Simulation of strongly correlated fermions in two spatial dimensions with fermionic projected entangled-pair states

Philippe Corboz, Román Orús, Bela Bauer, and Guifré Vidal

Phys. Rev. B 81, 165104 – Published 8 April 2010

Abstract

We explain how to implement, in the context of projected entangled-pair states (PEPSs), the general procedure of fermionization of a tensor network introduced in P. Corboz and G. Vidal, Phys. Rev. B 80, 165129 (2009). The resulting fermionic PEPS, similar to previous proposals, can be used to study the ground state of interacting fermions on a two-dimensional lattice. As in the bosonic case, the cost of simulations depends on the amount of entanglement in the ground state and not directly on the strength of interactions. The present formulation of fermionic PEPS leads to a straightforward numerical implementation that allowed us to recycle much of the code for bosonic PEPS. We demonstrate that fermionic PEPS are a useful variational ansatz for interacting fermion systems by computing approximations to the ground state of several models on an infinite lattice. For a model of interacting spinless fermions, ground state energies lower than Hartree-Fock results are obtained, shifting the boundary between the metal and charge-density wave phases. For the $t − J$ model, energies comparable with those of a specialized Gutzwiller-projected ansatz are also obtained.

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Received 3 December 2009

DOI:https://doi.org/10.1103/PhysRevB.81.165104

Authors & Affiliations

Philippe Corboz¹, Román Orús¹, Bela Bauer², and Guifré Vidal¹

¹School of Mathematics and Physics, The University of Queensland, Queensland 4072, Australia
²Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland

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Vol. 81, Iss. 16 — 15 April 2010

