A mean field approach to the Ising chain in a transverse magnetic field

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A mean field approach to the Ising chain in a transverse magnetic field

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2017 Eur. J. Phys. 38 045404 (http://iopscience.iop.org/0143-0807/38/4/045404) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 155.210.176.73 This content was downloaded on 09/05/2017 at 11:17 Please note that terms and conditions apply.

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European Journal of Physics Eur. J. Phys. 38 (2017) 045404 (9pp)

https://doi.org/10.1088/1361-6404/aa6cd4

A mean field approach to the Ising chain in a transverse magnetic field C Osácar1,3 and A F Pacheco2 1 2

Facultad de Ciencias, Universidad de Zaragoza, E-50009 Zaragoza, Spain Facultad de Ciencias and BIFI, Universidad de Zaragoza, E-50009 Zaragoza, Spain

E-mail: [email protected] and [email protected] Received 16 January 2017, revised 21 March 2017 Accepted for publication 12 April 2017 Published 8 May 2017 Abstract

We evaluate a mean field method to describe the properties of the ground state of the Ising chain in a transverse magnetic field. Specifically, a method of the Bethe–Peierls type is used by solving spin blocks with a self-consistency condition at the borders. The computations include the critical point for the phase transition, exponent of magnetisation and energy density. All results are obtained using basic quantum mechanics at an undergraduate level. The advantages and the limitations of the approach are emphasised. Keywords: Ising chain, mean field, transverse magnetic field 1. Introduction In physics, when one is dealing with a difficult many-body problem, a possible approach is to analyse the behaviour of one body—or few body—assuming that the influence of all the other individuals is approximated by an averaged effective interaction, called the mean field; thus one reduces the many-body problem to a few-body problem. This is the basic idea of the mean field (MF) approach which was introduced in the area of magnetic phase transitions by Curie and Weiss [1]. There are several ways of implementing this idea. Two examples of the application of the MF strategy are the approximation of Hartree and Hartree–Fock [2] in atomic and nuclear physics, and the equal load sharing approximation in the fibre-bundle models of fracture [3]. In this paper, we study the properties of the ground state of an infinite Ising chain in a transverse magnetic field, which is also called the transverse Ising model (TIM). The TIM is the quantum counterpart [4] in one spatial dimension of the classical 2-dim Ising model of statistical mechanics. It was exactly solved in [5] and it provides an excellent testing ground for various approximation techniques [6]. 3

Author to whom any correspondence should be addressed.

0143-0807/17/045404+09$33.00 © 2017 European Physical Society Printed in the UK

1

Eur. J. Phys. 38 (2017) 045404

C Osácar and A F Pacheco

The TIM is a quantum system whose Hamiltonian, without any loss of generality, can be expressed in the form:

H=

N

å

i =-N

⎤ ⎡ y - s (i) - sx (i) sx (i + 1) ⎥ , ⎦ ⎣⎢ 2 z

N¥

(1)

where the operators sz (i ) and sx (i ) are the Pauli matrices

sz =

(

)

( )

1 0 , s = 0 1 . x 0 -1 1 0

( 2)

Thus, in the TIM, the intensity of the ferromagnetic term is 1, and the intensity of the external transverse field is (y 2 ). In the ground state, for y=0, the spontaneous symmetry breaking chooses between the two orientations +x and -x; if +x is the preferred orientation, the parameter of order, or magnetisation, ⟨ sx⟩, has a value equal to 1. Here, the system is completely ordered. As the value of y grows in the interval 0 < y < yc , the order parameter falls gradually to zero and for y > yc , ⟨ sx⟩ = 0 and there is no order any more. Hence this model illustrates what is an order-disorder phase transition. In the TIM, the exact value of the ground state energy density is given by:

r ( y) =

-1 4p

ò0

2p

( y 2 + 4 + 4y cos x ) 2 dx. 1

( 3)

The second derivative of this function diverges at the critical point:

y = yc = 2.

(4)

And the magnetisation, for y < yc , adopts the form: b ⎛ ⎛ y ⎞2 ⎞ ⎟ ⎜ ⟨ sx ( y)⟩ = ⎜1 - ⎜ ⎟ ⎟ , ⎝ yc ⎠ ⎠ ⎝

b=

1 . 8

(5)

