ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor...

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ITensor E.M. Stoudenmire Apr 2018 - Sao Paulo

Transcript of ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor...

Page 1: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

ITensor

E.M. Stoudenmire Apr 2018 - Sao Paulo

Page 2: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

Tensor Renormalization Group

The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors

This algorithm was proposed by Levin and Nave (cond-mat/0611687)

It is called a "renormalization group" because it decimates the tensor network in a heirarchical fashion

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Page 3: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

Classical Partition Function

A key application of TRG is computing properties of classical lattice models

We will use it to evaluate the partition function, but can also evaluate local observables, and even conformal field theory scaling dimensions!

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Page 4: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

Classical Ising Model

The partition function of the classical Ising model is

Z =X

�1�2�3...

e�E(�1,�2,�3,...)/T

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Where is the energy function

and is the temperature

E(�1,�2,�3, . . .)<latexit sha1_base64="mzpSmIoTvDW+7mg/6CoTpkR1fW8=">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</latexit><latexit sha1_base64="mzpSmIoTvDW+7mg/6CoTpkR1fW8=">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</latexit><latexit sha1_base64="mzpSmIoTvDW+7mg/6CoTpkR1fW8=">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</latexit><latexit sha1_base64="mzpSmIoTvDW+7mg/6CoTpkR1fW8=">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</latexit>

T<latexit sha1_base64="A2m5U9Mmaofve7by/jknpDOtcz4=">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</latexit><latexit sha1_base64="A2m5U9Mmaofve7by/jknpDOtcz4=">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</latexit><latexit sha1_base64="A2m5U9Mmaofve7by/jknpDOtcz4=">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</latexit><latexit sha1_base64="A2m5U9Mmaofve7by/jknpDOtcz4=">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</latexit>

The spin variables take the values � = +1,�1<latexit sha1_base64="52d3V0gIqkCY6rn6f+XSBwXk/TY=">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</latexit><latexit sha1_base64="52d3V0gIqkCY6rn6f+XSBwXk/TY=">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</latexit><latexit sha1_base64="52d3V0gIqkCY6rn6f+XSBwXk/TY=">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</latexit><latexit sha1_base64="52d3V0gIqkCY6rn6f+XSBwXk/TY=">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</latexit>

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Page 5: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

One Dimensional Ising Model

In one dimension, the energy of the Ising model is

E(�1,�2,�3, . . . ,�N ) = �1�2 + �2�3 + �3�4 + . . .+ �N�1<latexit sha1_base64="L/JU06l8MUbgCsdp7fJeumfzCJ4=">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</latexit><latexit sha1_base64="L/JU06l8MUbgCsdp7fJeumfzCJ4=">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</latexit><latexit sha1_base64="L/JU06l8MUbgCsdp7fJeumfzCJ4=">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</latexit><latexit sha1_base64="L/JU06l8MUbgCsdp7fJeumfzCJ4=">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</latexit>

So anti-aligned (antiferromagnetic) configurations have lowest energy

Taking periodic boundary conditions, so interaction between first and last Ising spin

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Page 6: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

Transfer Matrix Trick

To compute the partition function, use following trick

Z =

X

{�}

exp

�1

T

X

n

�n�n+1

!

