The mapping to classical statistical mechanics: single site models

This chapter will discuss the reason for the central importance of the quantum Ising and rotor models in the theory of quantum phase tran-sitions, quite apart from any experimental motivations. It turns out that the quantum transitions in these models in d dimensions are inti-mately connected to certain well-studied finite temperature phase tran-sitions in classical statistical mechanics models in D = d + 1 dimen-sions [134, 483, 382, 164, 537]. We will then be able to transfer much of the sophisticated technology developed to analyze these classical models to the quantum models of interest here.

We will discuss this mapping here in the simplest context of d = 0, D = 1: we will consider single site quantum Ising and rotor models, and explicitly discuss their mapping to classical statistical mechanics models in D = 1 (the cases d > 0 will then be discussed in Chapter 3). These very simple classical models in D = 1, actually do not have any phase transitions. Nevertheless, it is quite useful to examine them thoroughly as they do have regions in which the correlation ‘length’ ξ becomes very large: the properties of these regions are very similar to those in the vicinity of the phase transition points in higher dimensions. In particular, we will introduce the central ideas of the scaling limit and universality in this very simple context. We will then go on to map the classical models to equivalent zero-dimensional quantum models and demonstrate that this mapping becomes exact in the scaling limit.

The following sections will actually carry out the quantum-to-classical mapping in reverse. With the benefit of hindsight, we will begin by ex-amining certain D = 1 classical statistical mechanics model and show that they are intimately related to single-site quantum Ising and rotor models. The classical models we shall study are the D = 1, N -component classical spin ferromagnets, and are surely familiar to most

18 The mapping to classical statistical mechanics: single site models readers in other contexts. We will consider the N = 1, 2, 3 case in the following sections in turn. The models with N > 3 are very similar to the case N = 3.

For a traditional, ‘classical’ perspective of these models, the reader is referred to the review by Thompson [490].

2.1 The classical Ising chain

Here we will consider the D = 1, N = 1 classical spin ferromagnet, more commonly known as the ferromagnetic Ising chain [242]. This chain has the partition function

Z = X

{σ^zi=±1}

exp (−H) (2.1)

where σ_i^z are Ising spins on sites i of a chain which take the values±1, and H is given by

H =−K XM

i=1

σ_i^zσ_i+1^z − h XM i=1

σ_i^z. (2.2)

In all our discussion of classical statistical mechanics models we absorb its ‘temperature’ into the definition of the coupling constants, as we have done above for K and h; in contrast, the temperature of quantum-mechanical models will always be explicitly indicated, and we will reserve the symbol T for it—as we will see below, the total length of the classical model will determine T . There are a total of M Ising spins (M large), and for convenience we have also added a uniform magnetic field h acting on all the spins. We will assume periodic boundary conditions, and therefore σ^z_M+1≡ σ^z1.

We will evaluate the partition function exactly following the original solution of Ising [242]. The trick is to writeZ as a trace over a matrix product, with one matrix for every site on the chain. Notice that the partition functions involves the exponential of a sum of terms on the sites of the chain: rewrite this as the product of exponentials of each term, and we easily obtain

Z = X

{σ^zi}

YM i=1

T1(σ^z_i, σ^z_i+1)T2(σ_i^z) (2.3)

where T1(σ^z₁, σ₂^z) = exp(Kσ^z₁σ^z₂) and T2(σ^z) = exp(hσ^z). Now notice that (2.3) has precisely the structure of a matrix product, if we interpret

2.1 The classical Ising chain 19 the two possible values of σ_i^zas the index labeling the rows and columns of a 2× 2 matrix T¹; T2has only one index and so should be interpreted as a diagonal matrix. So we have

Z = Tr (T¹T2T1T2· · · M times · · ·) (2.4) where the summation over the{σi^z} has been converted to a matrix trace because of the periodic boundary conditions, and

T1=

The matrix T1T2is identified as the ‘transfer matrix’ of the Ising chain H (Eqn (2.2)), the nomenclature suggesting that it transfers the trace over spins from each site to its neighbor. We can manipulate (2.4) into

Z = Tr (T¹T2)^M

= Tr

T₂^1/2T1T₂^1/2M

= ǫ^M₁ + ǫ^M₂ (2.6)

where ǫ1,2 are the eigenvalues of the symmetric matrix T₂^1/2T1T₂^1/2=

With these eigenvalues, (2.6) leads to an exact result for the free energy F =− ln Z. We will return to interpreting this result for F momentarily.

