itensor - usp › ~gtlandi › tutorial02.pdf · tensor renormalization group the tensor...

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ITensor

E.M. Stoudenmire Apr 2018 - Sao Paulo

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

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

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

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

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=">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</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=">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><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=">AAAC+nicdVHLbhMxFHWGVwmvFJZsDCOksiCMIyS6Qarohg1SkZq2UjyNPM7N1IrtGdmeVpEz/4JYgdjyHWyBv8GZzqJtkitZOjr3HB9f36yUwrok+deJbt2+c/fe1v3ug4ePHj/pbT89skVlOAx5IQtzkjELUmgYOuEknJQGmMokHGez/WX/+ByMFYU+dPMSUsVyLaaCMxeoce/j57GnVuSKnXr6osYtpqURCmr8AcOpf7OzUfH67WE97sVJP2kKrwLSghi1dTDe7nylk4JXCrTjklk7IknpUs+ME1xC3aWVhZLxGcthFKBmCmzqm2Fr/CowEzwtTDja4Ya96vBMWTtXWVAq5s7szd6SXNcbVW66m3qhy8qB5pdB00piV+Dlz+GJMMCdnAfAuBHhrZifMcO4C//bpRoueKEU0xNPZ+DqEUk9BW0rA8ssv4gJNUznYcD6ujozbEVNZSONyWKNurl+sNaAgyMe4A1J7DzflBSMV1xhp+TmBlfB0aBPkj758i7e22+3u4Weo5doBxH0Hu2hT+gADRFH39Fv9Af9jRbRt+hH9PNSGnVazzN0raJf/wEMLfZL</latexit>

Where the matrix M is defined as

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

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=">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><latexit sha1_base64="X3ckg6JoI47kgr3U9yuaJ38cqco=">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</latexit>

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

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

Two-dimensional Ising Model

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

E(�1,�2, . . .) =X

hiji

�i�j

<latexit sha1_base64="HpQriEvhYwHgQ8R9PFtmVzLMn8Q=">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</latexit><latexit sha1_base64="HpQriEvhYwHgQ8R9PFtmVzLMn8Q=">AAADBHicdVFdaxQxFM2OX3X92uqjL8FBqCDLZBH0RSgUwQcfKrhtYWcY7mTuTtNNMkOSqSzTffV3+AN8Unz1d+ij/hKz01lou9sLgcO55+Qk92aVFNZF0Z9ecOPmrdt3tu72791/8PDRYPvxgS1rw3HMS1maowwsSqFx7ISTeFQZBJVJPMxme8v+4SkaK0r9yc0rTBQUWkwFB+epdPDh3U5sRaEgZS9ph0Yeybx09gV967lapU0sQRcSqaAnNDYtXqzkYgVO0kEYDaO26DpgHQhJV/vpdu9rnJe8Vqgdl2DthEWVSxowTnAf0Y9rixXwGRQ48VCDQps07bcX9LlncjotjT/a0Za96GhAWTtXmVcqcMf2am9JbupNajd9kzRCV7VDzc+DprWkrqTLGdJcGOROzj0AboR/K+XHYIA7P+l+rPEzL5UCnTfxDN1iwpImRm1rg8us5ixkqxkuLqszA2vqbvQhO9ugbq8fbTRQ7whH9JokOC2uS/LGCy6/U3Z1g+vgYDRk0ZB9fBXu7nXb3SJPyTOyQxh5TXbJe7JPxoST7+Q3+Uv+BV+Cb8GP4Oe5NOh1nifkUgW//gMogvk8</latexit><latexit sha1_base64="HpQriEvhYwHgQ8R9PFtmVzLMn8Q=">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</latexit><latexit sha1_base64="HpQriEvhYwHgQ8R9PFtmVzLMn8Q=">AAADBHicdVFdaxQxFM2OX3X92uqjL8FBqCDLZBH0RSgUwQcfKrhtYWcY7mTuTtNNMkOSqSzTffV3+AN8Unz1d+ij/hKz01lou9sLgcO55+Qk92aVFNZF0Z9ecOPmrdt3tu72791/8PDRYPvxgS1rw3HMS1maowwsSqFx7ISTeFQZBJVJPMxme8v+4SkaK0r9yc0rTBQUWkwFB+epdPDh3U5sRaEgZS9ph0Yeybx09gV967lapU0sQRcSqaAnNDYtXqzkYgVO0kEYDaO26DpgHQhJV/vpdu9rnJe8Vqgdl2DthEWVSxowTnAf0Y9rixXwGRQ48VCDQps07bcX9LlncjotjT/a0Za96GhAWTtXmVcqcMf2am9JbupNajd9kzRCV7VDzc+DprWkrqTLGdJcGOROzj0AboR/K+XHYIA7P+l+rPEzL5UCnTfxDN1iwpImRm1rg8us5ixkqxkuLqszA2vqbvQhO9ugbq8fbTRQ7whH9JokOC2uS/LGCy6/U3Z1g+vgYDRk0ZB9fBXu7nXb3SJPyTOyQxh5TXbJe7JPxoST7+Q3+Uv+BV+Cb8GP4Oe5NOh1nifkUgW//gMogvk8</latexit>

