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Approaching the Kosterlitz-Thouless Transition for the Classical XY Model with Tensor Networks

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Approaching the Kosterlitz-Thouless Transition for the Classical XY Model with Tensor Networks Approaching the Kosterlitz-Thouless transition for the classical XY model with tensor networks Laurens Vanderstraeten,1, ∗ Bram Vanhecke,1 Andreas M. L¨auchli,2 and Frank Verstraete1 1Department of Physics and Astronomy, University of Ghent, Krijgslaan 281, 9000 Gent, Belgium 2Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria We apply variational tensor-network methods for simulating the Kosterlitz-Thouless phase transi- tion in the classical two-dimensional XY model. In particular, using uniform matrix product states (MPS) with non-abelian O(2) symmetry, we compute the universal drop in the spin stiffness at the critical point. In the critical low-temperature regime, we focus on the MPS entanglement spectrum to characterize the Luttinger-liquid phase. In the high-temperature phase, we confirm the exponen- tial divergence of the correlation length and estimate the critical temperature with high precision. Our MPS approach can be used to study generic two-dimensional phase transitions with continuous symmetries. I. INTRODUCTION transition (in the absence of a local order parameter) is the spin stiffness ρ(T ), which discontinuously drops [7] In contemporary theoretical physics the interplay be- from a finite value in the critical phase to zero in the tween symmetry and dimensionality is widely appre- gapped phase; from the renormalization analysis, the size ciated for giving rise to fascinating physical phenom- of the drop is known to be given by ena. Historically, one of the crucial results in estab- 2Tc lishing this viewpoint was the phase transition in the lim ρ(T ) = ; (1) − π two-dimensional classical XY model. This model is in- T !Tc troduced by placing continuous angles fθig on a square but the value of Tc is not known exactly. When ap- lattice and characterizing their interactions by the hamil- proaching the critical point from the gapped side, the tonian BKT transition is characterized by an exponential diver- X X gence of the correlation length [5,8], H = − cos (θi − θj) − h cos(θi); hiji i b + ξ(T ) / exp p ;T ! Tc ; (2) T − Tc where hiji labels all nearest-neighbour pairs and we in- troduce an external magnetic field h for further reference. with b a non-universal parameter. The first ingredient for understanding the phase dia- The BKT phase transition in the XY model has been gram of the XY model (without magnetic field) is the a notoriously hard case for numerical simulations, be- Mermin-Wagner theorem [1,2]: No conventional long- cause of the exponentially diverging correlation length range order can exist at any finite temperature in this and the ensuing logarithmic finite-size corrections around model, because of the proliferation of spin-wave exci- the phase transition. Nonetheless, shortly after the the- tations in two dimensions. Still, one expects a phase oretical work, Monte-Carlo simulations [9, 10] provided transition in this system. At low temperatures, the sys- considerable evidence for the correctness of the theory tem can be described by a simple continuum field the- and rough estimates for the transition temperature. In ory with algebraically decaying correlation functions. A recent years, these estimates were significantly improved high-temperature expansion, however, suggests an ex- [11, 12], obtaining a value around Tc ≈ 0:8929, in agree- ponential decay of correlations in this system at suffi- ment with high-temperature expansions [13]. Still, these ciently high temperature. As Berezinskii [3] and Koster- results depend heavily on assumptions about the loga- litz and Thouless [4] (BKT) have showed, the phase tran- rithmic finite-size corrections and an improved extrapo- sition between the low- and high-temperature regime is lation [14] places the transition temperature at a higher driven by the unbinding of vortices and is, therefore, value of Tc ≈ 0:8935. Crucially, the best estimates use of a topological nature. The transition is captured by the universal value for the spin stiffness at the phase tran- a renormalization-group (RG) analysis [5,6], where in sition for pinpointing the critical point. arXiv:1907.04576v1 [cond-mat.stat-mech] 10 Jul 2019 the critical phase the long-wavelength properties are de- Tensor networks [15, 16] provide an original frame- scribed by a free bosonic field theory with continuously work for capturing the symmetry-dimensionality inter- varying critical exponents, until vortices become relevant play, both from the theoretical and the numerical side. and drive the system into a gapped phase at a criti- Although orginally devised for capturing the entangle- cal