ALGORITHMIC PERSPECTIVE ON STRONGLY CORRELATED …ALGORITHMIC PERSPECTIVE ON STRONGLY CORRELATED SYSTEMS Lecture: Introduction to Matrix Product States Fall 2015 Professor: Bryan Clark

ALGORITHMIC PERSPECTIVE ON STRONGLY CORRELATED

SYSTEMS

Lecture: Introduction to Matrix Product States

Fall 2015 Professor: Bryan Clark

Last Time: Exact Diagonalization Matrix Product States

Today: More Matrix Product States

h |�1�2�3...�ni = M�1M�2M�3 ...M�n

Overlap: h | iX

�1,�2,...�n

h |�1�2�3...�nih�1�2�3...�n| i

=X

�1,�2...�n

(M�1M�2M�3 ...M�n)(M�1M�2M�3 ...M�n)T

=X

�1,�2...�n

(M�1M�2M�3 ...M�n)(M�nTM�n�1TM�n�2T ...M�1T )

=X

�1,�2...�n�1

M�1M�2M�3 ...M�n�1X

�n

(M�nM�nT )M�n�1TM�n�2T ...M�1T

A=

X

�1,�2...�n�2

M�1M�2M�3 ...(X

�n�1

M�n�1AM�n�1T )M�n�2T ...M�1T

Overlap: h | i

Overlap: h | i

Overlap: h | i

Multiply matrices first?

Overlap: h | i

A

Overlap: h | i

A

A

Suppose I want to start in the other direction?

Tr

"X

�1,�2...�n

(M�1M�2M�3 ...M�n)(M�nTM�n�1TM�n�2T ...M�1T )

#

X

�1,�2...�n

(M�1M�2M�3 ...M�n)(M�nTM�n�1TM�n�2T ...M�1T )

Tr

"X

�1,�2...�n

(M�nTM�n�1TM�n�2T ...M�1T )(M�1M�2M�3 ...M�n)

#

A

Tr

"X

�1,�2...�n

(M�nTM�n�1TM�n�2T ...M�1T )(M�1M�2M�3 ...M�n)

#

A

=X

�1,�2...�n�1

M�1M�2M�3 ...M�n�1X

�n

(M�nM�nT )M�n�1TM�n�2T ...M�1T

A

Canonization h |�1�2�3...�ni = M�1M�2M�3 ...M�n

M�1UU†M�2M�3 ...M�nGauge Freedom:

Q: How should we use our gauge freedom?

=(8 x 4) (4 x 4) (4 x 4)

U D VM"

M#

8 x 4

4 x 4

4 x 4

SVD

= I4 x 8

8 x 4

4 x 4

(left canonical)

U#

U#

U#

U" U"

U"

U"TU" + U#TU# = I

Canonization h |�1�2�3...�ni = M�1M�2M�3 ...M�n

M�1UU†M�2M�3 ...M�nGauge Freedom:

SVD

=I

(right canonical)

Q: How should we use our gauge freedom?

M" M# =

4 x 8

4 x 4 4 x 4

(4 x 4) (4 x 4) (4 x 8)

U D VV " V #

V"

V#

V "V "T + V #V #T = I

Mixed canonical

Matrix Product Operator

h�1�2�3�4�5|H|�01�

02�

03�

04�

05i