text: 2 general perturbation methods have been developed. Such methods are similar to those employed for the Maxwell equations and are typically based on the multipolar expansion [4], involving several approximations and assumptions: (a) Linearization of the ﬁeld equations. (b) Far ﬁeld approximation. (c) No retardation in the source. (d) Symmetry of plane wave or spherical wave. (e) Transverse-traceless (TT) gauge. These approximations have of course a limited validity and in certain cases are not adequate. In our recent work [5] we proposed a method which requires only the linearity assumption (a) and relaxes the others, and we used it to compute the emission of a Weber bar. In practice, we computed exactly the retarded potential integrals, working in the harmonic gauge, without imposing the TT gauge. We found an explicit expression for the metric tensor, and not only for the irradiated energy-momentum, and stressed the importance of appropriate boundary conditions imposed at the ends of the bar. We compared the response to the ﬁeld of a “generalized” two-rings detector with the response computed in the (local) TT gauge, and found that they are the same, because the longitudinal components of the metric which are present in the harmonic gauge have no physical eﬀects on the detector. In this work we apply the general method of [5] to a problem which is relatively new and physically interesting, namely the computation of the gravitational emission of a plasma wave. The wave is modelled through an analogy with an oscillating string, whose length is larger than the wavelength of the gravitational emission. We discuss the validity and the limitations of this analogy, considering the concrete example of an Alfven wave in the solar corona and we obtain analytical and numerical results for the far ﬁeld and the near ﬁeld (the latter to be published in a separate paper). The crucial ﬁrst step of our method is the explicit construction of a full conserved T µν tensor (Sect. ). In the usual solution procedure of the linearized Einstein equations for far-ﬁeld waves, the only relevant components of the source are the mass-energy density T 00 and its multipoles [1, 4]. This is true, however, only if all the approximations listed above are valid. If we are instead interested, for instance, into the near ﬁeld or into sources with internal interference, we will need all the components of T µν. Furthermore, we would like to stress that even if one works in the multipole approximation and assigns the T 00 component in such a way that the conservation of total momentum and angular

text: general

text: perturbation

text: methods

text: have

text: been

text: developed.

text: Such

text: methods

text: are

text: similar

text: to

text: those

text: employed

text: for

text: the

text: Maxwell

text: equations

text: and

text: are

text: typically

text: based

text: on

text: the

text: multipolar

text: expansion

text: involving

text: several

text: approximations

text: and

text: assumptions:

text: (a)

text: Linearization

text: of

text: the

text: ﬁeld

text: equations.

text: (b)

text: Far

text: ﬁeld

text: approximation.

text: (c)

text: No

text: retardation

text: in

text: the

text: source.

text: (d)

text: Symmetry

text: of

text: plane

text: wave

text: or

text: spherical

text: wave.

text: (e)

text: Transverse-traceless

text: (TT)

text: gauge.

text: These

text: approximations

text: have

text: of

text: course

text: a

text: limited

text: validity

text: and

text: in

text: certain

text: cases

text: are

text: not

text: adequate.

text: In

text: our

text: recent

text: work

text: we

text: proposed

text: a

text: method

text: which

text: requires

text: only

text: the

text: linearity

text: assumption

text: (a)

text: and

text: relaxes

text: the

text: others,

text: and

text: we

text: used

text: it

text: to

text: compute

text: the

text: emission

text: of

text: a

text: Weber

text: bar.

text: In

text: practice,

text: we

text: computed

text: exactly

text: the

text: retarded

text: potential

text: integrals,

text: working

text: in

text: the

text: harmonic

text: gauge,

text: without

text: imposing

text: the

text: TT

text: gauge.

text: We

text: found

text: an

text: explicit

text: expression

text: for

text: the

text: metric

text: tensor,

text: and

text: not

text: only

text: for

text: the

text: irradiated

text: energy-momentum,

text: and

text: stressed

text: the

text: importance

text: of

text: appropriate

text: boundary

text: conditions

text: imposed

text: at

text: the

text: ends

text: of

text: the

text: bar.

text: We

text: compared

text: the

text: response

text: to

text: the

text: ﬁeld

text: of

text: a

text: “generalized”

text: two-rings

text: detector

text: with

text: the

text: response

text: computed

text: in

text: the

text: (local)

text: TT

text: gauge,

text: and

text: found

text: that

text: they

text: are

text: the

text: same,

text: because

text: the

text: longitudinal

text: components

text: of

text: the

text: metric

text: which

text: are

text: present

text: in

text: the

text: harmonic

text: gauge

text: have

text: no

text: physical

text: eﬀects

text: on

text: the

text: detector.

text: In

text: this

text: work

text: we

text: apply

text: the

text: general

text: method

text: of

text: to

text: a

text: problem

text: which

text: is

text: relatively

text: new

text: and

text: physically

text: interesting,

text: namely

text: the

text: computation

text: of

text: the

text: gravitational

text: emission

text: of

text: a

text: plasma

text: wave.

text: The

text: wave

text: is

text: modelled

text: through

text: an

text: analogy

text: with

text: an

text: oscillating

text: string,

text: whose

text: length

text: is

text: larger

text: than

text: the

text: wavelength

text: of

text: the

text: gravitational

text: emission.

text: We

text: discuss

text: the

text: validity

text: and

text: the

text: limitations

text: of

text: this

text: analogy,

text: considering

text: the

text: concrete

text: example

text: of

text: an

text: Alfven

text: wave

text: in

text: the

text: solar

text: corona

text: and

text: we

text: obtain

text: analytical

text: and

text: numerical

text: results

text: for

text: the

text: far

text: ﬁeld

text: and

text: the

text: near

text: ﬁeld

text: (the

text: latter

text: to

text: be

text: published

text: in

text: a

text: separate

text: paper).

text: The

text: crucial

text: ﬁrst

text: step

text: of

text: our

text: method

text: is

text: the

text: explicit

text: construction

text: of

text: a

text: full

text: conserved

text: T

text: µν

text: tensor

text: (Sect.

text: In

text: the

text: usual

text: solution

text: procedure

text: of

text: the

text: linearized

text: Einstein

text: equations

text: for

text: far-ﬁeld

text: waves,

text: the

text: only

text: relevant

text: components

text: of

text: the

text: source

text: are

text: the

text: mass-energy

text: density

text: T

text: and

text: its

text: multipoles

text: This

text: is

text: true,

text: however,

text: only

text: if

text: all

text: the

text: approximations

text: listed

text: above

text: are

text: valid.

text: If

text: we

text: are

text: instead

text: interested,

text: for

text: instance,

text: into

text: the

text: near

text: ﬁeld

text: or

text: into

text: sources

text: with

text: internal

text: interference,

text: we

text: will

text: need

text: all

text: the

text: components

text: of

text: T

text: µν.

text: Furthermore,

text: we

text: would

text: like

text: to

text: stress

text: that

text: even

text: if

text: one

text: works

text: in

text: the

text: multipole

text: approximation

text: and

text: assigns

text: the

text: T

text: component

text: in

text: such

text: a

text: way

text: that

text: the

text: conservation

text: of

text: total

text: momentum

text: and

text: angular


text: 12 The ﬁrst term, however, represents a static ﬁeld, and will be ignored. The oscillation velocity of the mass m is only in the x-direction, and is the time derivative of the displacement u(z, t), i (cid:48) computed at the coordinate z where the mass m is at time t : i i i (cid:18) (cid:19) (cid:48) ∂ − 1| − (cid:48) | v (z , t ) = u z , t r x (t ) (30) i,x i i i i i ∂t c Taking into account eq. (1), where for the internal points (unlike for the boundary) we can take → immediately the limit ε 0, we obtain to the desired approximation (cid:18) (cid:19) (cid:16) (cid:17) (cid:48) (cid:39) − jπz ω i v (z , t ) ωu cos sin ωt + r + k z (31) i,x i i 0 z i 2L c where k = ωn is the wavenumber vector for the gravitational wave. Note that since the string c can be long compared to a wavelength, the phase k z cannot be neglected. z i The total ﬁeld is given by a sum over mass points, which is converted to an integral with the → substitution m σdz = Mstr dz. The integral to be evaluated is i 2L (cid:90) (cid:18) (cid:19) (cid:18) (cid:19) L jπz jπ jπL cos(k L) z I = dz cos sin(Φ + k z) =2 sin − sin Φ (32) str 2L z 2 (jπ)2 4L2k2 − z L where, here and in the following − Φ = ωt kr (33) The resulting metric element will be denoted with the subscript “S”, to distinguish it from the boundary contribution: (cid:18) (cid:19) −8GM jπ jπ cos(k L) h¯00(r, t) = str sin n ωu − z sin Φ (34) S c3r 2 x 0 (jπ)2 4L2k2 z ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: The

text: ﬁrst

text: term,

text: however,

text: represents

text: a

text: static

text: ﬁeld,

text: and

text: will

text: be

text: ignored.

text: The

text: oscillation

text: velocity

text: of

text: the

text: mass

text: m

text: is

text: only

text: in

text: the

text: x-direction,

text: and

text: is

text: the

text: time

text: derivative

text: of

text: the

text: displacement

text: u(z,

text: t),

text: i

text: computed

text: at

text: the

text: coordinate

text: z

text: where

text: the

text: mass

text: m

text: is

text: at

text: time

text: t

text: i

text: i

text: i

label: equation

text: v

text: (z

text: t

text: u

text: z

text: t

text: r

text: x

text: (t

text: i,x

text: i

text: i

text: i

text: i

text: i

text: ∂t

text: c

text: Taking

text: into

text: account

text: eq.

text: where

text: for

text: the

text: internal

text: points

text: (unlike

text: for

text: the

text: boundary)

text: we

text: can

text: take

text: immediately

text: the

text: limit

text: ε

text: we

text: obtain

text: to

text: the

text: desired

text: approximation

text: jπz

text: ω

text: i

text: v

text: (z

text: t

text: ωu

text: cos

text: sin

text: ωt

text: r

text: k

text: z

text: i,x

text: i

text: i

text: z

text: i

text: c

text: where

text: k

text: ωn

text: is

text: the

text: wavenumber

text: vector

text: for

text: the

text: gravitational

text: wave.

text: Note

text: that

text: since

text: the

text: string

text: c

text: can

text: be

text: long

text: compared

text: to

text: a

text: wavelength,

text: the

text: phase

text: k

text: z

text: cannot

text: be

text: neglected.

text: z

text: i

text: The

text: total

text: ﬁeld

text: is

text: given

text: by

text: a

text: sum

text: over

text: mass

text: points,

text: which

text: is

text: converted

text: to

text: an

text: integral

text: with

text: the

text: substitution

text: m

text: σdz

text: Mstr

text: dz.