temperature Tc. The quantity that characterizes the ment in strongly-correlated quantum lattice systems, ten- sor networks are increasingly being applied to problems in statistical mechanics. Since the framework is entirely different from traditional approaches such as Monte- ∗ [email protected] Carlo sampling, it can shed a new, entanglement-based, 2 light on statistical-mechanics problems. Indeed, tensor II. MPS FOR THE XY TRANSFER MATRIX networks encode all physical properties of a given system into local tensors, and they allow to understand the re- A. Partition function lation between physical symmetries of the local degrees of freedom and the global properties of the system in a We start by writing down the partition function for the transparent way. One particular example is the classifi- XY model as a tensor network. The partition function cation of symmetry-protected topological phases in one- at a given inverse temperature β = 1=T is given by dimensional quantum systems, which is brought back to the symmetry properties of the local tensors that make Z Y dθi Y Y Z = eβ cos(θi−θj ) eβh cos(θi): up a matrix product state (MPS) [17, 18]. This was ex- 2π tended to two dimensions, where the topological order i hiji i of a wavefunction can be related to the symmetry prop- erties of the local tensor in a projected entangled-pair In order to arrive at a tensor-network representation, state (PEPS) [19]. On the numerical side, physical sym- we introduce a duality transformation [6] that maps the metries can be explicitly incorporated in tensor-network above partition function to a representation in terms of algorithms and lead to an improved efficiency and per- bosonic degrees of freedom on the links. Such a map is formance [20{23]. obtained by introducing the following decomposition on every link in the lattice The encoding of symmetries in tensor networks is per- N x cos(θi−θj ) X in(θi−θj ) e = lim In(x)e ; formed most elegantly when working directly in the ther- N!1 modynamic limit, because one can purely focus on the n=−N symmetry properties of the bulk tensors without bother- where I (x) are the modified Bessel functions of the first ing about what happens at the boundaries of the system. n kind. Then, by integrating over all the θ's, the partition Yet, applying uniform tensor networks with an explicit function is transformed into encoding of the physical symmetries seem to break down when considering critical phases. For example, in a criti- N ! Y X Y cal phase a uniform MPS [24] typically favours an artifi- ns;3;ns;4 Z = lim Inl (β) Fn ;n N!1 s;1 s;2 cial breaking of a continuous symmetry, where the asso- l2L nl=−N s ciated order parameter decreases very slowly as the bond dimension is increased. The apparent reason for this ar- where F is a four-index tensor tificial symmetry breaking is that MPS have a built-in Z dθ limitation for the amount of entanglement in the state, F n3;n4 = eβh cos θeiθ(n1+n2−n3−n4): n1;n2 which makes it energetically favourable to break a con- 2π tinuous symmetry. This seems to imply that uniform The first product runs over all the links in the lattice, and tensor networks fail to capture the essential properties of s labels all the sites in the lattice. We can now represent critical phases with a continuous symmetry. this partition function as a network of tensors, In this paper, we explore this question in more de- tail by investigating the precise sense in which uniform O O O O MPS capture critical phases with a continuous symme- try. As explained, the XY model serves as the paradig- O O O O matic example of a system where the absence of symme- Z = ; try breaking leads to a critical phase, and, therefore, we O O O O take the XY model as our test case. In contrast to ear- lier tensor-network approaches for the XY model [25], we O O O O use uniform MPS methods for transfer matrices [26{28] as a means for characterizing the BKT phase transition. In the first two sections we explain the duality transfor- where every tensor O is given by mation [6] that allows us to define a row-to-row transfer matrix, approximate its fixed point as a uniform MPS n1 4 !1=2 and to compute local observables. In the next section, Y n3;n4 n4 O n2 = In (β) F we focus on the spin stiffness as the characteristic quan- i n1;n2 tity in the BKT phase transition. Afterwards, we use i=1 n3 the Luttinger-liquid formalism to characterize the criti- cal phase. Finally, in the last section, we focus on the and the virtual legs ni have infinite dimension. In prac- gapped phase and locate the critical temperature with tice, however, it will be possible to truncate these indices high precision. without loss of accuracy. We have introduced arrows on 3 the legs to indicate the signs in which the ni's appear in establishing an optimization problem for the tensor A: the F tensor above.
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