Correlation Inequalities for Classical and Quantum XY Models

Costanza Benassi, Benjamin Lees, and Daniel Ueltschi

Abstract We review correlation inequalities of truncated functions for the classical and quantum XY models. A consequence is that the critical temperature of the XY model is necessarily smaller than that of the Ising model, in both the classical and quantum cases. We also discuss an explicit lower bound on the critical temperature of the quantum XY model.

Keywords Classical XY model • Correlation inequalities • Lattice systems • Quantum XY model • Spin systems

1 Setting and Results

The goal of this survey is to recall some results of old that have been rather neglected in recent years. We restrict ourselves to the cases of classical and quantum XY models. Correlation inequalities are an invaluable tool that allows to obtain the monotonicity of spontaneous magnetisation, the existence of infinite volume limits, and comparisons between the critical temperatures of various models. Many correlation inequalities have been established for the planar rotor (or classical XY) model, with interesting applications and consequences in the study of the phase diagram and the Gibbs states [1–7]. Some of these inequalities can also be proved for its quantum counterpart [8–11]. Let  be a finite set of sites. The classical XY model (or planar rotor model) is a model of interacting spins on such a lattice. The configuration space of the system is 1 defined as ˝ Dffxgx2 W x 2 S 8x 2 g: each site hosts a unimodular vector lying on a unit circle. It is convenient to represent the spins by means of angles,

C. Benassi () • D. Ueltschi Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK e-mail: [email protected]; [email protected] B. Lees Department of Mathematics, University of Gothenburg, Gothenburg, Sweden e-mail: [email protected]

© Springer International Publishing AG 2017 15 A. Michelangeli, G. Dell’Antonio (eds.), Advances in Quantum Mechanics, Springer INdAM Series 18, DOI 10.1007/978-3-319-58904-6_2 16 C. Benassi et al. namely

 1  x D cos x (1)  2  x D sin x (2) with x 2 Œ0; 2. The energy of a configuration  2 ˝ with angles  Dfxgx2 is X Y Y cl./ 1  1 2  2; H D JA x C JA x (3) A x2A x2A

i R  ˇ with JA 2 for all A  . The expectation value at inverse temperature of a functional f on the configuration space is Z 1 ˇ cl ./ cl   H ./; h f i;ˇ D cl d e f (4) Z;ˇ R R R R ˇ cl./ 2 2 where Zcl D de H is the partition function and d D ::: Q ;ˇ 0 0 dx x2 2 . 1 We now define the quantum XY model. We restrict ourselves to the spin- 2 case. qu As before, the model is defined on a finite set of sites ; the Hilbert space is H D 2 2 i ˝x2C . The spin operators acting on C are the three hermitian matrices S , i D 1; 2; 3, that satisfy S1; S2 D iS3 and its cyclic permutations, and .S1/2 C .S2/2 C . 3/2 3 1 S D 4 . They are explicitly formulated in terms of Pauli matrices: Â Ã Â Ã Â Ã 1 1 01 2 1 0 i 3 1 10 S D ; S D ; S D : (5) 2 10 2 i 0 2 0 1

The hamiltonian describing the interaction is X Y Y qu 1 1 2 2 ; H D JA Sx C JA SA (6) A x2A x2A

i i 1 i with Sx D S ˝ nfxg.ThefJAgA are nonnegative coupling constants. The Gibbs state at inverse temperature ˇ is

1 ˇ qu O qu O  H ; h i;ˇ D qu Tr e (7) Z;ˇ

qu qu ˇH qu with Z;ˇ D Tr e the partition function and O any operator acting on H . The first result holds for both classical and quantum models. Correlation Inequalities for Classical and Quantum XY Models 17

1; 2 0  Theorem 1.1 Assume that JA JA  for all A  . The following inequalities hold true for all X; Y  , and for all ˇ>0. D E D E D E Y Y cl Y cl Y cl Classical W  1  1   1  1  0; x x ;ˇ x ;ˇ x ;ˇ x2X x2Y x2X x2Y D E D E D E Y Y cl Y cl Y cl  1  2   1  2 Ä 0: x x ;ˇ x ;ˇ x ;ˇ x2X x2Y x2X x2Y D E D E D E Y Y qu Y qu Y qu Quantum W S1 S1  S1 S1  0; x x ;ˇ x ;ˇ x ;ˇ x2X x2Y x2X x2Y D E D E D E Y Y qu Y qu Y qu S1 S2  S1 S2 Ä 0: x x ;ˇ x ;ˇ x ;ˇ x2X x2Y x2X x2Y