For y > yc , ⟨ sx ( y)⟩ = 0 . In section 2 we develop the MF approximation for this model. In section 3 we compare the MF results with the exact results expressed in equations (3)–(5). In section 4 we present a discussion and argue about the interest of this content in a quantum mechanics undergraduate course for students of physics or chemical physics. Our specific MF strategy uses finite spin blocks with self-consistency conditions at the borders. For a reference dealing with finite spin blocks and symmetry in quantum mechanics see [7]. Finally, in the appendix, we reproduce another type of mean field approach which is simultaneously variational [8]. 2. Mean field strategy Conceptually, it is very important to remember that a system with a finite number of degrees of freedom cannot exhibit a phase transition like that of order-disorder mentioned for the TIM in section 1. This means that in the ground state of a block formed by n=1, 2, etc spins, the expectation value of the operator sx no matter which is the value of y, is null. Note the difference with what it was said in section 1 for the infinite chain, where for 0 < y < yc , the magnetisation is different from 0. Let us check this idea using two explicit examples of finite blocks. For a 1-spin block, n=1, the spin can be  or  and its Hamiltonian is: 2

Eur. J. Phys. 38 (2017) 045404


y h1 = - sz (1). 2 Thus, the minimum energy eigenstate of h1 is ∣ y⟩1 = , and ⟨ ∣sx (1)∣⟩ = 0 . In the case of the block n=2, the vector space is formed by four elements: ∣ A⟩ =  ,

∣ B⟩ =  ,

∣C⟩ =

1 ( + ) , 2

∣ D⟩ =

1 ( - ) 2

(6)

( 7)

∣ A⟩, ∣ B⟩ and ∣ C ⟩ are symmetric under the right–left interchange, and ∣ D⟩ is antisymmetric. Its Hamiltonian is:

y h 2 = - (sz (1) + sz (2)) - sx (1) · sx (2). 2

(8)

Therefore, the Hamiltonian matrix is

⎛ y -1 0 ⎜ [h 2] = ⎜- 1 - y 0 0 -1 ⎜⎜ 0 ⎝ 0 0 0

0⎞ ⎟ 0 ⎟. 0 ⎟⎟ 1⎠

(9)

As this matrix shows, h2 can only mix ∣ A⟩ with ∣ B⟩. The minimum energy eigenstate of h2 is in the 2×2 top-left sector of [h 2], and hence the ground state ∣ y⟩2 is a linear combination of the type ∣ y⟩2 = a1  + a 2 . Again, the expectation value of sx is null:

⟨ a1  + a 2 ∣sx ( j )∣a1  + a 2 ⟩ = 0

j = 1, 2.

(10)

Then the question is: can we build up a simple method able to describe the phase transition existing in the TIM? The answer is yes. The mean field strategy provides just what we seek. With this aim we define new primed Hamiltonians in the form

h1¢ = - ssx (1) -

y sz (1) - ssx (1) , 2

(11)

h 2¢ = - ssx (1) -

y (sz (1) + sz (2)) - sx (1) · sx (2) - ssx (2) . 2

(12)

Thus, for a given finite block of arbitrary size n, its primed Hamiltonian h¢ is built by adding to h two new terms of the type (-ssx (1)) and (-ssx (n )) at the borders of the blocks. The added terms represent the interaction of the rest of spins with the block and the positive constant s will represent the value of the magnetisation existing in the chain.

h n¢ = - ssx (1) -

y n å sz ( j) 2 j=1

n-1

å sx ( j) · sx ( j + 1) - ssx (n).

(13)

j=1

Now we recognise the essence of the mean field strategy: we have to solve a problem of n spins under the effect of a field created by the rest of the chain; this result will depend on s. At the end, we will impose a self consistency condition such that the value of the operator sx acting on the spin in any position of the block should be equal to the value of the constant s assumed ab initio

⟨ y ¢n∣sx∣y ¢n⟩ = s.

(14)

The hope is that, as in any approximation method of this type, when n grows, the results of the method will approximate better to the exact values. 3

Eur. J. Phys. 38 (2017) 045404


We start with n=1. In the space {  ,  } the representation of h1¢ is the following matrix

⎛ - y - 2s ⎞ ⎟ [h1¢] = ⎜⎜ 2 y ⎟. ⎝- 2s 2 ⎠

(15)

Note that h1¢ has now mixed  and . Its minimum energy eigenvalue is:

y2 + 4s 2 . 4 The corresponding eigenvector is l1¢ = -

∣ y¢⟩1 =

1 1 + k2

(16)

( + k ).

(17)

⎞ y2 + 4s 2 ⎟⎟. 4 ⎠

(18)

Where k is the constant

k=

1 ⎛ y ⎜- + 2s ⎜⎝ 2

And as

⟨ y1¢ ∣ sx ∣ y1¢⟩ =

2k 1 + k2

(19)

the self-consistency condition reads:

s=

2k . 1 + k2

(20)

This procedure followed for n=1 is the same followed for any n. Consider now the case n=2. The representation of h 2¢ in the vector space (7) is the following matrix

⎛ y - 1 - 2s ⎜ - y - 2s [h 2¢] = ⎜ - 1 ⎜ ⎜- 2 s - 2 s - 1 ⎝ 0 0 0

0⎞ ⎟ 0 ⎟. ⎟ 0⎟ 1⎠

(21)

Observing the structure of [h 2¢], ∣ C ⟩ is now mixed with ∣ A⟩ and ∣ B⟩, and the ground state ∣ y¢2⟩ is now a linear combination of the type

∣ y¢⟩2 = b1  + b 2  + b3

1 ( + ) , 2

(22)

where b1, b2 and b3 are constants. The expectation value of sx ( j) is now:

⟨ y ¢2∣sx ( j )∣y ¢2⟩ =

2 b3 (b1 + b 2) ,

j = 1, 2.