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Figure 1
(Color online) (a) Diagrammatic representation of the tensor $Ψ$ with coefficients $Ψ_{i_{1} i_{2} \dots i_{9}}$ for a state $| Ψ ⟩ ∊ V^{\otimes 9}$ of a $3 \times 3$ square lattice $L$ . This tensor is expressed in terms of a PEPS made of a set of 9 tensors ${A^{[\vec{r}]}}$ , one for each site $\vec{r} ∊ L$ . (b) Bulk tensor $A^{[\vec{r}]}$ with components $A_{u l s d r}^{[\vec{r}]}$ . Notice that the legs corresponding to indices $u$ , $l$ , $s$ , $d$ , and $r$ emerge from the circle in anticlockwise order. (c) Hermitian conjugate $A^{[\vec{r}] †}$ of the PEPS tensor $A^{[\vec{r}]}$ , represented as its mirror image. Notice that the legs corresponding to indices $r$ , $d$ , $s$ , $l$ , and $u$ of ${(A^{[\vec{r}] †})}_{r d s l u}$ emerge again in anticlockwise order. (d) Hermitian conjugates of $Ψ$ and its PEPS representation.Reuse & Permissions
Figure 2
(Color online) (a) Scalar product $⟨ Ψ | Ψ ⟩$ written in terms of tensors $Ψ$ and $Ψ^{†}$ . (b) The same scalar product, but written in terms of a PEPS together with its Hermitian conjugate. (c)–(d) Using the jump move of Fig. 3, this tensor network can be modified so that each tensor $A^{[\vec{r}]}$ is drawn next to its Hermitian conjugate $A^{[\vec{r}]}$ . (e) Tensor network $E$ in terms of reduced tensors $a^{[\vec{r}]}$ . (f) $a^{[\vec{r}]}$ defined in terms of $A^{[\vec{r}]}$ and $A^{[\vec{r}] †}$ , Eq. (6).Reuse & Permissions
Figure 3
(Color online) Jump move: The graphical representation of a tensor network is not unique. In particular, a line can be dragged over a circle without changing the tensor network that is represented. This property, trivial in tensor networks for bosonic systems, will have a less obvious analog in the fermionic case.Reuse & Permissions
Figure 4
(Color online) (a) Tensor network $E$ , made of reduced tensors $a^{[\vec{r}]}$ , corresponding to the scalar product $⟨ Ψ | Ψ ⟩$ in an $L \times L$ lattice $L$ with open boundary conditions. (b) Environment $E^{[{\vec{r}}_{1} {\vec{r}}_{2}]}$ for two contiguous sites ${\vec{r}}_{1}, {\vec{r}}_{2} ∊ L$ , obtained from $E$ by removing tensors $a^{[{\vec{r}}_{1}]}$ and $a^{[{\vec{r}}_{2}]}$ . (c) Approximate environment $G^{[{\vec{r}}_{1} {\vec{r}}_{2}]}$ consisting of six tensors ${E_{α}}$ . (d) Tensor $o$ of coefficients for a two-site local operator $\hat{o}$ , Eq. (7). (e) Tensor network made of the approximate environment $G^{[{\vec{r}}_{1} {\vec{r}}_{2}]}$ and tensors $A^{[{\vec{r}}_{1}]}$ , $A^{[{\vec{r}}_{1}] †}$ , $A^{[{\vec{r}}_{2}]}$ , $A^{[{\vec{r}}_{2}] †}$ , and $o$ . Its contraction produces an approximation to $⟨ Ψ | \hat{o} | Ψ ⟩$ .Reuse & Permissions
Figure 5
(Color online) A tensor network is fermionized by following two rules (see text): (a) all tensors in the network are chosen to be parity preserving ( $P$ is the parity operator acting on the indices of tensor $T$ ); (b) line crossings are replaced with a special gate, a fermionic swap gate $X$ .Reuse & Permissions
Figure 6
(Color online) (a) Fermionic PEPS for the state $| Ψ ⟩ ∊ V^{\otimes 9}$ of a $3 \times 3$ lattice $L$ , obtained from the bosonic PEPS in Fig. 1a by replacing line crossings with fermionic swap gates. The coefficients $Ψ_{i_{1} i_{2} \dots i_{9}}$ are obtained by contracting this fermionic PEPS using standard tensor multiplication techniques (see text). The presence of fermionic swap gates introduces a complex structure of minus signs in $Ψ_{i_{1} i_{2} \dots i_{9}}$ . (b) Bulk tensor $A^{[\vec{r}]}$ , which is chosen to be parity preserving, Eq. (25). (c) Hermitian conjugate of $A^{[\vec{r}]}$ . (d) Scalar product $⟨ Ψ | Ψ ⟩$ written as a tensor network of reduced tensors $a^{[\vec{r}]}$ defined in (e). Notice that, in a $L \times L$ lattice, $O (L^{3})$ fermionic swap gates pervade the PEPS and yet, thanks to the jump move, Fig. 10, it is possible to write the scalar product $⟨ Ψ | Ψ ⟩$ in terms of only $O (L^{2})$ fermionic swap gates and in such a way that all of them are near a pair $(A^{[\vec{r}]}, A^{[\vec{r}] †})$ and can therefore be absorbed into the definition of the reduced tensor $a^{[\vec{r}]}$ . (For a detailed derivation, replace line crossings with fermionic swap gates in Fig. 2). As a result, there are no fermionic swap gates left in $E$ , and therefore this tensor network can be contracted using the techniques employed for bosonic PEPS.Reuse & Permissions
