Gauge Theory for Quantum Spin Glasses

J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by 54.167.13.156 on 09/16/22 Journal of the Physical Society of Japan Vol. 75, No. 1, January, 2006, 014001 #2006 The Physical Society of Japan Gauge Theory for Quantum Spin Glasses Satoshi M ORITA, Yukiyasu O ZEKI1 and Hidetoshi NISHIMORI Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551 1 Department of Applied Physics and Chemistry, The University of Electro-Communications, Chofugaoka, Chofu, Tokyo 182-8585 (Received August 22, 2005; accepted October 21, 2005; published December 26, 2005) The gauge theory for random spin systems is extended to quantum spin glasses to derive a number of exact and/or rigorous results. The transverse Ising model and the quantum gauge glass are shown to be gauge invariant. For these models, an identity is proved that the expectation value of the gauge invariant operator in the ferromagnetic limit is equal to the one in the classical equilibrium state on the Nishimori line. As a result, a set of inequalities for the correlation function are proved, which restrict the location of the ordered phase. It is also proved that there is no long-range order in the two-dimensional quantum gauge glass in the ground state. The phase diagram for the quantum XY Mattis model is determined. KEYWORDS: spin glass, gauge theory, correlation function, quantum spin system DOI: 10.1143/JPSJ.75.014001 1. Introduction The problem of spin glasses has been attracting continued attention.1,2) After the pioneering work by Edwards and Anderson (EA),3) Sherrington and Kirkpatrick have exactly solved the mean-field model by assuming the replica symmetry.4) Parisi has proposed the replica symmetry breaking solution and established the theoretical picture that the low-temperature phase is composed of infinitely many stable states with ultrametric structure.5) A topic of active investigations in recent years concerns the properties of short-range systems. Numerical studies have provided strong evidence for the existence of the spin glass (SG) transition for both the EA model6–9) and the XY gauge glass10–13) in three dimensions but against it for the two-dimensional EA model.8,9) For the two-dimensional gauge glass, although the long-range SG order has been denied rigorously,14) it is still possible that the system has a quasi long-range order in which the SG correlation decays in a power low. There remains the controversy about the existence of this order: some numerical studies have supported the absent of a finite-temperature transition13,15,16) but some groups argue against such a conclusion.17,18) Analytical calculations for spin glasses in finite dimensions are difficult because of randomness and frustration. However, a method using the gauge symmetry of the system is well-known as a powerful technique.19,20) This method provides various rigorous results, for instance, the exact internal energy and an upper bound for the specific heat in the special region of the phase diagram. Another noteworthy result is a set of inequalities for the correlation function, which restrict the topology of the phase diagram. In this relation, it has been suggested that the phase boundary between the ferromagnetic (FM) and SG phases is vertical by modifying the probability distribution.21) These results are generalized to a wider class of systems including the usual Ising SG and the Zq and XY gauge glasses.22) Although the gauge theory provides us with surprising results, its targets have so far been limited to classical spin systems. In the present paper, we generalize this theory so that it applies to quantum spin systems. A difficulty of this generalization is the fact that we must define the gauge transformation for spins without violating the commutation rule. We circumvent this problem by using a rotational operator on the Hilbert space as the gauge transformation. This paper consists of six sections. In the next section, we formulate the gauge transformation in two quantum spin glasses, the transverse