<latexit sha1_base64="id/z8s97O0/RQMpH5/TRsSBoTns=">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</latexit><latexit sha1_base64="id/z8s97O0/RQMpH5/TRsSBoTns=">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</latexit><latexit sha1_base64="id/z8s97O0/RQMpH5/TRsSBoTns=">AAADCnicdVHLbhMxFHWGVwmvFJZsLEZIRYhoHCHRDVKlblgW1LQV8SjyOHcmVmzPyPa0RO78AR/BN7ACseUn2LCFz8CZRKJt0itZOjr3HB/73qySwrok+dWJbty8dfvO1t3uvfsPHj7qbT8+smVtOAx5KUtzkjELUmgYOuEknFQGmMokHGez/UX/+BSMFaU+dPMKUsUKLXLBmQvUuPfhI36Lqa3V2FNPrSgUo02DKXyqMJWQux2aG8b9K9L4w2ap1Hgp/A+8fkkaakQxdS+6416c9JO28DogKxCjVR2Mtztf6KTktQLtuGTWjkhSudQz4wSX0HRpbaFifMYKGAWomQKb+vbzDX4emAnOSxOOdrhlLzo8U9bOVRaUirmpvdpbkJt6o9rlu6kXuqodaL4MymuJXYkXk8QTYYA7OQ+AcSPCWzGfsjAsF+bdpRrOeKkU0xNPZ+CaEUk9BW1rA4ssfx4Tapguwgeby+rMsDU1la00Jucb1O31g40GHBzxAF+TxE6L65KC8YIr7JRc3eA6OBr0SdIn71/He/ur7W6hp+gZ2kEEvUF76B06QEPE0Tf0G/1Bf6PP0dfoe/RjKY06K88TdKmin/8Avjv8xw==</latexit><latexit sha1_base64="id/z8s97O0/RQMpH5/TRsSBoTns=">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</latexit>

=X

{�}

Y

n

e�(�n�n+1)/T = Tr�MN

<latexit sha1_base64="gBLgquG2EqOroxdZM2Ry0ZZJe54=">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</latexit><latexit sha1_base64="gBLgquG2EqOroxdZM2Ry0ZZJe54=">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</latexit><latexit sha1_base64="gBLgquG2EqOroxdZM2Ry0ZZJe54=">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</latexit><latexit sha1_base64="gBLgquG2EqOroxdZM2Ry0ZZJe54=">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</latexit>

M��0 = e�(��0)/T<latexit sha1_base64="lbKpRExynMv9mVra891FBEYhrS0=">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</latexit><latexit sha1_base64="lbKpRExynMv9mVra891FBEYhrS0=">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</latexit><latexit sha1_base64="lbKpRExynMv9mVra891FBEYhrS0=">AAAC+nicdVHLbhMxFHWGVwmvFJZsDCOksiCMIyS6Qarohg1SkZq2UjyNPM7N1IrtGdmeVpEz/4JYgdjyHWyBv8GZzqJtkitZOjr3HB9f36yUwrok+deJbt2+c/fe1v3ug4ePHj/pbT89skVlOAx5IQtzkjELUmgYOuEknJQGmMokHGez/WX/+ByMFYU+dPMSUsVyLaaCMxeoce/j57GnVuSKnXr6osYtpqURCmr8AcOpf7OzUfH67WE97sVJP2kKrwLSghi1dTDe7nylk4JXCrTjklk7IknpUs+ME1xC3aWVhZLxGcthFKBmCmzqm2Fr/CowEzwtTDja4Ya96vBMWTtXWVAq5s7szd6SXNcbVW66m3qhy8qB5pdB00piV+Dlz+GJMMCdnAfAuBHhrZifMcO4C//bpRoueKEU0xNPZ+DqEUk9BW0rA8ssv4gJNUznYcD6ujozbEVNZSONyWKNurl+sNaAgyMe4A1J7DzflBSMV1xhp+TmBlfB0aBPkj758i7e22+3u4Weo5doBxH0Hu2hT+gADRFH39Fv9Af9jRbRt+hH9PNSGnVazzN0raJf/wEMLfZL</latexit><latexit sha1_base64="lbKpRExynMv9mVra891FBEYhrS0=">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</latexit>

Where the matrix M is defined as

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Page 7: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

Transfer Matrix Trick

Can visualize the transfer-matrix form of Z as a tensor network

Z = Tr�MN

�<latexit sha1_base64="R1U+n/kx/9f977tFV2VeOPpDBr4=">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</latexit><latexit sha1_base64="R1U+n/kx/9f977tFV2VeOPpDBr4=">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</latexit><latexit sha1_base64="R1U+n/kx/9f977tFV2VeOPpDBr4=">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</latexit><latexit sha1_base64="R1U+n/kx/9f977tFV2VeOPpDBr4=">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</latexit>

