2. Projected Entangled-Pair States#

2.1. Construction of PEPS#

Recall MPS form:

(1)#\[| \mathrm{MPS} \rangle = \sum\limits_{a_1, \cdots, a_{L-1}} \sum\limits_{\sigma_1, \cdots, \sigma_{L}} M_{1, a_1}^{\sigma_1} | \sigma_1 \rangle \otimes M_{a_1, a_2}^{\sigma_2} | \sigma_2 \rangle \otimes \cdots \otimes M_{a_{L-1}, 1}^{\sigma_L} | \sigma_L \rangle\]

It can be viewed as

(2)#\[\begin{split}\sum M^{\sigma_1}_{1, a_1} M^{\sigma_2}_{b_1, a_2} \cdots M^{\sigma_L}_{b_{L-1}, 1} | \sigma_1 \sigma_2 \cdots \sigma_L \rangle \langle a_1 b_1 a_2 b_2 \cdots a_{L-1} | j_1 j_1 j_2 j_2 \cdots j_{L-1} j_{L-1} \rangle \\ = \sum (M_{1, a_1}^{\sigma_1} | \sigma_1 \rangle \langle a_1|) \otimes \sum (M^{\sigma_2}_{b_1 a_2} | \sigma_2 \rangle \langle b_1 a_2 |) \otimes \cdots (\sum | j_1 j_1 \rangle \otimes | j_2 j_2 \rangle \cdots)\end{split}\]

Define projection operator

(3)#\[\hat P = \sum\limits_{\sigma, a, b} M^{\sigma}_{a, b} | \sigma \rangle \langle a, b |\]

and the entangled-pair state between two sites

(4)#\[| \phi \rangle = \sum\limits_{j = 0}^{D-1} | jj\rangle\]

then we get

\[| \mathrm{MPS} \rangle = \hat P | \phi \rangle \otimes \hat P | \phi \rangle \cdots\]

We can see from the simple construction that 1D PEPS maps the entangled-pair virtual space \((\mathbb C^\mathrm{D})^{\otimes 2}\) to the physical space \(\mathbb C^d\).

More generally, 1D PEPS can be natually extended to higher-dimensional ones. Generally a PEPS can be written as

(5)#\[| \Psi_A \rangle = \sum\limits_{k_{(1,1)}, \cdots, k_{(M,N)}}^{d} \mathcal F ([A_{(1,1)}]^{k_{(1,1)}} \cdots [A_{(M,N)}]^{k_{(M,N)}}) | k_{(1,1)} k_{(1,2)} \cdots k_{(M,N)} \rangle\]

where

  • \((h,v)\) are indices of sites;

  • The matrix \([A_{(h,v)}]\) has four virtual (or auxiliary) indices \(l, r, u, d\) (denoting the direction) and one physical index \(k\);

  • \(\mathcal F\) is the notation of some contraction of matrix.

From the projection view, one can define a projection operator \(\mathcal{P}: (\mathbb C^D)^{\otimes 4} \to \mathbb C^d\):

(6)#\[\mathcal{P} = [A_{(h,v)}]_{lrud}^k | c_{(h,v)}^k \rangle \langle \alpha^l_{(h,v)} \beta^r_{(h,v)} \gamma^{u}_{(h,v)} \delta^d_{(h,v)} |\]

where

  • Physical states are denoted with alphabet \(a,b,c \cdots\)

  • Auxiliary or virtual states are denoted by Greek alphabet \(\alpha, \beta, \gamma, \delta\) for left, right, up and down directions in each site.

The entangled-pair states between two sites (should be normalized) are written as:

(7)#\[\begin{split}| \phi^h_{(h, v)} \rangle = \sum\limits_{i=1}^D | \gamma_{(h+1, v)}^i \delta_{(h,v)}^i \rangle \\ | \phi^v_{(h,v)} \rangle = \sum\limits_{i=1}^D | \alpha_{(h,v+1)}^i \beta_{(h,v)}^i \rangle\end{split}\]

in the notation of wave function, we denote \(| abc \cdots \rangle \equiv \cdots \hat c^\dagger \hat b^\dagger \hat a^\dagger | \mathrm{vac} \rangle\) and \((| abc \rangle)^\dagger = \langle abc |\). This is worth being noted while considering Fermion.

Example: GHZ State

For boson system that \(d = D = 2\). If we define the projection operator

\[\mathcal P = | 0 \rangle \langle 0000 | + | 1 \rangle \langle 1111|\]

Then if one of the virtual state is chosen to \(0\) or \(1\), then all of the physical indices and virtual indices should be identical. So this PEPS defines a GHZ state.

Example: Classical Partition Function

Let a Hamiltonian \(H[s] = \sum\limits_{\langle ij \rangle} h[s_i, s_j]\), where \(i, j\) are indices of site position. Then if we define the projection coefficient

\[[A_i]^k_{lrud} = \exp[-\dfrac{\beta}{4} (h[s_i,s_l] + h[s_i,s_r] + h[s_i,s_u] + h[s_i,s_d])]\]

Then it defines a PEPS

\[| \Psi \rangle = \exp(-\dfrac{\beta \hat H}{2}) (| \uparrow \rangle + | \downarrow\rangle)^{\otimes N}\]

and the inner product of \(| \Psi \rangle\) is proportional to partition function \(Z(\beta)\). The PEPS therefore defines a classical thermal state.

Also since at critical temperature, the 2D spin correlation function has a \(-\dfrac{1}{4}\) polynominal decay behavior, therefore PEPS can handle polynominally-decaying correlations of system.

2.2. Calculating Properties#

Recall that in MPS, we implement left or right normalization procedure, and then calcualting ground states become a common eigenvalue problem. However, due to the loop property of PEPS, we cannot exactly use the left/right canonical algorithm. Although we can directly calculate the overlap and expectation value by a general eigenvalue problem:

\[\begin{split}\langle \Psi | \Psi \rangle = A^\dagger \mathcal N A \\ \langle \Psi | \hat H | \Psi \rangle = A^\dagger \mathcal H A \\ \mathcal H v = \lambda \mathcal N v\end{split}\]

However, the coefficient matrix of \(\mathcal H\) and \(\mathcal N\) will grow exponentially, and the well-definition of \(\mathcal N\) is not clear. Hence approximate method of property calculation about PEPS should be developed.

2.3. Fermion PEPS#

Before expanding PEPS formalism into fermion, we should note that the major difference between fermion and boson is that fermion has exchange antisymmetry so that for two sites \(i\) and \(j\), the direct product will change sign:

\[| i \rangle \otimes | j \rangle = (-1)^{P_i \cdot P_j} |j \rangle \otimes | i \rangle\]

in which \(P_i\) is the Parity of state \(i\). Parity, in this context, can be primitively understood as whether or not the partical number of a certain state is odd or even. If partical number is odd, parity is \(1\); else, it is \(0\).

More mathematically, for a super vector space \(V\), it has a direct sum decomposition to \(V^0 \oplus V^1\) according to the parity, so does its dual space \(V^* = V^{*0} \oplus V^{*1}\).

Apart from electronic state, we can also define parity for a general tensor. Consider the tensor at space \(V \otimes \cdots\), then the parity is just that added from the vectors in each space.

References:

[1] , [2] : Introductory Review

[3] : PEPS Construction

[4] : Fermionic MPS, more mathematically.

[5] , [6]: Fermionic PEPS