Now, we show how the above approach can also lead to exact infor-mation on correlation functions. For simplicity, we will consider only the case h = 0 (the generalization to non-zero h is not difficult), and describe the two-point spin correlator

σ^z_iσ^z_j

Going through exactly the same steps as those in the derivation of (2.6) we see that

where we have assumed that j≥ i, and σ^z(without a site index) is also interpreted as a 2× 2 diagonal Pauli matrix ˆσ^z in (1.6). The trace in

20 The mapping to classical statistical mechanics: single site models (2.10) can be evaluated in closed form in the basis in which T1is diagonal.

The eigenvectors of T1 are the states in (1.8) and the corresponding eigenvalues are ǫ1 = 2 cosh(K) and ǫ2 = 2 sinh(K). Now using the matrix elements h→ |σ^z| →i = h← |σ^z| ←i = 0 and h→ |σ^z| ←i = h→

|σ^z| ←i = 1 we get from (2.6) and (2.10) σ_i^zσ_j^z

=ǫ^M₁ ^−j+iǫ^j₂⁻ⁱ+ ǫ^M₂ ^−j+iǫ^j₁⁻ⁱ

ǫ^M₁ + ǫ^M₂ (2.11)

The equations (2.10) and (2.11) are our main results on the Ising chain with an arbitrary number of sites, M . While simple, they contain a great deal of useful information, as we will now show; much of the structure we will extract below generalizes to more complex models.

Let us examine the form of the correlations in (2.11) in the limit of an infinite chain (M → ∞); then we have

σ_i^zσ^z_j

= (tanh(K))^j⁻ⁱ (2.12)

It is useful for the following discussion to label the spins not by the site index i, but by a physical length co-ordinate τ ; we have chosen the symbol τ , rather than the more conventional x, because we will shortly interpret this ‘length’ as the imaginary time direction of a quantum problem. So if we imagine that the spins are placed on a lattice of spacing a, then σ^z(τ )≡ σj^z where

τ = ja. (2.13)

With this notation, we can write (2.12) as

hσ^z(τ )σ^z(0)i = e^{−|τ |/ξ} (2.14) where the correlation length, ξ, is given by

1 ξ = 1

aln coth(K). (2.15)

We emphasize that the symbol ξ always represents the actual correlation length at h = 0; the actual correlation length for h6= 0 will, of course, be different. In the large K limit, the correlation length becomes much larger than the lattice spacing, a:

ξ a ≈1

2e^2K≫ 1 K≫ 1. (2.16)

In the sequel, we shall primarily be interested in physics on the scale of order ξ, in the regime where ξ is much greater than a. It is precisely in this situation that the concepts of the scaling limit and universality become useful, and they are introduced in the following subsections.

2.1 The classical Ising chain 21 2.1.1 The scaling limit

The simplest way to think of the scaling limit is to first divide all lengths into “large” and “small” lengths. For the Ising chain, we take the cor-relation length ξ, the observation scale τ , and the system size

Lτ ≡ Ma, (2.17)

as our large lengths, and the lattice spacing, a, as the only small length.

The scaling limit of an observable is then defined as its value when all corrections involving the ratio of small to large lengths are neglected.

There are two conceptually rather different, but equivalent, ways of thinking about the scaling limit. We can either send the small length a to zero while keeping the large lengths fixed (as particle physicists are inclined to do) or send all the large lengths to infinity while keeping a fixed (as is more common among condensed matter physicists). The physics can only depend upon the ratio of lengths, so it is clear that the two methods are equivalent. We shall choose among these points of view at our convenience, and show that it is often very useful to straddle this cultural divide and use the insights of both perspectives.