Ising model and the quantum gauge glass (QGG), and show that these models have gauge symmetry. In §3, we prove an identity for a gauge invariant operator. This identity is valid even when the system parameters depend on time following the time-dependent Schrödinger equation. In §4, we derive a set of inequalities for correlation functions and order parameters. These results restrict the location of the FM phase or the Kosterlitz– Thouless (KT) phase in the phase diagram. In §5, we extend these inequalities to the ground state. The resulting inequalities for the order parameters show that the FM order does not exist at zero-temperature in the two-dimensional QGG. In §6, we consider the quantum XY Mattis model and determine its phase diagram. The last section is devoted to summary. 2. Gauge Transformation for Random Quantum Spin Systems 2.1 Transverse Ising model First, let us consider the random-bond Ising model in a transverse field. The Hamiltonian for this model is written as X X H¼
Ji j
iz
zj
h
ix ; ð1Þ i hi ji where
i is the Pauli matrix at site i. Although we treat spin1=2 systems in this paper, one can straightforwardly generalize all the results to spin-S systems. There is no restriction in the spatial dimensionality or lattice structure. The exchange interaction Ji j is a quenched random variable. One of the useful probability distributions for Ji j is the binary distribution PðJi j Þ ¼ p ðJi j
JÞ þ ð1
pÞ ðJi j þ JÞ: ð2Þ It is convenient for later arguments to rewrite this distribution as 014001-1 PðJi j Þ ¼ eK p i j ; 2 cosh K p Kp ¼ 1 p log ; 2 1
p ð3Þ J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by 54.167.13.156 on 09/16/22 J. Phys. Soc. Jpn., Vol. 75, No. 1 S. MORITA et al. where i j ¼ Ji j =J is the sign of the exchange interaction Ji j . Another useful distribution is the Gaussian distribution   1 ðJi j
J0 Þ2 PðJi j Þ ¼ pffiffiffiffiffiffiffiffiffiffi exp
; ð4Þ 2J 2 2J 2 2 where J0 and J denote the average and variance, respectively. For quantum spin systems, the classical gauge transformation, which simultaneously changes the sign of all components, is not valid because the commutation rule ½
ix ;
iy  ¼ 2i
iz is changed to ½
ix ;
iy  ¼
2i
iz . Thus we define a gauge transformation for spins using a unitary operator as Y U :
i ! G
i G
1 ; G¼ Gi ; 8 < 1i   i Gi ¼ : exp

ix 2 i ði ¼ þ1Þ ði ¼
1Þ ð5Þ U: ! ð
ix ; i
iy ; i
iz Þ: ð6Þ ð7Þ The transverse-field term in eq. (1) does not change by the gauge transformation. The Hamiltonian (1) is clearly invariant under the successive operations of V and U: ðUVÞH ¼ H. However, the distribution function of bond configuration is changed, for the J Ising model, as PðJi j Þ ! e K p i j  i  j : 2 cosh K p Thus this model can be quantized straightforwardly by replacing the elements of the above spin vectors by the Pauli matrices. The Hamiltonian of the QGG is therefore written explicitly as   Xn H ¼
J cos !i j
ix
xj þ
iy
yj hi ji ð12Þ  o y x x y
sin !i j
i
j

i
j : The phase factor !i j 2 ½0; 2Þ is a quenched random variable whose probability distribution is of cosine type Pð!i j Þ ¼ A difference of gauge transformations between quantum and classical systems is the transformation rule of
ix . The gauge transformation for the bond variables fJi j g is the same as in classical systems, namely V : Ji j ! Ji j i  j : hi ji ; where i is a classical gauge variable at site i and takes two values 1. If i ¼
1,
iy;z !

iy;z and
ix !
ix . Equivalently we can write ð
ix ;
iy ;
iz Þ x composed of x and y components Si ¼ ð SSyi Þ and rotational 
sin  i matrix in the XY plane, RðÞ ¼ ð cos Þ as sin  cos  X t Hcl ¼
J Si Rð!i j ÞS j : ð11Þ ð8Þ It is important for the following argument that this transformed distribution is proportional to the Boltzmann factor of a classical system. Similarly, the Gaussian distribution (4) is changed as !   Ji2j þ J02 1 J0 p ffiffiffiffiffiffiffiffiffiffi PðJi j Þ ! exp 2 Ji j i  j : ð9Þ exp
