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 renormalization 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 ∈ 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 |Ψ({mi})i = TijkmTemfg ... (2) in each site. Each index {i, j, . . . } 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, {mi} are the physical −βH = −βH(i, j, . . . ) with local interactions, we can M-dimensional indexes, and {i, j, . . . } 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- hΨ|H|Ψi network models. ≥ 0 (3) Motivated by the classical-quantum mapping in which hΨ|Ψi 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 variational 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 hΨ|Ψi = 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 hΨ|H |Ψi = 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 A† (SVD), Mij;kl = Uij,nΛn,n0 Vn0,kl. Finally, we approximate M by only considering√ the biggest Dc singular√ val- The tensor trace denotes summation over all indices, 1 A 2 A† 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 indexes of Dc dimensions, and the transformation gives us 3

quantum rotor lattice by adding an interaction term

X Lˆ2 1 X H = α − nˆ nˆ (13) 2I g α β α hα,βi

where hi, ji is a summation over nearest neighbors. For the case N = 2, parametrizen î = (cos(θi), sin(θi)), and ∂ eimiθi Lî = −i with eigenstates given by |mii = √ . ∂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 → ∞ for which hnî ·nˆji ∼ |xi−xj | 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 hnî · nˆji = 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 ˆ on each site.n ˆ can be seen as de- α α Lˆ2 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. To each particle (or rotor) we assign a momen- tump ˆα with componentsp â,α satisfying the canonical where the overline indicates summation over near neigh- commutation relations: bors of the site α. Let the physical variable of our tensor ansatz be the set of quantum numbers {m} of Lˆ. Note [ˆna,α, pˆb,β] = iδα,βδa,b. (10) that the m’s can take any integer value; however, for the low-energy physics we expect the lowest energy states to It is convenient to define the rotor angular momentum contribute the most. For this and numerical reasons, we ˆ ˆ εcab ˆ La,b,α =n â,αpˆb,α − pâ,αnˆb,α and Lc,α = 2 La,b,α so will have a cutoff on the physical index of the tensors to that the commutation relations become be the lowest d-quantum numbers of angular momentum, Lˆ2 m ∈ {− d−1 , − d−1 + 1, ··· , d−1 }. Defining Oα,0 = α , [Lâ,α, Lˆb,β] = iεabcLˆc,αδα,β, α 2 2 2 2I α,1 q 1 α,2 q 1 O = − cos(θα), and O = − sin(θα), we see [Lâ,α, nˆb,β] = iεabcnˆc,αδα,β, (11) g g that we will have 3 types of tensors besides in the ex- [ˆna,α, nˆb,β] = 0. T pectation value of the Hamiltonian (Eq.9): With the angular momentum, we can assign to each rotor 2 X −k k a kinetic energy term, resulting in a total Hamiltonian of α,0 = T ⊗ T k, the form T 2I k r ˆ2 1 k 0 X Lα α,1 X 0 k H = (12) T = − hk| cos(θα)|k iT ⊗ T , (15) kin 2I g α k,k0 r 1 X k 0 where we assumed that each rotor has the same moment α,2 = − hk| sin(θ )|k0iT ⊗ T k . T g α of inertia I. We obtain the Hamiltonian for the O(N) k,k0 4

obtain for a totally real tensor T an onsite energy and a 2.5 bond energy of the order 10100. In the future we hope to finish and increase the efficiency of the code. 2 To quantify the approximation done in the first step of the TERG with the singular cut value, we calculated the 1.5 average relative error between the approximation tensor and the actual tensor for each entry. We started with a randomize real tensors, then calculated the average rela- 1 tive error of an entry by calculating the relative error between each entry and then getting the average. Repeat- 0.5 ing the process 100 times and taking another average, we

Average relative error of entries found that the approximation seems fairly accurate. Re- 0 sults are shown in Fig. 5. It was also checked “by hand” 0 5 10 15 that no particular entry relative error were far away from Dc this average. V. CONCLUSION FIG. 5. A plot showing the average relative error of the entries between the approximated tensor and the actual tensor in the second step of TRG as a function of the singular value cut Dc. Besides TRG and TERG, many other methods based The black line is a piecewise linear fit. on tensor-network algorithms have been develop to study strongly correlated systems in low-dimensions such as the multiscale renormalization ansatz (MERA)[6], and lin- R 2π ikθ R 2π ikθ earized tensor-renormalization group (LTRG)[7]. In this Using cos(θ)e dθ = πδk,±1 and sin(θ)e dθ = 0 0 paper, we discussed how tensor-renormalization tech- ±iπδk,±1, Eq. 15 becomes niques give us a method to study 2-dimensional quan- r tum lattice models. However, tensor-renormalization can 1 1 X k k α,1 = − (T ⊗ T k+1 + T ⊗ T k−1), (16) also be used to study 1-dimensional quantum systems T 2 g k through algorithms such as the tensor-entanglemente- r i 1 X k k filtering renormalization group (TEFRG) [3]. As dis- α,2 = − (T ⊗ T k+1 − T ⊗ T k−1). T 2 g cussed in Ref. [5], the 1-dimensional O(2) quantum rotor k exhibits a Kosterlitz-Thouless transition. Recent work [8] have shown that the thermodynamic properties of the This is all we need to perform the TERG method. classical 2-dimensional XY model as calculated with the use of tensor-renormalization agree very well with results from Kosterlitz [9], with high temperature expansions IV. PRELIMINARY RESULTS [10], and Monte Carlo calculations [11]. It is left for future work to study the Kosterlitz-Thouless transition of We tried a TPS like the one in Eq. 2 where we let T be the O(2) quantum rotor chain with the use of tensor- a real tensor with bond dimensionality D = 2, and with renormalization techniques such as TEFRG. rotation by 90◦ symmetry. Unfortunately, the implementation is a preliminary one and may contain mistakes. Even for 6 iteration of the TRG, the SVD step breaks ACKNOWLEDGMENTS down due to overflow error. The current implementation lacks the minimization step which can be done with the I would like to thank Prof. Kardar for his MIT 8.334 scipy.optimize library. For a 23 ×23 lattice, after a 24 dec- class, as well as Tim Hsie for making me aware of the imation (performing the TRG approximation 4 times) we existence of the tensor-renormalization group.

[1] M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 [7] W. Li, S.-J. Ran, S.-S. Gong, Y. Zhao, B. Xi, F. Ye, and (2007). G. Su, Phys. Rev. Lett. 106, 127202 (2011). [2] Z.-C. Gu, M. Levin, and X.-G. Wen, Phys. Rev. B 78, [8] J. F. Yu, Z. Y. Xie, Y. Meurice, Y. Liu, A. Denbleyker, 205116 (2008). H. Zou, M. P. Qin, J. Chen, and T. Xiang, Phys. Rev. [3] Z.-C. Gu and X.-G. Wen, Phys. Rev. B 80, 155131 E 89, 013308 (2014). (2009). [9] J. M. Kosterlitz, Journal of Physics C: Solid State Physics [4] S. R. White, Phys. Rev. Lett. 69, 2863 (1992). 7, 1046 (1974). [5] S. Sachdev, Quantum Phase Transitions (Cambridge [10] H. Arisue, Phys. Rev. E 79, 011107 (2009). University Press, 2011). [11] M. Hasenbusch and K. Pinn, Journal of Physics A: Math- [6] G. Vidal, Phys. Rev. Lett. 99, 220405 (2007). ematical and General 30, 63 (1997).