Repulsive Casimir forces at quantum criticality

# Repulsive Casimir forces at quantum criticality

P. Jakubczyk Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland    M. Napiórkowski Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland    T. Sȩk Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
July 27, 2019
###### Abstract

We study the Casimir effect in the vicinity of a quantum critical point. As a prototypical system we analyze the -dimensional imperfect (mean-field) Bose gas enclosed in a hypercubic container of extension and subject to periodic boundary conditions. The thermodynamic state is adjusted so that , where is the thermal de Broglie length, and denotes microscopic lengthscales. Our exact calculation indicates that the Casimir force in the above specified regime is generically repulsive and decays either algebraically or exponentially, with a non-universal amplitude.

## I Motivation

Casimir-type interactions Casimir48 (); Fisher78 (); Krech94 (); Mostepanienko97 (); Kardar99 (); Brankov00 (); Bordag01 (); Gambassi09 () are nowadays recognized in a multum of systems spanning from biological membranes to cosmology. The QED and condensed-matter contexts are those, where the theoretical predictions concerning the existence and properties of Casimir forces found firm experimental confirmation Garcia99 (); Garcia02 (); Fukuto05 (); Ganshin06 (); Hartlein08 (). Herein we specify to the latter context, where a fluctuating medium is enclosed in a hypercubic box of spatial extension . We assume throughout the study. The spectrum of fluctuations (both thermal and quantum) of the medium is constrained by boundary conditions imposed by the confining walls, and, as a result, the free energy acquires a contribution depending on the separation . It therefore becomes favorable to either increase, or decrease the distance , resulting in an effective interaction between the boundary walls. The experimentally confirmed cases usually correspond to situations, where the Casimir force is attractive. A well-known exception is the case of two materials characterized by different dielectric properties Dzyaloshinskii61 () . On the theory side, exact results place severe restrictions on the possibility of obtaining Casimir repulsion in QED models Kenneth06 (); Rahi10 (). Detours around these restriction invoke out-of equilibrium systems Bimonte11 (); Messina11 (); Kruger11 (). The situation is more complex in the case of condensed-matter systems, where, at least theoretically, one may change the character of the force by varying boundary conditions.

In the present study, we consider an exactly soluble model of interacting bosons at finite, but asymptotically low temperature, in a thermodynamic state corresponding to the vicinity of a quantum critical point. We show that in a specific limit the Casimir force is repulsive and decays as a power of the separation even for the periodic boundary conditions which generically yield Casimir attraction. This gives a hint on the possible regime of parameters, where a repulsive Casimir force might be detectable experimentally in a system involving Bose-Einstein condensation, and, potentially a wider class of quantum-critical systems.

The essential ingredient of the critical Casimir effect is the interplay between two large lengthscales: and the bulk correlation length . As long as , the effective interaction between the walls decays exponentially with setting the decay scale. If, however, the system is tuned sufficiently close to a (bulk) critical point, or is in a phase exhibiting soft excitations, one has , and . The crossover between the above two regimes [ and ] is governed by a scaling function, showing universal properties.

The situation becomes more complex for , where, in addition to , and , the thermal de Broglie length becomes macroscopic. We assume here that a phase transition may by tuned by a non-thermal control parameter (such as density, pressure, or chemical composition). Considering the Casimir forces in the low- limit one identifies three regimes differentiated by the hierarchy of the macroscopic scales , , and . The standard thermal regime is recovered for . For the case , where one performs the limit before sending the system size to infinity, by virtue of the quantum-classical mapping, one expects the system properties to be similar to those of the thermal regime, albeit in elevated dimensionality. Finally, there is the possibility of the thermal length being squashed between the scales characterizing the system size, namely

 L≫λ≫D≫lmic. (1)

Here denotes any microscopic length present in the system. To our knowledge, the limit defined by Eq. (1) was not addressed so far, and this is not very simple to give a prediction for the asymptotics of the Casimir force relying solely on general arguments. Note that Eq. (1) implies that the thermodynamic limit cannot be taken the usual way, keeping temperature fixed. Instead, while increasing temperature has to be reduced so that the condition remains fulfilled. On the other hand, from a realistic (and experimental) point of view, the hierarchy of Eq. (1) corresponds to a perfectly well defined regime.