In the quantum case, similar inequalities hold for Schwinger functions, see [11] for details. The proofs are given in Sects. 3 and 4 respectively. These inequalities are known as Ginibre inequalities—first introduced by Griffiths for the Ising model [12] and systematised in a seminal work by Ginibre [13], which provides a general framework for inequalities of this form. Ginibre inequalities for the classical XY model have then been established with different techniques [1, 3–5, 13]. The equivalent result for the quantum case has been proved with different approaches [8–11]. An extension to the ground state of quantum systems with spin 1 was proposed in [11]. A straightforward corollary of this theorem is monotonicity with respect to coupling constants, as we see now. 1; 2 0  ;  Corollary 1.2 Assume that JA JA  for all A  . Then for all X Y  , and for all ˇ>0

@ Y 1 cl Classical W h  i;ˇ  0; @J1 x Y x2X @ Y 1 cl h  i;ˇ Ä 0: @J2 x Y x2X @ Y Quantum W h S1iqu  0; @J1 x ;ˇ Y x2X @ Y h S1iqu Ä 0: @J2 x ;ˇ Y x2X

Interestingly this result appears to be not trivially true for the quantum Heisen- berg ferromagnet. Indeed a toy version of the fully SU(2) invariant model has been provided explicitly, for which this result does not hold (nearest neighbours 18 C. Benassi et al. interaction on a three-sites chain with open boundary conditions) [14]. The question whether this result might still be established in a proper setting is still open. On the other hand, Ginibre inequalities have been proved for the classical Heisenberg ferromagnet [1, 2, 4]. Monotonicity of correlations with respect to temperature does not follow straightforwardly from the corollary. This can nonetheless be proved for the classical XY model. 1 2  Theorem 1.3 Classical model: Assume that JA jJAj for all A  , and that 2 0 ;  JA D whenever jAj is odd. Then for all A B  , we have D E @ Y cl  1  0: @ˇ x ;ˇ x2B

cl Let us restrict to the two-body case and assume that H is given by X   cl  1 1   2 2 : H D Jxy x y C xy x y x;y2

Then if jxyjÄ1 for all x; y,

@ Y h  1icl  0: (8) @J z ;ˇ xy z2A

Notice that this theorem has a wider range of applicability than Corollary 1.2:in the theorem above, the coupling constants along one of the directions are allowed to be negative (though not too negative), while in the corollary the nonnegativity of all coupling constants a is necessary hypothesis. This result has been proposed and discussed in various works [4, 6, 13]—see Sect. 3 for the details. Unfortunately we lack a quantum equivalent of these statements. We conclude this section by remarking that correlation inequalities in the quan- tum case can be applied also to other models of interest. For example, we consider a certain formulation of Kitaev’s model (see [15] for its original formulation and [16] 2 for a review of the topic). Let   Z be a square lattice with edges E. Each edge Kitaev 2 of the lattice hosts a spin, i.e. the Hilbert space of this model is H D˝e2E C . The Kitaev hamiltonian is X Y X Y Kitaev 1 3; H D Jx Se C JF Se (9) x2 e2EW F eF x2e where F denotes the faces of the lattice, i.e. the unit squares which are the building Z2 ; i i 1 blocks of , Jx JF are ferromagnetic coupling constants and Se D S ˝ Ene. Kitaev H has the same structure as hamiltonian (6) so Ginibre inequalities apply as Correlation Inequalities for Classical and Quantum XY Models 19 well. It is not clear, though, whether this might lead to useful results for the study of this specific model. Another relevant model is the plaquette orbital model that was studied in [17, 18]; ; i i interactions between neighbours x y are of the form SxSy, with i being equal to 1 or 3 depending on the edge.

2 Comparison Between Ising and XY Models

We now compare the correlations of the Ising and XY models and their respective Is  critical temperatures. The configuration space of the Ising model is ˝ Df1; 1g , that is, Ising configurations are given by fsxgx2 with sx D˙1 for each x 2 .We Is consider many-body interactions, so the energy of a configuration s 2 ˝ is X Y HIs .s/ D J s I (10) ;fJAg A x A x2A we assume that the system is “ferromagnetic”, i.e. the coupling constants JA  0 are nonnegative. The Gibbs state at inverse temperature ˇ is