text: The

text: integral

text: to

text: be

text: evaluated

text: is

text: i

text: L

text: jπz

text: jπ

text: jπL

text: cos(k

text: L)

text: z

text: I

text: dz

text: cos

text: sin(Φ

text: k

text: z)

text: sin

text: sin

text: Φ

text: str

text: z

text: z

text: L

text: where,

text: here

text: and

text: in

text: the

text: following

text: Φ

text: ωt

text: kr

text: The

text: resulting

text: metric

text: element

text: will

text: be

text: denoted

text: with

text: the

text: subscript

text: “S”,

text: to

text: distinguish

text: it

text: from

text: the

text: boundary

text: contribution:

text: jπ

text: jπ

text: cos(k

text: L)

text: t)

text: str

text: sin

text: n

text: ωu

text: z

text: sin

text: Φ

text: S

text: x

text: z

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: 17 in u. The element T 03, which for the string is a pure boundary term containing the end-masses, is zero for the plasma to ﬁrst order. The spatial components T ij, which in the case of the string account for the elastic forces, are given for the plasma by the electromagnetic stress tensor. The only time-dependent element which does not vanish to ﬁrst order is T 13, because B = 0 and when y B2 and B2 appear, one should remember that B is of ﬁrst order in u (compare (47)) and B is x z x z constant in space and time, in the bulk of the wave. In conclusion, we can write          ρc2 ρc∂u 0 0   ∂t     ρc∂u 0 0 BxBz  T µν (x, t) = ∂t µ0 (42) pl 0 0 0 0 0 BxBz 0 0 µ0 The conservation equation ∂ T µν = 0 with ν = 2 is trivially veriﬁed, the one with ν = 3 µ pl gives ∂ (B B /µ ) = 0, which is also satisﬁed in our hypotheses. For ν = 0 we obtain a mass x x z 0 conservation equation. For ν = 1 we obtain the equation of motion of a small plasma volume under the eﬀect of the restoring force, according to the original Alfven equation ∂2u −B ∂B z x ρ = (43) ∂t2 µ ∂z 0 This equation can also be derived from the general equations of magneto-hydrodynamics, under suitable simplifying assumptions [15]. Let us make the following ansatz for the instantaneous displacement u(t, z) of the plasma and for the x-component of the magnetic ﬁeld: u(z, t) = u cos(k z) cos ωt (44) 0 j B (z, t) = B sin(k z) cos ωt (45) x 0x j ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: in

text: u.

text: The

text: element

text: T

text: which

text: for

text: the

text: string

text: is

text: a

text: pure

text: boundary

text: term

text: containing

text: the

text: end-masses,

text: is

text: zero

text: for

text: the

text: plasma

text: to

text: ﬁrst

text: order.

text: The

text: spatial

text: components

text: T

text: ij,

text: which

text: in

text: the

text: case

text: of

text: the

text: string

text: account

text: for

text: the

text: elastic

text: forces,

text: are

text: given

text: for

text: the

text: plasma

text: by

text: the

text: electromagnetic

text: stress

text: tensor.

text: The

text: only

text: time-dependent

text: element

text: which

text: does

text: not

text: vanish

text: to

text: ﬁrst

text: order

text: is

text: T

text: because

text: B

text: and

text: when

text: y

text: and

text: appear,

text: one

text: should

text: remember

text: that

text: B

text: is

text: of

text: ﬁrst

text: order

text: in

text: u

text: (compare

text: and

text: B

text: is

text: x

text: z

text: x

text: z

text: constant

text: in

text: space

text: and

text: time,

text: in

text: the

text: bulk

text: of

text: the

text: wave.

text: In

text: conclusion,

text: we

text: can

text: write

text: ρc∂u

text: ∂t

text: ρc∂u

text: BxBz

text: T

text: µν

text: (x,

text: t)

text: ∂t

text: µ0

text: pl

text: BxBz

text: µ0

text: The

text: conservation

text: equation

text: T

text: µν

text: with

text: ν

text: is

text: trivially

text: veriﬁed,

text: the

text: one

text: with

text: ν

text: µ

text: pl

text: gives

text: (B

text: B

text: /µ

text: which

text: is

text: also

text: satisﬁed

text: in

text: our

text: hypotheses.

text: For

text: ν

text: we

text: obtain

text: a

text: mass

text: x

text: x

text: z

text: conservation

text: equation.

text: For

text: ν

text: we

text: obtain

text: the

text: equation

text: of

text: motion

text: of

text: a

text: small

text: plasma

text: volume

text: under

text: the

text: eﬀect

text: of

text: the

text: restoring

text: force,

text: according

text: to

text: the

text: original

text: Alfven

text: equation

text: −B

text: ∂B

text: z

text: x

text: ρ

text: µ

text: ∂z

text: This

text: equation

text: can

text: also

text: be

text: derived

text: from

text: the

text: general

text: equations

text: of

text: magneto-hydrodynamics,

text: under

text: suitable

text: simplifying

text: assumptions

text: Let

text: us

text: make

text: the

text: following

text: ansatz

text: for

text: the

text: instantaneous

text: displacement

text: u(t,

text: z)

text: of

text: the

text: plasma

text: and

text: for

text: the

text: x-component

text: of

text: the

text: magnetic

text: ﬁeld:

text: u(z,

text: t)

text: u

text: cos(k

text: z)

text: cos

text: ωt

text: j

text: B

text: (z,

text: t)

text: B

text: sin(k

text: z)

text: cos

text: ωt

text: x

text: j

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 19 1.´10-34 A 5.´10-35 B 5.0´106 1.0´107 1.5´107 2.0´107 2.5´107 z -5.´10-35 FIG. 4: A: Numerical evaluation of the metric component h¯13 generated by a plasma wave along an arc at · distance r = 2.5 107 m from the origin, as a function of the z-coordinate (with y = 0). The parameters of the wave are given in Table 1. The phase of the time-dependent factor sin Φ is equal to 5π/12. The component h¯13 is the retarded integral of T13, which does not signiﬁcantly depend on the square of the amplitude u. The result of the analytical computation (eq. (49)) is not shown, because it is almost superposed to the numerical plot; there is a diﬀerence, however, which is shown ampliﬁed by a factor 10 (B-curve). This diﬀerence can be attributed to a ﬁeld component proportional to 1/r2, not present in the standard analytical 1/r far-ﬁeld solution. Note that while for a Weber bar ﬁeld components of this kind are negligible at a distance larger than one wavelength, in the present case they persist even at a distance of 80 wavelengths. In the last expression we have replaced the product χζ by the cross-section A, and eliminated − A and the magnetic ﬁeld thanks to eq. (47). Remember that Φ = (ωt kr). Like for the string, the ﬁeld amplitude is independent from the frequency and proportional to M/L. A comparison between this formula and the result of a numerical evaluation of h¯13 is given in Fig. 4. V. CONCLUSIONS AND OUTLOOK This section collects the main open questions, to be addressed in future work. They include both physical and mathematical aspects, many of which have been already pointed out in the main text. 1. Correspondence of our general metric with the metric computed from the quadrupole current (cid:28) formula, in the limit of a string much shorter than the gravitational wavelength (L λ). This correspondence must take into account the mass renormalization caused by the con- (cid:0) (cid:1) straint, which leads to a diminution of the eﬀective mass by a factor of the order of 103. − With our choice of coordinates, the angular momentum density jjk = 1 xjT 0k xkT 0j has c non-zero j13 components for the vibrating string. Gravitational radiation can be expressed ##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##

font: default

text: A

text: B

text: z

text: FIG.

text: A:

label: caption

text: Numerical

text: evaluation

text: of

text: the

text: metric

text: component

text: generated

text: by

text: a

text: plasma

text: wave

text: along

text: an

text: arc

text: at

text: distance

text: r

text: m

text: from

text: the

text: origin,

text: as

text: a

text: function

text: of

text: the

text: z-coordinate

text: (with

text: y

text: The

text: parameters

text: of

text: the

text: wave

text: are

text: given

text: in

text: Table

text: The

text: phase

text: of

text: the

text: time-dependent

text: factor

text: sin

text: Φ

text: is

text: equal

text: to

text: The

text: component

text: is

text: the

text: retarded

text: integral

text: of

text: which

text: does

text: not

text: signiﬁcantly

text: depend

text: on

text: the

text: square

text: of

text: the

text: amplitude

text: u.

text: The

text: result

text: of

text: the

text: analytical

text: computation

text: (eq.

text: is

text: not

text: shown,

text: because

text: it

text: is

text: almost

text: superposed

text: to

text: the

text: numerical

text: plot;

text: there

text: is

text: a

text: diﬀerence,

text: however,

text: which

text: is

text: shown

text: ampliﬁed

text: by

text: a

text: factor

text: (B-curve).

text: This

text: diﬀerence

text: can

text: be

text: attributed

text: to

text: a

text: ﬁeld

text: component

text: proportional

text: to

text: not

text: present

text: in

text: the

text: standard

text: analytical

text: far-ﬁeld

text: solution.

text: Note

text: that

text: while

text: for

text: a

text: Weber

text: bar

text: ﬁeld

text: components

text: of

text: this

text: kind

text: are

text: negligible

text: at

text: a

text: distance

text: larger

text: than

text: one

text: wavelength,

text: in

text: the

text: present

text: case

text: they

text: persist

text: even

text: at

text: a

text: distance

text: of

text: wavelengths.

text: In

text: the

text: last

text: expression

text: we

text: have

text: replaced

text: the

text: product

text: χζ

text: by

text: the

text: cross-section

text: A,

text: and

text: eliminated

text: A

text: and

text: the

text: magnetic

text: ﬁeld

text: thanks

text: to

text: eq.

text: Remember

text: that

text: Φ

text: (ωt

text: kr).

text: Like

text: for

text: the

text: string,

text: the

text: ﬁeld

text: amplitude

text: is

text: independent

text: from

text: the

text: frequency

text: and

text: proportional

text: to

text: M/L.

text: A

text: comparison

text: between

text: this

text: formula

text: and

text: the

text: result

text: of

text: a

text: numerical

text: evaluation

text: of

text: is

text: given

text: in

text: Fig.

text: V.

text: CONCLUSIONS

text: AND

text: OUTLOOK

text: This

text: section

text: collects

text: the

text: main

text: open

text: questions,

text: to

text: be

text: addressed

text: in

text: future

text: work.

text: They

text: include

text: both

text: physical

text: and

text: mathematical

text: aspects,

text: many

text: of

text: which

text: have

text: been

text: already

text: pointed

text: out

text: in

text: the

text: main

text: text.

text: Correspondence

text: of

text: our

text: general

text: metric

text: with

text: the

text: metric

text: computed

text: from

text: the

text: quadrupole

text: current

text: formula,

text: in

text: the

text: limit

text: of

text: a

text: string

text: much

text: shorter

text: than

text: the

text: gravitational

text: wavelength

text: (L

text: λ).