In what follows, we analyze the Casimir forces in the limit of Eq. (1), employing a specific microscopic model of interacting bosons, the so-called imperfect Bose gas (IBG), which exhibits a phase transition to a Bose-Einstein-condensed phase for at any . The transition can be tuned by varying the chemical potential, which acts as the non-thermal control parameter. The model is susceptible to an exact analytical treatment within the grand-canonical formalism.

## Ii Model

We consider a system of spinless, interacting bosons at a fixed temperature and the chemical potential . The system is enclosed in a hypercubic box of volume and is governed by the Hamiltonian

 ^H=∑kℏ2k22m^nk+a2V^N2, (2)

where we use the standard notation. The repulsive interaction term () may be recovered from a 2-particle potential in the Kac limit , i.e. for vanishing interaction strength and diverging range. After imposing periodic boundary conditions, the grand canonical partition function is cast in the convenient form Napiorkowski11 ():

 (3)

where

 ϕ(s)=−s22aβ+sμa+1V∑ndln(1−es−βϵkd)− (4) ∑nd1Dλd−1gd+12(es−βϵkd).

Here , , , are the Bose functions, and the contour parameter is negative. The occurrence of the factor in the exponential in Eq. (3) assures that the saddle-point approximation becomes exact for :

 limV→∞1VlogΞ(T,L,D,μ)=βμ22a−ϕ(¯s). (5)

It follows that for the problem of evaluating the partition function becomes reduced to solving the stationary-point equation

 ϕ′(¯s)=0 (6)

for . More explicitly Eq. (6) reads:

 μa−¯saβ=1λd−1Dgd−12(e¯s)+2V∞∑n=1e¯s−πλ2D2n21−e¯s−πλ2D2n2+ 1Ve¯s1−e¯s+2λd−1D∞∑k=1ek¯skd−12∞∑n=1e−kπλ2D2n2. (7)

Bulk properties of the model defined by Eq. (2) were studied rigorously since 1980s Davies72 (); Berg84 (); Lewis86 (); Zagrebnov01 (). The limit and the (bulk) quantum critical behavior were addressed in Ref. Jakubczyk13 (). For in the phase diagram spanned by and there is a line of second-order phase transitions to the phase hosting the Bose-Einstein condensate. The critical line extends down to , where it ends with a quantum critical point. The transition at falls into the universality class of the spherical model Berlin52 (). The Casimir forces corresponding to the above model were investigated in Refs. Napiorkowski11 (); Napiorkowski13 () in the thermal regime, where . Here we analyze the opposite case defined by the relation (1).

In addition to and , the model involves the lengthscale

 Lμ=(a|μ|−1)1/d, (8)

which may be large or small compared to and . The microscopic scale is set by

 l=(2πmah2)1/(d−2), (9)

which is defined for . The quantity plays the role of the microscopic scale occurring in Eq. (1). Observe, that both the above lengthscales involve the interaction coupling and therefore are not present for the perfect Bose gas.

In the thermal regime the excess surface grand-canonical free energy (per unit area) is extracted by a subtraction of the bulk contribution from the full grand-canonical free energy . One obtains:

 ωs(T,D,μ)=limL→∞[Ω(T,L,D,μ)Ld−1−Dωb(T,μ)]. (10)

The Casimir force (per unit area) is then evaluated via

 F=−∂ωs(T,D,μ)∂D. (11)

In the regime considered in the present paper [Eq. (1)] one can take the alternative approach amounting to calculating the derivative and neglecting a constant (-independent) term identified with the bulk contribution. The character of the solution to Eq. (6) in the asymptotic regime specified by (1) crucially depends on the dimensionality . Here we primarily focus on the interval , where the term in Eq. (II) is singular at . This guarantees that when (keeping all the other lengthscales fixed), the solution remains separated from zero. In consequence, the terms in Eq. (II) can be dropped. The last contribution to Eq. (II) is bounded from above by and is also negligible as compared to the other terms. The solution of Eq. (II) then boils down to analyzing the equation

 μa−¯saβ=1λd−1Dgd−12(e¯s) (12)

in the asymptotic regimes, where the correlation length (see below) is asymptotically large.