(23)

Therefore the self-consistency condition is:

2 b3 (b1 + b 2) = s.

(24)

In n=1 (19) and n=2 (23), we see that the possibility of having a non-null magnetisation has required the mixing of states induced by the new terms added at the borders of the blocks. 4

Eur. J. Phys. 38 (2017) 045404


Figure 1. Energy density versus y for different sizes n.

Figure 2. ⟨ sx⟩ versus y for different sizes n. The thick line is the exact result.

3. Results The self-consistency condition demands the computation of ⟨ sx⟩ in a position of the block under study. For n  3, the positions are not equivalent depending on the position of the site with respect to the centre. Although the results are not very sensitive to that position, the figures here presented have been calculated using the site nearest to the centre of the block. Once the value of s fulfills the self-consistency condition, for any size of the block n and for each value of y, we have access to the magnitudes given by this MF method. Specifically, the energy density and magnetisation are plotted in figures 1 and 2 respectively, and the values of yc and β are collected in table 1. The magnetisation value is 1 for y=0 and goes to 0 at y = yc . Beyond this point the self-consistency is only possible for s=0. In other words, the critical value yc marks the end of the magnetisation of the system. Therefore, for y > yc , h¢ º h and all magnitudes are derived from the resolution of blocks without surface effects. In figure 1 we have plotted the ground state energy density versus y for different sizes n and also the exact result of the model. For the different n, the energy density or energy per 5

Eur. J. Phys. 38 (2017) 045404


Figure 3. Second derivative of the energy density for n=2 (dotted line), n=5 (dot

dashed line) and exact solution (solid line).

Table 1. Result of yc and β versus n, and exact values.

n

yc

β

1 2 3 4 5 6 Exact

4 3.41828 3.04425 2.87035 2.72985 2.64452 2

0.5 0.456385 0.428716 0.415700 0.397413 0.392710 0.125

spin is the ratio between l¢n and n. In this figure note that for small y, the MF curves are under the exact result, but for large y they are over the exact result. This means that this form of implementing the mean feld method does not provide either lower or upper bounds to the ground energy of the model. Other approximation methods do provide rigorous bounds for the ground state energy; see, for example, [9, 10] and [11] for variational methods which provide upper bounds, and [12] for lower bounds. In this sense, the MF approach described in the appendix, which involves only one site and has a variational nature, does provide an upper bound for ρ. In figure 3 we have plotted the second derivative of the energy density versus y. Note that in the MF curves the second derivatives show vertical spikes at their corresponding yc. These spikes are provoked by the discontinuities existing in the first derivative.

4. Final comments As mentioned above, we think that the content of this paper may be of interest for quantum mechanics undergraduate courses, for students of physics or chemical physics. A modest home-work task such as reproducing our numerical results for say n = 2, 3 or 4, may be quite illuminating for his (her) comprehension of this subject. 6

Eur. J. Phys. 38 (2017) 045404


Figure A1. Energy density for the alternative mean field approach (thin line) and exact

solution (solid line).

Two concepts to grasp after having implemented the MF approach are the following: first, to understand that this is a rough approximation to a model that can be exactly solved; and second, that it constitutes a toy representation of a 2nd order phase transition. Computationally speaking, the student may acquire a good command of the algebra of spin systems. This includes he computation of eigenvalues and eigenvectors of Hermitian matrices, the implementation of the self-consistency condition, and the graphical representation and comparison of the results with the exact ones. We have used MATHEMATICA, but any other platform, like Maple, which allows symbolic calculations is useful when one builds the Hamiltonian matrices of blocks of an arbitrary size n. We quote three modern references related to learning quantum mechanics by means of computer algebra. See [13, 14] and [15]. From a didactic point of view, a symbolic representation is simpler to relate with the theory that a purely numeric one and allows the student to focus on what he (she) wants to do instead of how it is done. Nevertheless, it is not difficult to reproduce the above results in a purely numeric platform as, for example, MATLAB. We are well aware that the method developed here is not competitive for quantitative computations. See, for example, the slow convergence of the results as n increases in table 1 and figures 1 and 2. We would like to conclude remarking that the virtues of this method, naturalness and simplicity, do not guarantee at all the accuracy of the numerical results.

Acknowledgments AFP acknowledges José V García Esteve for useful comments.

