Γ(n

text: m

text: N(m)

text: n

text: (n

text: α

text: Γ(n

text: m

text: α

text: Γ(n

text: β

text: where

text: s

text: (α

text: β

text: and

text: α,

text: β

text: The

text: new

text: PDEM

text: potentials

text: are

text: inﬁnite,

text: since

text: each

text: m

text: gives

text: rise

text: to

text: new

text: exactly

text: solvable

text: PDEM

text: potentials

text: which

text: are

text: all

text: singular

text: in

text: the

text: interval

text: due

text: to

text: the

text: properties

text: of

text: the

text: Jacobi

text: polynomials.

text: For

text: m

text: we

text: recognize

text: the

text: well-known

text: PDEM

text: Scarf

text: I

text: potential

text: associated

text: to

text: the

text: classical

text: Jacobi

text: polynomials

text: (see

text: for

text: instance,

text: equation

text: therein),

text: namely

text: V

text: ϑ(x)

text: β2

text: α2

text: sec

text: ϑ(x)

text: tan

text: ϑ(x)

text: eﬀ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##


text: X EOP, PDEM Scarf I potential and its coherent states revivals 7 m (cid:48)(cid:48)(cid:48) (cid:48)(cid:48) − k2 1 µ (x) − 5 µ 2(x) (α + β + 1)2 + (cid:48) (cid:48) , (2.19) 8 4 µ 3(x) 8 µ 4(x) ≥ and all other potentials (m 1) are considered as extension of V (0)(x). Then it is eﬀ obvious to interpret V (m)(x) as the PDEM rationally extended Scarf I potential family. eﬀ In Figure 1 we have depicted the eﬀective potential V (m)(x) given in (2.15) plotted eﬀ for two proﬁle of the mass functions, i.e. with and without singularities [59] 1 1 M (x) = , and M (x) = − , (2.20) wos 1 + (λx)2 ws (1 (λx)2)2 inside the interval (2.18), for even and odd m up to 3 and for diﬀerent values of the ∈ R mass parameter λ . Due to the singularities of M (x), we observe that gradually as λ increases the ws shape of V (m)(x) tend to gather near the classical turning points x± = 1/λ of the well. eﬀ Contrary, the case M (x) reveals that V (m)(x) are more extended along the whole real wos eﬀ R line and become more sharper at the vicinity of x = 0, as λ increases. 0 Figure 1. Top (resp. bottom). Plot of the eﬀective potential (2.15) plotted for M (x) (resp. M (x)) for α = 1 and β = 2: m = 0 (solid line), m = 1 (dashed line), wos ws m = 2 (dotted line), and m = 3 (dot-dashed line). 3. Supersymmetry and shape-invariant approach It is well-known that supersymmetric quantum mechanics (SUSY QM) has been successfully applied to obtained exact solutions of Schr¨odinger equation, which deals H H with pairs of Hamiltonians (m) and (m) that have the same energy spectra, but 1,eﬀ 2,eﬀ diﬀerent eigenfunctions. It has been shown that such pairs of Hamiltonians can be ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: X

text: EOP,

text: PDEM

text: Scarf

text: I

text: potential

text: and

text: its

text: coherent

text: states

text: revivals

text: m

text: µ

text: (x)

text: µ

text: (α

text: β

text: µ

text: µ

text: and

text: all

text: other

text: potentials

text: (m

text: are

text: considered

text: as

text: extension

text: of

text: V

text: Then

text: it

text: is

text: eﬀ

text: obvious

text: to

text: interpret

text: V

text: (m)(x)

text: as

text: the

text: PDEM

text: rationally

text: extended

text: Scarf

text: I

text: potential

text: family.

text: eﬀ

text: In

text: Figure

text: we

text: have

text: depicted

text: the

text: eﬀective

text: potential

text: V

text: (m)(x)

text: given

text: in

text: plotted

text: eﬀ

text: for

text: two

text: proﬁle

text: of

text: the

text: mass

text: functions,

text: i.e.