= M<latexit sha1_base64="X3ckg6JoI47kgr3U9yuaJ38cqco=">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</latexit><latexit sha1_base64="X3ckg6JoI47kgr3U9yuaJ38cqco=">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</latexit><latexit sha1_base64="X3ckg6JoI47kgr3U9yuaJ38cqco=">AAACuXicdVFdSyMxFE1HXbXqrh+PvgQHwacyKQsuiCD44ougaFXoDHInvZ1mm2SGJKOUsT/BV331b+2/2XTsg9p6IXA495ycm9y0kMK6KPrXCBYWl34sr6w219Y3fv7a3Nq+sXlpOHZ4LnNzl4JFKTR2nHAS7wqDoFKJt+nwdNK/fUBjRa6v3ajAREGmRV9wcJ66Oqbn95th1IrqorOATUFIpnVxv9V4i3s5LxVqxyVY22VR4ZIKjBNc4rgZlxYL4EPIsOuhBoU2qepZx3TfMz3az40/2tGa/eioQFk7UqlXKnAD+7U3Ief1uqXr/0kqoYvSoebvQf1SUpfTycNpTxjkTo48AG6En5XyARjgzn9PM9b4yHOlQPeqeIhu3GVJFaO2pcFJVvUUstiAzvwDx5/VqYEZdSxracie5qjr69tzDdQ7wjb9Jgkesu+SvPGDy++Ufd3gLLhpt1jUYpe/w5PT6XZXyC7ZIweEkUNyQs7IBekQTjLyTF7Ia3AUQDAI/r5Lg8bUs0M+VWD/A26H3Ck=</latexit><latexit sha1_base64="X3ckg6JoI47kgr3U9yuaJ38cqco=">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</latexit>

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Page 8: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

Transfer Matrix Trick

One-dimension case exactly solvable

1) Diagonalize M to get its eigenvalues: {�1,�2}<latexit sha1_base64="iiHX0RV+FeGrVJbw08xrHGModrg=">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</latexit><latexit sha1_base64="iiHX0RV+FeGrVJbw08xrHGModrg=">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</latexit><latexit sha1_base64="iiHX0RV+FeGrVJbw08xrHGModrg=">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</latexit><latexit sha1_base64="iiHX0RV+FeGrVJbw08xrHGModrg=">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</latexit>

2) Observe that:

Z = Tr�MN

�= �N

1 + �N2

<latexit sha1_base64="7uspXJTnHdMVNTg3qhit0S9Sehg=">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</latexit><latexit sha1_base64="7uspXJTnHdMVNTg3qhit0S9Sehg=">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</latexit><latexit sha1_base64="7uspXJTnHdMVNTg3qhit0S9Sehg=">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</latexit><latexit sha1_base64="7uspXJTnHdMVNTg3qhit0S9Sehg=">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</latexit>

Using basis-invariance property of the trace

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Page 9: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

Two-dimensional Ising Model

Here we're primarily interested in the two-dimensional Ising model

E(�1,�2, . . .) =X

hiji

�i�j

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© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Page 10: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

Two-dimensional Transfer Tensor

Again possible to define Z by replacing transfer matrix, with a "transfer tensor"