## Iii Results for d=3

For the case of the term in Eq. (12) displays a logarithmic singularity at ; for . Eq. (12) finds asymptotic solution in the following form:

 ¯s≈⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩lDlnlDin Regime I −e−λ2DL3μin Regime II −λ2lL3μ−lDg1(e−λ2l/L3μ)in Regime III . (13)

Regime I corresponds to the condition ; Regime II to ; and, finally, Regime III is defined by .

The crucial observation is that the quantity is related to the correlation length by the formula Napiorkowski11 (); Napiorkowski12 ()

 ξ=κλ|¯s|−1/2, (14)

where is a numerical constant.

It follows that the singularity of occurring at the quantum critical point (in the limit ) is effectively cut off by the system width in Regime I, by the thermodynamic fields in Regime III, and be a combination of the thermodynamic and geometric parameters in Regime II. This gives rise to the rich behavior predicted for the Casimir force (bee below).

We now use Eq. (5) to compute the grand-canonical free energy and take the derivative . Neglecting a constant, which is attributed to the bulk term in the free energy, we obtain the following expressions for the Casimir force:

 βF(T,D,μ)≈⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩12lD2λ2ln2(lD)in Regime I1L3μe−λ2D/L3μin Regime II12lD2λ2g21(e−λ2/L3μ)in Regime III. (15)

The above result indicates that the effective force is always repulsive, and, except for Regime II, decays as a power of . The logarithmic correction in Regime I and the exponential behavior in Regime II are specific to . The obtained behavior is clearly different from that occurring in the thermal regime (), where the force is attractive and characterized by a universal amplitude wherever the interaction is long ranged (i.e. in the immediate vicinity of the transition or in the low-T phase). The present setup places the thermodynamic lengthscale in between the macroscopic, geometric quantities and , which has a far-reaching consequence for the properties of the Casimir interaction. In addition, the scale can be adjusted at will, leading to the emergence of the three asymptotic regimes defined above. Also note that the Bose condensate, manifesting itself with the solution (at finite) never appears in the analysis. This is because may not be made asymptotically large without sending temperature to zero (see Eq. (1). In consequence, the condensate appears only in the strict limit of infinite and . The results of this section are translated to the thermodynamic variables , and summarized in Fig. 1.

## Iv Results for 2<d<3

It is interesting to follow the evolution of the system, and the Casimir forces in particular, when continuously reducing the dimensionality parameter from 3 towards the other physical value . The case is special because the function exhibits a logarithmic singularity at . For we have:

 gd−12(e¯s)=Γ(3−d2)|¯s|d−32+ζ(d−12)+…. (16)

The asymptotic form of the saddle-point equation (12) admits the following solutions:

 ¯s≈⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩−C1l2d−25−dλ−2d−35−dD−25−din Regime I −C2L2d3−dμλ−2d−13−dD−23−din Regime II −ld−2(λ2L−dμ+λ3−dD−1g)in % Regime III , (17)

where we introduced , , and . The three emergent asymptotic regimes are defined by the condition

 Lμ≫(λd−1D)1/dRegime ILμ≪(λd−1D)1/dμ>0 Regime IILμ≪(λd−1D)1/dμ<0 Regime III. (18)

For this reduces to the previously obtained condition.

From Eq. (5) we evaluate the free energy and extract the Casimir force by taking the -derivative. The result reads:

 βF(T,D,μ)≈⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩12C′21l(d−1)(d−2)5−dλ−2d−15−dD−45−d (I)C2Ld(d−1)3−dμλ−2d−13−dD−23−d(II)12ld−2g2d−12(eβμ)λ−2(d−2)D−2(III) (19)

with . The Casimir force is repulsive in all the three asymptotic regimes and decays with a power of . Note a difference as compared to , where the logarithms and exponents appeared as consequence of the form of the asymptotic behaviour of the Bose function at . As , the power describing the decay of the force in Regime II [Eq. (19)] diverges, which gives rise to the exponential behavior in . The result is translated back to the thermodynamic variables and and depicted in Fig. 1.

## V Note on the case d=2

We now comment on the Casimir force in . This is a special case since the microscopic length defined in Eq. (9) does not exist. It makes no physical sense (nor mathematical) to consider the limit of the scales and becoming macroscopic without specifying the microscopic length. The only choice possible in is to take as given by Eq. (8). The absence of the quantity in manifests itself in the non-existence of a solution to the saddle-point equation (6) in the parameter range corresponding to Regime I, where is large.