To complete the definition of the scaling limit, we also have to discuss the manner in which the parameters K and h must be treated. From (2.15), we see that K can be expressed in terms of the ratio of lengths ξ/a; we can use this to eliminate explicit dependence upon K, and then the scaling limit is specified by the already specified ξ/a→ ∞ limit. It remains to discuss the behavior of h. In general, there is no a priori way of determining this and one has to examine the structure of the correlation functions to determine the appropriate limit. Let us guess the answer here by a physical argument. The scaling limit involves the study of large K, when the spin correlation length becomes large. Under these conditions, spins a few lattice spacings apart invariable point in the same direction, and should there be sensitive to the mean magnetic field h per unit length. This is measured by eh, defined by

eh ≡ h

a. (2.18)

So we take the scaling limit a → 0 while keeping ˜h fixed; any other choice would result in a limiting theory with spins under the influence of a field with either infinite or vanishing strength. Alternatively stated, we have chosen 1/˜h, a quantity with the dimensions of length, as one of our large length scales.

We have assembled all the necessary steps for the scaling limit.

Ex-22 The mapping to classical statistical mechanics: single site models press any observable in terms of the physical length τ , replace the num-ber of sites M by Lτ/a, solve (2.15) to express K in terms of ξ/a, and use (2.18) to replace h by eh. Then take the limit a→ 0 at fixed τ, L^τ, ξ, and eh.

We first describe the results for the free energy F . The quantity with the finite scaling limit should clearly be the free energy density,F,

F = −(ln Z)/Ma. (2.19)

First, from (2.8) we get in the scaling limit ǫ1,2≈ where E0 = −K/a is the ground state energy per unit length of the chain in zero external field.

In a similar manner, we can take the scaling limit of the correlation function in (2.11), which recall was in zero external field eh = 0. We obtain

hσ^z(τ )σ^z(0)i = e^{−|τ |/ξ}+ e^−(L^τ^{−|τ |)/ξ}

1 + e^−L^τ^/ξ . (2.22) The results (2.21), (2.22) are the main conclusions of this subsection.

2.1.2 Universality

The assertion of universality is that the results of the scaling limit are not sensitive to the precise microscopic model being used. This is can be seen as the formal consequence of the physically reasonable requirement that correlations at the scale of the large ξ should not depend upon the details of the interactions on the scale of the lattice spacing, a.

Let us describe this by an explicit example. Suppose, instead of using the model H in (2.2), we worked with a Hamiltonian H1 with both first (K1) and second (K2) neighbor exchanges between the Ising spins σ^z. This model can also be solved by the transfer matrix methods (one needs a basis of 4 sites corresponding to the 4 states of two near-neighbor spins, and the transfer matrix is 4× 4), but we will not present the explicit solution here. From the solution we can determine the correlation length,

2.1 The classical Ising chain 23 ξ of H1, which will be a function of both K1 and K2. Now, as in Section 2.3, express the free energy density in terms of ξ, and take the limit a → 0 at fixed ξ, L^τ, and ˜h. The implication of universality is that the result will be precisely identical to (2.21), with E0given by the ground state energy density of H1 in zero field: E0 =−(K¹+ K2)/a.

The reader is invited to check this assertion for this simple example.

We can make the above assertion more precise by introducing the concept of a universal scaling function. We write (2.21) in the form

F = E⁰+ 1 Lτ

Φ_F

Lτ

ξ , ehLτ

, (2.23)

where Φ_F is the universal scaling function, whose explicit value can be easily deduced by comparing with (2.21). Notice that the arguments of Φ_F are simply the two dimensionless ratios that can be made out of the three large lengths at our disposal: Lτ, ξ, and 1/˜h. The prefactor, 1/Lτ, in front of Φ_F is necessary because the free energy density has dimensions of inverse length.

As its name implies, the Φ_F is independent of microscopic details. In contrast, E0, the ground state energy of the Ising chain, clearly depends sensitively on the values of the microscopic exchange constants, and is therefore identified as a non-universal additive contribution toF.

