Long range order in the ground state of quantum interacting rotors in two dimensions

Physica A 391 (2012) 5918–5925

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Long range order in the ground state of quantum interacting rotors in two dimensions Jacek Wojtkiewicz ∗ Department for Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Hoża 74, 00-682 Warszawa, Poland

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Article history: Received 13 March 2012 Received in revised form 31 May 2012 Available online 23 June 2012 Keywords: Phase transitions Quantum lattice systems Long-range order Reflection positivity

abstract It has been rigorously proved that the ground state of interacting quantum rotors in two dimensions exhibits the magnetic long-range order in the ferromagnetic case. The assumptions are that the interaction is between nearest neighbors and it is strong enough, and that rotors are sufficiently heavy. The proof is based on the Reflection Positivity technique. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The nature of ground states of large quantum systems, orderings and phase transitions therein – the theory of quantum phase transitions – is as rich and interesting as the finite-temperature counterparts. It has played an important role in condensed matter physics over the past two decades [1,2]. In this paper, the nature of ground state(s) of the system of isotropic interacting quantum rotors in two dimensions is examined. Its Hamiltonian (1) can be considered as the natural extension of the classical XY model. The Hamiltonian (1) as well as its extensions and modifications appear naturally in the description of such systems as thin films of liquid helium and thin layers of superconductors [2], two-dimensional arrays of Josephson junctions [3], and molecular layers of diatomic molecules adsorbed on a surface [4,5]. There are certain known rigorous theorems concerning finite-temperature phase transitions in the system of interacting quantum rotors. It has been proven that there is no LRO in positive temperatures in one or two dimensions [6]. This result can be viewed as an analog of the famous Mermin–Wagner theorem for quantum spin systems [7]. On the other hand, for three-dimensional systems, the existence of LRO in sufficiently low temperatures (under certain assumptions on the coupling constant and moment of inertia) has been proved in Ref. [6]. This result has been obtained by suitable adaptation of the Reflection Positivity technique, developed in Ref. [8] for quantum spin systems. On physical grounds one can expect that the ground state is magnetically ordered [1,2]. (The proof of non-existence of orderings in a 2d system of rotors works only in strictly positive temperatures—it tells nothing about the ground state.) The goal of my paper is to present the proof of the existence of magnetic LRO in the ground state of ferromagnetically interacting quantum rotors in d = 2. The proof is based on an analogy between d = 3, T > 0 and d = 2, T = 0 cases for isotropic quantum spin systems. After the proof of LRO in Heisenberg models in the d = 3, T > 0 case had appeared [8], it was shown that one can take certain zero-temperature limits in this procedure and obtain the existence of LRO in the ground state of corresponding twodimensional systems [9,10]. A similar strategy has been used in my paper: it turned out that one can take zero-temperature

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0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.06.028

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limits in the method used in Ref. [6] and prove the existence of LRO in the ground state of interacting quantum rotors in two dimensions. The paper is organized as follows. In Section 2, the model is defined and basic results are formulated. In Section 3, the strategy of the proof of LRO in d = 3, T > 0 used in Ref. [6] is described in some details, as my paper relies on certain modifications and extensions of methods in Ref. [6]. In Section 4, the zero-temperature limit is taken and the existence of LRO in the ground state of two-dimensional rotors is proved. The sufficient condition for ordering is that the quantity IJ is larger than a certain computable constant (I is the moment of inertia of rotor, J is the coupling constant of interaction between rotors). Section 5 summarizes the results obtained and describes some open problems. 2. The model and basic results Denote by Λ the finite subset of the simple cubic lattice in d dimensions: Λ ⊂ Zd . We assume that Λ is a (discrete) hypercube with periodic boundary conditions and that the number of sites along every edge is even. The system of interacting rotors is defined by the following Hamiltonian: H =−

1  ∂2 2I x∈Λ ∂φ

2 x

−J



cos(φx − φy ).