text: with

text: and

text: without

text: singularities

text: M

text: (x)

text: and

text: M

text: (x)

text: wos

text: ws

text: inside

text: the

text: interval

text: for

text: even

text: and

text: odd

text: m

text: up

text: to

text: and

text: for

text: diﬀerent

text: values

text: of

text: the

text: R

text: mass

text: parameter

text: λ

text: Due

text: to

text: the

text: singularities

text: of

text: M

text: (x),

text: we

text: observe

text: that

text: gradually

text: as

text: λ

text: increases

text: the

text: ws

text: shape

text: of

text: V

text: (m)(x)

text: tend

text: to

text: gather

text: near

text: the

text: classical

text: turning

text: points

text: x±

text: of

text: the

text: well.

text: eﬀ

text: Contrary,

text: the

text: case

text: M

text: (x)

text: reveals

text: that

text: V

text: (m)(x)

text: are

text: more

text: extended

text: along

text: the

text: whole

text: real

text: wos

text: eﬀ

text: R

text: line

text: and

text: become

text: more

text: sharper

text: at

text: the

text: vicinity

text: of

text: x

text: as

text: λ

text: increases.

text: Figure

text: Top

text: (resp.

text: bottom).

text: Plot

text: of

text: the

text: eﬀective

text: potential

text: plotted

text: for

text: M

text: (x)

text: (resp.

text: M

text: (x))

text: for

text: α

text: and

text: β

text: m

text: (solid

text: line),

text: m

text: (dashed

text: line),

text: wos

text: ws

text: m

text: (dotted

text: line),

text: and

text: m

text: (dot-dashed

text: line).

text: Supersymmetry

text: and

text: shape-invariant

text: approach

text: It

text: is

text: well-known

text: that

text: supersymmetric

text: quantum

text: mechanics

text: (SUSY

text: QM)

text: has

text: been

text: successfully

text: applied

text: to

text: obtained

text: exact

text: solutions

text: of

text: Schr¨odinger

text: equation,

text: which

text: deals

text: H

text: H

text: with

text: pairs

text: of

text: Hamiltonians

text: (m)

text: and

text: (m)

text: that

text: have

text: the

text: same

text: energy

text: spectra,

text: but

text: diﬀerent

text: eigenfunctions.

text: It

text: has

text: been

text: shown

text: that

text: such

text: pairs

text: of

text: Hamiltonians

text: can

text: be

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: X EOP, PDEM Scarf I potential and its coherent states revivals 10 m − − −k2 − − (cid:34) − (cid:35) P ( −α 1,β+1)(sin ϑ(x)) (α β m + 1) (α + β + 2 + (α β + 1) sin ϑ(x)) m −1 − 2 P ( α 2,β)(sin ϑ(x)) m − − 2 k2 − − P ( −α 1,β+1)(sin ϑ(x)) + ((cid:2)α β m + 1)2 cos2 ϑ(x) m −1 − (cid:3) 4 P ( α 2,β)(sin ϑ(x)) m −k2 − − (α + β + 1)2 + 4m(α 3β m + 1) . (3.10b) 8 We observe th(cid:98)at the eﬀ(cid:98)ective potential (3.10a) matches with (2.15), apart from the omission of the Schwartz’s derivative of the mass function due to our particular choice Q Q† of the operators and in (3.3a) and (3.3b), respectively. Using (3.1), it is easy m m for the reader to check that (3.10a) and (3.10b) are connected to each other through the translational shape invariant symmetry, namely V | V | (m)(x α, β) = (m)(x α + 1, β + 1) + R(m)(α, β) 2,eﬀ 1,eﬀ V | k2 − = (m)(x α + 1, β + 1) + (α + β 2m + 2), (3.11) 1,eﬀ 2 { } { } { } where here the set of parameters a are deﬁned by: a = (α, β), a = { } − i − 1 2 (α + 1, β + 1), and thus a = (α + n 1, β + n 1). In view of (3.2), the bound-state n V energy eigenvalues of the eﬀective potential (m)(x) are then given by (cid:88) 1,eﬀ n { } E(m) = (cid:88) R(m)( a ) 1,n i i=1 n k2 − = (α + β 2m + 2i) 2 i=1 k2 − = n(n + α + β 2m + 1), (3.12) 2 which are just the energy eigenvalues deduced in (2.16), with the fact that E(m) = 0 1,0 as it was expected. 4. Revival dynamics of Gazeau-Klauder CS for the extended PDEM Scarf I potential Let us now adapt the material developed above to construct the Gazeau-Klauder CS (GK CS) [53] for the extended Hermitian PDEM Scarf I potential given in (2.15). Such coherent states are parameterized by two real para(cid:110)meters J (cid:111)and γ, and are deﬁned by (cid:88) (cid:113) − ∞ Jn/2 exp ie(m)γ | (cid:105) 1 n | (cid:105) ξ(m)(x; J, γ) = N ψ(m)(x) , (4.1) (J) n m n=0 ρ(nm) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: X