text: This

text: correspondence

text: must

text: take

text: into

text: account

text: the

text: mass

text: renormalization

text: caused

text: by

text: the

text: con-

text: straint,

text: which

text: leads

text: to

text: a

text: diminution

text: of

text: the

text: eﬀective

text: mass

text: by

text: a

text: factor

text: of

text: the

text: order

text: of

text: With

text: our

text: choice

text: of

text: coordinates,

text: the

text: angular

text: momentum

text: density

text: jjk

text: xjT

text: xkT

text: has

text: c

text: non-zero

text: components

text: for

text: the

text: vibrating

text: string.

text: Gravitational

text: radiation

text: can

text: be

text: expressed

text: ##LTFigure##

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: ##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: ##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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 where ρ is straightforward: S − − − ρ (x, t) = σδ(x u)θ [(L + z w )(L z + w )] (7) S l r and σ is the mass for unit length of the string. (Remember the conventions (3) and that a δ(y) ∇ · is always understood.) The density ρ satisﬁes the continuity equation ∂ρS + (ρ v) = 0. The S ∂t S boundary density ρ will be determined by imposing the conservation conditions ∂ T µν = 0. B µ The deﬁnition of the elements T 31 and T 33 is based on standard elasticity theory (see for instance [12]). By analogy with the case of a Weber bar we set − ∂u − − − T 31 (x, t) = T δ(x u)θ [(L + z w )(L z + w )] (8) str l r ∂z and − − − − T 33 (x, t) = T δ(x u)θ [(L + z w )(L z + w )] (9) str l r where T is the tension in the string. The deﬁnition of the element T 30, which is a pure str boundary term, will be given in Sect. . Note that the functions w and w can be omitted when the quantities ρ , T 31 or T 33 are l r S multiplied by quantities of order ε, because, for instance, − − (cid:48) δ(L + z w ) = δ(L + z) w δ (L + z) + O(w2) (10) l l l B. Implementing the conservation equation with ν = 1 Let us now impose the ν = 1 component of the conservation equation ∂ T µν = 0, i.e. µ ∂T 01 ∂T 11 ∂T 21 ∂T 31 + + + = 0 (11) c∂t ∂x ∂y ∂z ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: ρ

text: is

text: straightforward:

text: S

text: ρ

text: (x,

text: t)

text: σδ(x

text: u)θ

text: [(L

text: z

text: w

text: )(L

text: z

text: w

text: S

text: l

text: r

text: and

text: σ

text: is

text: the

text: mass

text: for

text: unit

text: length

text: of

text: the

text: string.

text: (Remember

text: the

text: conventions

text: and

text: that

text: a

text: δ(y)

text: is

text: always

text: understood.)

text: The

text: density

text: ρ

text: satisﬁes

text: the

text: continuity

text: equation

text: ∂ρS

text: (ρ

text: v)

text: The

text: S

text: ∂t

text: S

text: boundary

text: density

text: ρ

text: will

text: be

text: determined

text: by

text: imposing

text: the

text: conservation

text: conditions

text: T

text: µν

text: B

text: µ

text: The

text: deﬁnition

text: of

text: the

text: elements

text: T

text: and

text: T

text: is

text: based

text: on

text: standard

text: elasticity

text: theory

text: (see

text: for

text: instance

text: By

text: analogy

text: with

text: the

text: case

text: of

text: a

text: Weber

text: bar

text: we

text: set

text: ∂u

text: T

text: (x,

text: t)

text: T

text: δ(x

text: u)θ

text: [(L

text: z

text: w

text: )(L

text: z

text: w

text: str

text: l

text: r

text: ∂z

text: and

text: T

text: (x,

text: t)

text: T

text: δ(x

text: u)θ

text: [(L

text: z

text: w

text: )(L

text: z

text: w

text: str

text: l

text: r

text: where

text: T

text: is

text: the

text: tension

text: in

text: the

text: string.

text: The

text: deﬁnition

text: of

text: the

text: element

text: T

text: which

text: is

text: a

text: pure

text: str

text: boundary

text: term,

text: will

text: be

text: given

text: in

text: Sect.

text: Note

text: that

text: the

text: functions

text: w

text: and

text: w

text: can

text: be

text: omitted

text: when

text: the

text: quantities

text: ρ

text: T

text: or

text: T

text: are

text: l

text: r

text: S

text: multiplied

text: by

text: quantities

text: of

text: order

text: ε,

text: because,

text: for

text: instance,

text: δ(L

text: z

text: w

text: δ(L

text: z)

text: w

text: δ

text: (L

text: z)

text: l

text: l

text: l

text: B.

text: Implementing

text: the

text: conservation

text: equation

text: with

text: ν

text: Let

text: us

text: now

text: impose

text: the

text: ν

text: component

text: of

text: the

text: conservation

text: equation

text: T

text: µν

text: i.e.

text: µ

text: ∂T

text: ∂T

text: ∂T

text: ∂T

text: c∂t

text: ∂x

text: ∂y

text: ∂z

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 (cid:90) (cid:90) (cid:26) (cid:27) (cid:89) (cid:2) (cid:3) − N 1 ikδ − − − − − d [q] e(cid:126)i S[q]q 2 = dq q2 exp (cid:126) q2 q2... q2 ... q2 − q2 (8) m k m 2 i 1 (cid:82) m N 1 f k=1 (Remember that actually q = q = 0.) The denominator d [q] e(cid:126)i S[q] is given by the same i f integral, without q2 in front of the exponential. The exponential is completely factorized m (cid:82) (cid:8) (cid:9) and all integrals are simpliﬁed in the ratio, except the one in the variable dq . We obtain (cid:10) (cid:11) m (cid:82) (cid:8) (cid:9) − (cid:126) dq q2 exp ikδq2 q2 = m m − 2(cid:126) m = (9) m dq exp ikδq2 ikδ m 2(cid:126) m In the continuum limit this is divergent. This result is not an anomaly of the continuum limit of our system, because other quantities have ﬁnite averages. For comparison, one can − evaluate the average of exp( kq2 ), which still gives a Gaussian integral; one easily obtains m a ﬁnite result, which vanishes in the continuum limit. On the other hand, if we compute the average of a quantity like q2 / (1 + q2 ), which tends to a ﬁnite value for large q , we obtain m m m a ﬁnite and non-zero result in the continuum limit; to see this, one can exploit the known integral (cid:90) (cid:0) (cid:1) (cid:26) (cid:20)(cid:18) (cid:19) (cid:21)(cid:27) − √ − √ x2 exp i ax2 − 1 i 1√ i dx 2 = πeia 1 + erf + a + π (10) 2 x2 + 1 2 2 a → and consider that the “Error function” erf(x) is ﬁnite for a 0, and tends to -1. In · → our case a = const. δ, and δ 0 in the continuum limit. The normalization factor √ → − at the denominator behaves like √1 , therefore for δ 0 the term 1√ i π gives a ﬁnite δ a contribution and the rest goes to zero. In conclusion, the quantum averages computed with the path integral tell us consistently that each intermediate coordinate q tends to m grow without limit. Quantum-mechanically, the system runs away to large q, far from the (cid:104) (cid:105) classical solution. Actually, the divergence of q2 for M = 0 had been already predicted m from the “explosion” of the ﬂuctuations in eq.s (6), (7). IV. THE “HYPERPLANE” S = 0 IN THE SPACE OF CONFIGURATIONS The crucial question now is: which conﬁgurations in the path integral contribute to this divergent result? Certainly not those near the stationary point of the action, q(t) = 0. We make the following conjecture: The main contributions to the path integral come from ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: N

text: ikδ

text: d

text: [q]

text: S[q]q

text: dq

text: exp

text: m

text: k

text: m

text: i

text: m

text: N

text: f

text: (Remember

text: that

text: actually

text: q

text: q

text: The

text: denominator

text: d

text: [q]

text: S[q]

text: is

text: given

text: by

text: the

text: same

text: i

text: f

text: integral,

text: without

text: in

text: front

text: of

text: the

text: exponential.

text: The

text: exponential

text: is

text: completely

text: factorized

text: m

text: and

text: all

text: integrals

text: are

text: simpliﬁed

text: in

text: the

text: ratio,

text: except

text: the

text: one

text: in

text: the

text: variable

text: dq

text: We

text: obtain

text: m

text: dq

text: exp

text: m

text: m

text: m

text: m

text: dq

text: exp

text: ikδ

text: m

text: m

text: In

text: the

text: continuum

text: limit

text: this

text: is

text: divergent.

text: This

text: result

text: is

text: not

text: an

text: anomaly

text: of

text: the

text: continuum

text: limit

text: of

text: our

text: system,

text: because

text: other

text: quantities

text: have

text: ﬁnite

text: averages.

text: For

text: comparison,

text: one

text: can

text: evaluate

text: the

text: average

text: of

text: exp(

text: which

text: still

text: gives

text: a

text: Gaussian

text: integral;

text: one

text: easily

text: obtains

text: m

text: a

text: ﬁnite

text: result,

text: which

text: vanishes

text: in

text: the

text: continuum

text: limit.

text: On

text: the

text: other

text: hand,

text: if

text: we

text: compute

text: the

text: average

text: of

text: a

text: quantity

text: like

text: which

text: tends

text: to

text: a

text: ﬁnite

text: value

text: for

text: large

text: q

text: we

text: obtain

text: m

text: m

text: m

text: a

text: ﬁnite

text: and

text: non-zero

text: result

text: in

text: the

text: continuum

text: limit;

text: to

text: see

text: this,

text: one

text: can

text: exploit

text: the

text: known

text: integral

text: exp

text: i

text: i

text: 1√

text: i

text: dx

text: πeia

text: erf

text: a

text: π

text: a

text: and

text: consider

text: that

text: the

text: “Error

text: function”

text: erf(x)

text: is

text: ﬁnite

text: for

text: a

text: and

text: tends

text: to

text: In

text: our

text: case

text: a

text: const.

text: δ,

text: and

text: δ

text: in

text: the

text: continuum

text: limit.

text: The

text: normalization

text: factor

text: at

text: the

text: denominator

text: behaves

text: like

text: √1

text: therefore

text: for

text: δ

text: the

text: term

text: 1√

text: i

text: π

text: gives

text: a

text: ﬁnite

text: δ

text: a

text: contribution

text: and

text: the

text: rest

text: goes

text: to

text: zero.

text: In

text: conclusion,

text: the

text: quantum

text: averages

text: computed

text: with

text: the

text: path

text: integral

text: tell

text: us

text: consistently

text: that

text: each

text: intermediate

text: coordinate

text: q

text: tends

text: to

text: m

text: grow

text: without

text: limit.

text: Quantum-mechanically,

text: the

text: system

text: runs

text: away

text: to

text: large

text: q,

text: far

text: from

text: the

text: classical

text: solution.

text: Actually,

text: the

text: divergence

text: of

text: for

text: M

text: had

text: been

text: already

text: predicted

text: m

text: from

text: the

text: “explosion”

text: of

text: the

text: ﬂuctuations

text: in

text: eq.s

text: IV.

text: THE

text: “HYPERPLANE”

text: S

text: IN

text: THE

text: SPACE

text: OF

text: CONFIGURATIONS

text: The

text: crucial

text: question

text: now

text: is:

text: which

text: conﬁgurations

text: in

text: the

text: path

text: integral

text: contribute

text: to

text: this

text: divergent

text: result?