2J 2 J 2J 2 To simplify the arguments, we focus on the binary distribution (3) and (8), hereafter. It is straightforward to apply the same methods to the Gaussian distribution. 2.2 Quantum gauge glass Next, we consider the quantum gauge glass (QGG). Similarly to the transverse Ising model, the gauge transformation is defined by the rotation operator. To properly define the quantum version of gauge glass, let us first consider the Hamiltonian of the classical gauge glass (CGG), X Hcl ¼
J cosði
 j
!i j Þ: ð10Þ hi ji eK p cos !i j ; 2I0 ðK p Þ a periodic Gaussian (Villain) type rffiffiffiffiffiffiffi 1   Kp X K p ð!i j
2nÞ2 Pð!i j Þ ¼ exp
2 n¼
1 2 ð13Þ ð14Þ or a binary type Pð!i j Þ ¼ p ð!i j Þ þ ð1
pÞð!i j
Þ: ð15Þ Equation (12) is a special case of the XY model with Dzyaloshinskii–Moriya interactions. This Hamiltonian is written as X X H ¼
J
i 
j
Ji j ð
i 
j Þz ; ð16Þ hi ji hi ji where the second term is the random Dzyaloshinskii–Moriya interaction. If we set new parameters,   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 Ji j 2 2 ~ Ji j ¼ J þ Ji j ; !i j ¼
tan ; ð17Þ J the above Hamiltonian is rewritten as   X n J~i j cos !i j
ix
xj þ
iy
yj H¼
hi ji  o
sin !i j
ix
yj

iy
xj : ð18Þ This is equal to eq. (12) except that the interaction depends on bonds. In the CGG, the gauge transformation for spins is defined by the shift of spin variables as i ! i
i , where i 2 ½0; 2Þ is the gauge variable. Using the same notation as in eq. (11), this transformation is expressed as U : Si ! Rð
i ÞSi : ð19Þ Thus we use this definition of gauge transformation for the QGG. Using a rotational operator on the Hilbert space, we define   Y i i z U :
i ! G
i G
1 G ¼ exp

i : ð20Þ 2 i The transformation rule for the transposed vector t
i is defined as This Hamiltonian can be rewritten using the spin vector 014001-2 U : t
i ! t
i Rð i Þ ¼ G t
i G
1 : ð21Þ J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by 54.167.13.156 on 09/16/22 J. Phys. Soc. Jpn., Vol. 75, No. 1 S. MORITA et al. The gauge transformation of random variables is the same as in the classical case, V : !i j ! !i j
i j: þ   X hQiFz ¼ fi j g ð22Þ Under the gauge transformation UV, the Hamiltonian is invariant because X t ðUVÞH ¼
J
i Rð i ÞRð!i j
i þ j ÞRð
j Þ
j hi ji ð23Þ ¼ H;  eK p P ð29Þ i j  i  j hjQji: The last part of the right-hand side is rewritten in terms of cl ðK p Þ as P X eK p i j i  j hjQji fg ¼ X eK p P i j  i  j !  ð30Þ Tr cl ðK p ÞQ : fg jÞ : ð24Þ This transformed distribution is proportional to the Boltzmann factor for the CGG. If we choose the Gaussian or binary type, the Boltzmann factor for the Villain or J model appears, respectively. 3. X NB pÞ fg where we used the property of rotation matrices, Rð ÞRðÞ ¼ Rð þ Þ. The probability distribution (13) is changed as eK p cosð!i j
i þ Pð!i j Þ ! 2I0 ðK p Þ 1 2N ð2 cosh K Identity for Gauge Invariant Operators The gauge symmetry of the Hamiltonian yields a useful identity for gauge invariant operators. First, let us suppose that the system was initially in the perfect FM state jFz i in the transverse Ising model. This state appears in the FM limit, T ¼ 0, p ¼ 1, h ¼ 0. The gauge transformation operator G defined in eq. (5) changes this state as Therefore, we obtain   hQiFz ¼ X fi j g P  eK p P i j i  j 2N ð2 cosh K p ÞNB Tr cl ðK p ÞQ: ð31Þ Since Tr cl ðK p ÞQ is invariant under the transformation V, this is identical to the right-hand side of eq. (26) The above result can be generalized to the case that the transverse field hðtÞ depends on time, following the classical example.23) We consider the zero-temperature time evolution following the Schrödinger equation. Using the time ordered product, the time evolution operator is written as  Zt  0 0 Ut ¼ T exp
i Hðt Þ dt : ð32Þ 0 GjFz i ¼ ji; ji ¼ ji1 ji2    jiN : ð25Þ If i ¼ þ1, jii denotes the state with up spin in the z direction, and if i ¼
1, the spin at site i is down. Using the property of jFz i in (25), we prove the following identity for a gauge-invariant operator Q which satisfies Q ¼ ðUVÞQ or equivalently VQ ¼ G
1 QG, i   h hQiFz ¼ hQi cl ðK p Þ ; ð26Þ where hi cl ðK p Þ is the expectation value for the classical equilibrium state on the Nishimori line (NL), that is, hQi cl ðK p Þ cl ðK p Þ ¼ Tr ¼ e cl ðK p ÞQ; Kp Tr e P Kp 