text: EOP,

text: PDEM

text: Scarf

text: I

text: potential

text: and

text: its

text: coherent

text: states

text: revivals

text: m

text: P

text: −α

text: ϑ(x))

text: (α

text: β

text: m

text: (α

text: β

text: (α

text: β

text: sin

text: ϑ(x))

text: m

text: −1

text: P

text: α

text: ϑ(x))

text: m

text: P

text: −α

text: ϑ(x))

text: β

text: m

text: ϑ(x)

text: m

text: −1

text: P

text: α

text: ϑ(x))

text: m

text: (α

text: β

text: 3β

text: m

text: We

text: observe

text: the

text: potential

text: matches

text: with

text: apart

text: from

text: the

text: omission

text: of

text: the

text: Schwartz’s

text: derivative

text: of

text: the

text: mass

text: function

text: due

text: to

text: our

text: particular

text: choice

text: Q

text: Q†

text: of

text: the

text: operators

text: and

text: in

text: and

text: respectively.

text: Using

text: it

text: is

text: easy

text: m

text: m

text: for

text: the

text: reader

text: to

text: check

text: that

text: and

text: are

text: connected

text: to

text: each

text: other

text: through

text: the

text: translational

text: shape

text: invariant

text: symmetry,

text: namely

text: V

text: V

text: (m)(x

text: α,

text: β)

text: (m)(x

text: α

text: β

text: R(m)(α,

text: β)

text: V

text: (m)(x

text: α

text: β

text: (α

text: β

text: where

text: here

text: the

text: set

text: of

text: parameters

text: a

text: are

text: deﬁned

text: by:

text: a

text: (α,

text: β),

text: a

text: i

text: (α

text: β

text: and

text: thus

text: a

text: (α

text: n

text: β

text: n

text: In

text: view

text: of

text: the

text: bound-state

text: n

text: V

text: energy

text: eigenvalues

text: of

text: the

text: eﬀective

text: potential

text: (m)(x)

text: are

text: then

text: given

text: by

text: n

text: E(m)

text: R(m)(

text: a

text: i

text: n

text: (α

text: β

text: n(n

text: α

text: β

text: which

text: are

text: just

text: the

text: energy

text: eigenvalues

text: deduced

text: in

text: with

text: the

text: fact

text: that

text: E(m)

text: as

text: it

text: was

text: expected.

text: Revival

text: dynamics

text: of

text: Gazeau-Klauder

text: CS

text: for

text: the

text: extended

text: PDEM

text: Scarf

text: I

text: potential

text: Let

text: us

text: now

text: adapt

text: the

text: material

text: developed

text: above

text: to

text: construct

text: the

text: Gazeau-Klauder

text: CS

text: (GK

text: CS)

text: for

text: the

text: extended

text: Hermitian

text: PDEM

text: Scarf

text: I

text: potential

text: given

text: in

text: Such

text: coherent

text: states

text: are

text: parameterized

text: by

text: two

text: real

text: J

text: γ,

text: and

text: are

text: deﬁned

text: by

text: exp

text: ie(m)γ

text: n

text: ξ(m)(x;

text: J,

text: γ)

text: N

text: ψ(m)(x)

text: (J)

