Transverse Ising Spin Glass and Random Field Systems

Abstract

Transverse Ising systems with randomly frustrated interactions or random fields have an intriguing nature that a phase transition peculiar in randomly frustrated systems is driven by tunable thermal fluctuations as well as quantum fluctuations. The first part of Chap. 6 discusses transverse Ising spin glasses in great detail, where systems are characterised by randomly competing interactions and a uniform transverse field. It includes analytic and numerical studies of the Sherrington-Kirkpatrick model in a transverse field, numerical studies of the Edwards-Anderson model in a transverse field in two and three dimensions, and the mean field theory of Ising spin glasses with random p-body interactions in a transverse field. One of the interesting problem in quantum spin glasses is the stability of the state with replica symmetry breaking in quantum fluctuations. This problem is also mentioned. The second part of Chap. 6 focuses on the random field transverse Ising models. Phase diagrams at zero and finite temperatures are studied by the mean field theory. Appendices of this chapter include a brief note on the infinite-range vector (Heisenberg) spin glass model, mapping of the transverse Ising spin glass Hamiltonian to an effective classical Hamiltonian, mapping of random field Ising ferromagnet in a transverse field to a random Ising antiferromagnet in a longitudinal and transverse fields, and mathematical details of the replica method in transverse Ising spin glasses.

Appendix 6.A

6.1.1 6.A.1 The Vector Spin Glass Model

The Sherrington-Kirkpatrick model generalised to quantum spins (quantum vector spin glass) was introduced by Bray et al. [44], who applied the replica method to the Hamiltonian given by

$$ H = -\sum_{ij}J_{ij} \boldsymbol{S}_{i} \cdot \boldsymbol{S}_{j}, $$

(6.A.1)

where the sum is over all pair of spins (the interaction is long-range), and the exchange interaction J _ij are independent random variables, with a symmetric Gaussian distribution

$$ P(J_{ij}) = \biggl( \frac{N}{2\pi J^{2}} \biggr)^{1/2} \exp \biggl( {-}\frac{NJ_{ij}^{2}}{2J^{2}} \biggr). $$

(6.A.2)

The spin operators satisfy the standard commutation relations,

$$ \bigl[S_{i}^{\alpha},S_{j}^{\beta}\bigr] = 2i \delta_{ij} \varepsilon_{\alpha\beta\gamma}S_{k}^{\gamma}. $$

(6.A.3)

Bray et al. [44] used replica method to handle the quenched disorder of the spin glass problem writing

$$ \overline{ \ln Z} = \lim_{n \to0} \frac{\overline{Z^{n}}-1}{n}. $$

(6.A.4)

The partition function of the vector spin glass model can be written as

$$ Z^{n} = \mathrm{Tr}\, P \exp \Biggl[ \beta \int_{0}^{1} d\tau \,\sum_{ij} J_{ij} \sum _{\alpha=1}^{n} \boldsymbol{S}_{i}^{\alpha}( \tau) \cdot \boldsymbol{S}_{j}^{\alpha}(\tau) \Biggr]. $$

(6.A.5)

Performing the Gaussian average, one gets

$$ \overline{ Z^{n}} = \mathrm{Tr}\, P \exp \biggl[ \frac{\beta^{2}J^{2}}{2N^{2}} \int_{0}^{1}d\tau \int_{0}^{1}d \tau' \,\sum_{\alpha, \beta} \sum _{i,j} \bigl[ \bigl(\boldsymbol{S}_{i}^{\alpha}( \tau)\cdot \boldsymbol{S}_{j}^{\alpha} (\tau)\bigr) \bigl( \boldsymbol{S}_{i}^{\beta}\bigl(\tau'\bigr)\cdot \boldsymbol{S}_{j}^{\beta}\bigl(\tau'\bigr)\bigr)\bigr] \biggr]. $$

(6.A.6)