z
z ij ij i j P 
z
z ij ij i j ð27Þ : To prove the identity (26), we apply the gauge transformation for the configuration of randomness of eq. (7) appearing on left-hand side of eq. (26). This operation does not change its value because the transformation V of eq. (7) only changes the order of the summation over i j . Thus the left-hand side of eq. (26) is rewritten as P   X eK p i j i  j hQ iF z ¼ NB hFz jVQjFz i fi j g ð2 cosh K p Þ ð28Þ P X eK p i j i  j ¼ NB hjQji; fi j g ð2 cosh K p Þ where we used the assumption that the operator Q is gauge invariant, VQ ¼ G
1 QG. Since the expectation value on the left-hand side does not depend on , the summation over  and division by 2N yield Since the time dependence of the Hamiltonian does not invalidate the gauge symmetry, this operator is also gauge invariant ðUVÞUt ¼ GðVUt ÞG
1 ¼ Ut : ð33Þ Examples of the gauge invariant operator include the transverse magnetization
ix ðtÞ ¼ Uty
ix Ut , the autocorrelation function
iz ð0Þ
iz ðtÞ and the interaction term H0 ðtÞ of the Hamiltonian (the first term on the right-hand side of eq. (1)). For the QGG, we can prove a similar identity i   h hQiFx ¼ hQi cl ðK p Þ : ð34Þ Here one should note that cl ðK p Þ is different from the normal density operator. If we choose the cosine-type distribution (13), cl ðK p Þ is defined by the Boltzmann factor for the CGG as P K cosð!i j
i þ j Þ Tr e p i j j ih j P ðK Þ ¼ ; ð35Þ cl p K cosð!i j
i þ j Þ Tr e p i j where j i ¼ GjFx i and Tr stands for integration over i from 0 to 2. Since the state vector j i does not diagonalize the Hamiltonian for the quantum gauge glass, cl ðK p Þ is not equal to the density operator ðK p Þ ¼ e
H : Tr e
H ð36Þ The identities (26) and (34) show that the expectation value of gauge invariant operator in the FM limit is equal to the one in the classical equilibrium state on the NL. The equivalence of the two states has already been proved in the dynamical gauge theory for classical systems.23,24) The 014001-3 J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by 54.167.13.156 on 09/16/22 J. Phys. Soc. Jpn., Vol. 75, No. 1 S. MORITA et al. present results are generalization of these dynamical cases to quantum systems. Note that the zero-temperature time evolution for quantum systems is deterministic in contrast to the stochastic dynamics for the classical SG. 4. NL p = pc T PM Tc Correlation Function and Order Parameter Using the identities proved in the previous section, we can derive a class of inequalities for the correlation function. First, we treat the transverse Ising model. Since the correlation function is not invariant under the gauge transformation, let us consider the following gauge-invariant quantity Q ¼
iz
zj h
iz
zj iK;h ; K;h K;h Here hi  j icl K p is the correlation function for the classical Ising system with the same configuration fi j g and no external field. By taking the absolute value of both sides of this equation, we find  h  D i   z zE    i  j cl  ; 

ð39Þ   i j Kp K;h where we used the fact that correlation function h
iz
zj iK;h does not exceed unity. Similarly, we can prove  D i E   h    hi  j icl   sgn
z
z ð40Þ Kp i j   K;h 3 2 cl h  i 6 D i j EK p 7 6 D 1 E 4 5¼4
iz
zj
iz
zj K;h 3 7 5  1: ð41Þ FM SG p FM h Fig. 1. The phase diagram of the transverse Ising model. The paramagnetic (PM), the ferromagnetic (FM) and spin glass (SG) phases meet at the multicritical point (MCP) in the plane h ¼ 0. The dashed line at p ¼ pc sets a bound for the existence of the FM phase also for h 6¼ 0. U ð42Þ ¼ t Si S j
i t Si Rð
=2ÞS j : ¼ ð
ix
xj þ
iy
yj Þ
ið
ix
yj

iy
xj Þ: ij ¼G i jG ij for the ð43Þ The gauge transformation U changes this operator according to
1 ¼ e
ið i
jÞ i j: ð44Þ The first factor is the same as the correlation function appearing in the classical gauge glass except for a minus sign. Usually, the correlation function for XY-like systems is defined as the expectation value of
ix
xj þ
iy
yj , which corresponds to cosði