text: Certainly

text: not

text: those

text: near

text: the

text: stationary

text: point

text: of

text: the

text: action,

text: q(t)

text: We

text: make

text: the

text: following

text: conjecture:

text: The

text: main

text: contributions

text: to

text: the

text: path

text: integral

text: come

text: from

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 regions of the functional conﬁguration space in which S is constant, although not stationary, in such a way that these contributions interfere constructively. For most usual dynamical systems these regions with constant action, reminiscent of equipotential surfaces in a force ﬁeld, have lower dimensionality and zero measure, so they cannot really contribute to the path integral. We shall show, however, that in the case of the massless oscillator they are full-dimensional subspaces of the conﬁguration space. As a ﬁrst step towards the general case, we give a characterization of these regions in the zero action case S = 0. The region with zero action in functional space is somewhat easier to visualize, because it is “connected” to the classical trajectory q(t) = 0, i.e. geometrically to the origin of space. We can picture this region as an inﬁnite-dimensional hyperplane through the origin; for all the conﬁgurations q(t) lying on this hyperplane the action is constantly zero, but it changes if we move out of the hyperplane, since only the origin is a stationary (cid:104) (cid:105) point where δS = 0 in any direction. The value of q2 is not limited on the hyperplane. m The functional integration on this hyperplane yields a signiﬁcant contribution to the path integral, because the functional measure of this set is not zero and there is no destructive in- terference between neighboring paths. This contribution is weighed with the overall constant (cid:126) factor eiS0/ = 1 (S = 0), both in the numerator and denominator of (5). Note that if this 0 (cid:104) (cid:105) (cid:104) (cid:105) was the only signiﬁcant contribution to q2 from the total path integral, then q2 would be m m real, while the explicit calculation shows that it is imaginary. This means, if our conjecture has to be consistent, that in the total path integral there are other non-classical contributions (cid:82) (cid:82) (cid:54) (cid:126) (cid:126) of the same kind, proportional to overall factors eiS1/ , eiS2/ . . . (S , S ... = 0), and which 1 2 (cid:126) (cid:126) do not simplify in the ratio d[q]eiS/ q 2/ d[q]eiS/ . It is possible to write some algebraic m relations which must hold among these contributions; this will be done in a forthcoming paper. V. PROOF THAT THE HYPERPLANE S = 0 IS INFINITE-DIMENSIONAL In order to show that the hyperplane S = 0 is inﬁnite-dimensional we ﬁrst need to relax the conditions q = q = 0 on the paths. This may look counterintuitive for the classi- i f (cid:104) (cid:105) cal paths, but is technically allowed in the path integral for the transition element q2 m because the result does not change if we add in the numerator and denominator an inte- (cid:82) (cid:82) (cid:82) (cid:82) gration over q and q . Being the exponential completely factorized, we can just rewrite i f dq dq d [q] e(cid:126)i S[q]q 2 instead of d [q] e(cid:126)i S[q]q 2 in eq. (8), and similarly for the denom- i f m m ##LTLine## ##LTLine##

text: regions

text: of

text: the

text: functional

text: conﬁguration

text: space

text: in

text: which

text: S

text: is

text: constant,

text: although

text: not

text: stationary,

text: in

text: such

text: a

text: way

text: that

text: these

text: contributions

text: interfere

text: constructively.

text: For

text: most

text: usual

text: dynamical

text: systems

text: these

text: regions

text: with

text: constant

text: action,

text: reminiscent

text: of

text: equipotential

text: surfaces

text: in

text: a

text: force

text: ﬁeld,

text: have

text: lower

text: dimensionality

text: and

text: zero

text: measure,

text: so

text: they

text: cannot

text: really

text: contribute

text: to

text: the

text: path

text: integral.

text: We

text: shall

text: show,

text: however,

text: that

text: in

text: the

text: case

text: of

text: the

text: massless

text: oscillator

text: they

text: are

text: full-dimensional

text: subspaces

text: of

text: the

text: conﬁguration

text: space.

text: As

text: a

text: ﬁrst

text: step

text: towards

text: the

text: general

text: case,

text: we

text: give

text: a

text: characterization

text: of

text: these

text: regions

text: in

text: the

text: zero

text: action

text: case

text: S

text: The

text: region

text: with

text: zero

text: action

text: in

text: functional

text: space

text: is

text: somewhat

text: easier

text: to

text: visualize,

text: because

text: it

text: is

text: “connected”

text: to

text: the

text: classical

text: trajectory

text: q(t)

text: i.e.

text: geometrically

text: to

text: the

text: origin

text: of

text: space.

text: We

text: can

text: picture

text: this

text: region

text: as

text: an

text: inﬁnite-dimensional

text: hyperplane

text: through

text: the

text: origin;

text: for

text: all

text: the

text: conﬁgurations

text: q(t)

text: lying

text: on

text: this

text: hyperplane

text: the

text: action

text: is

text: constantly

text: zero,

text: but

text: it

text: changes

text: if

text: we

text: move

text: out

text: of

text: the

text: hyperplane,

text: since

text: only

text: the

text: origin

text: is

text: a

text: stationary

text: point

text: where

text: δS

text: in

text: any

text: direction.

text: The

text: value

text: of

text: is

text: not

text: limited

text: on

text: the

text: hyperplane.

text: m

text: The

text: functional

text: integration

text: on

text: this

text: hyperplane

text: yields

text: a

text: signiﬁcant

text: contribution

text: to

text: the

text: path

text: integral,

text: because

text: the

text: functional

text: measure

text: of

text: this

text: set

text: is

text: not

text: zero

text: and

text: there

text: is

text: no

text: destructive

text: in-

text: terference

text: between

text: neighboring

text: paths.

text: This

text: contribution

text: is

text: weighed

text: with

text: the

text: overall

text: constant

text: factor

text: (S

text: both

text: in

text: the

text: numerator

text: and

text: denominator

text: of

text: Note

text: that

text: if

text: this

text: was

text: the

text: only

text: signiﬁcant

text: contribution

text: to

text: from

text: the

text: total

text: path

text: integral,

text: then

text: would

text: be

text: m

text: m

text: real,

text: while

text: the

text: explicit

text: calculation

text: shows

text: that

text: it

text: is

text: imaginary.

text: This

text: means,

text: if

text: our

text: conjecture

text: has

text: to

text: be

text: consistent,

text: that

text: in

text: the

text: total

text: path

text: integral

text: there

text: are

text: other

text: non-classical

text: contributions

text: of

text: the

text: same

text: kind,

text: proportional

text: to

text: overall

text: factors

text: (S

text: S

text: and

text: which

text: do

text: not

text: simplify

text: in

text: the

text: ratio

text: d[q]eiS/

text: q

text: d[q]eiS/

text: It

text: is

text: possible

text: to

text: write

text: some

text: algebraic

text: m

text: relations

text: which

text: must

text: hold

text: among

text: these

text: contributions;

text: this

text: will

text: be

text: done

text: in

text: a

text: forthcoming

text: paper.

text: V.

text: PROOF

text: THAT

text: THE

text: HYPERPLANE

text: S

text: IS

text: INFINITE-DIMENSIONAL

text: In

text: order

text: to

text: show

text: that

text: the

text: hyperplane

text: S

text: is

text: inﬁnite-dimensional

text: we

text: ﬁrst

text: need

text: to

text: relax

text: the

text: conditions

text: q

text: q

text: on

text: the

text: paths.

text: This

text: may

text: look

text: counterintuitive

text: for

text: the

text: classi-

text: i

text: f

text: cal

text: paths,

text: but

text: is

text: technically

text: allowed

text: in

text: the

text: path

text: integral

text: for

text: the

text: transition

text: element

text: m

text: because

text: the

text: result

text: does

text: not

text: change

text: if

text: we

text: add

text: in

text: the

text: numerator

text: and

text: denominator

text: an

text: inte-

text: gration

text: over

text: q

text: and

text: q

text: Being

text: the

text: exponential

text: completely

text: factorized,

text: we

text: can

text: just

text: rewrite

text: i

text: f

text: dq

text: dq

text: d

text: [q]

text: S[q]q

text: instead

text: of

text: d

text: [q]

text: S[q]q

text: in

text: eq.

text: and

text: similarly

text: for

text: the

text: denom-

text: i

text: f

text: m

text: m

text: ##LTLine##

text: ##LTLine##


text: 17 (cid:80) Remark. In the balanced subspace, the amplitude of any vector on Π and Πcross for all j in boundary − j,j+1 the Hamiltonian (Eq. 18) vanishes. The Hamiltonian then is simply H = 2n 1 Π . j=1 j,j+1 M (cid:105) M (cid:105) For now we take s = 1 and denote by and the Motzkin number M by M . (cid:105) 2n,1 2n 2n,1 2n Take the state φ in the balanced subspace deﬁned by (cid:105) M (cid:105) (cid:105) (cid:105) · · · (cid:105) φ = α + α e + α e + + α − e − (22) 0 2n 1 1 2 2 (M2n 1) (M2n 1)(cid:80) (cid:105) M − | | where e is the ith excited states of H in th(cid:88)e balanced subspace. Clearly, 2n 1 α 2 = 1 and i i=0 i M − (cid:104) | | (cid:105) 2n 1 | | (cid:104) | | (cid:105) (cid:80) φ H φ = αi 2 ei H ei i=1 − { (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) } where we recall that H = 2n 1 R R + L L + ϕ ϕ in the balanced subspace. Let us take (cid:105) j=1 | j,|j+1≤ φ to have a small overlap with the ground state, say α 2 1/2. Then 0 (cid:88) (cid:88) − − (cid:104) | | (cid:105) ≥ M2n 1 | | (cid:104) | | (cid:105) ≥ 1 M2n 1(cid:104) | | (cid:105) φ H φ α 2 e H e e H e . (23) i 1 1 1 1 2 i=1 i=1 (cid:105) We choose φ to be (cid:88) (cid:105) ≡ √ 1 (cid:105) φ e2πiθ˜A˜p m , (24) p M 2n mp where A˜ is the area under the Motzkin walk m , θ˜ is a constant we will specify later and the p p sum is over all Motzkin walks. The overlap with the ground state is (cid:88) M − (cid:104)M | (cid:105) 1 2n 1 φ = e2πiθ˜A˜p. (25) 2n M 2n m=0 → ∞ As n , the random walk converges to a Wiener process [38] and a random Motzkin walk converges to a Brownian excursion [39]. We wish to scale the random walk such that it takes place O on [0, 1], which is the standard form and gives (1) mean and variance. To this end, we ﬁrst describe the standard Brownian excursion. Let B (t) be a standard Brownian motion on [0, 1] with B (0) = 0. A standard (normalized) Brownian excursion, Bex (t), on the ≥ ∈ interval [0, 1] is deﬁned by B (t) conditioned on B (t) 0 and B (1) = 0 for t [0, 1] (see [40]). Let (cid:90) Bex (t) be a Brownian excursion and [40, p. 84] B ≡ 1 ex Bex (t) dt 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Remark.