Using the Hubbard-Stratonovitch transformation (6.3.18), one can simplify the above expression. One can then express the free energy in terms of the self-interaction R ^αα(τ,τ′), spin glass order parameter Q ^αβ(τ,τ′) and the quadrupolar order parameter Q ^αα(τ,τ′). In the paramagnetic phase, the spin glass order parameter and the quadrupolar order parameter vanish, and one can derive an effective Hamiltonian as in the case of a quantum Ising system. One then expands the total free energy in powers of Q ^αβ and obtains the value of T _c , setting the coefficient (Q ^αβ)² equal to zero, one gets the value of T _c . Bray and Moore established the existence of spin glass transition for all S and estimated the value of T _c (S) given by the condition

$$ J \chi_{\mathrm{loc}}=1; \quad \chi_{\mathrm{loc}} = \beta \int _{0}^{1}d\tau \int_{0}^{1} d\tau'\, \bigl\langle TS^{z}(\tau) S^{z}\bigl( \tau'\bigr) \bigr\rangle. $$

(6.A.7)

In the extreme quantum case S=1/2, one can obtain, following Bray et al., the value of transition temperature from the paramagnetic phase o the spin glass ordered phase as

$$ k_{B}T_{c} \sim J/4\sqrt{3}. $$

(6.A.8)

6.1.2 6.A.2 The Effective Classical Hamiltonian of a Transverse Ising Spin Glass

Let us consider the quantum transverse Ising spin glass Hamiltonian given by (6.3.1)

$$ H = H_{0} + V = -\varGamma\sum_{i}S_{i}^{x} - \sum_{ij}J_{ij}S_{i}^{z}S_{j}^{z}. $$

(6.A.9)

Let us consider the configuration averaged n-replicated partition function

$$ \overline{ Z^{n}} = \overline{ \bigl[\exp (-\beta H) \bigr]^{n}}. $$

(6.A.10)

One can now transform the above n-replicated partition function in the following form

$$ \overline{ Z^{n}} = \mathrm{Tr}\, \overline{ \exp \Biggl(-\beta \sum _{\alpha=1}^{n} H(\alpha)\Biggr)} $$

(6.A.11)

where H(α) is the Hamiltonian of the α-th replica and it is separated in the following form

$$ H(\alpha) = H_{0}(\alpha) + V(\alpha) = -\varGamma\sum _{i}S_{i\alpha}^{x} - \sum _{ij}J_{ij}S_{i\alpha}^{z}S_{j \alpha}^{z}. $$

(6.A.12)

Applying the Trotter formula (cf. Sect. 3.1), one now gets

$$ \overline{Z^{n}}= \lim_{M \to\infty} \prod _{\alpha=1}^{n} \biggl[ \exp \biggl( {-} \frac{\beta}{M} H_{0}(\alpha) \biggr) \exp \biggl( - \frac{\beta}{M} V(\alpha) \biggr) \biggr]^{M}. $$

(6.A.13)

Similarly as in the pure transverse Ising system, one introduces complete sets of eigenvectors of the operator $S_{\alpha}^{z}$ and using the relation

$$ \langle S|\exp \bigl(\gamma S^{x}\bigr)\bigl|S' \bigr \rangle= \bigl[(1/2)\sinh2\gamma\bigr]^{1/2}\exp \biggl( \frac{SS'}{2}\ln\coth\gamma \biggr), $$

(6.A.14)

one gets

$$ \overline{Z^{n}} = \lim_{M \to\infty} \mathrm{Tr}\, \overline{ \exp \Biggl(\,\sum_{\alpha=1}^{n} H_{\mathrm{eff}}( \alpha)\Biggr) } $$

(6.A.15)

where the effective classical Hamiltonian

$$ H_{\mathrm{eff}} = \sum_{l=1}^{M} \Biggl( K_{M} \sum_{i=1}^{N} S( \alpha)_{i,l} S(\alpha)_{i,l+1}+ \sum _{\langle ij \rangle} K_{ij}^{(M)} S_{i,l}( \alpha) S_{j,l}(\alpha)+ \ln c_{M} \Biggr), $$

(6.A.16)

with

One can then perform the Gaussian averaging. This indicates an effective (d+2)-dimensional classical Hamiltonian [387].