 j Þ for classical systems. However, this expression is not useful for the gauge theory because it does not separate into gauge variables and spin operators after transformation by U as Uð
ix
xj þ
iy
yj Þ ¼ cosð eiði
 j Þ ¼ Si  S j
i ðSi  S j Þz This motivates us to define a correlation operator quantum gauge glass as, hc PM K;h If the probability p of the FM interaction is less than the critical probability pc at the multicritical point for the classical Ising system, the right-hand side of the inequality  (39) vanishes in the limit i
j ! 1. Thus the correlation function for the transverse Ising model on the left-hand side is also equal to zero. Therefore the region of the FM phase is restricted to the range p > pc (Fig. 1). Since the transverse field represents quantum fluctuations, the correlation function for the transverse Ising model should be reduced from classical system with h ¼ 0, which is the physical origin of the above-mentioned restriction. Next, let us consider how to define a correlation function of the QGG which has convenient properties for the gauge theory. In the CGG, a useful correlation function is defined in terms of an exponential function as eiði
 j Þ . This is rewritten using the notation of eq. (12) as ij PM ð37Þ where hiK;h denotes thermal average with temperature
1 ¼ J=K under a transverse field h. Substitution of the above quantity into eq. (26) yields D E D E cl
iz
zj ¼ i  j K p
iz
zj : ð38Þ 2 MCP i
sinð
i x x j Þð
i
j
þ
iy
yj Þ x y j Þð
i
j

iy
xj Þ: In addition, one can easily prove that E i   hD x x
i
j þ
iy
yj ij K ¼ ð45Þ ð46Þ K since the QGG is invariant under the following transformation: ð
ix ;
iy ;
iz Þ ! ð
iy ;

ix ;
iz Þ; !i j !
!i j : ð47Þ Thus we choose to discuss the correlation operator (43). To derive an inequality similar to eq. (39), it is useful to consider Q¼ y i j h i j iK : ð48Þ It is easy to prove that this quantity is gauge invariant because of the property of i j given in eq. (44). Substituting this quantity into eq. (34), we obtain i   h
ið i
j Þ cl ; ð49Þ ij K ¼ e i j Kp K where hicl K p stands for the expectation value for the CGG on the NL. By taking the absolute value of both sides of this equation, we find i   h cl   ð50Þ  i j K    e
ið i
j Þ K p  : Similarly, we obtain 014001-4 J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by 54.167.13.156 on 09/16/22 J. Phys. Soc. Jpn., Vol. 75, No. 1     sgn 2 4 eið ij K i
jÞ S. MORITA et al.  h
ið  e cl Kp 3 " i
5¼ i cl  Kp ;  1:   !  e
i i  cl 2 Kp of inequality (50) is estimated as   i e j i
jÞ  cl 2 Kp  cl 2 Kp   i
j  ! 1 ðVHÞ G
1 jx; !i ¼ x G
1 jx; !i: ð52Þ ij K  
ið e ð54Þ Vjx; !i ¼ G
1 jx; !i: mðK; K p Þ2  mcl ðK p ; K p Þ: ð55Þ If the parameter K p is smaller than the critical point K clpc for the classical system, the right-hand side vanishes. Consequently, the FM phase for the quantum gauge glass lies in the region satisfying K p > K clpc . This result is consistent with the intuitive picture that quantum effects reduce long-range order. If the spatial dimensionality of the system is equal to the lower critical dimension, d ¼ 2, there is no long-range order but quasi long-range order. The Kosterlitz–Thouless (KT) phase25) exists when both K and K p are sufficiently large. The ordering tendency of the KT phase is observed by the correlation length  in the paramagnetic (PM) phase as  
ji
jj=m ðK;K p Þ ; ð56Þ ij K  e 2  
ið i
j Þ cl  e Kp h i cl cl ¼ e
ið i
j Þ K p  e
ji
jj=m ðK p ;K p Þ : ð57Þ   From the square of the inequality (50), the limit i
j ! 1 yields 2cl m ðK p ; K p Þ: ð58Þ Thus, the KT phase for the QGG is also restricted to the region K p > K clpc . 5. Ground State Property of Quantum Gauge Glass Next we consider the ground state property of the QGG. It is necessary to consider the transformation rule for eigenstates of the Hamiltonian. Let us denote an eigenstate of the Hamiltonian with the eigenvalue x by jx; !i, Hjx; !i ¼ x jx; !i: ð59Þ Since the invariance of the Hamiltonian (23) can be rewritten as ð62Þ This rule is derived straightforwardly, if the state is not degenerate. When the state is degenerate, the rule is not unique. However, it is reasonable and has no problem, if we use this rule as the transformation rule. Now, we consider the ground sate. Let us denote the ground state of the Hamiltonian by jg; !i. The above transformation rule leads to the gauge transformation of the average of any operator Q in the ground state; Vhg; !jQjg; !i ¼ hg; !jGðVQÞG
