My garden free of trees, but behind the fence there a few trees growing.
I like the idea - to make undeground barrier, I thought about this too.
What can I use as a barrier? What material ?

Lots of things can be used as a barrier. You can buy expensive bamboo barrier (google it) and be absolutely sure you will stop everything. I have no clue how deep ivy rhizomes are, but paving stones 8X16X1 inches at Home Depot are relatively inexpensive, and my guess is that if you bury them completely they will do a good job. You can use any plastic or metal sheeting you can find, plastic thicker than 10 mils. You can use wood, and it will last a few years. You could also use no barrier, patrol the border once a year, and put your spade down one spade depth all along the border, severing all rhizomes.

None of the glyphosate "weed" poisons is acceptable to an organic grower and should not even be mentioned on an organic gardening forum.
These Ivies are very persistant plants and the single best method of eliminating them is digging them out. They seem to have the ability to shut off sections of the plant that have been sprayed with "weed" poisons so the poison does not reach other parts of the plant.
I have used the 8 x 16 x 1 paving blocks and find that both Ivy and Quack Grass will find the seams between the blocks and send roots into the planting beds.

Having battled this at one house I used to rent...I know what you are talking about!
The only SURE way to get rid of it is to dig it out...and each tiny little piece that has a node will need to be removed or it will re-sprout. I would suggest that you dig out what you can & then patrol the area for close to a year digging out all new starts...cause you will miss some & it will come back.
I put down that black plastic border to try to contain mine, but it isn't deep enough. I'd suggest something either metal or plastic that is stiff and that you can bury at least 12" deep to keep it from coming under...but you will still have to control any that trys to come over the top.
Talk to the neighbor & see if they are willing to get rid of this stuff!

Thank you all for responds. Problem is - behind my fence is a space what belongs to nobody, and a little force neighbor's fence. The only way - to build a barrier.

After cardboard is laid, much of the roots will die. When one of the roots does not die and manages to punch through, your choices are restricted. You can not dig it up without ruining the seal, and if you pull it, there will be more coming from that spot. It is the downside of the very organic method of cardboard+mulch.

Yes, touching up with a paint brush is not organic. I also still have more than half of the smallest bottle of Round up you can buy. I bought it in 1998, and I never sprayed it, only applied with a paint brush. With ivy and poison ivy it is a must-have backup.

In the case of poison ivy (my #1 enemy) you can not even dig the thing out without risking an ER visit, and the local largest PI vines are (or rather, were) thicker than my forearm and 100 ft long, and a root system to match. After you cut one such thing, no matter how much cardboard and mulch you put down, there will be dozens of shoots punching through within a 20 ft radius for a couple of years.

Alyonka, I got rid of a large bed of ivy last year. First, I used a weedwhacker to get rid of a lot of the foliage - put all the cut stuff in a plastic bag to be disposed of. Then I dug and pulled and tugged. In a few places the ivy broke off near ground level. I just waited until it regrew a bit then re-dug. I think it's important to have the soil nice and damp when you do this. Collect all the ivy for disposal.

The area behind you belongs to someone. Your county offices could identify the owner. I'd ask permission to rid that area of ivy also. I don't understand what you mean by "and a little force neighbor's fence."

Long story short, it's a big job but very doable. Hope you'll let us know how you do.

Rosie

Sorry, this part "little force" part - my ipad had changed it. What I meant "a little further"

Alyonka, keep pulling it, it's the best way for a small area. Grab a vine and try to get _all_of_it_ by pulling and shaking, all the side shoots. Let the vines dry without touching anything moist, the leaves will make nice mulch, but the dry vines need to be shredded/cut because they take more than a year to decompose and turn into strong ropes. As soon as it's dead it is a good organic matter for your soil. Don't plant anything important in that area for the first year so you can pull the rest of the ivy in case you miss some. Talk to your neighbors, they might be happy if somebody wants to eradicate the ivy and make that space between the fences an ivy DMZ. It is doable, I've spent hours and hours cleaning the nearby woods from english ivy and himalayan blackberries, simple pulling and digging works, no chemicals needed. Good luck! Udachi!

Spasibo. ;)
I still working hard pulling out roots of my garden. I decided to bury a barrier under the fence to block ivy from crawling to my yard. Planning on going to Home Depot.

Today my friend told me that English ivy doesn't go deep in the ground.
And he thinks its could be cypress tree, that we have around our yard..
Can you tell by picture what kind of roots is that.

Weed torch!