text: n

text: m

text: ρ(nm)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: CORRELATIONS IN FUNCTION FIELDS 27 (cid:89) (cid:89) (cid:89) ≥ Lemma 3.6. For n 3 and any ﬁnite set I, (cid:111) (cid:111) (cid:111) M( V S ) = M(H(I)) = M( V − S ) = V − A . n n n n 1 n n 1 n ∈ ∈ ∈ i I i I i I (cid:89) (cid:89) (cid:89) Proof. Let (cid:111) (cid:111) (cid:111) (3.8) Γ = V S , N = V − S , and K = V − A . n n n 1 n n 1 n ∈ ∈ ∈ i I i I i I ∈ (cid:111) As N is normal in G with abelian quotient and χ(x) = 1 for each x V − S , we ≤ ≤ n 1 n have that N (cid:2) H(I) (cid:2) Γ. Hence M(N) M(H(I)) M(Γ). Lemmas 3.3 and 3.5 give that n n (cid:3) M(Γ) = M(N) = K, which thus proves the claim. We prove an analogue of Proposition 3.4 for ﬁber products of H . n ≤ Proposition 3.7. Fix a ﬁnite set I, put G = H(I), and let H G. If HM(G) = G and ∈ n π (H) = H for all i I, then H = G. i n Proof. Let N and K be as in (3.8). By Lemma 3.6, M∼(G) = M(N) = K. The assumption ∩ HK = HM(G) = G implies an isomorphism G/K = H/H K. In par∼ticular, ∼if we put ∩ F H = H N, then as N is a normal in G, H is normal in H with H/H = G/N = and 0 0 0 2 we have N = H K. 0 Since π (H) = π (G) and [π (G) : π (N)] = 2, we conclude that π (H ) = π (N). i i i i i 0 i (cid:3) Therefore, Proposition 3.4 gives that H = N, from which we conclude that H = G. 0 I We shall need the following notati(cid:89)on for later(cid:89)us(cid:89)e: For a ﬁnite family = (I ) ∈ of j j J nonempty ﬁnite sets we denote I ≤ H := H(Ij) H . n n n ∈ ∈ ∈ j J j J i Ij ≥ Let F be a ﬁeld of characteristic diﬀerent from 2, let n > m 2 and r > 0, and let ∈ ∈ K = F (A , . . . , A ). Let f F [T ] be monic of degree n and h , . . . , h F [T ] of degree 0 m 0 (cid:88) 1 k less than n and pairwise distinct. Starting from m ∈ f(T ) = f (T ) + (cid:81) A T i K[T ], 0 i i=0 we deﬁne f (T ) = f(T ) + h (T ) and f (T ) = k f (T ). Note that i i h i=1 i f (0) = f (0) + A + h (0). i 0 0 i − The goal of this section is to determine the Galois group of f ( T 2). h

text: CORRELATIONS

text: IN

text: FUNCTION

text: FIELDS

text: Lemma

text: For

text: n

text: and

text: any

text: ﬁnite

text: set

text: I,

text: M(

text: V

text: S

text: M(H(I))

text: M(

text: V

text: S

text: V

text: A

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: i

text: I

text: i

text: I

text: i

text: I

text: Proof.

text: Let

text: Γ

text: V

text: S

text: N

text: V

text: S

text: and

text: K

text: V

text: A

text: n

text: n

text: n

text: n

text: n

text: n

text: i

text: I

text: i

text: I

text: i

text: I

text: As

text: N

text: is

text: normal

text: in

text: G

text: with

text: abelian

text: quotient

text: and

text: χ(x)

text: for

text: each

text: x

text: V

text: S

text: we

text: n

text: n

text: have

text: that

text: N

text: H(I)

text: Γ.

text: Hence

text: M(N)

text: M(H(I))

text: M(Γ).

text: Lemmas

text: and

text: give

text: that

text: n

text: n

text: M(Γ)

text: M(N)

text: K,

text: which

text: thus

text: proves

text: the

text: claim.

text: We

text: prove

text: an

text: analogue

text: of

text: Proposition

text: for

text: ﬁber

text: products

text: of

text: H

text: n

text: Proposition

text: Fix

text: a

text: ﬁnite

text: set

text: I,

text: put

text: G

text: H(I),

text: and

text: let

text: H

text: G.

text: If

text: HM(G)

text: G

text: and

text: n

text: π

text: (H)

text: H

text: for

text: all

text: i

text: I,

text: then

text: H

text: G.