## Vi Results for d>3

It is also interesting to examine the case , where the system may host a thermodynamically stable Bose-Einstein condensate for , but finite . For the function is finite for . In consequence, Eq. (12) has no solution at sufficiently low . This is because upon passing to the limit in Eq.  (II) vanishes, and the term gives a finite contribution. It must therefore be included in the anaylsis by replacing Eq. (12) with

 μa−¯saβ=1λd−1Dgd−12(e¯s)+1Ve¯s1−e¯s. (20)

The above equation is equivalent to the one arising in the bulk case for [see Ref. Napiorkowski13 ()] upon making the substitutions and . For we find a finite, unique solution to Eq. (20) provided . In the opposite case, the last term in Eq. (20) gives a finite contribution equal to the condensate density Napiorkowski12 (). This leads to the following result for the critical value of the chemical potential:

 μc(T,D)=aζ(d−12)1λd−1D, (21)

above which the Bose-Einstein condensate is present in the system. The transition between the phases may be induced by varying any of the parameters so that the geometric quantity may (for the presently relevant regime ) serve as the tuning parameter on equal footing with the thermodynamic ones. One may also observe, that may be related to the standard, thermodynamic critical value of the chemical potential Jakubczyk13 () via

 μc(T,D)=μ(d)c(T)λDζ(d−12)ζ(d2). (22)

Since , it follows that . At we have in Eq. (22), so that diverges and the condensate is ultimately suppressed in the regime .

For the determination of the Casimir force, we focus on the range of parameters, where the condensate is present (), making the setup clearly distinct from that discussed for . The calculation leading to an expression for is now analogous to the one performed in Ref.Napiorkowski13 () (where one makes the replacements , specified above). We obtain:

 |¯s(T,μ,D)|≈λd−1DV1ζ(d−12)μcμ−μc. (23)

This leads to the following expression for the Casimir force:

 βF(T,μ,D)=4πλd−3D3gd−12(e−πλ2/D2). (24)

which is again repulsive.

## Vii Summary

We have performed and exact study of Casimir forces occurring in the -dimensional imperfect Bose gas (interacting bosons in the Kac limit) in the regime, where the de Broglie length is squashed in between the lengthscales and characterizing the system geometric size (i.e. for ). We scanned the dependence of our results on the system dimensionality . We obtain a behavior of the Casimir force completely different from that established before in the thermal regime (i.e. for ) and also expected in the quantum regime by virtue of the quantum-classical mapping. The computed Casimir force turns out to be repulsive and decay as a power of the distance in most of the cases (with log-corrections in ). The peculiarity of our results may be traced back to the occurrence of an extra lengthscale () which is considered as macroscopic, and which is absent in the standard condensed-matter setup. We emphasise that the present model perfectly fits into the established classification once we restrict the thermal regime (i.e. treat as a microscopic scale). In this case it falls into the universality class of the -dimensional spherical model. The interplay between , , and the scale controlling the distance of the system from the (bulk) quantum critical point leads to the emergence of three regimes showing different asymptotic behavior of the Casimir force. An additional feature appears for , where the system admits a dimensional (”surface”) condensate stable as a thermodynamic phase for any finite . The transition to this phase may be tuned by varying , as well as . It is interesting to speculate about the generality of our results and their dependence on the details of the microscopic model. Clearly, our results (for ) do depend on the microscopic parameters. These however may be expressed via quantities of the dimensionality of length, which find natural analogues in other condensed-matter systems (in particular close to quantum criticality). We believe that (when expressed via these length parameters) our finding should apply at least to other system belonging to the universality class of the spherical model Dantchev96 (); Dohm09 (); Diehl12 (); Dantchev14 (); Diehl14 (). Another important question concerns the sensitivity of our results on the boundary conditions. Such a dependence is well known to occur in the thermal regime. We have checked that for von-Neuman boundary conditions the essential features of our results are unchanged.

###### Acknowledgements.
We thank Grzegorz Łach, Anna Maciołek and Piotr Nowakowski for discussions, and Hans Diehl for a useful correspondence. We acknowledge funding from the National Science Centre via grant 2014/15/B/ST3/02212.

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