1 X Is Is ˇH h f i D f .s/ e ;fJAg ; (11) ;fJAg;ˇ Is Z;fJ g;ˇ Is A s2˝

P ˇ Is Is Is  H;fJ g with f any functional on ˝ and Z D ˝Is e A is the partition ;fJAg;ˇ s2  function. The following result holds for both the classical [5] and the quantum case [9, 19]. 1; 2 0   Theorem 2.1 Assume that JA JA  for all A  . Then for all X  and all ˇ>0, Y Y  1 cl Is : Classical: h x i;ˇ Äh sxi; 1 ;ˇ fJAg x2X x2X Y Y 1 qu 2jXj Is ; Quantum: h Sxi;ˇ Ä h sxi;  ;ˇ fJA g x2X x2X

 2jAj 1 with JA D JA. A review of the proof of the classical case is proposed in Sect. 3. In the quantum 1 case, this statement for spin- 2 is a straightforward consequence of Corollary 1.2,but interestingly this result has been extended to any value of the spin [19]. We review the proof of this general case in Sect. 4. 20 C. Benassi et al.

1 We now consider the case of spin- 2 and pair interactions, that is, the hamiltonian is X qu . 1 1 2 2/: H D Sx Sy C Sx Sy (12) x;y2

We define the spontaneous magnetisation m.ˇ/ at inverse temperature ˇ by 1 X .ˇ/2 1 1 qu : m D lim inf 2 hSxSyi;ˇ (13) %Zd jj x;y2

qu qu We define the critical temperature for the model Tc D 1=ˇc as ˚ « ˇqu ˇ>0 .ˇ/ 0 ; c D sup W m D (14)

qu where ˇc 2 .0; 1. A consequence of Theorem 2.1 is the following. Corollary 2.2 The critical temperatures satisfy

qu 1 Ising: Tc Ä 4 Tc

The critical temperature of the Ising model in the three-dimensional cubic lattice Ising 4:511 0:001 cl has been calculated numerically and is Tc D ˙ [20]. It is Tc D qu 2:202˙0:001 [21] for the classical model and Tc D 1:008˙0:001 for the quantum model (S. Wessel, private communication). A major result of mathematical physics is the rigorous proof of the occurrence of long-range order in the classical and quantum XY models, in dimensions three and higher, and if the temperature is low enough [22, 23]. The method can be used to provide a rigorous lower bound on critical temperatures; the following theorem concerns the quantum model. Theorem 2.3 For the three-dimensional cubic lattice, the temperature of the quantum XY model satisfies

qu 0:323: Tc 

The best rigorous upper bound on the critical temperature of the three- Ising dimensional Ising model is Tc Ä 5:0010 [24]. Together with the above corollary and theorem, we get

0:323 qu 1 Ising 1:250: Ä Tc Ä 4 Tc Ä (15)

Proof (Theorem 2.3) We consider the XY model with spins in the 1–3 directions for convenience. We make use of the result [25, Theorem 5.1], that was obtained with Correlation Inequalities for Classical and Quantum XY Models 21 the method of reflection positivity and infrared bounds [22, 23]. Precisely, we use Eqs. (5.54), (5.57) and (5.63) of [25]. 8 q < 1 3 1 3  J hS S1 iqu  K 2 4 2 0 eq1 ˇ m.ˇ/  0 (16) : 1 1 qu I3 1 1 qu K3 hS0Se1 i  2 hS0Se1 i  ˇ

0 where e1 is a nearest neighbour of the origin, and J3; I3; K3; K3 are real numbers coming from explicit integrals. Their values are J3 D 1:15672; I3 D 0:349884; 0 K3 D 0:252731;andK3 D 0:105107. Notice that ˇ is rescaled by a factor 2 with respect toq [25], due to a different choice of coupling constants in the hamiltonian. 1 1 qu Let x D hS0Se1 i ; since we do not have good bounds on x, we treat it as an unknown. The magnetisation m.ˇ/ is guaranteed to be positive if x Ä t where t is 0 1 K3 J3 2 I3 K3 the zero of 4  ˇ  2 x;orx  rC,whererC is the largest zero of x  2 x  ˇ . At least one of these holds true when rC < t,thatis,when r Á 2 4 0 1 I3 I3 K3 < 1 1 2K3 2 2 C 4 C ˇ 2J3  ˇ J3 (17)

This is the case for 1=ˇ < 0:323 giving the upper bound Tc  0:323.

3 Proofs for the Classical XY Model

The proofs require several steps and additional lemmas. The following paragraphs  are devoted toQ a complete study of their proofs. Given local variables f xgx2,we  i  i  denote A D x2A x for A  .

3.1 Griffiths and FKG Inequalities, and Proof of Theorem 1.1

We start with Theorem 1.1. We describe the approach proposed in [1, 5], and use a similar notation. Their framework relies on some well known properties of the Ising model and on the so called FKG inequality.