Figure 7
(Color online) (a) The fermionic PEPS in Fig. 6 corresponds to a specific choice of fermionic order, which appears in the definition of the local basis of Eq. (12) and Jordan-Wigner transformation of Eq. (13). (b) Another possible fermionic PEPS, associated to another fermionic order. The two tensor networks are not equivalent in that they cannot be mapped into each other by jump moves alone (where one is not allowed to drag a line over the end of another open line), but the mapping is possible if we allow some additional fermionic exchanges to modify the order of the open lines. Importantly, it can be seen that the tensors ${A^{[\vec{r}]}}$ in both PEPS appear in the same way in any expectation value. In particular, they would be optimized using exactly the same figure of merits. This shows that the tensors ${A^{[\vec{r}]}}$ are independent of the choice of fermionic order.Reuse & Permissions
Figure 8
(Color online) (a) The state $| Ψ^{'} ⟩ = \hat{o} | Ψ ⟩$ is described by a tensor $Ψ^{'}$ of coefficients obtained by connecting tensors $Ψ$ and $o$ . The example corresponds to a $3 \times 4$ lattice and a choice of fermionic order such that the two sites on which $\hat{o}$ acts, which are nearest neighbors in $L$ , are not contiguous in the fermionic order (they occupy positions $k_{1} = 5$ and $k_{2} = 8$ ). The line crossings (fermionic swap gates) that appear in connecting tensors $Ψ$ and $o$ can be interpreted as changes in the fermionic order needed in order to bring the two sites on which $\hat{o}$ acts together, then bringing them back to their original position. (b) Expectation value $⟨ Ψ | \hat{o} | Ψ ⟩$ . (c) The same expectation value in terms of fermionic PEPS. (d) After some manipulation, an approximation of the expectation value is expressed in terms of an approximate two-site environment $G^{[{\vec{r}}_{1} {\vec{r}}_{2}]}$ and tensors $A^{[{\vec{r}}_{1}]}$ , $A^{[{\vec{r}}_{1}] †}$ , $A^{[{\vec{r}}_{2}]}$ , $A^{[{\vec{r}}_{2}] †}$ , and $o$ , as it was done for a bosonic system in Fig. 4. The only difference here is the presence of a few fermionic swap gates.Reuse & Permissions
Figure 9
(Color online) (a) PEPS for a state $| Ψ ⟩$ , of a $3 \times 3$ lattice, with odd parity $p = - 1$ . All PEPS tensors are parity preserving. One of the tensors has an additional index $j \equiv (- 1, 1)$ with fixed parity $p = - 1$ . (b) The tensor $o$ of coefficients $o_{i_{2} i_{1} j_{1} j_{2}}$ for a two-site operator such as a hopping term ${\hat{a}}^{†} \hat{b}$ in Eq. (30), that is the product of two parity changing operators, ${\hat{a}}^{†}$ and $\hat{b}$ , can be decomposed as the product of two parity preserving tensors $o^{'}$ and $o^{″}$ connected by an index $j \equiv (- 1, 1)$ with fixed parity $p = - 1$ . This decomposition is useful, e.g., in the computation of a correlator involving two distant sites, see Fig. 16. Notice that parity preserving tensor $o^{'}$ can also be used independently of $o^{″}$ in order to represent the parity changing operator $\hat{a}$ .Reuse & Permissions
Figure 10
(Color online) (a) As a result of rules 1 and 2, lines can jump over (parity invariant) tensors, with the fermionic swap gates traveling with the line crossings. (b) A parity changing tensor $\tilde{T}$ is represented with a parity preserving tensor $T$ with an extra index $j = (- 1, 1)$ that remains open. In this case the jump rule introduces an additional fermionic swap gate $X$ , involving $j$ and the jumping line, that reduces to a parity operator $P$ on the jumping line (see text). (c) A line can also be dragged over a set of open lines provided that the latter have even overall parity (in other words, provided they could be connected to a parity preserving tensor)Reuse & Permissions
