Small permute and contraction order changes in TRG and BTRG (#22)

* get rid of unnecessary permutes in `TRG`

* implement the optimal contraction ordering for `BTRG`

src/schemes/btrg.jl

@@ -54,11 +54,13 @@ function step!(scheme::BTRG, trunc::TensorKit.TruncationScheme)
     S1 = scheme.S1
     S2 = scheme.S2
 
-    @tensor opt = true scheme.T[-1 -2; -3 -4] := D[-1 1; 8] * S1[2; 1] * B[-2; 3 2] *
-                                                 S2[3; 4] *
-                                                 C[4; 5 -3] *
-                                                 S1[5; 6] * A[7 6; -4] * S2[8; 7]
-
+    @tensor scheme.T[-1 -2; -3 -4] := D[-1 7; 4] *
+                                      S1[1; 7] *
+                                      B[-2; 3 1] *
+                                      S2[3; 2] *
+                                      C[2; 8 -3] *
+                                      S1[8; 5] *
+                                      A[6 5; -4] * S2[4; 6]
     scheme.S1 = S1′
     scheme.S2 = S2′
     return scheme

src/schemes/trg.jl

@@ -17,17 +17,12 @@ function step!(scheme::TRG, trunc::TensorKit.TruncationScheme)
 
     U, S, V, _ = tsvd(scheme.T, ((1, 4), (2, 3)); trunc=trunc)
 
-    # Flip legs to their original domain (to mitigate space mismatch at the end)
-    U = permute(U, ((1,), (2, 3)))
-    V = permute(V, ((1, 2), (3,)))
-
     @plansor begin
-        C[-1; -2 -3] := U[-1; -2 1] * sqrt(S)[1; -3]
-        D[-1 -2; -3] := sqrt(S)[-1; 1] * V[1 -2; -3]
+        C[-1 -2; -3] := U[-1 -2; 1] * sqrt(S)[1; -3]
+        D[-1; -2 -3] := sqrt(S)[-1; 1] * V[1; -2 -3]
     end
 
-    # @plansor complains here, not sure why
-    @tensor scheme.T[-1 -2; -3 -4] := D[-1 1; 4] * B[-2; 3 1] * C[3; 2 -3] * A[4 2; -4]
+    @tensor scheme.T[-1 -2; -3 -4] := D[-1; 1 4] * B[-2; 3 1] * C[3 2; -3] * A[4 2; -4]
     return scheme
 end