Lemma 3.1 (Griffiths Inequalities for the Ising Model) Let f andQ g be ˝Is functionals on  such that they can be expressed as power series of x2A sx, A   with positive coefficients. Then

h f iIs  0I ;fJAg;ˇ h fgiIs hf iIs hgiIs : ;fJAg;ˇ ;fJAg;ˇ ;fJAg;ˇ 22 C. Benassi et al.

We do not provide the proof of this result—see [12, 13] for the original formulation and [26] for a modern description. An immediate consequence is the following. Corollary 3.2 Given f with the properties in Lemma 3.1, we have for any A  

@ h f iIs  0: ;fJAg;ˇ @JA

Another result which is very useful in this framework is the so called FKG  N inequality. We formulate it in a specific setting. Let IN D 0; 2 for some N 2 N. Any 2 IN is then a collection of angles D . 1;:::; N /. It is possible to introduce a partial ordering relation on IN as follows: for any ; 2 IN , Ä  if and only if i Ä i for all i 2f1;:::;Ng. A function f on IN is said to be increasing (or decreasing) if Ä  implies f . / Ä f ./ (or f . /  f ./)forall ; 2 IN . The following result holds. Q . / . / N . / Lemma 3.3 (FKG Inequality) Let d D p iD1d  i be a normalised  measure on IN, with d . i/ a normalised measure on 0; 2 ,p. /  0 for all 2 IN and

p. _ /p. ^ /  p. /p./; (18) where . _ /i D max. i;i/ and . ^ /i D min. i;i/. Then for any f and g increasing (or decreasing) functions on IN Z Z Z fgd  fd gd :

The inequality changes sign if one of the functions is increasing and the other is decreasing. We also skip the proof of this statement. We refer to [27] for the original result, to [5, 28] for the formulation above, and [26] for its relevance in the study of the Ising model. Before turning to the actual proof of the theorem, we introduce another useful lemma.

Lemma 3.4 Let fqxgx2 be a collection of positive increasing (decreasing) func- 0;  ; I  tions on 2 . Then for any 2 jj and any A  ,

qA. _ / C qA. ^ /  qA. / C qA. /:

We do not provide the proof here, see [5, 28] for more details. We can now discuss the proof of Theorem 1.1. Correlation Inequalities for Classical and Quantum XY Models 23

Proof (Theorem 1.1) Since the temperature does not play any rôle in this section, we set ˇ D 1 in the following and we drop any dependency on it. The main idea of the proof is to describe a classical XY spin as a pair of Ising spins and an angular 1 variable. The new notation for x 2 S is

 1 . / ; x D cos x Ux (19)  2 . / ; x D sin x Vx (20) ; 1; 1  . ;:::; / I with Ux Vx 2f g for all x 2 and D x1 x 2 jj. With this cl notation, it is possible to express H of Eq. (3) as the sum of two Ising hamiltonians with spins fUxgx2, fVxgx2 respectively: ! X Y Y cl . ; ; / 1 . / 2 . / H U V D JA cos x UA C JA sin x VA (21) A x2A x2A Is . / Is . /: D H; . / 1 U C H; . / 2 V (22) fcos AJAg fsin AJAg Q Q 1 . / . / 2 . / . / Let us introduce the notation: JA x2A cos x D JA , JA x2A sin x D K A , R R  R  Q 2 ::: 2 2 d D 0 0 x2  d x.Then R Is Is Is d Z Z cos. /X cos. /Y hUXUY i 1 1 ;fJA. /g ;fK A. /g ;fJA. /g h  i cl D R X Y H d ZIs ZIs ;fJA. /g ;fK A. /g R Is Is . / Is . / Is d Z; . / Z; . / cos XhUXi; . / cos Y hUY i; . /  fJA g fK AR g fJA g fJA g : d ZIs ZIs ;fJA. /g ;fK A. /g

The inequality above follows from Lemma 3.1. Moreover R Is Is Is Is d Z Z cos. /XhUXi sin. /Y hVY i 1 2 ;fJA. /g ;fK A. /g ;fJA. /g ;fK A. /g h  i cl D R : X Y H d ZIs ZIs ;fJA. /g ;fK A. /g

cos. /X and sin. /X are respectively decreasing and increasing on Ijj for any Is X  . Let us now consider hUXi . By Corollary 3.2, it is a decreasing ;fJA. /g I function on Ijj for any X  , since the coupling constants of H are ;fJA. /g Is decreasing in . Analogously, hVXi is an increasing function on Ijj for ;fK A. /g any X  . Theorem 1.1 is then a simple consequence of Lemma 3.3, with 2 d . x/ D  d x and

Is Is Z;f . /gZ;f . /g p. / D R JA K A : (23) d ZIs ZIs ;fJA. /g ;fK A. /g 24 C. Benassi et al.