1 jg; !i: where we used the identity mcl ¼ qcl resulting from the gauge theory on the NL for the CGG. Therefore we obtain ð61Þ Note that the effect of the operator V is restricted to the inside of the brackets ð  Þ. The state G
1 jx; !i is an eigenstate of the gauge-transformed Hamiltonian ðVHÞ. Therefore, one can derive the transformation rule of the eigenstate ¼ qcl ðK p ; K p Þ2 ¼ mcl ðK p ; K p Þ2 ; m ðK; K p Þ  ð60Þ we have an eigen-value equation Since the lower critical dimension dl is two for continuous spin systems, for d > 2, we expect a FM phase to exist at low temperature under small randomness. In this case, a two-spin correlation function tends to the square of magnetization when the two spins are sufficiently separated,     2 i
j ! 1: ð53Þ i j K ! mðK; K p Þ ; The right-hand side follows: i h 
ið i
j Þ cl  2  e Kp GðVHÞG
1 ¼ H; ð51Þ # 1 ij K jÞ ð63Þ The thermal average h  iK in equations derived in the previous section can be replaced by the ground state expectation value. For example, i   h cl g; !j i j jg; ! ¼ e
ið i
j Þ K p g; !j i j jg; ! ð64Þ is derived instead of eq. (49), which provides the inequality for the order parameters in the ground state, mð1; K p Þ2  mcl ðK p ; K p Þ ð65Þ instead of eq. (55), and m ð1; K p Þ  2cl m ðK p ; K p Þ ð66Þ instead of eq. (58). In two dimensions, it has been shown that the FM long range order exists in the ground state of the pure quantum XY model (K p ¼ þ1).26) However, in the disordered regime (K p < þ1), the FM order must disappear since the FM order in the CGG model, the right-hand side of the inequality (65), does not exist. The only possibility in this regime is the existence of the KT phase, which is consistent with the inequality (66). 6. Phase Diagram of Quantum Mattis Model In this section, we introduce and discuss the properties of the quantum XY Mattis model which has no frustration. Using the gauge transformation, one can obtain the phase diagram for this model explicitly. One of the phase boundaries is determined by the critical point of the pure quantum system and the other by that of the classical one. This is an important difference from the classical nonfrustrated systems.27) Let us locate a quenched random variable !i at each site and define the phase factor !i j as !i j ¼ ! j
!i . The Hamiltonian for the quantum Mattis model is defined in terms of the pure quantum XY model, H0 , that is,   Y i!i z H ¼ G! H0 G
1 ; G ¼ exp

: ð67Þ ! ! 2 i i 014001-5 J. Phys. Soc. Jpn. Downloaded from journals.jps.jp by 54.167.13.156 on 09/16/22 J. Phys. Soc. Jpn., Vol. 75, No. 1 S. MORITA et al. Thus, the ground-state energy is always the same as that of H0 and the ground state is obtained by operating G! on the ground state of the pure system. This system has no frustration in this sense. It is easy to show that the Hamiltonian for this model is invariant under the gauge transformation, ðUVÞH ¼ H, where the gauge transformation for the configuration is defined as !i ! !i
i . Using eqs. (44) and (67), we immediately obtain ij K¼ eið!i
! j Þ i j 0;K ; ð68Þ where the angular brackets on the right-hand side denote the thermal average with respect to H0 . Note that the correlation function for the pure system h i j i0;K does not depend the quenched variable !i . Here, we assume that the distribution function for the quenched random variable is proportional to the Boltzmann factor of the pure classical XY model, P K cosð!i
! j Þ e p ij Pð!Þ ¼ : ð69Þ Z0cl ðK p Þ Hereafter, the subscript 0 and the superscript cl stand for pure and classical systems, respectively. Then the configuration average is equal to the thermal average for the pure classical XY model with coupling K p . Consequently, we find   ið!i