A�t�r�b�l = e�(�t�r+�r�b+�b�l+�l�t)/T<latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">AAADOHicdZHfatRAFMYn8V9dq24VvfFmMAiV4jZZhPZGqPTGywrdtrBJl5PZs9thZyZhZlJZ0oCP4nP4Jt55pXjrEzibZrFttgcCP77zfXMmc9JccGPD8Ifn37l77/6DtYedR+uPnzztbjw7MlmhGQ5YJjJ9koJBwRUOLLcCT3KNIFOBx+lsf9E/PkdteKYO7TzHRMJU8QlnYJ006n79eFrGhk8ljCxtQC8hXYKo6AeKp+W7zbZ3a0Vsq33Cf00swb7dPqxG3SDshXXRNkQNBKSpg9GG9y0eZ6yQqCwTYMwwCnOblKAtZwKrTlwYzIHNYIpDhwokmqSsn6qib5wyppNMu09ZWqtXEyVIY+YydU4J9szc7C3EVb1hYSe7SclVXlhU7HLQpBDUZnTx7nTMNTIr5g6Aae7uStkZaGDWbacTK/zCMilBjct4hrYaRkkZozKFxsWs8iKIYg1q6n6wuu5ONbTcsaitQXSxwl0f318ZoC4R9Oktk+B8etskF7yScjuNbm6wDUf9XhT2os/vg739Zrtr5BV5TTZJRHbIHvlEDsiAMPLLW/deeC/97/5P/7f/59Lqe03mOblW/t9/KgYLQQ==</latexit><latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">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</latexit><latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">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</latexit><latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">AAADOHicdZHfatRAFMYn8V9dq24VvfFmMAiV4jZZhPZGqPTGywrdtrBJl5PZs9thZyZhZlJZ0oCP4nP4Jt55pXjrEzibZrFttgcCP77zfXMmc9JccGPD8Ifn37l77/6DtYedR+uPnzztbjw7MlmhGQ5YJjJ9koJBwRUOLLcCT3KNIFOBx+lsf9E/PkdteKYO7TzHRMJU8QlnYJ006n79eFrGhk8ljCxtQC8hXYKo6AeKp+W7zbZ3a0Vsq33Cf00swb7dPqxG3SDshXXRNkQNBKSpg9GG9y0eZ6yQqCwTYMwwCnOblKAtZwKrTlwYzIHNYIpDhwokmqSsn6qib5wyppNMu09ZWqtXEyVIY+YydU4J9szc7C3EVb1hYSe7SclVXlhU7HLQpBDUZnTx7nTMNTIr5g6Aae7uStkZaGDWbacTK/zCMilBjct4hrYaRkkZozKFxsWs8iKIYg1q6n6wuu5ONbTcsaitQXSxwl0f318ZoC4R9Oktk+B8etskF7yScjuNbm6wDUf9XhT2os/vg739Zrtr5BV5TTZJRHbIHvlEDsiAMPLLW/deeC/97/5P/7f/59Lqe03mOblW/t9/KgYLQQ==</latexit>

�t

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© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Page 11: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

Two-dimensional Transfer Tensor

In terms of tensor "A", partition function Z is

Z =<latexit sha1_base64="FtQ57alXsaVhBWw8afZpcSHrVG4=">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</latexit><latexit sha1_base64="FtQ57alXsaVhBWw8afZpcSHrVG4=">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</latexit><latexit sha1_base64="FtQ57alXsaVhBWw8afZpcSHrVG4=">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</latexit><latexit sha1_base64="FtQ57alXsaVhBWw8afZpcSHrVG4=">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</latexit>