6.1.3 6.A.3 Effective Single-Site Hamiltonian for Long-Range Interacting RFTIM

To derive the effective single-site Hamiltonian we consider the Hamiltonian of (long-range ferromagnetic) random field Ising model in a transverse field

$$ H = -\frac{J}{N} \sum_{i \neq j} S_{i}^{z} S_{j}^{z} - \sum _{i}h_{i}S_{i}^{z}- \varGamma\sum_{i}S_{i}^{x} $$

(6.A.17)

where the random variable at each site satisfies a Gaussian distribution (6.7.19). The configuration averaged free energy of the system is given by

$$ F = -kT \overline{ \ln Z}, $$

(6.A.18)

where k is the Boltzmann constant and Z is the partition function for a particular realisation of the random fields. Using replica trick [39, 76, 136, 275] we can write the n-replicated free energy in the form

(6.A.19)

where α denotes the α-th replica, P denotes the time ordering, $H_{0}(\alpha) = - \varGamma\sum_{i}S_{\alpha i}^{x}$ and S ^z(τ)’s are operators in the interaction representation. We can now perform the configuration averaging to obtain

(6.A.20)

A Hubbard-Stratonovitch transformation simplifies the term

$$ \exp \Biggl[ \int_{0}^{\beta} d\tau \,\sum _{\alpha=1}^{n} \sum_{ij} \frac{J}{N} S_{\alpha i}^{z}(\tau) S_{\alpha j}^{z}( \tau) \Biggr] = \exp \Biggl[ \int_{0}^{\beta} d\tau \,\sum_{\alpha=1}^{n} \Biggl \vert \sqrt{ \frac{J}{N}} \sum_{i=1}^{N} S_{\alpha i}^{z}(\tau) \Biggr \vert ^{2} \Biggr] $$

(6.A.21)

(where the terms of order (1/N) are neglected), so that we obtain the configuration averaged n-replicated free energy

(6.A.22)

where x ^α s are dummy variables. In the N→∞ limit, one can readily obtain the saddle point configuration averaged free energy

$$ F = -kT \lim_{n \to0} \frac{1}{n} \Biggl[ -\frac{\beta}{2} \sum _{\alpha=1}^{n} x_{\alpha}^{2}+ \ln \mathrm{Tr} \exp (A) \Biggr] $$

(6.A.23)

where

(6.A.24)

The square term appearing in the above expression can be simplified using once again the Hubbard-Stratonovitch transformation to obtain

(6.A.25)

where s is a dummy variable. Finally, one obtains the form of free energy (with $x = m^{z}\sqrt{2J}$ and sδ=h) given by

(6.A.26)

We have thus reduced the many-body Hamiltonian (in the N→∞ limit) to an effective single-site problem, where the molecular field at each site is given by (2m ^z J+h) where h is distributed with a probability distribution P(h).

6.1.4 6.A.4 Mapping of Random Ising Antiferromagnet in Uniform Longitudinal and Transverse Fields to RFTIM

The equivalence between the transition in the random Ising antiferromagnet in uniform transverse and longitudinal fields (RIAFTL) to that in the random field transverse Ising model is obtained by employing semi-classical decimation of the one sublattice of the RIAFTL system, which neglects commutators between the spin operators. Here a partial trace is done over sites of one sub-lattice, e.g., that in which the site label i is odd. The original (reduced) Hamiltonian

$$ -\beta H = \sum_{i} \bigl(-K_{i,i+1}S_{i}^{z}S_{i+1}^{z}+ h_{i}S_{i}^{z}+ \varGamma S_{i}^{x} \bigr) = \sum_{i}H_{i} $$

(6.A.27)

is mapped into a new form

$$ -\beta H' = \sum_{i} \bigl(-K'_{2i,2i+2} S_{2i}^{z} S_{2i+2}^{z}+ h_{2i}' S_{2i}^{z}+ \varGamma'S_{i}^{x} \bigr) = \sum_{i}H_{2i}'. $$

(6.A.28)

The trace over S _2i+1 produces the factors

(6.A.29)

This can be written as

$$ \exp \bigl(A+BS_{2i}^{z}+ CS_{2i+2}^{z}+ DS_{2i}^{z}S_{2i+2}^{z}\bigr) $$

(6.A.30)

where matching of the expression for all four possible sets of values for $(S_{2i}^{z},S_{2i+2}^{z})$ gives A, B, C, D in terms of Γ, h _2i+1, K _2i,2i+1, K _2i+1,2i+2. For example

(6.A.31)

(6.A.32)

so that we arrive at the recursion relations (6.7.25), (6.7.26) and (6.7.27). For h≪K, one can evaluate B(K ₁,K ₂) (where K ₁ and K ₂ are two neighbouring bonds), using the simplified relations