! j Þ cl : ð70Þ ij K ¼ e 0;K p i j 0;K Similarly, from eq. (68), the spin-glass correlation function satisfies 2 2       ð71Þ  i j K  ¼  i j 0;K  :   Taking the limit i
j ! 1 of eqs. (70) and (71), we obtain mðK; K p Þ ¼ mcl ð72Þ 0 ðK p Þm0 ðKÞ; qðK; K p Þ ¼ m0 ðKÞ2 : ð73Þ For d > 2, there are three phases, PM, FM and Mattis spinglass (MSG) phases. Figure 2(a) shows the phase diagram. The phase boundary between PM and other phases is at K ¼ cl K0c and the one between FM and MSG is at K ¼ K0c . If d ¼ 2, the correlation length determines the phase structure. From eqs. (70) and (71), we find 1 1 1 ¼ þ m ðK; K p Þ cl  ðK Þ ðKÞ p 0 0 a) ð74Þ b) 0 0 K PM K PM K0c MSG 0 Kp cl K0c K0c FM RKT ∞ ∞ 0 Kp cl K0c UKT ∞ ∞ q ðK; K p Þ ¼ 0 ðKÞ ; 2 ð75Þ where q ðK; K p Þ denotes the spin-glass correlation length. Thus, similarly to the d > 2 case, three phases exist: (i) m < 1 and q < 1: paramagnetic phase (PM), (ii) m ¼ 1 and q ¼ 1: uniform KT phase (UKT) and (iii) m < 1 and q ¼ 1: random KT phase (RKT). The phase diagram is shown in Fig. 2(b). We note again that the location of the horizontal phase boundary is determined by the critical point of the quantum pure system, K0c , and the vertical one comes from that of the cl classical pure system, K0c . 7. Conclusions In this paper, we have investigated quantum spin glasses using the gauge theory. First, we considered the transverse Ising model and the QGG. To construct the gauge theory, we defined the gauge transformation by rotational operator on the Hilbert space. It is essential that interaction of these models is written in term of one or two components of spin operators. If a system has Heisenberg-type interactions, we can not define a gauge transformation which satisfies the commutation rule. Using the gauge theory, we obtained mainly two results. One is the identity for gauge invariant operators. The FM limit state and the classical equilibrium state on the NL provide the same expectation value for gauge invariant operators. We note that this result remains valid when we introduce time evolution following the Schrödinger equation. This result has already been pointed out for classical spin glasses with stochastic dynamics.23,24) We have shown that the same applies to quantum spin systems. The other result is a set of inequalities for the correlation function. These inequalities show that the correlation function for the quantum model never exceeds the classical counterpart on the NL. The corresponding classical system is determined by a transformation rule for probability distribution. As a result, the order parameter is smaller than the square of the classical one. Therefore the FM phase (or the KT phase) should lie within the corresponding classical one. This is natural intuitively since quantum effects reduce ordering tendency, but to prove it rigorously is a different and quite a non-trivial problem. Moreover, these results are valid even if the system is in the ground state. Thus FM long range order vanishes in the two-dimensional QGG although the ground state of the pure quantum XY model has FM order. Next, we determined the phase diagram for the quantum XY Mattis model. This model is not a real SG because of lack of frustration. It is interesting, nevertheless, that both quantum and classical phase transitions occur in a single system, which may serve as a starting point for investigations of more realistic quantum spin glasses with frustration. Acknowledgment Fig. 2. (a) The phase diagram for the quantum XY Mattis model in d > 2. There are three kinds of phases, the PM, the FM and the SG. (b) The phase diagram in d ¼ 2. There exist three kinds of phases, the PM, the uniform KT (UKT) and the random KT (RKT). This work was supported by the Grant-in-Aid for Scientific Research on Priority Area ‘‘Statistical-Mechanical Approach to Probabilistic Information Processing’’ by the Ministry of Education, Culture, Sports, Science and Technology. 014001-6 J. 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