text: In

text: the

text: balanced

text: subspace,

text: the

text: amplitude

text: of

text: any

text: vector

text: on

text: Π

text: and

text: Πcross

text: for

text: all

text: j

text: in

text: boundary

text: the

text: Hamiltonian

text: (Eq.

text: vanishes.

text: The

text: Hamiltonian

text: then

text: is

text: simply

text: H

text: Π

text: M

text: M

text: For

text: now

text: we

text: take

text: s

text: and

text: denote

text: by

text: and

text: the

text: Motzkin

text: number

text: M

text: by

text: M

text: Take

text: the

text: state

text: φ

text: in

text: the

text: balanced

text: subspace

text: deﬁned

text: by

text: M

text: φ

text: α

text: α

text: e

text: α

text: e

text: α

text: e

text: M

text: where

text: e

text: is

text: the

text: ith

text: excited

text: states

text: of

text: H

text: in

text: balanced

text: subspace.

text: Clearly,

text: α

text: and

text: i

text: i

text: M

text: φ

text: H

text: φ

text: αi

text: ei

text: H

text: ei

text: where

text: we

text: recall

text: that

text: H

text: R

text: R

text: L

text: L

text: ϕ

text: ϕ

text: in

text: the

text: balanced

text: subspace.

text: Let

text: us

text: take

text: φ

text: to

text: have

text: a

text: small

text: overlap

text: with

text: the

text: ground

text: state,

text: say

text: α

text: Then

text: φ

text: H

text: φ

text: α

text: e

text: H

text: e

text: e

text: H

text: e

text: i

text: We

text: choose

text: φ

text: to

text: be

text: φ

text: m

text: p

text: M

text: mp

text: where

text: A˜

text: is

text: the

text: area

text: under

text: the

text: Motzkin

text: walk

text: m

text: θ˜

text: is

text: a

text: constant

text: we

text: will

text: specify

text: later

text: and

text: the

text: p

text: p

text: sum

text: is

text: over

text: all

text: Motzkin

text: walks.

text: The

text: overlap

text: with

text: the

text: ground

text: state

text: is

text: M

text: φ

text: M

text: As

text: n

text: the

text: random

text: walk

text: converges

text: to

text: a

text: Wiener

text: process

text: and

text: a

text: random

text: Motzkin

text: walk

text: converges

text: to

text: a

text: Brownian

text: excursion

text: We

text: wish

text: to

text: scale

text: the

text: random

text: walk

text: such

text: that

text: it

text: takes

text: place

text: O

text: on

text: which

text: is

text: the

text: standard

text: form

text: and

text: gives

text: mean

text: and

text: variance.

text: To

text: this

text: end,

text: we

text: ﬁrst

text: describe

text: the

text: standard

text: Brownian

text: excursion.

text: Let

text: B

text: (t)

text: be

text: a

text: standard

text: Brownian

text: motion

text: on

text: with

text: B

text: A

text: standard

text: (normalized)

text: Brownian

text: excursion,

text: Bex

text: (t),

text: on

text: the

text: interval

text: is

text: deﬁned

text: by

text: B

text: (t)

text: conditioned

text: on

text: B

text: (t)

text: and

text: B

text: for

text: t

text: (see

text: Let

text: Bex

text: (t)

text: be

text: a

text: Brownian

text: excursion

text: and

text: p.

text: B

text: ex

text: Bex

text: (t)

text: dt

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 18 (θ) ( ) F f x A A 1.0 2.5 0.8 2.0 0.6 1.5 0.4 1.0 0.5 0.2 θ x 0.0 0.5 1.0 1.5 2.0 0 10 20 30 40 Figure 7: Left: Plot of the probability density of the area of a Brownian excursion f (x) on [0, 1]. Right: A Fourier transform of f (x) as deﬁned by Eq. 27. A (cid:104) (cid:105) B the Brownian excursion area. The moments of ex are given by − E B 4π2 k/2k! ≥ ekx = − K k 0, k Γ [(3k 1) /2] − where K = 1 and (cid:88) 0 2 − − k 1 3k 4 ≥ K = K − + K K − , k 1. k(cid:112) k(cid:112)1 j k j 4 j=1 (cid:112) (cid:2) (cid:3) (cid:0) (cid:1) It follows that E [Bex] = 4 πK = 1 π ≈ 0√.626657 and the standard of deviation σ = 2 1 2 2 5/12 − π/8 ≈ 0.1548144. In addition E Bekx ∼ 3 2k k k/2 as k → ∞. B 12e Let f (x) be the probability density function of . The analytical form of f (x) is [40, see A ex A Eq. 92] (cid:18) (cid:19) (cid:88) √ ∞ 2 6 − −5 4 ∈ ∞ f (x) = v2/3 e vj U , ; v x [0, ) (26) A x2 j 6 3 j j=1 | | with v = 2 a 3 /27x2 where a are the zeros of the Airy function, Ai (x), and U is the conﬂuent j j j hypergeometric function [41]. See Fig. 7 for the plot of f (x). A The total area3 under all strictly positive Motzkin walks of length n satisﬁes a recursion relation √ 3 From convergence to a Brownian motion, we expect t(cid:80)he height in the middle to be c(cid:48) n, where c(cid:48) can be calculated from our previous techniques. Indeed, the expected height of the Motzkin walk in the middle (i.e., at site n) is (cid:80) ∞ E [m] = m∞=0 m Mn2,m M2 m=0 n,m ≤ ≤ where as before we denote the height(cid:82)by 0 m n(cid:0) and Mn(cid:1),m is the(cid:114)number of walks that start from zero and end at height m in n steps. Using similar derivation leading to Eq. 13 we ﬁnd (cid:82) ∞ − √ √ E ≈ dα α3 exp 3α2/2 2 [m] 0∞ − n = 2 n . dα α2 exp ( 3α2/2) 3π 0 ##LTFigure## ##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##

text: (θ)

text: F

text: f

text: x

text: A

text: A

text: θ

text: x

text: Figure

text: Left:

text: Plot

text: of

text: the

text: probability

text: density

text: of

text: the

text: area

text: of

text: a

text: Brownian

text: excursion

text: f

text: (x)

text: on

text: Right:

text: A

text: Fourier

text: transform

text: of

text: f

text: (x)

text: as

text: deﬁned

text: by

text: Eq.

text: A

text: B

text: the

text: Brownian

text: excursion

text: area.

text: The

text: moments

text: of

text: ex

text: are

text: given

text: by

text: E

text: B

text: ekx

text: K

text: k

text: k

text: Γ

text: where

text: K

text: and

text: k

text: K

text: K

text: K

text: K

text: k

text: j

text: k

text: j

text: It

text: follows

text: that

text: E

text: [Bex]

text: πK

text: π

text: and

text: the

text: standard

text: of

text: deviation

text: σ

text: In

text: addition

text: E

text: Bekx

text: k

text: as

text: k

text: B

text: Let

text: f

text: (x)

text: be

text: the

text: probability

text: density

text: function

text: of

text: The

text: analytical

text: form

text: of

text: f

text: (x)

text: is

text: see

text: A

text: ex

text: A

text: Eq.

text: −5

text: f

text: (x)

text: e

text: vj

text: U

text: v

text: x

text: A

text: j

text: j

text: with

text: v

text: a

text: where

text: a

text: are

text: the

text: zeros

text: of

text: the

text: Airy

text: function,

text: Ai

text: (x),

text: and

text: U

text: is

text: the

text: conﬂuent

text: j

text: j

text: j

text: hypergeometric

text: function

text: See

text: Fig.

text: for

text: the

text: plot

text: of

text: f

text: (x).

text: A

text: The

text: total

text: under

text: all

text: strictly

text: positive

text: Motzkin

text: walks

text: of

text: length

text: n

text: satisﬁes

text: a

text: recursion

text: relation

text: From

text: convergence

text: to

text: a

text: Brownian

text: motion,

text: we

text: expect

text: height

text: in

text: the

text: middle

text: to

text: be

text: n,

text: where

text: can

text: be

text: calculated

text: from

text: our

text: previous

text: techniques.

text: Indeed,

text: the

text: expected

text: height

text: of

text: the

text: Motzkin

text: walk

text: in

text: the

text: middle

text: (i.e.,

text: at

text: site

text: n)

text: is

text: E

text: [m]

text: m

text: n,m

text: where

text: as

text: before

text: we

text: denote

text: the

text: m

text: and

text: is

text: of

text: walks

text: that

text: start

text: from

text: zero

text: and

text: end

text: at

text: height

text: m

text: in

text: n

text: steps.

text: Using

text: similar

text: derivation

text: leading

text: to

text: Eq.

text: we

text: ﬁnd

text: E

text: dα

text: α3

text: exp

text: [m]

text: 0∞

text: n

text: n

text: dα

text: α2

text: exp

text: 3π

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

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text: 20 (cid:88) (cid:88) (cid:88) (cid:104) | (cid:105) (cid:104) | (cid:105) 1 − (cid:104) | (cid:105) (cid:104) | (cid:105) φ R R φ = e2πiθ˜(A˜p A˜q) m R R m (29) j,j+1 q j,j+1 p 2M 2n j j mp.mq The change in the area is either one or zero. There are three types of nonzero contributions per j, j + 1 in Eq. 29: − (cid:104) | (cid:105) (cid:104) | (cid:105) (m ) = (m ) : e2πi θ˜(A˜p A˜q) m R R m = 1 q j,j+1 p j,j+1 q j,j+1 p − (cid:104) | (cid:105) (cid:104) | (cid:105) − (m ) = 0r (m ) = r0 : e2πi θ˜(A˜p A˜q) m R R m = ei2πθ˜ q j,j+1 p j,j+1 q j,j+1 p − (cid:104) | (cid:105) (cid:104) | (cid:105) − − (m ) = r0 (m ) = 0r : e2πi θ˜(A˜p A˜q) m R R m = e i2πθ˜. q j,j+1 p j,j+1 q j,j+1 p Note that there is no dependence on the actual values of A˜ and A˜ but only on their diﬀerence. (cid:88) (cid:88) (cid:104) (cid:16) p(cid:17)(cid:105) q (cid:88) Putting it together we ﬁnd that Eq. 29 gives (cid:104) | (cid:105) (cid:104) | (cid:105) 1 − ≈ 1 φ R R φ = a 1 cos 2πθ˜ 2π2θ˜2a j,j+1 j j M M 2n 2n j j j where a is the number of strings that have 0r or r0 at the position j, j + 1. An entirely similar j calculation gives (cid:88) (cid:88) (cid:104) (cid:16) (cid:17)(cid:105) (cid:88) (cid:104) | (cid:105) (cid:104) | (cid:105) 1 − ≈ 1 (cid:88) φ L L φ = (cid:88) b (cid:104)1 cos (cid:16)2πθ˜(cid:17)(cid:105) (cid:88) 2π2θ˜2b , j,j+1 j j M M 2n 2n j j j (cid:104) | (cid:105) (cid:104) | (cid:105) 1 − ≈ 1 φ ϕ ϕ φ = c 1 cos 2πθ˜ 2π2θ˜2c , j,j+1 j j M M 2n 2n j j j where b and c are the number of strings that have 0(cid:96), (cid:96)0 and 00, (cid:96)r at positions j, j + 1 j j respectively. √ √ − Summing up the foregoing equations and using θ˜ = 3− n 3/2 we obtain (cid:88)2 5/3 π/2 − (cid:104) | | (cid:105) 9π2 n 3 φ H φ = − (a + b + c ) . (30) j j j 10 3π M (cid:0) (cid:1) 2n j O O − We need to show that aj+bj+cj = (1) to get n 2 upper bound. It is clear that c = M − and that a ≈ b ≈ c . LastMly2n1/3 ≤ M − /M ≤(cid:0)1; th(cid:1)erefore j 2(n 1) j j j 2(n 1) 2n (cid:104) | | (cid:105) O − φ H φ = n 2 . ≥ If we take a general integer s 1 then using similar reasoning (cid:88) (cid:0) (cid:1) (cid:0) (cid:1) − (cid:104) | | (cid:105) ∼ 2π2sn 3 (cid:48) (cid:48) (cid:48) O − φ H φ a + b + c = n 2 . (31) M j j j 2n,s j ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: φ