(1)

⟨xy⟩

Here I denotes the moment of inertia of the rotor; J is the coupling constant. The case J > 0 corresponds to ferromagnetic coupling between rotors, and J < 0—the antiferromagnetic one. We consider below only the ferromagnetic case, as only in this case is it possible to apply the Reflection Positivity arguments. Finally, φx ∈ [0, 2π [ describes the position of the rotor at the site x. Equivalently, the position of the rotor at the site x can be described as a unit vector Sx , taking the values on the unit circle: Sx ∈ S1 . It can be written as Sx = [Sxx , Sxy ] = [cos φx , sin φx ]. The total spin is S=



Sx .

x∈Λ 2 The operator Tx = − 2I1 ∂ 2 is an operator of the kinetic energy of the rotor at site x, and the operator T , defined as

∂φx

T =−

1  ∂2

2I x∈Λ ∂φx2 is the operator of total kinetic energy. The Hilbert space of states of the rotor at site x ∈ Λ is Hx = L2 (S 1 ). The Hilbert space HΛ of the whole system is a tensor product:

HΛ = ⊗ Hx . x∈Λ

The partition function is defined in the standard manner as ZΛ = Tr exp(−β HΛ ) =



exp(−β Ei ),

i

where the sum is taken over all eigenstates of HΛ ; Ei are corresponding eigenenergies. The average of an observable A is 1 ⟨A⟩Λ = Tr[A exp(−β HΛ )]. ZΛ Thermodynamic limit: The phase transition is well defined only in the thermodynamic limit, i.e. Λ ↗ ∞. Order parameter: The simplest definition of the order parameter would be an average of the total spin. However, this definition is of little use for the zero field (i.e. as a measure of the spontaneous magnetization) as it is zero due to symmetry. The more physical definition is a zero-field limit of magnetization: M = limh→0 M (h). However, it is difficult to deal with. The quantity which is more easy to handle is (suitably scaled) an average of the square of the total spin. It follows that if the average of the square of spin is different from zero, then also the zero-field magnetization is non-zero (Griffiths theorem—see 1 S)2 ⟩ as a measure of the order parameter. Ref. [8]). So, we take the average ⟨( |Λ | 1 It has been shown in Ref. [6] that the average ⟨( |Λ S)2 ⟩ is different from zero if the temperature is sufficiently low and | dimension d is greater than 2. This result can be reformulated as:

 ⟨(S ) ⟩Λ ≡ x 2

1 

|Λ|

x∈Λ

2  Sxx

 =

Λ

1 

|Λ|

x∈Λ

2  cos φx

≥C >0 Λ

(2)

for some positive constant C , which is independent of |Λ|. The result of my paper is a proof that the inequality (2) holds also for T = 0 in d = 2 (the average is taken over the ground state).

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3. The proof of existence of LRO in the system of rotors for d ≥ 3 3.1. Sufficient criterion for ordering In this subsection, we review the result due to Dyson et al. [8], which gives a sufficient criterion for appearance of ordering in spin systems. It applies to both spin systems (like Heisenberg or XY models) and to interacting rotors. We write it in the last versions, similar to used by Pastur and Khoruzhenko [6] in a somewhat different situation (multidimensional rotors with an interaction different from the one considered by ours). Theorem 1 ([6]). Assume that the model defined in the previous section fulfills following conditions: 1. ‘‘Gaussian domination’’: There exists bounded and integrable function Bk such that for arbitrary β > 0 and k ̸= 0 the following inequality holds: 1 (Sˆ xk , Sˆ x−k ) ≤ Bk β

(3)

where 1 Sˆ xk = √



|Λ|

eik·x Sxx

x∈Λ

and (A, B) is two-point Duhamel function (DTP) [8]. 2. The average of the following double commutator is bounded from above:

|⟨[Sˆ xk , [H , Sˆ x−k ]]⟩| ≤ Ck .

(4)

Then for arbitrary β > βc , where βc is defined by an equality





1 2(2π )d

d



d k Bk Ck coth B

βc 2



Ck Bk

 =1

(5)

(where B = [−π , π]d ) the system exhibits LRO, i.e. the inequality (2) holds. Remarks. (i) The theorem above covers both spin systems and rotors. Of course, the bounds Bk i Ck are model-dependent, and should be calculated separately in every case. We will not review the version for quantum spins, as it is discussed exhaustively in Refs. [8–10]. Instead, we pass to consideration of system of rotors. (ii) The most difficult is point 1. It is the point where the Reflection Positivity plays a crucial role. The remaining point 2 is more easy. In the remaining part of this section we will prove the Theorem above, following Pastur and Khoruzhenko [6]. As a result, we will obtain inequalities (3)–(5) together with concrete forms of bounds Bk , Ck . We begin from the DLS lemma—the matrix inequality, being the basic technical tool. 3.2. Technical tool: the Dyson–Lieb–Simon lemma Theorem (The Special Case of Lemma 4.1 by Dyson et al. [8]). Let H1 be the finite dimensional Hilbert space and H = H1 ⊗ H1 . Let A, B, . . . be operators defined on H1 . Denote: A+ = A ⊗ Id,

(6)

A− = Id ⊗ A.

