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The Tensor-Entanglement Renormalization Group of the 2D Quantum Rotor Model

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The Tensor-Entanglement Renormalization Group of the 2D Quantum Rotor Model The tensor-entanglement renormalization group of the 2D quantum rotor model Rolando La Placa Department of Physics, Harvard University, Cambridge, MA, 02138 (Dated: May 16, 2014) We review the tensor-renormalization group (TRG)[1], and the tensor-entanglement renormaliza- tion group (TERG) [2]. Then, we use TERG to setup a variational optimization of the ground state of the O(2) quantum rotor model in a square lattice. I. INTRODUCTION mean-field theory cannot be applied in systems with long- range entanglement, such as phases found beyond Lan- Tensor-renormalization group (TRG) is a real- dau's symmetry breaking paradigm [2]. In some cases, space renormalization technique used to study any 2- \Tensor Product States"(TPS) are possible trial ground dimensional classical lattice systems with local interac- state wave functions. To define a TPS, consider a system tions [1]. It stems from the idea that any such system with quantum numbers mi 2 1; 2;:::;M residing over a can be rewritten as a tensor-network model. To describe square lattice. A TPS of this square lattice is a tensor-network model, we assign a p-rank tensor T ij::: X to every site of a lattice, where p- is the number of bonds m1 m2 jΨ(fmig)i = TijkmTemfg ::: (2) in each site. Each index fi; j; : : : g is D dimensional, i;j::: and can be seen as residing on an adjacent bond of the site (Fig. 1). For a classical lattice model Hamiltonian mi where Tijkm are complex numbers, fmig are the physical −βH = −βH(i; j; : : : ) with local interactions, we can M-dimensional indexes, and fi; j; : : : g are the indexes find a tensor T such that with bond dimensionality D. The bond dimensionality X X might vary according to the model being studied, e.g. Z = e−βH(i;j;::: ) = T T ··· : (1) ijkl emfg D = 2 is enough to reproduce the quantum Ising model, i;j;::: i;j;::: and the Heisenberg model [2]. Using a TPS as a ground A real space-renormalization transformation T ! T 0 can state ansatz, be carried out to study the critical behavior of tensor- hΨjHjΨi network models. ≥ 0 (3) Motivated by the classical-quantum mapping in which hΨjΨi the partition function of a d-dimensional quantum model can be mapped to the partition function of a d + 1 di- can be minimized to find the appropriate tensor compo- mensions classical model, we expect that a similar renor- nents. This is computationally hard, but TRG can be malization technique to be valid in the study of some used to reduce the complexity of the computation. 1-dimensional quantum systems. In fact, examples of This variational approach using TPS and TRG TRG-based calculations for the quantum spin-1/2 and can be seen as an attempt to generalized the varia- spin-1 chains can be found in Ref. [3]. tional optimization described by the \Matrix Product States"(MPS) and the density matrix renormalization group (DMRG). DMRG has been an effective method to study 1-dimensional models such as spin chains [4]. In Section III, we will discuss a TRG-based method called the tensor-entanglement renormalization group (TREG). TERG is an approximation scheme to perform the varia- tional calculation described above. Finally, we try to use TERG on the 2-dimensional O(2) quantum rotor lattice. II. TENSOR-ENTANGLEMENT RENORMALIZATION GROUP FIG. 1. A Tensor-network model on a square lattice. Given a translationally invariant Hamiltonian on a lat- P tice with sites α and local interactions, H = Hα, we α will try an ansatz for the ground state in the form of Additionally, tensor-networks models and TRG can shown in Eq. 2. By assumptions, Hα can be written in α,0 α,1 β;2 be used to perform variational calculations. Traditional terms of local operators: Hα = O + O O + ··· . 