text: R

text: R

text: φ

text: A˜q)

text: m

text: R

text: R

text: m

text: q

text: p

text: j

text: j

text: mp.mq

text: The

text: change

text: in

text: the

text: area

text: is

text: either

text: one

text: or

text: zero.

text: There

text: are

text: three

text: types

text: of

text: nonzero

text: contributions

text: per

text: j,

text: j

text: in

text: Eq.

text: (m

text: (m

text: θ˜(A˜p

text: A˜q)

text: m

text: R

text: R

text: m

text: q

text: p

text: q

text: p

text: (m

text: (m

text: θ˜(A˜p

text: A˜q)

text: m

text: R

text: R

text: m

text: ei2πθ˜

text: q

text: p

text: q

text: p

text: (m

text: (m

text: θ˜(A˜p

text: A˜q)

text: m

text: R

text: R

text: m

text: e

text: i2πθ˜.

text: q

text: p

text: q

text: p

text: Note

text: that

text: there

text: is

text: no

text: dependence

text: on

text: the

text: actual

text: values

text: of

text: A˜

text: and

text: A˜

text: but

text: only

text: on

text: their

text: diﬀerence.

text: q

text: Putting

text: it

text: together

text: we

text: ﬁnd

text: that

text: Eq.

text: gives

text: φ

text: R

text: R

text: φ

text: a

text: cos

text: 2πθ˜

text: j

text: j

text: M

text: M

text: j

text: j

text: j

text: where

text: a

text: is

text: the

text: number

text: of

text: strings

text: that

text: have

text: or

text: at

text: the

text: position

text: j,

text: j

text: An

text: entirely

text: similar

text: j

text: calculation

text: gives

text: φ

text: L

text: L

text: φ

text: b

text: cos

text: j

text: j

text: M

text: M

text: j

text: j

text: j

text: φ

text: ϕ

text: ϕ

text: φ

text: c

text: cos

text: 2πθ˜

text: j

text: j

text: M

text: M

text: j

text: j

text: j

text: where

text: b

text: and

text: c

text: are

text: the

text: number

text: of

text: strings

text: that

text: have

text: and

text: at

text: positions

text: j,

text: j

text: j

text: j

text: respectively.

text: Summing

text: up

text: the

text: foregoing

text: equations

text: and

text: using

text: θ˜

text: 3−

text: n

text: we

text: obtain

text: n

text: φ

text: H

text: φ

text: (a

text: b

text: c

text: j

text: j

text: j

text: 3π

text: M

text: j

text: O

text: O

text: We

text: need

text: to

text: show

text: that

text: aj+bj+cj

text: to

text: get

text: n

text: upper

text: bound.

text: It

text: is

text: clear

text: that

text: c

text: M

text: and

text: that

text: a

text: b

text: c

text: M

text: /M

text: j

text: j

text: j

text: j

text: O

text: φ

text: H

text: φ

text: n

text: If

text: we

text: take

text: a

text: general

text: integer

text: s

text: then

text: using

text: similar

text: reasoning

text: O

text: φ

text: H

text: φ

text: a

text: b

text: c

text: n

text: M

text: j

text: j

text: j

text: j

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: 3 [10]. Later density matrix renormalization group (DMRG) and its natural representation by matrix product states (MPS) gave successful and systematic recipes for truncating the Hilbert space based on ignoring zero and small singular values in specifying the states [11, 12]. In particular, for gapped one dimensional systems, the MPS ansatz, where the bond dimension is a constant, suﬃces [2]. DMRG and MPS have been tremendously successful in practice for capturing the properties of QMBS in physics and chemistry [13–16]. One wonders about the limitations of DMRG for one dimensional systems. In particular, are there “reasonable systems” that violate the area law (see below) with a pre-factor that scales faster than log (n), which is the correction one expects from relativistic conformal ﬁeld theory in the context of critical systems [17]? In relativistic conformal ﬁeld theory the energy gap must scales as 1/n. A. Quantum criticality, area laws and entanglement entropy In the last two decades, condensed matter physics has revealed a new type of phase transition that is driven, not by thermal motion, but by quantum ﬂuctuations, i.e., zero point motions due to Heisenberg’s uncertainty principle (see [18] for a review). The hallmark of criticality is divergence of correlation length; in quantum critical systems the order parameter ﬁeld depends on (imaginary) time as well as space [19]. In quantum systems, criticality is often a byproduct of a vanishing energy gap or long-range entanglement [1, pp. 5-6], where by a gapped system one means that the energy ≥ to the ﬁrst excited state ∆E c as the system size tends to inﬁnity, where c is a positive constant1. The interactions in QMBS are usually, to a good approximation, local [20]. This has been exploited to give the so called area laws where locality of interaction gives sub-volume scaling of the entanglement entropy (see [1] for a review). The rigorous proof of an general area law does not exist; however, Hastings proved that it holds for one dimensional gapped systems [2]. Later Wolf et al showed that a ﬁnite correlation length, measured in terms of mutual information, implies an area law and that an area law always holds at ﬁnite temperature [21]. One expects that an exponential decay of correlations would imply an area law, but the data hiding states seemed to challenge this intuition [22, 23]. Nevertheless, Brandão and Horodecki showed that in 1D, exponential decay of correlations implies an area law [24]. Swingle and Senthil [25], based on scaling arguments, argued that “physically reasonable” local Hamiltonians with unique ground states can violate the area law by at most a log (n) factor, which for a spin chain implies that log (n) is the maximum expected entanglement entropy scaling. Physically reasonable essentially means not ﬁne-tuned; for example, translational invariance or robustness against perturbations qualify as physically reasonable. There are various examples of ﬁne-tuned Hamiltonians that have larger, even linear, scaling of entanglement entropy with − the system’s size (see below). Previously, a rigorous example of a spin 1 chain, translationally invariant, local Hamiltonian with a unique ground state that saturates the log (n) bound was proposed [3]. Here we show that the generalization √of this model to integer spin s > 1 gives entanglement entropies that, to leading order, scale as n. Moreover, the other properties such as the uniqueness, frustration freeness of the ground state, locality and translational invariance of the Hamiltonian are preserved. B. Frustration free interactions The Hamiltonian that we will deﬁne in this work is frustration free (FF). We say the ground state of a Hamiltonian is FF or unfrustrated when it is also a common ground state of all of the 1 All ﬁnite Hamiltonians are gapped as they have a ﬁnite number of eigenvalues (ignoring degeneracies of the ground state). Vanishing of the spectral gap is necessary for a quantum phase transition to occur. ##LTLine## ##LTLine##

text: Later

text: density

text: matrix

text: renormalization

text: group

text: (DMRG)

text: and

text: its

text: natural

text: representation

text: by

text: matrix

text: product

text: states

text: (MPS)

text: gave

text: successful

text: and

text: systematic

text: recipes

text: for

text: truncating

text: the

text: Hilbert

text: space

text: based

text: on

text: ignoring

text: zero

text: and

text: small

text: singular

text: values

text: in

text: specifying

text: the

text: states

text: In

text: particular,

text: for

text: gapped

text: one

text: dimensional

text: systems,

text: the

text: MPS

text: ansatz,

text: where

text: the

text: bond

text: dimension

text: is

text: a

text: constant,

text: suﬃces

text: DMRG

text: and

text: MPS

text: have

text: been

text: tremendously

text: successful

text: in

text: practice

text: for

text: capturing

text: the

text: properties

text: of

text: QMBS

text: in

text: physics

text: and

text: chemistry

text: One

text: wonders

text: about

text: the

text: limitations

text: of

text: DMRG

text: for

text: one

text: dimensional

text: systems.

text: In

text: particular,

text: are

text: there

text: “reasonable

text: systems”

text: that

text: violate

text: the

text: area

text: law

text: (see

text: below)

text: with

text: a

text: pre-factor

text: that

text: scales

text: faster

text: than

text: log

text: (n),

text: which

text: is

text: the

text: correction

text: one

text: expects

text: from

text: relativistic

text: conformal

text: ﬁeld

text: theory

text: in

text: the

text: context

text: of

text: critical

text: systems

text: In

text: relativistic

text: conformal

text: ﬁeld

text: theory

text: the

text: energy

text: gap

text: must

text: scales

text: as

text: A.

text: Quantum

text: criticality,

text: area

text: laws

text: and

text: entanglement

text: entropy

text: In

text: the

text: last

text: two

text: decades,

text: condensed

text: matter

text: physics

text: has

text: revealed

text: a

text: new

text: type

text: of

text: phase

text: transition

text: that

text: is

text: driven,

text: not

text: by

text: thermal

text: motion,

text: but

text: by

text: quantum

text: ﬂuctuations,

text: i.e.,

text: zero

text: point

text: motions

text: due

text: to

text: Heisenberg’s

text: uncertainty

text: principle

text: (see

text: for

text: a

text: review).

text: The

text: hallmark

text: of

text: criticality

text: is

text: divergence

text: of

text: correlation

text: length;

text: in

text: quantum

text: critical

text: systems

text: the

text: order

text: parameter

text: ﬁeld

text: depends

text: on

text: (imaginary)

text: time

text: as

text: well

text: as

text: space

text: In

text: quantum

text: systems,

text: criticality

text: is

text: often

text: a

text: byproduct

text: of

text: a

text: vanishing

text: energy

text: gap

text: or

text: long-range

text: entanglement

text: pp.