Figure 11
(Color online) Multiplication of two tensors in a fermionic tensor network. (a) Example of two tensors $T_{1}$ and $T_{2}$ connected by two lines. (b) By means of a jump move, all lines standing between $T_{1}$ and $T_{2}$ can be dragged away. (c) Any fermionic swap gate involving the lines that connect $T_{1}$ and $T_{2}$ can be absorbed into, e.g., tensor $T_{1}$ , which is mapped into some ${\tilde{T}}_{1}$ . (d) Finally tensors ${\tilde{T}}_{1}$ and $T_{2}$ are multiplied together, producing tensor $T$ . The cost of all these manipulations is dominated by the last step.Reuse & Permissions
Figure 12
(Color online) (a) PEPS for a state $| Ψ ⟩$ of fermionic $3 \times 3$ lattice with cylindrical boundary conditions. (b) Tensor network for the scalar product $⟨ Ψ | Ψ ⟩$ . (c) Tensor network $E$ of reduced tensors ${a^{[\vec{r}]}}$ . As in a system with open boundary conditions, $E$ has no line crossings and can be contracted as in the bosonic case.Reuse & Permissions
Figure 13
(Color online) (a) PEPS for a state $| Ψ ⟩$ of fermionic $3 \times 3$ lattice with toroidal boundary conditions. (b) Tensor network for the scalar product $⟨ Ψ | Ψ ⟩$ . (c) Tensor network $E$ of reduced tensors ${a^{[\vec{r}]}}$ . On the torus, it is not possible to eliminate all line crossings in $E$ , which must contain fermionic swap gates and therefore differs from the bosonic case. However, TERG techniques (Refs. 18, 19) can still be applied by noting that the coarse graining of the reduced tensors ${a^{[\vec{r}]}}$ can take place independently of the fermionic swap gates, which are also coarse-grained in an obvious way. The (coarse-grained) fermionic swap gates only need to be absorbed into the rest of the tensor network at the latest coarse-graining step, when the torus has been reduced to only a few lattice sites.Reuse & Permissions
Figure 14
(Color online) (a) Environment $E^{[{\vec{r}}_{1} {\vec{r}}_{2}]}$ for two contiguous sites ${\vec{r}}_{1}, {\vec{r}}_{2} ∊ L$ , in terms of the reduced tensors $a$ and $b$ corresponding to the PEPS tensors $A$ and $B$ . (b) Approximate environment $G^{[{\vec{r}}_{1} {\vec{r}}_{2}]}$ expressed in terms of 10 tensors, corresponding to the 10 shaded regions in (a), as used in the CTM approach. (Refs. 20, 27) Some of the tensor are used in Fig. 16b. (c) Approximate environment $G^{[{\vec{r}}_{1} {\vec{r}}_{2}]}$ written in terms of six tensors ${E_{α}}$ , as used in the computation of the expectation value of a two-site observable $\hat{o}$ [Fig. 16a] and the update of tensors $A$ and $B$ during an imaginary-time evolution (Fig. 30).Reuse & Permissions
Figure 15
(Color online) Reduced tensor $a$ defined in terms of PEPS tensors $A$ and $A^{†}$ and four fermionic swap gates. (b) Reduced tensor $b$ defined in terms of PEPS tensors $B$ and $B^{†}$ and four fermionic swap gates.Reuse & Permissions
Figure 16
(Color online) (a) Computation of an approximation $⟨ Ψ | \hat{o} {| Ψ ⟩}_{χ}$ to the expectation value $⟨ Ψ | \hat{o} | Ψ ⟩$ of a local observable $\hat{o}$ acting on two contiguous sites. This tensor network only differs from the one in Fig. 4e for a bosonic system in the presence of 12 fermionic swap gates. (b) Computation of two-point correlators. For concreteness, we consider the expected value $⟨ {\hat{c}}_{i}^{†} c_{j} ⟩$ discussed in Eq. (41). The figure shows the tensor network representing $⟨ Ψ | {\hat{c}}_{i}^{†} {\hat{c}}_{j} {| Ψ ⟩}_{χ}$ , where sites $i, j$ are in the same row of $L$ but separated by four sites. Notice the line connecting the two tensors corresponding to operators ${\hat{c}}_{i}^{†}$ and ${\hat{c}}_{j}$ , which crosses a number of other lines introducing a number of fermionic swap gates. Since this line corresponds to an index $j = (- 1, 1)$ with well-defined parity $p = - 1$ , the fermionic swap gates simplify into $\hat{I} \otimes \hat{P}$ .Reuse & Permissions
Figure 17
(Color online) Phase diagram of the free-fermion model 40 as a function of chemical potential $λ$ and pairing potential $γ$ . For $γ > 0$ , $λ > 0$ the system is superconducting.Reuse & Permissions