Appendix In this appendix we show an alternative mean field approach to the TIM. This approach appears in [8]. It is based in a Hartree type approximation and therefore, it has a variational nature. In consequence, its results for the density energy of the ground state constitute a rigorous upper bound for the exact values of the model. 7

Eur. J. Phys. 38 (2017) 045404


Table A1. Values of the energy density for the alternative mean field approach and

exact values. y

Mean field

Exact

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

−1 −1.01563 −1.0625 −1.14063 −1.25 −1.39063 −1.5625 −1.76563 −2

−1 −1.01569 −1.06354 −1.14648 −1.27324 −1.45976 −1.67193 −1.89605 −2.12709

The ansatz for the ground state is

∣ Y⟩ =

N



( cos q ∣  ⟩ + sin q ∣  ⟩)i ,

N¥

(A.1)

i =-N

where θ is an angle to be fixed by the minimisation of the expectation value of the TIM Hamiltonian (1) per site; this value is

r ( y, q) =

y y (cos2 q - sin2 q ) - 4cos2 qsin2 q = cos (2q ) - 1 + cos2 (2q ) . 2 2

For y > 4, the minimum of r ( y, q ) is at q = 0, and for y  4 is at y cos (2q ) = - . 4 Inserting this value (A.3) in r ( y, q ) (equation (A.2)) we obtain

(A.2)

(A.3)

y2 . (A.4) 16 Note that this form of r ( y) does not show any discontinuity in its derivatives. The value of r ( y) in equation (A.4) and the exact value of r ( y) (equation (3)) are plotted in figure A1. As both curves for small y are close to each other, we have put some numerical results on table A1. Computing the expectation value of sx in the ansatz ji = (cos q ∣  ⟩ + sin q ∣  ⟩)i and using the value in equation (A.3) the value obtained for the magnetisation is r ( y) = - 1 -

1

⎛ ⎛ y ⎞2 ⎞ 2 ⟨ j ∣ sx ∣ j⟩ = ⎜1 - ⎜ ⎟ ⎟ . ⎝4⎠ ⎠ ⎝

(A.5)

That is, this approach predicts yc=4 and b = 0.5. These results are equal to those obtained with the other approach for n=1. References [1] Stanley H E 1971 Mean field theory of of magnetic phase transitions Introduction to Phase Transitions and Critical Phenomena (Oxford: Oxford University Press) Barber M N 1983 Phase Transitions and Critical Phenomena ed C Domb and J L Lebowitz (New York: Academia) and references therein 8

Eur. J. Phys. 38 (2017) 045404


[2] Bethe H A and Jackiw R 1997 Intermediate Quantum Mechanics 3rd edn (Reading, MA: Addison-Wesley) [3] Herrmann H J and Roux S (ed) 1990 Statistical Models for the Fracture of Disordered Media (Amsterdam: North Holland) [4] Kogut J B and Susskind L 1975 Hamiltonian formulation of Wilson’s lattice gauge theories Phys. Rev. D 11 395 [5] Pfeuty P 1970 The one-dimensional Ising model with a transverse field Ann. Phys. 57 79 [6] Fradkin E and Susskind L 1978 Order and disorder in gauge systems and magnets Phys. Rev. D 17 2637 [7] Osácar C and Pacheco A F 2009 Symmetry and degeneracy in quantum mechanics. Self-duality in finite spin systems Eur. J. Phys. 30 890–1 [8] Yankielowicz S 1976 Nonperturbative approach to quantum field theories: phase transitions and confinement Lectures given at 17th Scottish Universities Summer School in Physics (St. Andrews, Scotland, 1–21 August 1976) SLAC Preprint SLAC-PUB-1800 [9] Drell S D, Weinsten M and Yankielowicz S 1977 Quantum field theories in a lattice: variational methods for arbitrary coupling strengths and the Ising model in a transverse magnetic field Phys. Rev. D 16 1769 [10] Jullien R, Pfeuty P, Fields J N and Doniach S 1978 Zero-temperature renormalization method for quantum systems, Ising model in a transverse field in one dimension Phys. Rev. B 18 3568 [11] Fernandez-Pacheco A 1979 Comment on the SLAC renormalization group approach to the Ising chain in a transverse magnetic field Phys. Rev. D 19 3173 [12] Epele L, Fanchiotti H, Canal C G and Pacheco A F 1991 Lower bound for the ground energy of spin systems Physica A 173 500 [13] Crandall R E 1991 MATHEMATICA for Sciences (Quantum Mechanics) (Reading, MA: Addison Wesley) ch 5–2 [14] Steeb W H and Yardi Y 2010 Quantum Mechanics using Computer Algebra (Singapore: World Scientific) [15] Schmied R 2015 Introduction to computational quantum mechanics arXiv:1403.7050v2 [quant-ph]

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