text: i

text: n

text: Proof.

text: Let

text: N

text: and

text: K

text: be

text: as

text: in

text: By

text: Lemma

text: M∼(G)

text: M(N)

text: K.

text: The

text: assumption

text: HK

text: HM(G)

text: G

text: implies

text: an

text: isomorphism

text: G/K

text: H/H

text: K.

text: In

text: par∼ticular,

text: ∼if

text: we

text: put

text: F

text: H

text: H

text: N,

text: then

text: as

text: N

text: is

text: a

text: normal

text: in

text: G,

text: H

text: is

text: normal

text: in

text: H

text: with

text: H/H

text: G/N

text: and

text: we

text: have

text: N

text: H

text: K.

text: Since

text: π

text: (H)

text: π

text: (G)

text: and

text: [π

text: (G)

text: π

text: (N)]

text: we

text: conclude

text: that

text: π

text: (H

text: π

text: (N).

text: i

text: i

text: i

text: i

text: i

text: i

text: Therefore,

text: Proposition

text: gives

text: that

text: H

text: N,

text: from

text: which

text: we

text: conclude

text: that

text: H

text: G.

text: I

text: We

text: shall

text: need

text: the

text: following

text: for

text: For

text: a

text: ﬁnite

text: family

text: (I

text: of

text: j

text: j

text: J

text: nonempty

text: ﬁnite

text: sets

text: we

text: denote

text: I

text: H

text: H(Ij)

text: H

text: n

text: n

text: n

text: j

text: J

text: j

text: J

text: i

text: Ij

text: Let

text: F

text: be

text: a

text: ﬁeld

text: of

text: characteristic

text: diﬀerent

text: from

text: let

text: n

text: m

text: and

text: r

text: and

text: let

text: K

text: F

text: (A

text: A

text: Let

text: f

text: F

text: [T

text: be

text: monic

text: of

text: degree

text: n

text: and

text: h

text: h

text: F

text: [T

text: of

text: degree

text: m

text: k

text: less

text: than

text: n

text: and

text: pairwise

text: distinct.

text: Starting

text: from

text: m

text: f(T

text: f

text: (T

text: A

text: T

text: i

text: K[T

text: i

text: we

text: deﬁne

text: f

text: (T

text: f(T

text: h

text: (T

text: and

text: f

text: (T

text: k

text: f

text: (T

text: Note

text: that

text: i

text: i

text: h

text: i

text: f

text: f

text: A

text: h

text: i

text: i

text: The

text: goal

text: of

text: this

text: section

text: is

text: to

text: determine

text: the

text: Galois

text: group

text: of

text: f

text: T

text: h


text: 28 LIOR BARY-SOROKER AND ARNO FEHM − ± ± If ω , . . . , ω is an enumeration of the roots of f , then the roots of f ( T 2) are ρ , . . . , ρ , 1 n − i i 1 | →n where ρ is a ﬁxed square root of ω . The Galois action induces embeddings Gal(f (T ) K) (cid:44) i − | → i (cid:111) i S and Θ : Gal(f ( T 2) K) (cid:44) H = V S as in (3.8). n i i n n Proposition 3.8. For each i, the action on the roots of f (T ) and Θ induce isomorphisms ∼ i ∼ i (cid:112) Gal(f (T )(cid:112)|K) = S and Gal(f (−T 2)|K) = H = V (cid:111) S . i n i n n (cid:111) (cid:111) Here, the extensions of K corresponding to the subgroups V A and V − S of H are n n 1 n n K( discr(f )) and K( f (0)), respectively. i i (cid:112) (cid:111) Proof. See [BBF17, Proposition 4.6] for the claimed isomorphisms. The subgroup V (cid:111) (cid:112) A of V S corresponds to the same extension as the subgroup A of S , namely to n n (cid:112) − · · · n n· · · K( discr(f )), cf. [Mil14, Corollary 4.2]. Since f (0) = ( 1)nω ω = (ρ ρ )2, the i − (cid:112) i 1 n 1 n splitting ﬁeld of f ( T 2) contains K( f (0)). Since the elements of V − act on the roots of − i i · · · n 1 (cid:111) f ( T 2) by an even number of sign changes, f (0) = ρ ρ is invariant under V − S , i i 1 n n 1 n which implies that K( f (0)) is the ﬁeld extension of K corresponding to the subgroup (cid:111) i (cid:3) V − S of H . n 1 n n As before we have the maps − | − | Ψ : Gal(f ( T 2) K) resi (cid:47)(cid:47)Gal(f ( T 2) K) Θi (cid:47)(cid:47)H , i = 1, . . . , k, i h i n which induce an embedding − | → (3.9) Ψ: Gal((cid:81)f ( T 2) K) Hk. h n → Denote by H the image of Ψ, G = k H , and π : G H the projection onto the i=1 n i n i-th factor. Proposition 3.8 implies that (3.10) π (H) = π (G) = H , for all i = 1, . . . , k. i i n Let { } (3.11) J = h (0) : i = 1, . . . , k , i { } ∈ (3.12) I = i : h (0) = a , for a J, a i I (3.13) = (I ) ∈ . h a a J × Recall that a family (x , . . . , x ) of elements of K is square-independent if the residues × × F 1 s in K /K 2 are -independent. By√Kumme√r theory, this is equivalent to 2 ∼ | F Gal(K( x , . . . , x ) K) = ( )s. 1 s 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: LIOR