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

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=">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</latexit><latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">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</latexit><latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">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</latexit><latexit sha1_base64="oYYOpfVaGJTen1NTwlIKZoxgIAM=">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</latexit>

�t

�r

�b

�l A

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

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=">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>

�t

�r

�b

�l A

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Tensor Renormalization Group

The TRG strategy to compute Z is in two steps

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

�t

�r

�b

�l A0

�t

�l=F3 F4

�b

�r

F1 F2=

�t

�l

�r

�b

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Tensor Renormalization Group

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

�t

�r

�b

�l

�t

�l

V

�b

�r

US(SVD)'

�t

�l

V

�b

�r

U

= Sp

Sp

�t

�l

F3

�b

�r

F1=A0

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Tensor Renormalization Group

Insert these factorizations to rewrite the network as:

=

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

=

Tensor Renormalization Group

Then group blocks of four tensors as follows:

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

Tensor Renormalization Group

=

Effect of Step 1 followed by Step 2

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

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

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

Z =<latexit sha1_base64="FtQ57alXsaVhBWw8afZpcSHrVG4=">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</latexit><latexit sha1_base64="FtQ57alXsaVhBWw8afZpcSHrVG4=">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</latexit><latexit sha1_base64="FtQ57alXsaVhBWw8afZpcSHrVG4=">AAACunicdVFNTxsxEHW2H9D0AyjHXixWlXqK1lElOBQJiUuPIBFAZFdo1pkkVmzvyp6lipb8Ba702L/Vf4N3yQFIGMnS05v3/MaevNTKU5L870Rv3r57v7H5ofvx0+cvW9s7X899UTmJA1nowl3m4FEriwNSpPGydAgm13iRz46b/sUNOq8Ke0bzEjMDE6vGSgI11BU/5NfbcdJL2uKrQCxBzJZ1cr3T+ZeOClkZtCQ1eD8USUlZDY6U1LjoppXHEuQMJjgM0IJBn9XtsAv+PTAjPi5cOJZ4yz511GC8n5s8KA3Q1L/sNeS63rCi8UFWK1tWhFY+Bo0rzangzcv5SDmUpOcBgHQqzMrlFBxICv/TTS3+kYUxYEd1OkNaDEVWp2h95bDJqm9jkTqwk/DAxXN17mBFnepWGovbNer2+v5aAw+OuM9fSYKbyWtJwfjEFXYqXm5wFZz3eyLpidOf8dHxcrub7BvbYz+YYPvsiP1mJ2zAJJuyO3bP/ka/ojxS0exRGnWWnl32rCJ6AP3G3GA=</latexit><latexit sha1_base64="FtQ57alXsaVhBWw8afZpcSHrVG4=">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</latexit> =<latexit sha1_base64="W76Txko4GTsJPJi6KAToHkK52AA=">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</latexit><latexit sha1_base64="W76Txko4GTsJPJi6KAToHkK52AA=">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</latexit><latexit sha1_base64="W76Txko4GTsJPJi6KAToHkK52AA=">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</latexit><latexit sha1_base64="W76Txko4GTsJPJi6KAToHkK52AA=">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</latexit>

© Edwin Miles Stoudenmire, Flatiron Institute, 2018

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

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