The last step missing is to show that p. / defined as above fulfills hypothesis (18) of Lemma 3.3. This amounts to showing

ZIs ZIs  ZIs ZIs I (24) ;fK A. _ /g ;fK A. ^ /g ;fK A. /g ;fK A. /g ZIs ZI  ZIs ZIs : (25) ;fJA. _ /g ;fJA. ^ /g ;fJA. /g ;fJA. /g

Since the arguments to prove these inequalities are very similar, we prove explicitly only the first one. Equation (24) is equivalent to

! 1 ! Is  Is Z;f . /g Z;f . _ /g K A K A  1 (26) ZIs ZIs ;fK A. ^ /g ;fK A. /g

Notice that ! ! 1 HIs Is Is ;fK . _ /K . /g Is Z; . / Z; . / h e A A i; . / fK A g fK A _ g D fK A g : Is Is HIs Z Z ;fK . /K . ^ /g Is ;fK A. ^ /g ;fK A. /g h e A A i ;fK A. ^ /g

Thanks to Lemma 3.4, the functions whose expectation value we are computing above fulfill the hypothesis of Lemma 3.1 and Corollary 3.2. Then, applying Lemma 3.4 and Corollary 3.2,

Is Is H Is H Is h e ;fK A. _ /K A. /g i he ;fK A. /K A. ^ /g i ;fK A. /g ;fK A. /g (27) Is H Is he ;fK A. /K A. ^ /g i : ;fK A. ^ /g

Hence p. / has the required property.

3.2 Proof of Theorem 1.3

Let us now turn to Theorem 1.3. We follow the framework described in [4, 13]. cl 1 2  Lemma 3.5 Let H be the hamiltonian defined in (3).IfJA jJAj for all A  2 0 and JA D for jAj odd, then there exist non negative coupling constants fKMgM2Z such that X cl H./ D KM cos .M  /; (28) M2Z P Z . ; ;:::; /   where, given M 2 ,MD m1 m2 m ,M D x2 mx x. Correlation Inequalities for Classical and Quantum XY Models 25

Proof The statement follows from the two following identities: 1 . / . / . . / . //; cos cos D 2 cos  C cos C (29) 1 . / . / . . / . //; sin sin D 2 cos   cos C (30)

8 ; 2 Œ0; 2. A necessary step for this lemma and for Theorem 1.3 is duplication of variables [13]: we consider two sets of angles (i.e. spins) on the lattice instead of just one, and denote them by fxgx2 and fNxgx2. The hamiltonian for the fNxg is X   N cl./N N1 1 N2 2 H D JA NA C JA NA A X D KN M cos.M  /:N (31) M2Z

 N Ni Here, fNxg are related to f xg as in Eqs. (1)and(2). The JA are non negative coupling N1 N2 0 N constants with JA jJAj and fKMg are as in Lemma 3.5. A composite hamiltonian can be defined as

cl cl  HO .; /N DH./  HN ./N X     N N KM CKM . / . /N KM KM . / . /N D 2 cos M  C cos M  C 2 cos M   cos M  M2M (32)  In the following we always suppose KM  KN M for all M 2 Z .The expectation value of any functional f .; /N can be written as Z 1  N ˇHO .;/N .; /:N h f iHO ;ˇ D d d e f (33) ZH;ˇZHN ;ˇ

Lemma 3.6 Suppose f .; /N belongs to the cone generated by cos.M  / ˙ cos.M  /N ,M 2 Z, i.e. f can be written as product, sum or multiplication by a positive scalar of objects of that form. Then

0: h f iHO ;ˇ  (34)

Proof Firstly, notice that

Z Yn ddN .cos.Ms  / ˙ cos.Ms  /N  0 (35) sD1 26 C. Benassi et al.

 for any M1;:::;Mn 2 Z and any sequence of .˙/. This follows from

cos.M  / C cos.M  /N D 2 cos.M  ˚/cos.M  ˚/;N (36) cos.M  /  cos.M  /N D 2 sin.M  ˚/sin.M  ˚/;N (37)

˚ 1 . N / ˚N 1 . N / with i D 2 i C i and i D 2 i  i . The integral (35) can be formulated as

Z ÂZ Ã2 d˚d˚N F.˚/F.˚/N D d˚F.˚/  0; (38) with F.˚/ an appropriate product of sines, cosines and positive constants.