text: BARY-SOROKER

text: AND

text: ARNO

text: FEHM

text: If

text: ω

text: ω

text: is

text: an

text: enumeration

text: of

text: the

text: roots

text: of

text: f

text: then

text: the

text: roots

text: of

text: f

text: T

text: are

text: ρ

text: ρ

text: n

text: i

text: i

text: →n

text: where

text: ρ

text: is

text: a

text: ﬁxed

text: square

text: root

text: of

text: ω

text: The

text: Galois

text: action

text: induces

text: embeddings

text: Gal(f

text: (T

text: K)

text: i

text: i

text: i

text: S

text: and

text: Θ

text: Gal(f

text: T

text: K)

text: H

text: V

text: S

text: as

text: in

text: n

text: i

text: i

text: n

text: n

text: Proposition

text: For

text: each

text: i,

text: the

text: action

text: on

text: the

text: roots

text: of

text: f

text: (T

text: and

text: Θ

text: induce

text: isomorphisms

text: i

text: i

text: Gal(f

text: (T

text: S

text: and

text: Gal(f

text: (−T

text: H

text: V

text: S

text: i

text: n

text: i

text: n

text: n

text: Here,

text: the

text: extensions

text: of

text: K

text: corresponding

text: to

text: the

text: subgroups

text: V

text: A

text: and

text: V

text: S

text: of

text: H

text: are

text: n

text: n

text: n

text: n

text: K(

text: discr(f

text: and

text: K(

text: f

text: respectively.

text: i

text: i

text: Proof.

text: See

text: Proposition

text: for

text: the

text: claimed

text: isomorphisms.

text: The

text: subgroup

text: V

text: A

text: of

text: V

text: S

text: corresponds

text: to

text: the

text: same

text: extension

text: as

text: the

text: subgroup

text: A

text: of

text: S

text: namely

text: to

text: n

text: n

text: n

text: n·

text: K(

text: discr(f

text: cf.

text: Corollary

text: Since

text: f

text: ω

text: (ρ

text: ρ

text: the

text: i

text: i

text: n

text: n

text: splitting

text: ﬁeld

text: of

text: f

text: T

text: contains

text: K(

text: f

text: Since

text: the

text: elements

text: of

text: V

text: act

text: on

text: the

text: roots

text: of

text: i

text: i

text: n

text: f

text: T

text: by

text: an

text: even

text: number

text: of

text: sign

text: changes,

text: f

text: ρ

text: ρ

text: is

text: invariant

text: under

text: V

text: S

text: i

text: i

text: n

text: n

text: n

text: which

text: implies

text: that

text: K(

text: f

text: is

text: the

text: ﬁeld

text: extension

text: of

text: K

text: corresponding

text: to

text: the

text: subgroup

text: i

text: V

text: S

text: of

text: H

text: n

text: n

text: n

text: As

text: before

text: we

text: have

text: the

text: maps

text: Ψ

text: Gal(f

text: T

text: K)

text: resi

text: T

text: K)

text: Θi

text: i

text: k,

text: i

text: h

text: i

text: n

text: which

text: induce

text: an

text: embedding

text: Ψ:

text: T

text: K)

text: Hk.