A�t�r�b�l = e�(�t�r+�r�b+�b�l+�l�t)/T<latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">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</latexit><latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">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</latexit><latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">AAADOHicdZHfatRAFMYn8V9dq24VvfFmMAiV4jZZhPZGqPTGywrdtrBJl5PZs9thZyZhZlJZ0oCP4nP4Jt55pXjrEzibZrFttgcCP77zfXMmc9JccGPD8Ifn37l77/6DtYedR+uPnzztbjw7MlmhGQ5YJjJ9koJBwRUOLLcCT3KNIFOBx+lsf9E/PkdteKYO7TzHRMJU8QlnYJ006n79eFrGhk8ljCxtQC8hXYKo6AeKp+W7zbZ3a0Vsq33Cf00swb7dPqxG3SDshXXRNkQNBKSpg9GG9y0eZ6yQqCwTYMwwCnOblKAtZwKrTlwYzIHNYIpDhwokmqSsn6qib5wyppNMu09ZWqtXEyVIY+YydU4J9szc7C3EVb1hYSe7SclVXlhU7HLQpBDUZnTx7nTMNTIr5g6Aae7uStkZaGDWbacTK/zCMilBjct4hrYaRkkZozKFxsWs8iKIYg1q6n6wuu5ONbTcsaitQXSxwl0f318ZoC4R9Oktk+B8etskF7yScjuNbm6wDUf9XhT2os/vg739Zrtr5BV5TTZJRHbIHvlEDsiAMPLLW/deeC/97/5P/7f/59Lqe03mOblW/t9/KgYLQQ==</latexit><latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">AAADOHicdZHfatRAFMYn8V9dq24VvfFmMAiV4jZZhPZGqPTGywrdtrBJl5PZs9thZyZhZlJZ0oCP4nP4Jt55pXjrEzibZrFttgcCP77zfXMmc9JccGPD8Ifn37l77/6DtYedR+uPnzztbjw7MlmhGQ5YJjJ9koJBwRUOLLcCT3KNIFOBx+lsf9E/PkdteKYO7TzHRMJU8QlnYJ006n79eFrGhk8ljCxtQC8hXYKo6AeKp+W7zbZ3a0Vsq33Cf00swb7dPqxG3SDshXXRNkQNBKSpg9GG9y0eZ6yQqCwTYMwwCnOblKAtZwKrTlwYzIHNYIpDhwokmqSsn6qib5wyppNMu09ZWqtXEyVIY+YydU4J9szc7C3EVb1hYSe7SclVXlhU7HLQpBDUZnTx7nTMNTIr5g6Aae7uStkZaGDWbacTK/zCMilBjct4hrYaRkkZozKFxsWs8iKIYg1q6n6wuu5ONbTcsaitQXSxwl0f318ZoC4R9Oktk+B8etskF7yScjuNbm6wDUf9XhT2os/vg739Zrtr5BV5TTZJRHbIHvlEDsiAMPLLW/deeC/97/5P/7f/59Lqe03mOblW/t9/KgYLQQ==</latexit>

�t

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© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Page 12: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

Tensor Renormalization Group

The TRG strategy to compute Z is in two steps

First, use the SVD to factorize the A tensor two ways:

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�t

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F1 F2=

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© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Page 13: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

Tensor Renormalization Group

Each factorization can be derived from an SVD by grouping the square roots of singular values with unitary tensors:

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US(SVD)'

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© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Page 14: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

Tensor Renormalization Group

Insert these factorizations to rewrite the network as:

=

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Page 15: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

=

Tensor Renormalization Group

Then group blocks of four tensors as follows:

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Page 16: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

Tensor Renormalization Group

=

Effect of Step 1 followed by Step 2

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Page 17: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

Tensor Renormalization Group

Two steps of TRG algorithm:

Step #1:�t

�r

�b

�l A0

�t

�l=F3 F4

�b

�r

F1 F2=

�t

�l

�r

�b

Step #2: = A1

Can be repeated to turn network of A1's into A2's etc.© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Page 18: ITensor - USP › ~gtlandi › tutorial02.pdf · Tensor Renormalization Group The tensor renormalization group or TRG is an elegant algorithm to contract a lattice of tensors This

Tensor Renormalization Group

After iterating enough steps (how many?) entire network become a single A tensor AN

AN

Tracing the AN tensor gives the partition function

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© Edwin Miles Stoudenmire, Flatiron Institute, 2018

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ITensor Code Activity

Read through the code tutorial/trg/trg.cc

= An+1

Your job: Step 2 – recombine factors, defining new A

F3

F1

F4

F2

An =F3 F4

F1 F2

=

Step 1 – factor two ways (approximately)

x

x

00

y00

yy

x

00

y00

x

y00

x

00

y

x

00

00

00

00Try following parameters:

T = 3.0 maxm = 20 topscale = 8

./trg [log(Z)/Ns = 0.149216]

( (

© Edwin Miles Stoudenmire, Flatiron Institute, 2018