(7)

and

Then for arbitrary self-adjoint operators in the real matrix representations A, C1 , . . . , Cl and arbitrary collection of real numbers h1 , . . . , hl we have







l  Tr exp A + A − (Ci+ − Ci− − hi )2

+

−

i =1

For the proof, see Ref. [8].







 +

−

≤ Tr exp A + A −

l  i=1

 (Ci − Ci ) +

− 2

.

(8)

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Fig. 1. Division of the system into two identical subsystems Λ1 , Λ2 by the symmetry plane Π .

Remarks. 1. The minus sign at square terms containing Ci operators is crucial. Moreover, it is a trivial observation that one can replace any of the square terms −(Ci+ − Ci− )2 by −γi (Ci+ − Ci− )2 , where γi is a positive constant. 2. The DLS lemma covers the case where the dimension of H1 is finite. In the case of interacting rotors, we need a suitable generalization of this lemma to the infinite-dimensional case. It has been done in Ref. [6] using the theory of Wiener processes. It turns out that one can give a simpler proof [11]. 3.3. Reflection positivity for rotors 3.3.1. Proof of the inequality (3) Let us rewrite the original Hamiltonian (1) in a slightly different manner. The interaction term can be written as cos(φx − φy ) = cos φx cos φy + sin φx sin φy

=−

1

 (cos φx − cos φy )2 + (sin φx − sin φy )2 + 1.

2

So the rotor Hamiltonian takes the form (modulo unimportant constant term): H =−

1  ∂2 2I x∈Λ ∂φ

2 x

+

J  2 ⟨xy⟩

 (cos φx − cos φy )2 + (sin φx − sin φy )2 .

(9)

Now, let us introduce the more general Hamiltonian: H ( h) = −

1  ∂2 2I x∈Λ ∂φ

2 x

+

J  2 ⟨xy⟩

(cos φx − hx − cos φy + hy )2 + (sin φx − sin φy )2



(10)

where h ≡ {hx } is the set of arbitrary real numbers, or – equivalently – an arbitrary real function, defined on Λ. Let Z (h) denote the partition function for the Hamiltonian defined by (10). Let us divide the system into two identical halves Λ1 , Λ2 so that Λ = Λ1 ∪ Λ2 ; Λ1 is a mirror image of Λ2 under reflection in the Π plane which does not contain sites, and vice versa. (See Fig. 1 for an illustration). In a more explicit manner: Let us choose one of the coordinate axes (say, the 1 , where N is the number of sites 1-st one). Let us number the 1-st coordinates of lattice sites as: −N2+1 , . . . , − 12 , 12 , . . . , N − 2 d on the cube edge (so |Λ| = N ; remember we assume that N is an even number). This way, the subset Λ1 contains all sites, whose first coordinate is negative, and Λ2 -sites, whose first coordinate is positive. The plane of symmetry Π is a hyperplane, defined by the equation x1 = 0. The Hilbert space H of states of all systems is a tensor product of two identical spaces: H = H1 ⊗ H1 , where Hi is a space of states of the subsystem defined on Λi . Now, let us write the Hamiltonian in the form corresponding to the division of Λ described above: HΛ = H1 + H2 + HI ,

(11)

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J. Wojtkiewicz / Physica A 391 (2012) 5918–5925

where: Hi = −

1  ∂2 2I x∈Λ ∂φ i

2 x

+

J



2 ⟨xy⟩:x,y∈Λ i

  (cos φx − hx − cos φy + hy )2 + (sin φx − sin φy )2 ,

(12)

(i = 1, 2), and HI =

J



2 ⟨xy⟩:x∈Λ ,y∈Λ 1 2

  (cos φx − hx − cos φy + hy )2 + (sin φx − sin φy )2 .