In a similar manner, we can introduce a universal scaling function of the two-point correlation function. We have

hσ^z(τ )σ^z(0)i = Φ^σ

τ Lτ,Lτ

ξ , ehLτ

(2.24) where Φσ is another universal scaling function, and there is now no non-universal additive constant. Again Φσ is a function of all the inde-pendent dimensionless combinations of large lengths; there is no prefac-tor because the correlaprefac-tor is clearly dimensionless. We can read off the value of Φσ(y1, y2, 0) by comparing (2.24) with (2.22), but determining the full function Φ(y1, y2, y3) requires knowledge of the lattice correlator in the presence of a non-zero h, which is somewhat tedious to obtain. A simpler method will become apparent in the following subsection.

2.1.3 Mapping to a quantum model: Ising spin in a transverse field

We will show the statistical mechanics of the Ising chain can be mapped onto the quantum mechanics of a single Ising spin [483, 164]. Further,

24 The mapping to classical statistical mechanics: single site models as stated in the introduction to this chapter, correlators of the quantum spin will precisely reproduce the scaling limit of the classical Ising chain.

Let us return to the expressions (2.4), (2.5), and write the transfer matrices T1, T2in terms of ratios of “large” to small length scales. We have

T1 = e^K(1 + e^−2Kσˆ^x)

≈ e^K(1 + (a/2ξ)ˆσ^x)

≈ exp (a(−E⁰+ (1/2ξ)ˆσ^x) . T2 = exp

aehˆσ^z

, (2.25)

where ˆσ^x,zare the Pauli matrices in (1.6). Notice that both T1,2have the form eâO, where O is some operator, acting on the| ↑, ↓i states, which is independent of a. Using the fact that eâO¹eâO²= eâ(O¹^+O²⁾(1 +O(a²)), we can write (2.4) in the limit a→ 0 as

T1T2 ≈ exp(−aH^Q)

Z = (T¹T2)^M ≈ Tr exp(−H^Q/T ) (2.26) where

HQ= E0−∆

2σˆ^x− ˜hˆσ^z, (2.27) with

T ≡ 1 Lτ

, ∆≡ 1

ξ. (2.28)

We have introduced the fundamental quantum Hamiltonian HQ. It de-scribes the dynamics of a single Ising quantum spin, whose Hilbert space consists of the two states | ↑, ↓i, and which is under the influence of a longitudinal field ˜h, and a transverse field ∆; it is the single site ver-sion of (1.5) with an additional longitudinal field. Notice, from the first relation in (2.26), that the transfer matrix of the classical chain H is the quantum evolution operator e^−H^Q^τ over an imaginary time τ = a, the lattice spacing: so the transfer from one site to the next is similar to evolution in imaginary time, and length co-ordinates for the classical chain translate into imaginary time co-ordinates for the quantum model HQ. The energy ∆ is also the gap between the ground and excited state of HQin zero (longitudinal) field, and it is precisely equal to the inverse of the correlation length of the classical Ising chain, as expected from the length to time mapping. Further, the partition function of the quantum spin is taken at a temperature T which precisely equals the inverse of

2.1 The classical Ising chain 25 the total length of the classical chain. These correspondences between a gap of a quantum system and a correlation length of the corresponding classical model along the ‘time’ direction, and between the temperature of the quantum system and the total length of the classical model, are extremely general, and will apply to essentially all of the models we shall consider in this book.

We can use (2.26) and (2.27) to quickly evaluate the free energy of the quantum spin, F = −T ln Z. The eigenenergies of H^Q are E0± q

(∆/2)²+ eh², and we have

F = E⁰− T ln

2 cosh

(∆/2)²+ ˜h²

(2.29) which agrees precisely with the scaling limit of the classical Ising chain (2.21). Indeed, the single spin quantum Hamiltonian HQ is precisely the theory describing the universal scaling properties of the entire class of classical Ising chains with short range interactions. Statements of this type are often shortened to “HQ is the scaling theory of H”.