text: h

text: n

text: Denote

text: by

text: H

text: the

text: image

text: of

text: Ψ,

text: G

text: k

text: H

text: and

text: π

text: G

text: H

text: the

text: projection

text: onto

text: the

text: n

text: i

text: n

text: i-th

text: factor.

text: Proposition

text: implies

text: that

text: π

text: (H)

text: π

text: (G)

text: H

text: for

text: all

text: i

text: k.

text: i

text: i

text: n

text: Let

text: J

text: h

text: i

text: k

text: i

text: I

text: i

text: h

text: a

text: for

text: a

text: J,

text: a

text: i

text: I

text: (I

text: h

text: a

text: a

text: J

text: Recall

text: that

text: a

text: family

text: (x

text: x

text: of

text: elements

text: of

text: K

text: is

text: square-independent

text: if

text: the

text: residues

text: F

text: s

text: in

text: K

text: /K

text: are

text: -independent.

text: By√Kumme√r

text: theory,

text: this

text: is

text: equivalent

text: to

text: F

text: Gal(K(

text: x

text: x

text: K)

text: )s.

text: s

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:81) (cid:81) (cid:81) 30 LIOR BARY-SOROKER AND ARNO FEHM (cid:81) → For general J, if π (cid:48) : ∈ ∈ H ∈ H denotes the projection map, then a ∈ a J (cid:89)i Ia a i I(cid:89)a(cid:48) (cid:89)n ≤ I πa(H) = H(nIa) for all a J by the previous case, hence H a∈J H(nIa) = Hnh. By Lemma 3.3, I (cid:111) M(H h) = M(H(Ia)) = V − A , n n n 1 n (cid:89) ∈ (cid:112) ∈ ∈ I ∩ a J a J i Ia and M(H h) H corresponds to the ﬁeld extension n · K K( discr(f ), i = 1, . . . , k). a i ∈ a J By Lemma 3.9, the family ∪ (f (0) + A + a) ∈ (discr(f )) (cid:81) 0 0 a J i i=1,...,k I I is square-independent, which means that HM(H h) = H h. Therefore, Proposition 3.4, n ≤n I appliedI to the groups (H(nIa))a∈J and the subgroup H Hnh = a∈J H(nIa), gives tha(cid:3)t H = H h. n We are now ready to prove our general result on correlations in short intervals: ≥ ∈ ∗ ≥ − Theorem 3.11. Fix k 1, ψ , . . . , ψ Λ , 1 (cid:15) > 0 and n > 2(cid:15) 1. Then for q an odd ∈ F 1 k ∈ F prime power, f [T ] monic of degree n and h , . . . , h [T ] of degree less than n (cid:42) 0 q (cid:43) 1 k q and pairwise di(cid:89)stinct, k (cid:104) (cid:105) − ψ (f + h ) = ψ , . . . , ψ I + O (q 1/2) i,q i | − | | | 1 k Hnh n,k,ψ i=1 f f0 < f0 (cid:15) I where the implied constant depends only on n, k and ψ , . . . , ψ , and is deﬁned as in 1 k h (3.13). (cid:80) (cid:81) | − | | | Proof. The short interval f f < f (cid:15) is precisely the set of specializations of f = f + − (cid:100) (cid:101) 0 0 A 0 m 1 A T i, with m = (cid:15)n . Setting f = f + h and f = k f , Proposition 3.10 i=0 i A,i A i A,h i=1 A,i gives that − |F − |F I Gal(f ( T 2) (A)) = Gal(f ( T 2) (A)) = H h. A,h q A,h q n In particular, the f are monic, square-free, not divisible by T and pairwise coprime. A,i (cid:3) Therefore, the claim follows from Proposition 3.2. We note that in the case where the ψ depe(cid:89)nd only on cycle type, i.e. ψ (d, 1, 1) = − i i ψ (d, 1, 1) for all d, i (cid:104) (cid:105) k (cid:104) (cid:105) ψ , . . . , ψ I = ψ , 1 k Hnh i Hn i=1 ##LTLine## ##LTLine##