Let us now turn to hf iHO ;ˇ. Since the partition function is always positive, we can focus on Z ddNeˇHO .;/N f .; /:N (39)

By a Taylor expansion of eˇHO .;/N and by the properties of f , this can be expressed as a sum with positive coefficients of integrals in the form (35). Hence the nonnegativity of the expectation value. We have now all we need to prove Theorem 1.3. Proof (Theorem 1.3) In order to prove the first statement of the theorem we use the  1 formulation of the hamiltonian described in Lemma 3.5. Moreover, since A can be clearly expressed as the sum (with positive coefficients) of terms of the form cos.M  /, M 2 Z, it is enough to prove that for any M; N 2 Z

@ .M / cl @ hcos  i;ˇ KN (40) cl cl cl Dhcos.M  /cos.N  /i;ˇ hcos.M  /i;ˇhcos.N  /i;ˇ  0:

O  Consider now the hamiltonian H introduced above and hiHO ;ˇ the correspond- ing Gibbs state. From Lemma 3.6 we have    . / . /N . / . /N 0: h cos M   cos M  cos N   cos N  iHO ;ˇ  (41)

If we take the limit KN M % KM , we find twice the expression in Eq. (40). Hence the result. Let us now turn to the second statement of the theorem. In the case of two-body cl interaction H assumes the form (8), which, with a notation resembling the one introduced in Lemma 3.5 can be explicitly formulated as X cl./  .  / C .  / H D Kxy cos x  y C Kxy cos x C y (42) x;y2 Correlation Inequalities for Classical and Quantum XY Models 27 with J   K˙ xy 1  : xy D 2  xy (43)

˙ Z Clearly Kxy is analogous to the KM introduced in Lemma 3.5 for M 2 such that all its elements are zero except mx D 1; my D˙1.Thenwehave

@ 1 C xy @ 1  xy @ hAi cl D hAi cl C hAi cl : (44) @ H 2 @  H 2 @ C H Jxy Kxy Kxy

Due to Eq. (40) the expression above is the sum of two positive terms, hence it is positive.

3.3 Proof of Theorem 2.1

In this section we discuss the proof of Theorem 2.1 for the classical XY model. We use some of the concepts introduced in Sect. 3.2. The present proof has been proposed in [1, 5]. Proof (Theorem 2.1) As for the proof of Theorem  1.1, we express the XY spins 0;  by means of two Ising spins and an angle in 2 —see Eqs. (19)and(20)forthe explicit expression of the spins and (22) for the new formulation of the hamiltonian cl H. With the same notation: R Is Is Is d Z Z cos. /XhUXi 1 ;fJA. /g ;fK A. /g ;fJA. /g h i cl D R X H d ZIs ZIs ;fJA. /g ;fK A. /g R Is Is Is d Z; . / Z; . / max 2Ijj hUXi; . / (45) Ä fJA R g fK A g fJA g d ZIs ZIs ;fJA. /g ;fK A. /g Is : DhUAi; 1 fJAg

4 Proof for the Quantum XY Model

We now discuss the proof of Theorem 1.1 in the quantum case. This theorem has been proved for pair interaction in [8], and it has been proposed independently in various works for more generic interactions [9–11]. We describe here the simpler approach proposed in [11]. Since the temperature does not play any role from now on, we set ˇ D 1 and omit any dependencyQ on it in the following. As for the classical i i case we introduce the notation SA D x2A Sx. 28 C. Benassi et al.

Proof (Theorem 1.1) For the proof it is convenient to perform a unitary transforma- tion on the hamiltonian (6) and consider its version with interactions along the first and third directions of spin, namely X qu 1 1 3 3 ; H D JASA C JASA (46) A

3 2  with JA D JA for all A  . The proof of this theorem uses some techniques similar to the ones introduced for the classical Theorem 1.3. These were indeed introduced by Ginibre [13]ina general framework. As for the classical case, it is convenient to duplicate the model. We introduce a new doubled Hilbert space HN D H ˝ H. Given an operator O acting on H we define two operators acting on HN,

O˙ D O ˝ 1 ˙ 1 ˝ O: (47)

qu The hamiltonian we consider for the doubled system is H;C: X qu qu 1 1 qu 1. 1 / 3. 3 / H;C D H ˝  C  ˝ H D JA SA C C JA SA C (48) A

