Order parameter conditions from mutual information and symmetry conditions
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b O R DER PAR AMETER C ONDITIONS FROM MUTUAL INFOR MATIONAND SYMMETRY C ONDITIONS
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Ivan Arraut
The Open University of Hong Kong30 Good Shepherd Street,Homantin, Kowloon, Hong Kong. [email protected]
Wing Chi Yu
Department of Physics,City University of Hong Kong,Kowloon, Hong Kong.City Univeristy of Hong Kong Shenzhen Research Institute,Shenzhen, China. [email protected]
February 5, 2021 A BSTRACT
The mutual information method has demonstrated to be very useful for deriving the potential orderparameter of a system. Although the method suggests some constraints which help to define thisquantity, there is still some freedom in the definition. The method then results inefficient for caseswhere we have order parameters with a large number of constants in the expansion, which happenswhen we have many degenerate vacuums. Here we introduce some additional constraints based onthe existence of broken symmetries, which help us to reduce the arbitrariness in the definitions ofthe order parameter in the proposed mutual information method.
When the symmetry of a system is spontaneously broken, a vacuum degeneracy emerges naturally [1]. This is a con-sequence of the fact that the vacuum itself does not respect some of the symmetries of the Lagrangian (Hamiltonian).These are called broken symmetries and their generators are called broken generators. There are many systems in thenature able to develop this condition called spontaneous symmetry breaking [2, 3]. The phenomena has several applica-tions in condensed matter systems as well as in high-energy physics. Tied to this phenomena, is the Nambu-Goldstonetheorem, which establishes that the number of broken symmetries are equal to the number of gapless particles appear-ing in the system and called Nambu-Goldstone bosons [1, 2]. When the system experience spontaneous symmetrybreaking, an order parameter appears. The order parameter is just a field showing some long correlation in the system.One way to find whether a system is able to develop long range order or not, is by using the concept of mutual informa-tion [4, 5, 6, 7, 8]. Once it is established that the system develops long range order, then an order parameter field shouldbe associated to this phenomena. This field can be taken as an expansion of n -terms with n arbitrary constants. Theconstants are partially fixed by the trace-less conditions of the order parameter with respect to the degenerate vacuum[6]. This constraint is good for the cases where there are only a few constants to fix in the expansion. However, for thecases where we have several constants, the trace-less condition alone is of limited scope. In [6], besides the tracelesscondition, an arbitrarily imposed normalization’s condition over the constants is assumed. Although in [6] it is assumedno-knowledge about the symmetries in the system, we can still assume their existence. It is known that the combina-tion of the order-parameter plus broken generators, can create some additional constraints over the order-parameters ofthe system [9, 10, 11]. In this paper, by assuming the general existence of broken symmetries, we add two important(and additional) constraints. The first one corresponds to the fact that for every m -set of vacuums | > m connectedby a set of broken symmetries Q m , we have the condition | > = e − iω | > = e − iω | > = ... = e − i ( m − ω | > m ,with ω defining the phase difference between the degenerate vacuums. This means that each vacuum differs at mostby a phase. The phase difference between neighbors vacuums will be defined in this paper. The second conditioncorresponds to the cyclic property of vacuums connected by broken symmetries, which suggests that the successive PREPRINT - F
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5, 2021action of a broken generator over a series of vacuums, connected each other through the action of the same generator,will bring back the initial vacuum after n successive applications of the broken generator. These two conditions com-plement correctly the mutual information method and they are enough for determining the form of the order parameterand they automatically include the traceless condition. The paper is organized as follows: In Sec. (2), we analyzethe relation between mutual information and spontaneous symmetry breaking; in addition we analyze the previouslyproposed constraints which partially fix the order parameter of a system. In Sec. (3), we introduce one extra-constraintrelated to the existence of broken symmetries and we demonstrate that the traceless condition is just a special case ofthe constraints. In Sec. (4), we verify a few systems where we are able to predict