(13)

Now, let us denote: A+ = −β H1 ,

A− = −β H2 ,

Cx+ = cos φx ,

Cy− = cos φy ,

γx = β J

(14)

where indices at C operators are x, y instead of i as in the formula (8). Moreover, y is an image of x under reflection with the symmetry plane Π . Observe now that the form of the Hamiltonian given by formulas (11)–(13) fulfills all assumptions of the DLS lemma if −β J < 0, i.e. J > 0. (This is the reason that we consider only ferromagnetic coupling of rotors.) For operators A+ , A− , Cx+ , Cx− defined above by (14), an inequality (8) holds [6,11]. Now, using the following facts: 1. We assume periodic boundary conditions. This implies that we have sufficiently many possibilities in choosing the symmetry plane Π . More precisely, we have Nd symmetry planes, where N d = |Λ|. 2. Generalized DLS lemma. 3. Standard RP arguments [6,8]. One obtains the main inequality: ZΛ (h) ≤ ZΛ (0)

(15)

for an arbitrary h function. Expanding the inequality (15) around h = 0 one obtains

  ZΛ (λh) dλ d

= 0 or equivalently

λ=0

d2

 

Z (λh) 2 Λ

dλ

  FΛ (λh) dλ d

=0

(16)

λ=0

≤ 0 or equivalently λ=0

d2 dλ

 

F (λh) 2 Λ

≥0

(17)

λ=0

(in formulas (16) and (17), FΛ denotes the free energy: FΛ = − ln ZΛ /β ). An inequality (17) implies that for an arbitrary function h the following inequality holds:



   1  (cos φx − cos φy )(hx − hy ), (cos φx − cos φy )(hx − hy ) ≤ |hx − hy |2 ; β J ⟨xy⟩ ⟨xy⟩ ⟨xy⟩

(18)

where (·, ·) at l.h.s. denotes the Duhamel two-point function (DTF) [8]. Next, let us take first: hx = √1|Λ| Re eik·x , and then: hx = √1|Λ| Im eik·x . Summing both sides of these inequalites, we obtain desired inequality (3), where the function Bk is Bk =

1 2JE (k)

,

(19)

where in turn E (k) =

d  (1 − cos kj ). j =1

So we have proved the first point, namely the inequality (3).

(20)

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3.3.2. Proof of the inequality (4) Now, we have to calculate the average of double commutator (4). It is a simple task. First, we have

[Sˆx k , [Tx , Sˆx −k ]] = [Sxx , [Tx , Sxx ]] (the derivative appearing in the kinetic energy operator acts on angles and does not affect the exponential terms eikx ). Now, for arbitrary function F (φx ) we have:

2 1  ′ F (φx ) . (21) 2I Take now F (φx ) = cos(φx ). We see that the double commutator is a bounded function—the upper bound is 1. So its average is bounded too and we can take simply [F (φx ), [Tx , F (φx )]] =

1

C (k) =

(22)

2I as an upper bound of the double commutator. 3.3.3. Final of the proof of existence of LRO





x Assume now that we are given the bounds Bk and Ck . It has been shown in Refs. [6,8] that the average Sˆkx Sˆ− k can be

estimated in the following manner in terms of Bk and Ck :



x Sˆkx Sˆ− k

1 ≤ Bk Ck coth 2



   β Ck 2

Bk

,

k ̸= 0.

(23)

This inequality applies to both quantum spins and rotors. Now, inserting bounds (19) and (22) to the inequality (23), we get



1



x Sˆkx Sˆ− k ≤



 

1

coth β

4IJE (k)

2

JE (k)

 ≡

I

1 2

Gβ (k).

(24)

In the formula above, we have denoted

 Gβ (k) =

 

1 4IJE (k)

coth β

JE (k) I

 .

(25)

Now, with the use of the Parseval/Plancherel identity, we obtain    x x 1 1  x 2 x |Λ| = (Sx ) + (Sxy )2 = Sˆk Sˆ−k = |Sˆ0x |2 + Sˆkx Sˆ− k. 2 2 x∈Λ k k̸=0 As we have (by symmetry)    x 2  1 1   x 2 (Sx ) + (Sxy )2 = (Sx ) = |Λ| 2 x∈Λ 2 x∈Λ and



1 

|Λ|

2 Sxx

=

x∈Λ

1

|Λ|

|Sˆ0x |2 =

1 2

−

1 

|Λ|

x Skx S− k,

k̸=0

then we obtain a lower bound for order parameter:



1 

|Λ|

2  Sxx

=

x∈Λ



1

−

2

1 

|Λ|

 x Skx S− k

k̸=0

≥

1 2

 1−

1 

|Λ|

 Gβ (k) .