(6.A.33)

(6.A.34)

(6.A.35)

(6.A.36)

where

$$ \varOmega_{\pm} = \bigl[ \varLambda_{\pm}^{2}+ \varGamma^{2}\bigr]^{1/2}, \qquad \varLambda_{\pm}= K_{1} \pm K_{2}. $$

(6.A.37)

Hence

$$ B(K_{1},K_{2}) = \frac{1}{4} \ln \biggl[ \frac{\cosh(\varOmega_{+}-h \frac{\varLambda_{+}}{\varOmega_{+}}) \cosh(\varOmega_{-}-h \frac{\varLambda_{-}}{\varOmega_{-}}) }{ \cosh(\varOmega_{+}+h \frac{\varLambda_{+}}{\varOmega_{+}}) \cosh(\varOmega_{-}+h \frac{\varLambda_{-}}{\varOmega_{-}}) } \biggr]. $$

(6.A.38)

If we now use the relation (for small h)

$$ \ln \biggl[ \frac{\cosh(\alpha+ \gamma h)}{ \cosh(\alpha-\gamma h)} \biggr] = 2\gamma h \tanh\alpha+ \cdots $$

(6.A.39)

we get

$$ B(K_{1},K_{2}) = -\frac{h}{2} \biggl[ \frac{\varLambda_{+}}{ \varOmega_{+}} \tanh\varOmega_{+} + \frac{\varLambda_{-}}{\varOmega_{-}} \tanh \varOmega_{-} \biggr] + O\bigl(h^{2}\bigr), $$

(6.A.40)

etc.

6.1.5 6.A.5 Derivation of Free Energy for the SK Model with Antiferromagnetic Bias in a Transverse Field

We show the derivation of the free energy per spin for the system to be described by the Hamiltonian.

(6.A.41)

Carrying out the Suzuki-Trotter decomposition, we have the replicated partition function.

(6.A.42)

(6.A.43)

where α and k denote the replica and Trotter indices. M is the number of the Trotter slices and β is the inverse temperature. The disorder J _ij obeys

(6.A.44)

In other words, the J _ij follows

(6.A.45)

We should notice that j ₀>0, J=0 is pure ferromagnetic transverse Ising model, whereas j ₀<0, J=0 corresponds to pure antiferromagnetic transverse Ising model. Then, by using $\int_{-\infty}^{\infty}Dx\,\mathrm{e}^{ax}=\mathrm{e}^{a^{2}/2}$, $Dx \equiv dx \,\mathrm{e}^{-x^{2}/2}/\sqrt{2\pi}$, we have the average of the replicated partition function as

(6.A.46)

where the bracket was defined as ≪⋯≫=∫∏ _ij dJ _ij P(J _ij )(⋯). To take a proper thermodynamic limit, we use the scaling

(6.A.47)

For this rescaling of the parameters, the averaged replicated partition function ${\ll} Z_{M}^{n} {\gg}$ reads

(6.A.48)

We next assume the replica symmetry and static approximations such as

(6.A.49)

(6.A.50)

(6.A.51)

Then, we should notice the relation:

(6.A.52)

(6.A.53)

To take into account the above relations, we obtain in the limit of N→∞ as

(6.A.54)

Therefore, the following f is regarded as free energy per spin by the definition of replica theory

(6.A.55)

6.1.5.1 6.A.5.1 Saddle Point Equations

For simplicity, we define

(6.A.56)

(6.A.57)

Then, we have the following simplified free energy

(6.A.58)

The saddle point equations are derived as follows [159, 401].

(6.A.59)

(6.A.60)

(6.A.61)

(6.A.62)

6.1.5.2 6.A.5.2 At the Ground State

We first should notice that $\tilde{q}$ is always larger than q. In fact, we can easily show that

(6.A.63)

Then, we consider the limit of β→∞. If $\tilde{q}-q=\varepsilon\geq0$ is of order 1 object, the free energy f diverges in the limit of β→∞ as $(\beta\tilde{J})^{2} (q^{2}-\tilde{q}^{2})/4$. Therefore, we conclude that $q = \tilde{q}$ should be satisfied in the limit of β→∞ and we obtain the saddle pint equation at the ground state as

(6.A.64)

(6.A.65)

(6.A.66)