the form of the order parameter byusing the method proposed in this paper. Finally, in Sec. (5), we conclude. The mutual information method, developed in [6] , suggests that the existence of long range order can be deduced byevaluating a quantity which depends on the von-Neumann entropy. For understanding the method, we can start bydefining a Hamiltonian for a quantum many-body system as ˆ H = X i ˆ h i , (1)where ˆ h i corresponds to the local Hamiltonian for the site i . Consider now a local interaction with another block j ,separated from i by the distance | i − j | . The reduced density matrix of a block i is defined as [6] < µ ′ | ρ i | µ > = tr (ˆ a iµ ′ ρ ˆ a + iµ ) . (2)The trace is done over all the degrees of freedom, except those corresponding to the block i itself. The density matrixis defined as usual by the expression ρ = | Ψ >< Ψ | , where | Ψ > is the ground state of the Hamiltonian. Additionally, ˆ a i corresponds to the annihilation operator acting over the states | µ > on the site i and satisfying the commutation oranticommutation relations. Similarly, it is possible to define the reduced density matrix for two blocks as < µ ′ ν ′ | ρ i ∪ j | µν > = tr (ˆ a iµ ′ ˆ a jν ′ ρ ˆ a + jν ˆ a + iµ ) . (3)Due to the probability interpretation of the reduced density matrix’s diagonal elements, it is a matter of convention tonormalize the density matrices as follows tr ( ρ i ) = tr ( ρ j ) = tr ( ρ i ∪ j ) = 1 . (4)The density matrices can be diagonalized in the following form ρ i = X µ p µ | ψ iµ >< ψ iµ | , ρ j = X ν p ν | ψ iν >< ψ iν | , ρ i ∪ j = X µν q µν | φ iν >< φ iν | . (5)Additionally, we can define the von-Neumann entropy as follows [12] S = − X µ p µ log p µ . (6)This quantity measures the amount of entanglement between the block i and the rest of the system. If we want toevaluate only the correlation between a pair of blocks i and j , then we define the mutual information as S ( i | j ) = S ( ρ i ) + S ( ρ j ) − S ( ρ i ∪ j ) . (7)It has been proved before that this quantity can be used for analyzing the critical phenomena [6]. In fact, the non-vanishing mutual information as it is defined in eq. (7) and defined between two distant blocks, means that there existsa long-range correlation between the blocks, here defined correspondingly as i and j . Here we will not demonstrate itbut the details can be found in [4, 5, 6]. 2 PREPRINT - F
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It is possible to construct a local order operator from the spectra of the density matrix [6]. We can then define the orderparameter as ˆ φ = X <µ,ν> (cid:0) ω µν ˆ a + iµ ˆ a iν + ω ∗ µν ˆ a + iν ˆ a iµ (cid:1) . (8)Here the coefficients ω µν (and its conjugate) can be either, diagonal or non-diagonal. If µ = ν , then we have diagonalcoefficients and we only need the subindex µ and in addition ω µν = ω ∗ µν for this special case. The order parameter isdiagonal or non-diagonal depending on the basis where it is expanded. In particular, it depends on the relation betweenthe basis and the definition of the different vacuums in the same basis. Independent on whether or not the selectedbasis makes the order parameter diagonal, the order parameter has to satisfy the traceless condition tr ( ˆ φ ) = 0 = X µ,ν p µν ω µν = 0 (9)This condition becomes simplified when the order parameter is diagonal. In such a case, the expression is simply X µ ω µ p µ = 0 . (10)Note that µ ≤ ζ , where ζ is the rank of the density matrix. The traceless conditions cannot be used if the range ofthe matrix ρ is equal to one. This condition, is able to fix some relation between the ω µ -coefficients. If we have n -coefficients of this type, then the traceless condition reduces the number of independent coefficients to n − . The traceless condition is useful when the amount of coefficients to fix is not so large. This happens when we haveonly a few degenerate vacuums. However, when we have several degenerate vacuums, then we will also have severalcoefficients ω µ . Here we formulate additional conditions besides the traceless one, based on the fact that the actionof a broken symmetry generator is to map one vacuum into another one, taking into account that we are dealing withvacuum degeneracy. Consider that we can break the symmetry of the system spontaneously by selecting one among all the possible vacuums.Consider now a broken generator defined as Q a . Then we can define the condition Q a | > = | ¯0 > . (11)Here | > = | ¯0 > (inequivalent vacuums) but both vacuums are at the same energy level. In general, we can decomposethe order parameter in two components as follows ˆ φ = ˆ φ + i ˆ φ . (12)We can conventionally define the component ˆ φ as the one with zero vacuum expectation value for some arbitrarilyselected vacuum. Let’s fix such a vacuum to be | > , and then < | ˆ φ | > = 0 , < | ˆ φ | > = 0 . (13)The components can be selected such that they are related to each other through the broken symmetry generator Q a as [ Q a , ˆ φ ] = i ˆ φ . (14)This result suggests another way to say that the effect of the broken symmetry is to map one vacuum into the other. Ifwe have n -degenerate vacuums connected by the same broken generator, then by applying n -times Q a to the vacuumstate, we just get the initially fixed vacuum as follows ( Q a ) n | > = | > . (15)3 PREPRINT - F
