We study the phase structure of the random-plaquette Z 2 lattice gauge model in three dimensions. In this model, the"gauge coupling" for each plaquette is a quenched random variable that takes the value β with the probability 1 − p and −β with the probability p. This model is relevant for the recently proposed quantum memory of toric code. The parameter p is the concentration of the plaquettes with "wrong-sign" couplings −β, and interpreted as the error probability per qubit in quantum code.In the gauge system with p = 0, i.e., with the uniform gauge couplings β, it is known that there exists a second-order phase transition at a certain critical "temperature", T (≡ β −1 ) = T c = 1.31, which separates an ordered(Higgs) phase at T < T c and a disordered(confinement)
Quantum phase transition from the Néel to the dimer states in an antiferromagnetic (AF) Heisenberg model on square lattice is studied. We introduce a control parameter ␣ for the exchange coupling that connects the Néel ͑␣ =0͒ and the dimer ͑␣ =1͒ states. We employ the CP 1 representation of the s = 1 2 spin operator and integrate out the half of the CP 1 variables at odd sites to obtain a CP 1 nonlinear model. The effective coupling constant is a function of ␣, and at ␣ = 0, the CP 1 model is in the ordered phase that corresponds to the Néel state of the AF Heisenberg model. A phase transition to the dimer state occurs at a certain critical value of ␣ C as ␣ increases. In the Néel state, the dynamical composite U(1) gauge field in the CP 1 model is in a Higgs phase, and low-energy excitations are gapless spin waves. In the dimer phase, a confinement phase of the gauge theory with s = 1 excitations is realized. For the critical point, we argue that a deconfinement phase, which is similar to the Coulomb phase in three spatial dimensions, is realized and s = 1 2 spinons appear as low-energy excitations.Quantum phase transition (QPT) is one of the most interesting problem in these days. 1 It is often argued that the simple Ginzburg-Landau theory does not apply to certain classes of the QPTs. In this paper we study the s = 1 2 antiferromagnetic (AF) Heisenberg model on two-dimensional square lattices with nonuniform exchange couplings,where x denotes site of the spatial lattice, j is the direction index ͑j =1,2͒, and S ជ x is the spin operator at site x. We rename the even lattice sites x = ͑o , i͒, where o denotes odd site and the index i =1,2,3, and 4 specifies its four nearestneighbor (NN) even sites (see Fig. 1).The exchange couplings J xj = J oi are position dependent, and we explicitly consider the following case, which corresponds to the dimer configuration:where 0 ഛ ␣ ഛ 1 is a control parameter that connects the uniform Heisenberg model to the dimer model. It is not so difficult to derive the CP 1 field-theory model 2,3 from Eq. (1). 4 The spin operator S ជ x can be expressed in terms of the CP 1 variable z x = ͑z x 1 , z x 2 ͒ t aswhere ជ are the Pauli matrices and the CP 1 constraint ͚ a=1,2 ͉z x a ͉ 2 = 1 guarantees the magnitude of the localized spin as 1 2 .From our assignment of J oi (2), it is obvious that J o1 is larger than the others. We use the path-integral formalism and parametrize the CP 1 variable z o by refering to z o1 ϵ z e ,where p o is a parameter, e i is a phase factor and z e = i 2 z e * . At vanishing temperature ͑T͒, spins tend to point antiparallel their NN spins, and then the parameter p o can be treated as a small parameter. We expand ͱ 1−͉p o ͉ 2 Ӎ 1− 1 2 ͉p o ͉ 2 +¯and retain only terms up to quadratic of p o . Then we perform the Gaussian integration of p o to obtain an effective model of z e for which smooth configurations dominate at T =0.Calculation is rather long, but straightforward 4 and we obtain an effective field theory of the AF Heisenberg model under study,x is the Lagr...
In the present paper we shall study (2 + 1) dimensional Z N gauge theories on a lattice. It is shown that the gauge theories have two phases, one is a Higgs phase and the other is a confinement phase.We investigate low-energy excitation modes in the Higgs phase and clarify relationship between the Z N gauge theories and Kitaev's model for quantum memory and quantum computations. Then we study effects of random gauge couplings(RGC) which are identified with noise and errors in quantum computations by Kitaev's model. By using a duality transformation, it is shown that timeindependent RGC give no significant effects on the phase structure and the stability of quantum memory and computations. Then by using the replica methods, we study Z N gauge theories with time-dependent RGC and show that nontrivial phase transitions occur by the RGC.
We introduce a 3D compact U(1) lattice gauge theory having nonlocal interactions in the temporal direction, and study its phase structure. The model is relevant for the compact QED3 and strongly correlated electron systems like the t-J model of cuprates. For a power-law decaying long-range interaction, which simulates the effect of gapless matter fields, a second-order phase transition takes place separating the confinement and deconfinement phases. For an exponentially decaying interaction simulating matter fields with gaps, the system exhibits no signals of a second-order transition.
In the present paper, we shall study the 4-dimensional Z 2 lattice gauge model with a random gauge coupling; the random-plaquette gauge model(RPGM). The random gauge coupling at each plaquette takes the value J with the probability 1 − p and −J with p. This model exhibits a confinement-Higgs phase transition. We numerically obtain a phase boundary curve in the (p − T )-plane where T is the "temperature" measured in unit of J/k B . This model plays an important role in estimating the accuracy threshold of a quantum memory of a toric code. In this paper, we are mainly interested in its "self-duality" aspect, and the relationship with the random-bond Ising model(RBIM) in 2dimensions. The "self-duality" argument can be applied both for RPGM and RBIM, giving the same duality equations, hence predicting the same phase boundary. The phase boundary curve obtained by our numerical simulation almost coincides with this predicted phase boundary at the high-temperature region. The phase transition is of first order for relatively small values of p < 0.08, but becomes of second order for larger p. The value of p at the intersection of the phase boundary curve and the Nishimori line is regarded as the accuracy threshold of errors in a toric quantum memory. It is estimated as p = 0.110 ± 0.002, which is very close to the value conjectured by Takeda and Nishimori through the "self-duality" argument.
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