Figure 18
(Color online) Relative error of the ground-state energy of the free-fermion model (40) as a function of $λ$ , for different values of $γ$ and $D$ . The dimension $χ$ is 20 and 40 for $D = 2$ and $D = 4$ , respectively.Reuse & Permissions
Figure 19
(Color online) Upper panels: Correlation function $C (r) = ⟨ {\hat{c}}_{i}^{†} {\hat{c}}_{j} ⟩$ as a function of distance $r$ (in $x$ -direction), see Eq. (41), for two different values of $(γ, λ)$ of the free-fermion model (40). Middle panels: Absolute value of the correlation function $C (r)$ in semilogarithmic scale. Lower panels: The difference between the simulation result $C (r, D)$ and the exact result $C_{e x} (r)$ for different values of $D$ . Notice that better accuracies are obtained in the gapped phase, which corresponds to a less entangled ground state.Reuse & Permissions
Figure 20
(Color online) Relative error of the ground-state energy of the free-fermion model (40) as a function of $χ$ , the bond dimension of the environment, for $D = 4$ . In these examples the energy decreases monotonously with $χ$ , i.e., the energy from the simulation is always larger than the true ground-state energy.Reuse & Permissions
Figure 21
(Color online) Relative error of the ground-state energy of the free-fermion model (40) as a function of the number of renormalization steps in the CTM algorithm. The boundary tensors are initialized randomly, and the dimensions are $D = 4$ , and $χ = 32$ $(χ = 16$ for $λ = 1$ ). The environment in the gapped phase converges considerably faster than in the critical phase.Reuse & Permissions
Figure 22
(Color online) Comparison of the accuracies of the ground-state energy of the free-fermion model (40) obtained with the two different updates described in Appendix .Reuse & Permissions
Figure 23
(Color online) Phase diagram of the interacting spinless fermion model (42). $V$ is the interaction strength, and $n$ is the particle density (filling). The charge-density wave phase (checkerboard pattern) at exactly half filling $(n = 0.5)$ is separated from the metal phase by a first order phase transition, with an intermediate region corresponding to PS. With increasing $D$ , the boundary between the metal phase and the PS region moves away from the HF result from Ref. 29. Dotted and dashed lines are a guide to the eye. The uncertainty of the phase boundaries obtained with iPEPS is smaller than the symbol size (in $x$ direction).Reuse & Permissions
Figure 24
(Color online) Upper panel: Energy per site of the interacting fermion model (42) as a function of chemical potential $μ$ for $V = 2$ , obtained by iPEPS and HF. (Ref. 29) The first order phase transition between the metal phase and the CDW phase occurs at a value $μ^{*}$ where the two corresponding energies cross. Lower panel: Particle density $n$ as a function of chemical potential. At the first order phase transition point $μ^{*}$ , $n$ jumps from a certain value $n^{*}$ in the metal phase to $n = 0.5$ in the CDW phase. For densities in between $n^{*}$ and $n = 0.5$ the system exhibits phase separation. The numbers in brackets indicate the uncertainty in the last digit.Reuse & Permissions
Figure 25
(Color online) The convergence of the energy per site as a function of $χ$ is not monotonous. For large $χ$ the values do not seem to change significantly anymore. We associate an error bar to the values of the energy, depending on the convergence behavior in $χ$ . This error is smaller than the symbol sizes in Fig. 24.Reuse & Permissions
Figure 26
(Color online) Upper panel: Energy per site in units of $J$ of the $t - J$ model as a function of filling $n$ , with $t / J = 3$ . With increasing $D$ , the energies obtained by iPEPS are approaching the values from the VMC study in Ref. 38. Note that the chemical potential term has been subtracted from the energy. The error bars of the VMC results are of the order $10^{- 3} J$ . Lower panel: Relative deviation between the energies obtained by iPEPS and VMC. Full symbols indicate that iPEPS energy is lower than the VMC energy.Reuse & Permissions