The Gibbs state is denoted as

1 qu H;C ; hhOii D qu 2 Tr O e (49) .Z / for any operator O acting on HN. It follows from some straightforward algebra that 1 OP qu O qu P qu O P h i h i h i D 2hh  iiI (50) 1 .OP/ .O P O P /; ˙ D 2 C ˙ C   (51) for any O, P operators on H. 2 2 2 Just as C constitutes the “building block” for H,soC ˝ C is to HN.We C2 C2 1 1 3 3 can provide an explicit basis of ˝ such that SC, S, SC, S have all non negative elements: 1 1 j iDp .jCCiCji/; j iD p .jCCiji/; (52) C 2  2 1 1 j iDp .jCiCjCi/; j iDp .jCijCi/: (53) C 2  2 Correlation Inequalities for Classical and Quantum XY Models 29

Above by jCi and ji we denote the basis of C2 formed by eigenvectors of S3 with 1 1 ; eigenvalues 2 and  2 respectively, and ji jiDjii˝jji. It can be easily checked that the basis above has the required property. This result implies straightforwardly HN . 1/ . 1/ . 3/ . 3/ that there exists a basis of  such that Sx C, Sx , Sx C and Sx  have non negative element for all x 2 . Let us consider the truncated correlation function we are interested in: D E D E D E Y Y qu Y qu Y qu     S1 S1  S1 S1 D 1 hh S1 S1 ii: (54) x x  x  x  2 X  Y  x2X x2Y x2X x2Y

We can evaluate the right hand side of the equation above by a Taylor expansion:     X 1     .Zqu/2hh S1 S1 ii D Tr S1 S1 .Hqu /n (55)  X  Y  nŠ X  Y  ;C n0

qu Given the formulation of H;C as in Eq. (48) and relation (51), it is clear that it can be expressed as a polynomial with positive coefficients of operators with . 1 / . 1 / nonnegative elements. The same holds for SX  and SY . The trace of operators with nonnegative elements is non negative, hence the first inequality of the theorem. 2 The second inequality can be proved precisely in the same way (with SY substituted 3 . 3 / by SY ), by noticing that SY  has necessarily non positive elements. Let us now turn to Theorem 2.1. While in the classical case it is necessary to introduce an artificial framework, interestingly the proof for the quantum case does 1 not require such a construction. For spin- 2 the statement can be easily recovered by recalling that the classical Ising model can be recovered as a particular case of the quantum XY model (not of the classical one!). We review here a more general proof valid for any value of spin S [19]. Proof (Theorem 2.1) We reformulate the quantum Hamiltonian in order to have the interaction along the first and the third axis, as in Eq. (46).Weproveherethe following result, which is unitarily equivalent to the statement of the theorem: ˝ ˛ 3 qu S jXj Is : SX ;ˇ Ä hsAi; S jAj 3 ;ˇ (56) f JAg

From now on we set ˇ D 1 and drop all the dependencies on ˇ since it does not play i S 1 i any role. Let Sx D Sx be the rescaled spin operators. The models we compare are the following: X qu 1 1 3 3; H;S D JASA C JASA (57) A X Is 3 : H; 3 D JAsA (58) fJAg A 30 C. Benassi et al.

Clearly, (56) is equivalent to

3 qu Is : hSXi; ÄhsXi; 3 (59) S fJAg

This is what we aim to prove. Let us now build a composite system where each site of the lattice hosts a quantum degree of freedom and an Ising variable at the same H qu Is time. Let D H; C H; 3 ,i.e. S fJAg X H 1 1 3. 3/: D JASA C JA sA C SA (60) A

The Gibbs state is the natural one given the Gibbs states for the two separated 3 systems. We denote it by hi. We are interested in the expectation value hsX  SXi for some X  . Since the trace is invariant under unitary transformations, we can . 1; 2; 3/ . 1; 2; 3/ apply on each site the unitary Sx Sx Sx ! Sx sxSs sxSx and find X   X P 3 H 3 1 1 3 .1 3/  .1 / A JASACJAsA CSA Tr sX  SX e D Tr sX  SX e (61) s2˝ s2˝

The expression evaluated above is just the expectation value we are interested in multiplied by the partition function of the system—which is positive and therefore 3 not useful inP the evaluationQ of the sign of hsX  SXi. By a Taylor expansion and by nx 0 N   the property s2˝ x2A sx  with nx 2 for all x 2 and any A  ,itis clear that the expression above is nonnegative. This implies that

Is 3 qu 3 0: hsXi; 3 hSXi; DhsX  SXi  (62) fJAg S

This proves Eq. (59).