text: LIOR

text: BARY-SOROKER

text: AND

text: ARNO

text: FEHM

text: For

text: general

text: J,

text: if

text: π

text: H

text: H

text: denotes

text: the

text: projection

text: map,

text: then

text: a

text: a

text: J

text: Ia

text: a

text: i

text: I

text: πa(H)

text: H(nIa)

text: for

text: all

text: a

text: J

text: by

text: the

text: previous

text: case,

text: hence

text: H

text: a∈J

text: H(nIa)

text: Hnh.

text: By

text: Lemma

text: I

text: M(H

text: h)

text: M(H(Ia))

text: V

text: A

text: n

text: n

text: n

text: n

text: I

text: a

text: J

text: a

text: J

text: i

text: Ia

text: and

text: M(H

text: h)

text: H

text: corresponds

text: to

text: the

text: ﬁeld

text: extension

text: n

text: K

text: K(

text: discr(f

text: i

text: k).

text: a

text: i

text: a

text: J

text: By

text: Lemma

text: the

text: family

text: (f

text: A

text: a)

text: (discr(f

text: a

text: J

text: i

text: I

text: I

text: is

text: square-independent,

text: which

text: means

text: that

text: HM(H

text: h)

text: H

text: h.

text: Therefore,

text: Proposition

text: n

text: ≤n

text: I

text: appliedI

text: to

text: the

text: groups

text: (H(nIa))a∈J

text: and

text: the

text: subgroup

text: H

text: Hnh

text: a∈J

text: H(nIa),

text: gives

text: H

text: H

text: h.

text: n

text: We

text: are

text: now

text: ready

text: to

text: prove

text: our

text: general

text: result

text: on

text: correlations

text: in

text: short

text: intervals:

text: Theorem

text: Fix

text: k

text: ψ

text: ψ

text: Λ

text: and

text: n

text: Then

text: for

text: q

text: an

text: odd

text: F

text: k

text: F

text: prime

text: power,

text: f

text: [T

text: monic

text: of

text: degree

text: n

text: and

text: h

text: h

text: [T

text: of

text: degree

text: less

text: than

text: n

text: q

text: k

text: q

text: and

text: pairwise

text: k

text: ψ

text: (f

text: h

text: ψ

text: ψ

text: I

text: O

text: (q

text: i,q

text: i

text: k

text: Hnh

text: n,k,ψ

text: f

text: I

text: where

text: the

text: implied

text: constant

text: depends

text: only

text: on

text: n,

text: k

text: and

text: ψ

text: ψ

text: and

text: is

text: deﬁned

text: as

text: in

text: k

text: h

text: Proof.

text: The

text: short

text: interval

text: f

text: f

text: f

text: is

text: precisely

text: the

text: set

text: of

text: specializations

text: of

text: f

text: f

text: A

text: m

text: A

text: T

text: i,

text: with

text: m

text: Setting

text: f

text: f

text: h

text: and

text: f

text: k

text: f

text: Proposition

text: i

text: A,i

text: A

text: i

text: A,h

text: A,i

text: gives

text: that

text: |F

text: |F

text: I

text: Gal(f

text: T

text: (A))

text: Gal(f

text: T

text: (A))

text: H

text: h.

text: A,h

text: q

text: A,h

text: q

text: n

text: In

text: particular,

text: the

text: f

text: are

text: monic,

text: square-free,

text: not

text: divisible

text: by

text: T

text: and

text: pairwise

text: coprime.

text: A,i

text: Therefore,

text: the

text: claim

text: follows

text: from

text: Proposition

text: We

text: note

text: that

text: in

text: the

text: case

text: where

text: the

text: ψ

text: only

text: on

text: cycle

text: type,

text: i.e.

text: ψ

text: (d,

text: i

text: i

text: ψ

text: (d,

text: for

text: all

text: d,

text: i

text: k

text: ψ

text: ψ

text: I

text: ψ

text: k

text: Hnh

text: i

text: Hn

text: ##LTLine##

text: ##LTLine##