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5, 2021Then in general, each vacuum is connected to each other by one expression of the form | > = e − iφ | > , | > = e − iφ | > , | > = e − iφ | > , ...., | > n = e − ( n − iφ | > , | > = e − niφ | > , (16)where φ = αQ a represents the effect of the broken generators over the states. If we have m -broken generators definedas Q , Q , Q ,... Q m , each generator will reproduce similar mappings, as those defined in eqns. (11), (14), (15) and(16). If we have the n vacuums defined previously, then we can know the phase difference between vacuums in aexplicit form as φ = 2 πn , (17)for each broken generator acting over a set of vacuums. Then for the special case where we only have two vacuums,then φ = π because n = 2 . This situation appears for example for the case of ferromagnetism [6]. Note that for thegeneral expansion of the order parameter ˆ φ = X n ω n ˆ a + in ˆ a in , (18)the relation between the coefficients ω n is determined by the condition (17) because each value of n corresponds to adifferent vacuum. Then, for n = 2 , since φ = π , we have | > = −| > , (19)from the result (16). This condition applied to the standard expansion (18), means ω = − ω with the series expan-sion containing only two terms. When we have more than two vacuums, the conditions can be generalized and thecomplex coefficients ω n will be related to each other by a fixed phase. Note that the traceless condition defined in [6],corresponds to a special situation here. In addition, note that here the condition suggesting that the maximum valueof ω n is one, is reduced to the condition fixing only one of the magnitudes of ω i to one, namely, | ω i | = 1 . Then thedifference between the different coefficients will be reduced to only phase differences. In this way, then instead of thecondition (16), we can use ω = e − iφ ω , ω = e − iφ ω , ω = e − iφ ω , ...., | > n = e − ( n − iφ ω , ω = e − niφ ω , (20)expressing then the fact that all the ω i coefficients can be expressed as a common magnitude | ω | , times a phase. Thecyclic property given by the last term in this equation, which defines the result (17), gives us the following conditions X n | ω n | = n | ω | , (21)for the case of multiple vacuums, connected by broken symmetries and having the same energy level. If some vacuumsare connected by broken symmetries but in addition have different energy levels, chemical potentials should appear,generating then some gap for the possible Nambu-Goldstone bosons (if any) appearing in the system. We can deriveanother important conditions if we sum over all the coefficients ω n related to the different definitions of vacuum asfollows X n ω n = | ω | (cid:0) e − iφ + e − iφ + e − iφ + ... + e − niφ (cid:1) . (22)The previous expression can be simplified if we take into account that it represents a geometric series, and then X n ω n = | ω | (cid:18) − e − niφ e iφ − (cid:19) = | ω | (cid:18) e − iφ − e − ( n +1) iφ − e − iφ (cid:19) . (23)If the vacuums satisfy the cyclic property, then e − ( n +1) iφ = e − iφ ; and then we can conclude that X n ω n = 0 , (24)4 PREPRINT - F
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5, 2021which is a natural result equivalent to the traceless condition in [6]. This demonstrates that the traceless conditionis just a natural consequence of the fact that the vacuums are connected by some broken symmetry plus the cyclicproperty summarized in the definition e − niφ = 1 . Note that once the norm of ω is fixed, then the norm for allthe other coefficients is also fixed as far as we consider vacuums of the same energy level. Then once we knowhow many vacuums are at the same energy level, for example, by analyzing the reduced density matrix spectrum, alltheir coefficients ω µ will be defined. Regarding the order of magnitude of ω µ , by fixing the magnitude of | ω | = 1 ,immediately fixes the magnitude of the others (not the sign) coefficients connected by a broken generator. For anynumber of vacuums, the phase φ can be found from eq. (17). In this way for example, if we have three vacuums then n = 3 and φ = 2 π/ . In this way we have ω = − ω + √ iω , ω = − ω − √ iω . (25)This result satisfies evidently the result (24). Here we will analyze different cases where we can apply our method for getting in a natural way the correspondingorder parameters.