(26)

k̸=0

The sum appearing on the r.h.s. of the (26) can be approximated by the integral Id (β) =

1

(2π )d

 [−π,π ]d

dd k Gβ (k)

(27)

with the error tending to zero as the lattice size grows to infinity. An elementary examination of the Id (β) integral shows that it is divergent in dimensions 1 and 2, and that it is convergent for d ≥ 3. Moreover, it is monotonically decreasing in β . Equating the r.h.s. of (26) to zero, we obtain (in the thermodynamic limit) the condition for the upper bound of inverse critical temperature βc . It is given by the solution of the following transcendental equation: Id (βc ) ≡

1

(2π )d

 [−π,π ]d

dd k Gβc (k) = 1.

This way, we have proven Theorem 1 for rotors together with concrete estimation (28) of the critical temperature.

(28)

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4. Ordering in the ground state of d = 2 rotors An inequality (26) is valid for arbitrary β . Let us examine the limit β → ∞. Using the elementary inequality: 1

coth x ≤ 1 +

x

for x > 0

we obtain the following estimate for Gβ (k): Gβ (k) =

√

 

1

2 IJE (k)

coth β

JE (k) I





1

≤ √ √ 2 IJ

E (k)

1+

1

β



I JE (k)

 .

(29)

Take now the thermodynamic limit in the following order: limΛ→∞ limβ→∞ . Taking the limit limβ→∞ first, we see that if we take the sum over k in the function Gβ (k), then the sum of the last term in (29) will vanish—as it is a finite sum and is multiplied by the factor β1 . Let us apply this observation to the inequality (26). For β → ∞, the thermal average becomes the average over the ground state. Taking now the limit Λ → ∞, we obtain:



1 

|Λ|

2  Sxx

≥

x∈Λ

g .s.

1



2

1 1 1− √ 2 IJ (2π )d



1 dd k √ E (k)



.

(30)

Remarks 1. The result depends crucially on the ordering of limits: If we take the limit Λ → ∞ first, then we will obtain the trivial estimation. It is so because the I1 (β) and I2 (β) integrals are divergent for arbitrary β . So the limit β → ∞ of the Eq. (26) is trivial (and useless), too. Remark 2. Analogous trick has been used in [9,10] in the proof of LRO in the ground state 2d XXZ models. There exist also more direct method to prove ordering in the ground state of XXZ and Heisenberg models [12,13]. Let us denote the integral appearing at the r.h.s. of (30) as Id : Id =

1

(2π )d



1 dd k √ . E (k)

(31)

From the formula (30) we can deduce easily the sufficient condition for ordering. This condition can be formulated as Theorem 2. The ground state of the system of interacting rotors exhibits LRO if the r.h.s. of the formula (30) is greater than zero, i.e. if the moment of inertia of rotor and the coupling constant fulfill the condition 1 1 − √ Id ≥ 0, 2 IJ

or

IJ ≥

Id2 4

.

(32)

Remarks 3. The integral appearing in (32) is divergent in d = 1, so one could expect that the chain of rotors does not exhibit LRO—see Ref. [1]. In d ≥ 2, the integral is convergent. Its approximate value is I2 ≈ 0.909. So if the quantity IJ fulfills the condition (32) then the system of two-dimensional rotors will exhibit LRO in the ground state. (For d > 2, it is known from earlier papers [6].) 5. Summary, directions of further investigations The result of this paper is that if the quantity IJ is sufficiently large (larger than certain computable constant, given by the inequality (32)), then the ground state of interacting rotors exhibits the long-range order. One can ask question on the ordering of the ground state in the opposite situation, i.e. for quantity IJ being small. To my best knowledge, this is an open question. One can suspect that the LRO should be absent. Such an expectation is motivated by results of the paper [14], where a somewhat similar system has been analyzed, namely, the anharmonic crystal model. It is known that in 3d there is LRO in this model in the case of temperature sufficiently small and the oscillator mass being large [6]. On the other hand, in Ref. [14] it has been proved that the ground state of the anharmonic crystal is not ordered for all temperatures (including T = 0) if the mass of the oscillator is sufficiently small. One can expect similar behavior also in the rotor model. One can prove also the presence of the orderings in multidimensional rotors (where Sx ∈ Sν , ν > 1) by methods similar to those used in this paper, provided that the interaction takes the form for which it is possible to apply Reflection Positivity arguments.

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In the description of real systems, one often encounters much more complicated interactions than considered in this paper. For instance, in the Hamiltonian describing hydrogen particles adsorbed on a boron nitride surface, one should take into account dipolar and quadrupolar competing interactions, geometric frustration etc. [4]. Rotor systems with such interactions seem much harder to examine than the system considered in this paper. The author’s expectation is that they are at least as difficult to analyze as frustrated spin systems [15], where the number of rigorous results is very small. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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