The correspondence between HQ and H also extends to correlation functions. Let us define the time-ordered correlator, G of HQ in imagi-nary time by

G(τ1, τ2) =

ZTr e^−H^Q^/Tσˆ^z(τ1)ˆσ^z(τ2)

for τ1> τ2 1

ZTr e^−H^Q^/Tσˆ^z(τ2)ˆσ^z(τ1)

for τ1< τ2

, (2.30)

where ˆσ^z(τ ) is defined by the imaginary time evolution under the HQ: ˆ

σ^z(τ )≡ e^H^Q^τσˆ^ze^−H^Q^τ. (2.31) Now, upon carrying through the mapping described above for the free energy for the case of the correlation function, we find that

G(τ1, τ2) = lim

a→0hσ^z(τ1)σ^z(τ2)iH, (2.32) where we have emphasized by the subscript that the average on the right hand side is for the classical model with Hamiltonian H. The time-ordered functions appear in the quantum problem for the same reason we had to assume j≥ i in (2.10): as the transfer matrix evolves the system from ‘earlier’ sites to ‘later’ sites, the earlier ˆσ^z operators appear first in the trace.

The representation (2.30) also makes the origin of the mapping be-tween the quantum gap, ∆, and the classical correlation length, ξ, in

26 The mapping to classical statistical mechanics: single site models (2.28) quite clear. We can evaluate (2.30) at T = 0 by inserting a com-plete set of HQ eigenstates and obtain the general representation

G(τ1, τ2) =X

|h0|σ^z|ni|²e^−(Eⁿ^−E⁰⁾^|τ¹^−τ²^|, (2.33) where|ni are all the eigenstates of H^Q with eigenvalues En, and|0i is the ground state. For sufficiently large |τ¹− τ²|, the sum over n will be dominated by lowest energy state for which the matrix element is non-zero, and this gives an exponential decay of the correlation function over a ‘length’ ξ = 1/(E1− E⁰) = 1/∆. Of course, in the present simple system there are only a total of two states, but this result is clearly more general.

It is quite easy to evaluate (2.30) for HQ, and the direct quantum computation is much simpler than the use of the classical mapping in (2.32). We find where Φσ is precisely the same scaling function that appeared in (2.24), and can be computed from (2.30) to be

Φσ(y1, y2, y3) = 4y₃² It can be checked that the y3 = 0 case of this result agrees with the combination of (2.22) and (2.24).

2.2 The classicalXY chain and a O(2) quantum rotor We will consider the D = 1, N = 2 classical ferromagnet; this is also referred to as the XY ferromagnet. We generalize (2.1,2.2) to N = 2 by replacing σ^z_i by a two-component unit-length variable ni. This modifies (2.1) to

2.2 The classicalXY chain and a O(2) quantum rotor 27 where, as in the Ising case, we have added a uniform field h = (h, 0).

(3.3). It is convenient to parameterize the unit length classical spins, ni, by

n_i = (cos θi, sin θi) (2.38) where the continuous angular variables θi, run from 0 to 2π. In these variables, H takes the form

H =−K and the partition function is

Z =

We again assume periodic boundary conditions with θM+1≡ θ¹. Notice that in zero field, H remains invariant if all the spins are rotated by the same angle φ, θi → θⁱ + φ, and so our results will not depend upon the particular orientation chosen for h. The partition function can be evaluated by transfer matrix methods [156, 255] quite similar to those used for the Ising chain. Although we will not use such a method to obtain our results, we nevertheless describe the main steps for completeness. First writeZ in the form

Z =

where the symmetric transfer matrix operator ˆT is defined by hθ| ˆT|θ^′i = exp

and the trace is clearly over continuous angular variable θ. As in the Ising case, we have to diagonalize the transfer matrix ˆT by solving the eigenvalue equation corre-sponding eigenvalues λµ. Then the partition function Z is simply

Z =X

λ^M_µ (2.44)

28 The mapping to classical statistical mechanics: single site models where the sum extends over the infinite number of eigenvalues λµ. The solution of (2.43) is quite involved, and the present approach is a rather convoluted method of obtaining the universal properties of H.

Instead, it is useful to approach the problem with a little physical insight, and take the scaling limit at the earliest possible stage. We anticipate, from our experience with the Ising model, that the universal scaling behavior will emerge at large values of K. For this case, θi is not expected to vary much from one site to the next, suggesting that it should be useful to expand in terms of gradients of θi. So we define a continuous co-ordinate τ = ja, where a is the lattice spacing, and the label τ anticipates its eventual interpretation as the imaginary time

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