For the case of ferromagnetism, we define the unbroken vacuum as | Ψ > = 1 √ | ↑↑ ... ↑↑ > + | ↓↓ ... ↓↓ > ) (26)This situation corresponds to a spin chain, with all the atoms in the chain having a common spin either, up or down. Weconsider here these two possibilities as the two independent vacuums. If the symmetry of the system is spontaneouslybroken, the system will select any of the two possible vacuums. If we apply our formulation to this case, then n = 2 and then we get φ = π in agreement with eq. (17), and from eq. (18), we obtain ˆ φ = ω ˆ a + i ˆ a i + ω ˆ a + i ˆ a i , (27)Since n = 2 , then ω = − ω if we use the result (20). Since all the ω n -coefficients related to the existence of indepen-dent vacuums (but connected through a symmetry transformation) are related to each other by a phase transformation,they can be normalized to one and then we can define them unambiguosly. Then | ω | = | ω | . This case correspondsto the ferromagnetic case showed in [6]. Then the method proposed here works perfectly. Here we consider the Hamiltonian ˆ H = N X j =1 (cid:0) S xj S xj +1 + S yj S yj +1 + S zj S zj +1 (cid:1) . (28)This Hamiltonian is basically the same of the Dimer case [13] but excluding the next nearest neighbor interactions.Here S xj , S yj and S zj are spin-1/2 operators at the site j . This model is normally solved by using the Bethe-ansatzmethod [14]. This case has two different vacuums connected through the condition ω = ω ∗ . (29)If ω = x + iy , then ω = x − iy and then the two independent vacuums corresponding to the ground state areseparated by an angle tan (cid:18) θ (cid:19) = yx . (30)For this case, we identify two independent vacuums. The order parameter is then obtained from eq. (18) as5 PREPRINT - F
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5, 2021 ˆ φ = ω ˆ a + i ˆ a i + ω ∗ ˆ a + i ˆ a i . (31)With the definitions given for ω , we find that the previous expression corresponds to [6] ˆ φ = x ˆ O xi − y ˆ O yi . (32)Here ˆ O x,yi has a correspondence with the Pauli matrices ˆ σ x,y . This means that ˆ O x,yi represent the generators ofrotations in the spin space. Then immediately we can conclude that the order parameter defined in eq. (31) has twocomponents, corresponding to the two independent vacuums, connected each other through a combination of rotationaround x and rotation around y . This conclusion can be obtained from the expression (16), which represents theconnection between the number of coefficients ω i and the number of independent vacuums. The two independentvacuums can be defined as | > = e iθ | > , (33)with the angle θ defining the rotation around the z -axis. The same angle is defined in eq. (30). In this paper we have learned a different way to interpret and define the order parameters in systems where there is adegeneracy on the vacuum state and as a consequence, there exists the possibility of having spontaneous symmetrybreaking once a specific vacuum condition is selected. This new way to visualize the problem helps us to findadditional constraints for the definition of an order parameter. Our starting point is the expansion of the orderparameter in terms of the particle number operator as it was suggested in [6]. Here we have explained how toconstraint the coefficients ω i of the expansion based on the fact that a broken symmetry for the vacuum, maps oneof the vacuums toward another one. The order parameter itself can be expanded in a base with vectors defining thedifferent vacuums. It is for this reason that the vacuum expectation value of an order parameter does not vanish whenit is defined for a single vacuum. Other results, like the traceless condition for the order parameter emerges naturallyfrom this scenario. This condition is a natural consequence of the expansion of the order parameter in a base definingall the possible vacuums of the system. Acknowledgement
W. C. Yu acknowledges financial support from the National Natural Science Foundation of China (Grants No.12005179) and City University of Hong Kong (Grant No.9610438).
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