text: where

text: by

text: a

text: gapped

text: system

text: one

text: means

text: that

text: the

text: energy

text: to

text: the

text: ﬁrst

text: excited

text: state

text: ∆E

text: c

text: as

text: the

text: system

text: size

text: tends

text: to

text: inﬁnity,

text: where

text: c

text: is

text: a

text: positive

text: The

text: interactions

text: in

text: QMBS

text: are

text: usually,

text: to

text: a

text: good

text: approximation,

text: local

text: This

text: has

text: been

text: exploited

text: to

text: give

text: the

text: so

text: called

text: area

text: laws

text: where

text: locality

text: of

text: interaction

text: gives

text: sub-volume

text: scaling

text: of

text: the

text: entanglement

text: entropy

text: (see

text: for

text: a

text: review).

text: The

text: rigorous

text: proof

text: of

text: an

text: general

text: area

text: law

text: does

text: not

text: exist;

text: however,

text: Hastings

text: proved

text: that

text: it

text: holds

text: for

text: one

text: dimensional

text: gapped

text: systems

text: Later

text: Wolf

text: et

text: al

text: showed

text: that

text: a

text: ﬁnite

text: correlation

text: length,

text: measured

text: in

text: terms

text: of

text: mutual

text: information,

text: implies

text: an

text: area

text: law

text: and

text: that

text: an

text: area

text: law

text: always

text: holds

text: at

text: ﬁnite

text: temperature

text: One

text: expects

text: that

text: an

text: exponential

text: decay

text: of

text: correlations

text: would

text: imply

text: an

text: area

text: law,

text: but

text: the

text: data

text: hiding

text: states

text: seemed

text: to

text: challenge

text: this

text: intuition

text: Nevertheless,

text: Brandão

text: and

text: Horodecki

text: showed

text: that

text: in

text: exponential

text: decay

text: of

text: correlations

text: implies

text: an

text: area

text: law

text: Swingle

text: and

text: Senthil

text: based

text: on

text: scaling

text: arguments,

text: argued

text: that

text: “physically

text: reasonable”

text: local

text: Hamiltonians

text: with

text: unique

text: ground

text: states

text: can

text: violate

text: the

text: area

text: law

text: by

text: at

text: most

text: a

text: log

text: (n)

text: factor,

text: which

text: for

text: a

text: spin

text: chain

text: implies

text: that

text: log

text: (n)

text: is

text: the

text: maximum

text: expected

text: entanglement

text: entropy

text: scaling.

text: Physically

text: reasonable

text: essentially

text: means

text: not

text: ﬁne-tuned;

text: for

text: example,

text: translational

text: invariance

text: or

text: robustness

text: against

text: perturbations

text: qualify

text: as

text: physically

text: reasonable.

text: There

text: are

text: various

text: examples

text: of

text: ﬁne-tuned

text: Hamiltonians

text: that

text: have

text: larger,

text: even

text: linear,

text: scaling

text: of

text: entanglement

text: entropy

text: with

text: the

text: system’s

text: size

text: (see

text: below).

text: Previously,

text: a

text: rigorous

text: example

text: of

text: a

text: spin

text: chain,

text: translationally

text: invariant,

text: local

text: Hamiltonian

text: with

text: a

text: unique

text: ground

text: state

text: that

text: saturates

text: the

text: log

text: (n)

text: bound

text: was

text: proposed

text: Here

text: we

text: show

text: that

text: the

text: generalization

text: √of

text: this

text: model

text: to

text: integer

text: spin

text: s

text: gives

text: entanglement

text: entropies

text: that,

text: to

text: leading

text: order,

text: scale

text: as

text: n.

text: Moreover,

text: the

text: other

text: properties

text: such

text: as

text: the

text: uniqueness,

text: frustration

text: freeness

text: of

text: the

text: ground

text: state,

text: locality

text: and

text: translational

text: invariance

text: of

text: the

text: Hamiltonian

text: are

text: preserved.

text: B.

text: Frustration

text: free

text: interactions

text: The

text: Hamiltonian

text: that

text: we

text: will

text: deﬁne

text: in

text: this

text: work

text: is

text: frustration

text: free

text: (FF).

text: We

text: say

text: the

text: ground

text: state

text: of

text: a

text: Hamiltonian

text: is

text: FF

text: or

text: unfrustrated

text: when

text: it

text: is

text: also

text: a

text: common

text: ground

text: state

text: of

text: all

text: of

text: the

text: All

text: ﬁnite

text: Hamiltonians

text: are

text: gapped

text: as

text: they

text: have

text: a

text: ﬁnite

text: number

text: of

text: eigenvalues

text: (ignoring

text: degeneracies

text: of

text: the

text: ground

text: state).

text: Vanishing

text: of

text: the

text: spectral

text: gap

text: is

text: necessary

text: for

text: a

text: quantum

text: phase

text: transition

text: to

text: occur.

text: ##LTLine##

text: ##LTLine##


text: 23 is the process by which we ﬁrst embed a Dyck walk with a particular coloring assignment into a Motzkin space (i.e., adding zeros), then either cut a peak of a given color (i.e., any (cid:96)krk) or add a peak of a given color where there are two consecutive zeros. Therefore, H acts on the Dyck eff space Hs . D (cid:105) M (cid:105) | (cid:105) Lemma 3. The unique ground state of H is D(cid:115)s that satisﬁes = V Ds and is given by eff (cid:18) (cid:19) 2n,s (cid:88) (cid:88) (cid:112) n (cid:105) 1 2n (cid:105) Ds = s . (36) M2n,s m=0 2m s∈Ds m M (cid:105) |M (cid:105) Proof. We proved that is the unique ground state of H = H + H . Since H = | (cid:105) | (cid:105) 2n,s | (cid:105) move int 2n,s HV Ds = 0, H Ds = 0. We now prove that Ds is the unique ground state. For if it were | ef(cid:48)f(cid:105) (cid:48)(cid:105) (cid:105) | (cid:48)(cid:105) not, then H D = 0 for a state D other than Ds . But by construction H V D = 0, eff | (cid:48)(cid:105) M (cid:105) move therefore (H + H ) V D = 0. Since is the unique ground state of H, then we reach move int (cid:48)(cid:105) (cid:105) 2n,s a contradiction unless D = Ds . H deﬁnes a random Markov process. However, before describing the Markov process, we shall eff use the Projection Lemma [43], that in our notation reads Lemma. (Projection Lemma [43]) H acts on the ground subspace of H . If the spectral gap eff move of H and H are both poly (1/n) then the spectral gap of H is also poly (1/n) for small enough move eff (cid:15) (cid:15) > 0. Mathematically, (cid:0) (cid:1) O (cid:107) (cid:107) − (cid:15)2 H 2 ≤ ≤ (cid:15)λ (H ) − int(cid:107) (cid:107) λ (H ) (cid:15)λ (H ) . (37) 1 eff (cid:0) (cid:1) 1 (cid:15) 1 eff ∆ (H ) 2(cid:15) H move int − (cid:107) (cid:107) where we take ∆ (H ) = Ω n 2 > 2(cid:15) H . move int (cid:105) (cid:105) ↔ ↔ Since H ψ = 0 so long as ψ is symmetric under 0(cid:96)k (cid:96)k0 and 0rk rk0, the ground move | (cid:105) | (cid:105) ∈ subspace of H is spanned by V s for s Ds and is degenerate. Therefore we can subtract the M (cid:105)(cid:104)mMove Span ( ) from the Hilbert space and consider the Projection Lemma on the orthogonal 2n,s 2n,s complement, whereby the subspace with the smallest eigenvalue becomes the gap. From the ﬁrst inequality we have (cid:0) (cid:1) O (cid:107) (cid:107) ≥ − (cid:15)2 H 2 ∆ (H ) (cid:15)∆ (H ) − int(cid:107) (cid:107). (38) (cid:15) eff ∆ (H ) 2(cid:15) H move int (cid:28) − (cid:107) (cid:107) If we choose an (cid:15) n 3 then ∆ (H ) can be considered large with respect to (cid:15) H , which move (cid:0) (cid:1) int gives ≥ ≥ − O ∆ (H) ∆ (H ) (cid:15)∆ (H ) (cid:15)2n4 . (cid:15) eff ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: is

text: the

text: process

text: by

text: which

text: we

text: ﬁrst

text: embed

text: a

text: Dyck

text: walk

text: with

text: a

text: particular

text: coloring

text: assignment

text: into

text: a

text: Motzkin

text: space

text: (i.e.,

text: adding

text: zeros),

text: then

text: either

text: cut

text: a

text: peak

text: of

text: a

text: given

text: color

text: (i.e.,

text: any

text: or

text: add

text: a

text: peak

text: of

text: a

text: given

text: color

text: where

text: there

text: are

text: two

text: consecutive

text: zeros.

text: Therefore,

text: H

text: acts

text: on

text: the

text: Dyck

text: eff

text: space

text: Hs

text: D

text: M

text: Lemma

text: The

text: unique

text: ground

text: state

text: of

text: H

text: is

text: that

text: satisﬁes

text: V

text: Ds

text: and

text: is

text: given

text: by

text: eff

text: n

text: Ds

text: s

text: s∈Ds

text: m

text: M

text: |M

text: Proof.

text: We

text: proved

text: that

text: is

text: the

text: unique

text: ground

text: state

text: of

text: H

text: H

text: H

text: Since

text: H

text: move

text: int

text: HV

text: Ds

text: H

text: Ds

text: We

text: now

text: prove

text: that

text: Ds

text: is

text: the

text: unique

text: ground

text: state.

text: For

text: if

text: it

text: were

text: not,

text: then

text: H

text: D

text: for

text: a

text: state

text: D

text: other

text: than

text: Ds

text: But

text: by

text: construction

text: H

text: V

text: D

text: eff

text: M

text: move

text: therefore

text: (H

text: H

text: V

text: D

text: Since

text: is

text: the

text: unique

text: ground

text: state

text: of

text: H,

text: then

text: we

text: reach

text: move

text: int

text: a

text: contradiction

text: unless

text: D

text: Ds

text: H

text: deﬁnes

text: a

text: random

text: Markov

text: process.

text: However,

text: before

text: describing

text: the

text: Markov

text: process,

text: we

text: shall

text: eff

text: use

text: the

text: Projection

text: Lemma

text: that

text: in

text: our

text: notation

text: reads

text: Lemma.

text: (Projection

text: Lemma

text: H

text: acts

text: on

text: the

text: ground

text: subspace

text: of

text: H

text: If

text: the

text: spectral

text: gap

text: eff

text: move

text: of

text: H

text: and

text: H

text: are

text: both

text: poly

text: then

text: the

text: spectral

text: gap

text: of

text: H

text: is

text: also

text: poly

text: for

text: small

text: enough

text: move

text: eff

text: Mathematically,

text: O

text: H

text: (H

text: λ

text: (H

text: (H

text: eff

text: eff

text: (H

text: H

text: move

text: int

text: where

text: we

text: take

text: (H

text: Ω

text: n

text: H

text: move

text: int

text: Since

text: H

text: ψ

text: so

text: long

text: as

text: ψ

text: is

text: symmetric

text: under

text: and

text: the

text: ground

text: move

text: subspace

text: of

text: H

text: is

text: spanned

text: by

text: V

text: s

text: for

text: s

text: Ds

text: and

text: is

text: degenerate.