Figure 27
(Color online) Energy per site as a function of $χ$ . The energies for $D \leq 6$ do not seem to change significantly anymore for large $χ$ , whereas for $D = 8$ the energy is still increasing at the largest value of $χ$ used.Reuse & Permissions
Figure 28
(Color online) Definition of tensors $\bar{A}$ and $\bar{B}$ .Reuse & Permissions
Figure 29
(Color online) (a) Tensor $A^{•}$ is obtained from tensor $A$ by crossing the legs $s$ and $d$ . (b) Tensor $B^{•}$ is obtained from tensor $\bar{B}$ by crossing the legs $\bar{s}$ and $\bar{u}$ . (c) Tensors $W_{A^{•}}$ and $M_{A^{•}}$ are obtained by joining the $u, l, d$ indices and the $r, s$ indices of $A^{•}$ , and doing a singular value decomposition of the resultant matrix. The singular values are included in the definition of tensor $M_{A^{•}}$ . (d) Tensors $N_{B}$ and $W_{B}$ are obtained by joining indices $l, s$ and indices $u, r, d$ of $B$ together, then performing a singular value decomposition of the resulting matrix, then splitting the composed indices back apart. The singular values are included in tensor $N_{B}$ . (e) Tensors $W_{\bar{A}}$ and $M_{\bar{A}}$ are obtained by joining indices $\bar{u}, \bar{l}, \bar{d}$ and indices $\bar{r}, \bar{s}$ of $\bar{A}$ , then performing a singular value decomposition of the resulting matrix, then splitting the composed indices back apart. The singular values are included in tensor $M_{\bar{A}}$ . (f) Tensors $N_{B^{•}}$ and $U_{B^{•}}$ are obtained by joining indices $\bar{s}, \bar{l}$ and indices $\bar{u}, \bar{r}, \bar{d}$ of $B^{•}$ , then performing a singular value decomposition of the resulting matrix, then splitting the composed indices back apart. The singular values are included in tensor $N_{B^{•}}$ .Reuse & Permissions
Figure 30
(Color online) (a) Tensor $Z$ is obtained by contracting a tensor network that contains the approximate environment $G^{[{\vec{r}}_{1} {\vec{r}}_{2}]} = {E_{1}, \dots, E_{6}}$ , tensors $W_{A^{•}}$ , $W_{\bar{A}}$ , $U_{B}$ , and $U_{B^{•}}$ , and a few fermionic swap gates. (b) Tensor $Q$ is obtained from the contraction of $Z, M_{A^{•}}, N_{B}$ and the gate $g$ .Reuse & Permissions
Figure 31
(Color online) The updated tensors $M_{A^{•}}^{'}$ and $N_{B}^{'}$ are obtained by iteratively solving two linearized equations. Specifically, we iterate the following two steps until convergence: (a) given $N_{B}$ and $N_{B^{•}}$ , we obtain $M_{A^{•}}^{'}$ by solving the linear equation $M_{A^{•}}^{'} R = S$ , where $R$ and $S$ are obtained as indicated in the diagram. Then, we set $M_{A^{•}} = M_{A^{•}}^{'}$ , and compute $M_{\bar{A}} = M_{A^{•}}^{†}$ . (b) given $M_{A^{•}}$ and $M_{\bar{A}}$ , we obtain $N_{B}^{'}$ by solving the linear equation $N_{B}^{'} R = S$ , where $R$ and $S$ are again obtained as indicated in the diagram. Then, we set $N_{B} = N_{B}^{'}$ , and compute $N_{B^{•}} = N_{B}^{†}$ .Reuse & Permissions
Figure 32
(Color online) (a) Tensor $A^{•'}$ is obtained from the contraction of $W_{A^{•}}$ and $M_{A^{•}}^{'}$ . (b) The updated tensor $A^{'}$ is computed by crossing the $s$ and $d$ indices of $A^{•'}$ . (c) The updated tensor $B^{'}$ is obtained from the contraction of $N_{B}^{'}$ and $U_{B}$ .Reuse & Permissions
Figure 33
(Color online) (a) Local detail of an iPEPS expressed in terms of tensors $Γ_{A}$ and $Γ_{B}$ and four weight matrices $λ_{1}, \dots, λ_{4}$ , as required for the simplified update. (b)–(c) Relation with the usual iPEPS tensors $A$ and $B$ .Reuse & Permissions
Figure 34
(Color online) (a) Tensor $Θ$ and the tensor network that defines it. (b) Tensors ${\tilde{Γ}}_{A}^{•}$ and ${\tilde{Γ}}_{B}$ and weight matrix ${\tilde{λ}}_{1}$ , obtained from the singular value decomposition of $Θ$ .Reuse & Permissions
Figure 35
(Color online) (a) Tensor $Γ_{A}^{•'}$ is obtained by multiplying ${\tilde{Γ}}_{A}^{•}$ with the inverses of $λ_{2}$ , $λ_{3}$ and $λ_{4}$ . (b) Tensor $Γ_{B}^{'}$ is obtained by multiplying ${\tilde{Γ}}_{B}$ with the inverses of $λ_{2}$ , $λ_{3}$ and $λ_{4}$ .Reuse & Permissions

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