Acknowledgements The authors thank S. Bachmann and C.E. Pfister for useful comments.

References

1. H. Kunz, C.E. Pfister, P.A. Vuillermot, Correlation inequalities for some classical spin vector models. Phys. Lett. 54A, 428–430 (1975) 2. F. Dunlop, Correlation inequalities for multicomponent rotors. Commun. Math. Phys. 49, 247–256 (1976) 3. J.L. Monroe, Correlation inequalities for two-dimensional vector spin systems. J. Math. Phys. 16, 1809–1812 (1975) 4. J.L. Monroe, P.A. Pearce, Correlation inequalities for vector spin Models. J. Stat. Phys. 21, 615 (1979) 5. H. Kunz, C.E. Pfister, P.A. Vuillermot, Inequalities for some classical spin vector models. J. Phys. A Math. Gen. 9(10), 1673–1683 (1976) Correlation Inequalities for Classical and Quantum XY Models 31

6. A. Messager, S. Miracle-Sole, C. Pfister, Correlation inequalities and uniqueness of the equilbrium state for the plane rotator ferromagnetic model. Commun. Math. Phys. 58, 19–29 (1978) 7. J. Fröhlich, C.E. Pfister, Spin waves, vortices, and the structure of equilibrium states in classical XY model. Commun. Math. Phys. 89, 303–327 (1983) 8. G. Gallavotti, A proof of the Griffiths inequalities for the X-Y model. Stud. Appl. Math. 50, 89–92 (1971) 9. M. Suzuki, Correlation inequalities and phase transition in the generalised X-Y model. J. Math. Phys. 14, 837–838 (1973) 1 10. P.A. Pearce, J.L. Monoroe, A simple proof of spin- 2 X-Y inequalities. J. Phys. A Math. Gen. 12(7), L175 (1979) 11. C. Benassi, B. Lees, D. Ueltschi, Correlation inequalities for the quantum XY model. J. Stat. Phys. 164, 1157–1166 (2016) 12. R.B. Griffiths, Correlations in Ising ferromagnets. I. J. Math. Phys. 8, 478–483 (1967) 13. J. Ginibre, General formulation of Griffiths’ inequalities. Commun. Math. Phys. 16, 310–328 (1970) 14. C.A. Hurst, S. Sherman, Griffiths’ theorems for the ferromagnetic Heisenberg model. Phys. Rev. Lett. 22, 1357 (1969) 15. A.Y. Kitaev, Fault-tolerant quantum computation by anyons. Ann. Phys. 303(1), 2–30 (2003) 16. S. Bachmann, Local disorder, topological ground state degeneracy and entanglement entropy, and discrete anyons. arXiv:1608.03903 (2016) 17. S. Wenzel, W. Janke, Finite-temperature Néel ordering of fluctuations in a plaquette orbital model. Phys. Rev. B 80, 054403 (2009) 18. M. Biskup, R. Kotecký, True nature of long-range order in a plaquette orbital model. J. Statist. Mech. 2010, P11001 (2010) 19. P.A. Pearce, An inequality for spin-s X-Y ferromagnets. Phys. Lett. A 70(2), 117–118 (1979) 20. T. Preis, P. Virnau, W. Paul, J.J. Schneider, GPU accelerated Monte Carlo simulation of the 2D and 3D Ising model. J. Comput. Phys. 228(12), 4468–4477 (2009) 21. M. Hasenbusch, S. Meyer, Critical exponents of the 3D XY model from cluster update Monte Carlo. Phys. Lett. B 241(2), 238–242 (1990) 22. J. Fröhlich, B. Simon, T. Spencer, Infrared bounds, phase transitions and continuous symmetry breaking. Commun. Math. Phys. 50, 79–95 (1976) 23. F.J. Dyson, E.H. Lieb, B. Simon, Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. J. Stat. Phys. 18, 335–383 (1978) 24. J.O. Vigfusson, Upper bound on the critical temperature in the 3D Ising model. J. Phys. A Math. Gen. 18(17), 3417 (1985) 25. D. Ueltschi, Random loop representations for quantum spin systems. J. Math. Phys. 54(8), 083301 (2013) 26. S. Friedli, Y. Velenik, Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction. http://www.unige.ch/math/folks/velenik/smbook/index.html 27. C.M. Fortuin, P.W. Kasteleyn, J. Ginibre, Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22, 89–103 (1971) 28. C.J. Preston, A generalization of the FKG inequalities. Comm. Math. Phys. 36, 233–241 (1974)