text: Therefore

text: we

text: can

text: subtract

text: the

text: M

text: Span

text: from

text: the

text: Hilbert

text: space

text: and

text: consider

text: the

text: Projection

text: Lemma

text: on

text: the

text: orthogonal

text: complement,

text: whereby

text: the

text: subspace

text: with

text: the

text: smallest

text: eigenvalue

text: becomes

text: the

text: gap.

text: From

text: the

text: ﬁrst

text: inequality

text: we

text: have

text: O

text: H

text: (H

text: (H

text: eff

text: (H

text: H

text: move

text: int

text: If

text: we

text: choose

text: an

text: n

text: then

text: (H

text: can

text: be

text: considered

text: large

text: with

text: respect

text: to

text: H

text: which

text: move

text: int

text: gives

text: O

text: (H)

text: (H

text: (H

text: eff

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 (cid:18) (cid:19) ≡ 2n The number of all such walks is counted by the Catalan number C 1 . n n+1 n Catalan numbers are famous numbers in combinatorics. At the time of writing 212 diﬀerent combinatorial problems whose solutions are counted by the Catalan numbers have been catalogued by Richard Stanley [36]. ≈ We mention in passing that lim →∞ C 4n√ , which will be used later to prove the optimality n n n3/2 π of the proposed canonical path. Closely related paths are the Motzkin walks [35, p. 238, prob. 6.38]: Deﬁnition 2. A Motzkin walk (or path) on 2n steps is any path from (0, 0) to (0, 2n) with steps − (1, 0), (1, 1) and (1, 1) that never pass below the x-axis. Comment: A Motzkin walk on 2n steps is a any walk, made up of three types of steps: diagonal (cid:18) (cid:19) up, diagonal down and ﬂat. The walker starts at (x, y) = (0, 0) and ends(cid:80)at (x, y) = (2n, 0) such ≥ ≤ ≤ that the walker’s position at any intermediate lattice point (x , y ) has y 0 for all 0 k 2n. k k k 2n The number of all such walks is counted by the Motzkin numbers M = n C . Please 2n k=0 2k k see Fig. 1. ∈ { } Any Motzkin walk can equivalently be thought of as a string t (, ), 0 2n such that the string has a balanced set of right and left parenthesis with some number of zeros in between. Examples · · · · · · of such strings are (( ( ) )) that are in one to one correspondence with walks that have n up · · · steps followed by n down steps or (0)(0) (0) which correspondences to a zig-zag walk where a · · · step up, a ﬂat step, and step down is concatenated 2n/3 times or 00 0 which is a ﬂat walk where ≤ ≤ the position of the walker at any step is y = 0 for all 0 k 2n. k Now every step up in a Dyck or Motzkin walk has a unique ’matching’ step down to balance it to ultimately give y = 0. Suppose there are s colors available, then a Dyck walk of length 2m 2n can be colored sm diﬀerent ways. For example, take s = 2 then a walk can be n steps upward with alternating colors of blue and red which then will uniquely determine the coloring of the remaining · · · · · · n down steps. Combinatorially equivalent to this walk is the string [ ( [ [ ( ) ] ] ) ] where each left parenthesis/bracket has a unique right match. Any Motzkin walk initially has coordinates (x, y) = (0, 0). In the middle of the chain it will ≤ ≤ have a coordinate (x, y) = (n, m) for some 0 m n; here we denote m to be the “height” in the middle. To calculate the entropy of a half chain we will need to count the number of Motzkin walks that start at zero (x, y) = (0, 0) and reach height m in the middle. A theorem due to André (1887) counts related (Dyck like) lattice paths[37, p. 8]. ≤ ≤ Theorem. (Ballot problem) Let a, b be integers satisfying 1 b a. The number of lattice paths N (p) joining the origin O to the point (a, b) and not touching the diagonal x = y except at O is given by (cid:18) (cid:19) − N a b a + b (p) = (2) a + b b In other words, given a ballot at the end of which candidates P , Q obtain a, b votes respectively, − the probability that P leads Q throughout the counting of votes is a b. a+b First note that a = b + 1 gives the Catalan numbers. This theorem can also be interpreted as counting the number of Dyck walks that reach a given height for some ﬁxed x-coordinate. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: number

text: of

text: all

text: such

text: walks

text: is

text: counted

text: by

text: the

text: Catalan

text: number

text: C

text: n

text: n

text: Catalan

text: numbers

text: are

text: famous

text: numbers

text: in

text: combinatorics.

text: At

text: the

text: time

text: of

text: writing

text: diﬀerent

text: combinatorial

text: problems

text: whose

text: solutions

text: are

text: counted

text: by

text: the

text: Catalan

text: numbers

text: have

text: been

text: catalogued

text: by

text: Richard

text: Stanley

text: We

text: mention

text: in

text: passing

text: that

text: lim

text: C

text: which

text: will

text: be

text: used

text: later

text: to

text: prove

text: the

text: optimality

text: n

text: n

text: π

text: of

text: the

text: proposed

text: canonical

text: path.

text: Closely

text: related

text: paths

text: are

text: the

text: Motzkin

text: walks

text: p.

text: prob.

text: Deﬁnition

text: A

text: Motzkin

text: walk

text: (or

text: path)

text: on

text: steps

text: is

text: any

text: path

text: from

text: to

text: with

text: steps

text: and

text: that

text: never

text: pass

text: below

text: the

text: x-axis.

text: Comment:

text: A

text: Motzkin

text: walk

text: on

text: steps

text: is

text: a

text: any

text: walk,

text: made

text: up

text: of

text: three

text: types

text: of

text: steps:

text: diagonal

text: up,

text: diagonal

text: down

text: and

text: ﬂat.

text: The

text: walker

text: starts

text: at

text: (x,

text: y)

text: and

text: (x,

text: y)

text: such

text: that

text: the

text: walker’s

text: position

text: at

text: any

text: intermediate

text: lattice

text: point

text: (x

text: y

text: has

text: y

text: for

text: all

text: k

text: k

text: k

text: k

text: The

text: number

text: of

text: all

text: such

text: walks

text: is

text: counted

text: by

text: the

text: Motzkin

text: numbers

text: M

text: n

text: C

text: Please

text: k

text: see

text: Fig.

text: Any

text: Motzkin

text: walk

text: can

text: equivalently

text: be

text: thought

text: of

text: as

text: a

text: string

text: t

text: such

text: that

text: the

text: string

text: has

text: a

text: balanced

text: set

text: of

text: right

text: and

text: left

text: parenthesis

text: with

text: some

text: number

text: of

text: zeros

text: in

text: between.

text: Examples

text: of

text: such

text: strings

text: are

text: that

text: are

text: in

text: one

text: to

text: one

text: correspondence

text: with

text: walks

text: that

text: have

text: n

text: up

text: steps

text: followed

text: by

text: n

text: down

text: steps

text: or

text: which

text: correspondences

text: to

text: a

text: zig-zag

text: walk

text: where

text: a

text: step

text: up,

text: a

text: ﬂat

text: step,

text: and

text: step

text: down

text: is

text: concatenated

text: times

text: or

text: which

text: is

text: a

text: ﬂat

text: walk

text: where

text: the

text: position

text: of

text: the

text: walker

text: at

text: any

text: step

text: is

text: y

text: for

text: all

text: k

text: k

text: Now

text: every

text: step

text: up

text: in

text: a

text: Dyck

text: or

text: Motzkin

text: walk

text: has

text: a

text: unique

text: ’matching’

text: step

text: down

text: to

text: balance

text: it

text: to

text: ultimately

text: give

text: y

text: Suppose

text: there

text: are

text: s

text: colors

text: available,

text: then

text: a

text: Dyck

text: walk

text: of

text: length

text: can

text: be

text: colored

text: sm

text: diﬀerent

text: ways.

text: For

text: example,

text: take

text: s

text: then

text: a

text: walk

text: can

text: be

text: n

text: steps

text: upward

text: with

text: alternating

text: colors

text: of

text: blue

text: and

text: red

text: which

text: then

text: will

text: uniquely

text: determine

text: the

text: coloring

text: of

text: the

text: remaining

text: n

text: down

text: steps.

text: Combinatorially

text: equivalent

text: to

text: this

text: walk

text: is

text: the

text: string

text: where

text: each

text: left

text: parenthesis/bracket

text: has

text: a

text: unique

text: right

text: match.

text: Any

text: Motzkin

text: walk

text: initially

text: has

text: coordinates

text: (x,

text: y)

text: In

text: the

text: middle

text: of

text: the

text: chain

text: it

text: will

text: have

text: a

text: coordinate

text: (x,

text: y)

text: (n,

text: m)

text: for

text: some

text: m

text: n;

text: here

text: we

text: denote

text: m

text: to

text: be

text: the

text: “height”

text: in

text: the

text: middle.

text: To

text: calculate

text: the

text: entropy

text: of

text: a

text: half

text: chain

text: we

text: will

text: need

text: to

text: count

text: the

text: number

text: of

text: Motzkin

text: walks

text: that

text: start

text: at

text: zero

text: (x,

text: y)

text: and

text: reach

text: height

text: m

text: in

text: the

text: middle.

text: A

text: theorem

text: due

text: to

text: André

text: counts

text: related

text: (Dyck

text: like)

text: lattice

text: p.

text: Theorem.

text: (Ballot

text: problem)

text: Let

text: a,

text: b

text: be

text: integers

text: satisfying

text: b

text: a.

text: The

text: number

text: of

text: lattice

text: paths

text: N

text: (p)

text: joining

text: the

text: origin

text: O

text: to

text: the

text: point

text: (a,

text: b)

text: and

text: not

text: touching

text: the

text: diagonal

text: x

text: y

text: except

text: at

text: O

text: is

text: given

text: by

text: N

text: a

text: b

text: a

text: b

text: (p)

text: a

text: b

text: b

text: In

text: other

text: words,

text: given

text: a

text: ballot

text: at

text: the

text: end

text: of

text: which

text: candidates

text: P

text: Q

text: obtain

text: a,

text: b

text: votes

text: respectively,

text: the

text: probability

text: that

text: P

text: leads

text: Q

text: throughout

text: the

text: counting

text: of

text: votes

text: is

text: a

text: b.

text: a+b

text: First

text: note

text: that

text: a

text: b

text: gives

text: the

text: Catalan

text: numbers.

text: This

text: theorem

text: can

text: also

text: be

text: interpreted

text: as

text: counting

text: the

text: number

text: of

text: Dyck

text: walks

text: that

text: reach

text: a

text: given

text: height

text: for

text: some

text: ﬁxed

text: x-coordinate.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##