2 FIG. 2. Decomposition of a 4-tensor into two 3-tensors. For sublattice A , we decompose into S1 and S2. In sublattice B, the decomposition is into S3 and S4. FIG. 3. Tensor-renormalization transformation of a square lattice. Which means that we can rewrite: Start by dividing the square lattice into two sublattices m m A and B as shown in Fig. 3. Then, decompose the tensors X m1 1 m2 2 hΨjΨi = T T 0 0 0 0 T T 0 0 0 0 ··· (4) ijkm i j k m emfg e m f g T at each site of A sublattice in terms of two 3-rank 0 0 m1;:::;ii ;jj ;::: tensors S1 and S2, and do a similar decomposition of = tT r[T ⊗ T ⊗ · · · ] tensors in the B sublattice into 3-rank tensors S3 and S4 (Fig. 2). Note that the T tensors can be seen as 4-rank tensors in which each index has D2 dimensions. We have A X 1 2 Tijkl = SijnSnkl; (7) n m X α,0 m1 1 m2 B X 3 4 hΨjH jΨi = O T T 0 0 0 0 T ··· α m1m2 ijkm i j k m emfg Tijkl = SijnSnkl 0 0 m1;:::;ii ;jj ;::: n m X α,1 β;2 m1 1 m2 2 4 + O O T T 0 0 0 0 T ::: + ··· m1m2 m2m3 ijkm i j k m emfg with i; j; k; l each having D dimensions and n being D - 0 0 m1;:::;ii ;jj ;::: dimensional. With an approximation, we can reduce the α,0 α,1 β;2 dimensionality of n. Viewing A as a matrix M A , = tTr[T ⊗ T ··· ] + tTr[T ⊗ T ⊗ T ··· ] + ··· Tijkl ij;kl (5) we can factorize it into its singular value decomposition A A A Ay (SVD), Mij;kl = Uij;nΛn;n0 Vn0;kl. Finally, we approxi- mate M by only consideringp the biggest Dc singularp val- The tensor trace denotes summation over all indices, 1 A 2 Ay ues λn. Letting Sijn = λnUij;n and Snkl = λnV P m m α,a n;kl and we have defined T = m T ⊗ T and T = so that the approximate decomposition becomes. k 0 P Oα,aT ⊗ T k with Oα,a being the (k; k0) matrix k;k0 kk0 kk0 D element of Oα,a in the basis of quantum numbers k. Cal- Xc A ≈ S1 S2 : (8) culating the tensor traces is computationally expensive, Tijkl ijn nkl n but with the real-space renormalization we can decimate the tensor lattice and find an approximation with less The same process is done with the B sublattice. After degrees of freedom. We will find T~ s.t. this transformation we end up with lattice of the form shown in Fig. 3. By adding the degrees of freedom found in each small square in the new 3-rank tensors S lattice tTr[ ⊗ ⊗ · · · ] ≈ tTr[~ ⊗ ~ ⊗ · · · ] (6) T T T T (see Fig. 4), we obtain the desired T~ X ~ = S1 S2 S3 S4 : (9) with the tensor trace over T~ containing one-fourth of the Tqrst jkq rli ijs tkl i;j;k;l tensors in tTr[T ⊗ T ⊗ · · · ]. This approximation trans- formations T ! T~ is based on TRG [1]. ~ By our SVD approximation, the tensor Tqrst has in- dexes of Dc dimensions, and the transformation gives us 3 quantum rotor lattice by adding an interaction term X L^2 1 X H = α − n^ n^ (13) 2I g α β α hα,βi where hi; ji is a summation over nearest neighbors. For the case N = 2, parametrizen ^i = (cos(θi); sin(θi)), and @ eimiθi L^i = −i with eigenstates given by jmii = p . @θi 2π There are no naturally occurring quantum rotors in na- ture, but some systems such as antiferromagnets and su- FIG. 4. Adding the S tensors in a square of the approximated perfluids can be studied through this model. ~ lattice results in the renormalized T. For d > 1 the quantum rotor model exhibits a disor- dered/paramagnetic phase as g ! 1 for which hn^i ·n^ji ∼ jxi−xj j a new lattice with a quarter of the tensors of the orig- exp(− ξ ), and an ordered phase in the limit g ! 0 inal lattice. This decimation is iterated until there are with hn^i · n^ji = C. There is a critical gc in which the only few tensors in the tensor traces, making the calcula- quantum phase transition happens. In the d = 1 and tion of the expectation values tractable. Note that with N ≥ 3 case, there is no phase transition; however, the some labeling care this procedure can be generalized to N = 2 exhibits a phase transition but does not follow take into account the tensors of the form Tα,a, since a the long-distance behavior of d > 1. This is analogous square of tensors Tα,a; Tβ;b; Tγ;c; Tδ;d maps to a square to the Kosterlitz-Thouless transition of the classical XY T~α,a; T~β;b; T~γ;c; T~δ;d (A group of 4 tensors is "closed" un- model in d = 2. In fact, there is a mapping between the der RG). partition functions of the O(N) quantum rotor and the classical O(N) model [5]. III. QUANTUM ROTOR MODEL B. TREG of the O(2) quantum rotor model A. The quantum rotor Hamiltonian The local Hamiltonian for the 2-dimensional O(2) quantum rotor can be written as Consider a d-dimensional lattice with N-component unit operatorsn ^ on each site.n ^ can be seen as de- α α L^2 1X scribing the position of a particle constraint to move on H = α − (cos(θ ) cos(θ )+sin(θ ) sin(θ )) (14) α 2I g α β α β the surface of N-dimensional unit sphere located at the β site α.
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